Finite Group Databasebeta

← #29702108

Subgroups — #29702108

Order 15616.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454768C211yesmaximal, characteristic, complemented
2#21454387C211yesmaximal, characteristic
3#21446801C211yesmaximal, characteristic, complemented
4#21446595C211yesmaximal, characteristic, complemented
5#21446573C211yesmaximal, characteristic, complemented
6#21446568C211yesmaximal, characteristic, complemented
7#21446420C211yesmaximal, characteristic, complemented
8#21446420C211yesmaximal, characteristic, complemented
9#21446096C211yesmaximal, characteristic
10#21446096C211yesmaximal, characteristic
11#21445678C211yesmaximal, characteristic, complemented
12#21445666C211yesmaximal, characteristic
13#21445659C211yesmaximal, characteristic
14#21445629C211yesmaximal, characteristic
15C61×#3076C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#16839893—21no—
19#16839893—21no—
20#16839893—21no—
21Q8×#92418—21no—
22C2×#14519713C2211yescharacteristic
23C2×#14519713C2211yescharacteristic
24#16839789—21no—
25#16839789—21no—
26#16839789C2211yescharacteristic
27#16839789C2211yescharacteristic
28#16839789C2211yescharacteristic
29#16839789C2211yescharacteristic
30#16839786—21no—
31#16839786C2211yescharacteristic
32C2×C4⋊#92418—21no—
33C2×C4⋊#92418C2211yescharacteristic
34C2×#14519704C2211yescharacteristic
35C2×#14519704C2211yescharacteristic
36#16839718C2211yescharacteristic
37#16839718C2211yescharacteristic
38C4×#11758194C2211yescharacteristic
39C2×#14519687C2211yescharacteristic
40C2×C8×Dic61C2211yescharacteristic
41#16839608C2211yescharacteristic
42#16839527C2211yescharacteristic
43#16839527C2211yescharacteristic
44#16839521C2211yescharacteristic
45#16839521C2211yescharacteristic
46#16839521C2211yescharacteristic
47#16839521C2211yescharacteristic
48#16839521C2211yescharacteristic
49#16839521C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C61×#533C2211yescharacteristic
53C4×#41×C61C2211yescharacteristic
54C61×#479C411yescharacteristic, complemented
55C61×#479C411yescharacteristic, complemented
56C61×#478C411yescharacteristic, complemented
57C61×#478C411yescharacteristic, complemented
58C61×#477C411yescharacteristic, complemented
59C61×#476C411yescharacteristic, complemented
60C61×#475C411yescharacteristic, complemented
61C61×#474C411yescharacteristic, complemented
62C61×#427C2211yescharacteristic
63C61×#427C2211yescharacteristic
64C2×C4⋊C8×C61C2211yescharacteristic
65C61×#416C2211yescharacteristic
66C61×#416C2211yescharacteristic
67#14519784—22no—
68#14519784—22no—
69#14519784—41no—
70#14519784—41no—
71C2×#11758194—21no—
72C2×#11758194—21no—
73C2×#11758194C2311yescharacteristic
74C2×#11758194C2311yescharacteristic
75C2×#11758194C2311yescharacteristic
76C2×#11758194C2311yescharacteristic
77C22×#92418—21no—
78C22×#92418—21no—
79C22×#92418C2311yescharacteristic
80C22×#92418C2311yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83#14519713—41no—
84#14519713—41no—
85C4⋊#92418—21no—
86C4⋊#92418—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418—21no—
92C4⋊#92418—21no—
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418D411yescharacteristic
96C4⋊#92418D411yescharacteristic
97C4⋊#92418D411yescharacteristic
98C4⋊#92418D411yescharacteristic
99#14519704—22no—
100#14519704Q814yescomplemented
101#14519688—21no—
102#14519688—21no—
103#14519688—21no—
104#14519687—41no—
105C4×#92418—21no—
106C4×#92418—21no—
107C4×#92418—21no—
108C8×Dic61—41no—
109#14519684—21no—
110#14519684—21no—
111C2×#41×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C4⋊Q8×C61C2×C411yescharacteristic
115C4⋊Q8×C61C2×C411yescharacteristic
116#125×C61C2×C411yescharacteristic
117#125×C61C2×C411yescharacteristic
118C22⋊Q8×C61—21no—
119C22⋊Q8×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61C2×C411yescharacteristic
123C4×Q8×C61C2×C411yescharacteristic
124C4×D4×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61C2×C411yescharacteristic
127C4×D4×C61C2×C411yescharacteristic
128#117×C61—21no—
129#117×C61C2311yescharacteristic
130#117×C61C2311yescharacteristic
131#117×C61C2311yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#107×C61C2×C411yescharacteristic
135#107×C61C2×C411yescharacteristic
136#107×C61C2×C411yescharacteristic
137#107×C61C2×C411yescharacteristic
138#106×C61C2×C411yescharacteristic
139#106×C61C2×C411yescharacteristic
140#106×C61C2×C411yescharacteristic
141#106×C61C2×C411yescharacteristic
142C4⋊C8×C61C2×C411yescharacteristic
143C4⋊C8×C61C2×C411yescharacteristic
144C4⋊C8×C61C2×C411yescharacteristic
145C4⋊C8×C61C2×C411yescharacteristic
146#103×C61C2×C411yescharacteristic
147#103×C61C2×C411yescharacteristic
148#103×C61C2×C411yescharacteristic
149#103×C61C2×C411yescharacteristic
150#102×C61C2×C411yescharacteristic
151#102×C61C2×C411yescharacteristic
152#102×C61C2×C411yescharacteristic
153#102×C61C2×C411yescharacteristic
154#41×C61—21no—
155#41×C61—21no—
156#41×C61C2×Q812yes—
157#41×C61#4112yes—
158C2×Q8×C61—21no—
159C2×Q8×C61—21no—
160C2×Q8×C61C22×C411yescharacteristic
161C2×D4×C61—21no—
162C2×D4×C61C22×C411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61#4111yescharacteristic
167C22×C4×C61#4111yescharacteristic
168C22×C4×C61C2411yescharacteristic
169C22×Dic61—21no—
170C22×Dic61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C22×C411yescharacteristic
176C2×C8×C61C22×C411yescharacteristic
177C2×C8×C61C22×C411yescharacteristic
178C2×C8×C61C22×C411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61C22×C411yescharacteristic
188C4⋊C4×C61C22×C411yescharacteristic
189C4⋊C4×C61C22×C411yescharacteristic
190C4⋊C4×C61C22×C411yescharacteristic
191C4⋊C4×C61C22×C411yescharacteristic
192C4⋊C4×C61C22×C411yescharacteristic
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—21no—
197C22⋊C4×C61—21no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—21no—
202C42×C61—21no—
203C42×C61C22×C411yescharacteristic
204C42×C61C22×C411yescharacteristic
205C42×C61C2×D411yescharacteristic
206C42×C61C2×D411yescharacteristic
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61C2×Q812yes—
211#11758194—21no—
212#11758194—21no—
213#11758194—21no—
214#11758194—21no—
215#11758194—21no—
216#11758194—21no—
217#11758194—21no—
218#11758194—42no—
219#11758194—42no—
220#11758194C2×Q812yes—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—42no—
230C2×#92418—42no—
231C2×#92418C2×D411yescharacteristic
232C2×#92418C2×D411yescharacteristic
233C2×#92418C2×D411yescharacteristic
234C2×#92418C2×D411yescharacteristic
235C23×C61—21no—
236C23×C61C2×#4111yescharacteristic
237Q8×C61—21no—
238Q8×C61—42no—
239Q8×C61—81no—
240Q8×C61C4×Q812yes—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—21no—
244D4×C61C4×Q812yes—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—42no—
257C2×C4×C61—81no—
258C2×C4×C61C4×D411yescharacteristic
259C2×C4×C61C4×D411yescharacteristic
260C2×C4×C61C4×D411yescharacteristic
261C2×C4×C61C4×D411yescharacteristic
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×Q811yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C2×Dic61—22no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C8×C61—21no—
275C8×C61—21no—
276C8×C61—21no—
277C8×C61—21no—
278#92418—21no—
279#92418—21no—
280#92418—21no—
281#92418—21no—
282#92418—22no—
283#92418—41no—
284#92418—41no—
285#92418—41no—
286#22984—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—22no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61C2×C4×D411yescharacteristic
293C22×C61#55211yescharacteristic
294C22×C61#55411yescharacteristic
295C4×C61—22no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61C2×C4×Q811yescharacteristic
307C4×C61#52011yescharacteristic
308C4×C61#54611yescharacteristic
309C4×C61D4×Q811yescharacteristic
310Dic61—42no—
311Dic61—81no—
312#3076D6111yescharacteristic
313#2962—611no—
314#2906—611no—
315#1995—611no—
316#1950—611no—
317#1946—611no—
318#1945—611no—
319#1922—611no—
320#1922—611no—
321#1866—611no—
322#1866—611no—
323#1768—611no—
324#1765—611no—
325#1764—611no—
326#1759—611no—
327C2×C61—21no—
328C2×C61—21no—
329C2×C61—42no—
330C2×C61#236011yescharacteristic
331C2×C61#295511yescharacteristic
332C2×C61#295711yescharacteristic
333C2×#131—611no—
334C2×#131—611no—
335#533C2×D6111yescharacteristic
336C4×#41C2×D6111yescharacteristic
337#479Dic6111yescharacteristic, complemented
338#479Dic6111yescharacteristic, complemented
339#478Dic6111yescharacteristic, complemented
340#478Dic6111yescharacteristic, complemented
341#477Dic6111yescharacteristic, complemented
342#476Dic6111yescharacteristic, complemented
343#475Dic6111yescharacteristic, complemented
344#474Dic6111yescharacteristic, complemented
345#446—1221no—
346#446—1221no—
347#446—1221no—
348C8×Q8—1221no—
349C2×#108—611no—
350C2×#108—611no—
351#427C2×D6111yescharacteristic
352#427C2×D6111yescharacteristic
353#424—611no—
354#424—611no—
355#424—611no—
356#424—611no—
357#424—1221no—
358#424—1221no—
359#423—611no—
360#423—1221no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—1221no—
363C2×C4⋊C8C2×D6111yescharacteristic
364C2×#104—611no—
365C2×#104—611no—
366#416C2×D6111yescharacteristic
367#416C2×D6111yescharacteristic
368#405—611no—
369#405—611no—
370C4×M16—611no—
371C2×#97—611no—
372C2×C4×C8—611no—
373#376—611no—
374#341—611no—
375#341—611no—
376#339—611no—
377#339—611no—
378#339—611no—
379#339—611no—
380#339—611no—
381#339—611no—
382#336—611no—
383#336—611no—
384C61#2298411yescharacteristic, complemented, Sylow
385C2×#41C22×D6111yescharacteristic
386#131—1222no—
387#131—1222no—
388#131—2441no—
389#131—2441no—
390C2×M16—611no—
391C2×M16—611no—
392C2×M16—611no—
393C2×M16—611no—
394C2×M16—1221no—
395C2×M16—1221no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—1221no—
399C22×C8—1221no—
400C22×C8C22×D6111yescharacteristic
401C22×C8C22×D6111yescharacteristic
402C4⋊Q8C2×Dic6111yescharacteristic
403C4⋊Q8C2×Dic6111yescharacteristic
404#125C2×Dic6111yescharacteristic
405#125C2×Dic6111yescharacteristic
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8C2×Dic6111yescharacteristic
411C4×Q8C2×Dic6111yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4C2×Dic6111yescharacteristic
415C4×D4C2×Dic6111yescharacteristic
416#117—21no—
417#117C22×D6111yescharacteristic
418#117C22×D6111yescharacteristic
419#117C22×D6111yescharacteristic
420C2×C42—21no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42C22×D6111yescharacteristic
424#108—2441no—
425#108—2441no—
426#107C2×Dic6111yescharacteristic
427#107C2×Dic6111yescharacteristic
428#107C2×Dic6111yescharacteristic
429#107C2×Dic6111yescharacteristic
430#106C2×Dic6111yescharacteristic
431#106C2×Dic6111yescharacteristic
432#106C2×Dic6111yescharacteristic
433#106C2×Dic6111yescharacteristic
434C4⋊C8—611no—
435C4⋊C8—611no—
436C4⋊C8—611no—
437C4⋊C8—611no—
438C4⋊C8—1221no—
439C4⋊C8—1221no—
440C4⋊C8—1221no—
441C4⋊C8—1221no—
442C4⋊C8—1221no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447C4⋊C8—1221no—
448C4⋊C8C2×Dic6111yescharacteristic
449C4⋊C8C2×Dic6111yescharacteristic
450C4⋊C8C2×Dic6111yescharacteristic
451C4⋊C8C2×Dic6111yescharacteristic
452#104—614no—
453#104—1222no—
454#103C2×Dic6111yescharacteristic
455#103C2×Dic6111yescharacteristic
456#103C2×Dic6111yescharacteristic
457#103C2×Dic6111yescharacteristic
458#102C2×Dic6111yescharacteristic
459#102C2×Dic6111yescharacteristic
460#102C2×Dic6111yescharacteristic
461#102C2×Dic6111yescharacteristic
462#97—1221no—
463#97—1221no—
464#97—1221no—
465#97—2441no—
466C4×C8—1221no—
467C4×C8—1221no—
468C4×C8—1221no—
469C4×C8—2441no—
470#95—1221no—
471#95—1221no—
472#41—21no—
473#41—21no—
474#41Q8×D6112yes—
475#41#1175822612yes—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8C22×Dic6111yescharacteristic
479C2×D4—21no—
480C2×D4C22×Dic6111yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4#1175822411yescharacteristic
487C22×C4#1175822411yescharacteristic
488C22×C4C23×D6111yescharacteristic
489M16—612no—
490M16—1221no—
491M16—1221no—
492M16—1221no—
493M16—1221no—
494M16—1221no—
495M16—1221no—
496M16—1221no—
497M16—2442no—
498M16—2442no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—611no—
504C2×C8—611no—
505C2×C8—611no—
506C2×C8—611no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—2442no—
516C2×C8—2442no—
517C2×C8C22×Dic6111yescharacteristic
518C2×C8C22×Dic6111yescharacteristic
519C2×C8C22×Dic6111yescharacteristic
520C2×C8C22×Dic6111yescharacteristic
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4C22×Dic6111yescharacteristic
530C4⋊C4C22×Dic6111yescharacteristic
531C4⋊C4C22×Dic6111yescharacteristic
532C4⋊C4C22×Dic6111yescharacteristic
533C4⋊C4C22×Dic6111yescharacteristic
534C4⋊C4C22×Dic6111yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—612no—
546C42—1221no—
547C42—1221no—
548C42—1221no—
549C42D4×D6111yescharacteristic
550C42D4×D6111yescharacteristic
551C42C22×Dic6111yescharacteristic
552C42C22×Dic6111yescharacteristic
553C23—21no—
554C23C2×#1175822411yescharacteristic
555Q8—21no—
556Q8—42no—
557Q8—81no—
558Q8Q8×Dic6112yes—
559D4—22no—
560D4—22no—
561D4—21no—
562D4Q8×Dic6112yes—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—81no—
576C2×C4—1222no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4D4×Dic6111yescharacteristic
583C2×C4D4×Dic6111yescharacteristic
584C2×C4D4×Dic6111yescharacteristic
585C2×C4D4×Dic6111yescharacteristic
586C2×C4C23×Dic6111yescharacteristic
587C2×C4C2×D4×D6111yescharacteristic
588C2×C4C2×Q8×D6111yescharacteristic
589C2×C4C2×#1175822611yescharacteristic
590C2×C4#1451982311yescharacteristic
591C2×C4#1451982411yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—1221no—
597C8—1221no—
598C8—1221no—
599C8—1221no—
600C8—1222no—
601C8—2441no—
602C8—2441no—
603C8—2441no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22C2×D4×Dic6111yescharacteristic
610C22#1684045411yescharacteristic
611C22#1684047011yescharacteristic
612C4—22no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—4881no—
624C4—2442no—
625C4C2×Q8×Dic6111yescharacteristic
626C4#1684024511yescharacteristic
627C4#1684040311yescharacteristic
628C4#1684042511yescharacteristic
629C2—21no—
630C2—21no—
631C2—42no—
632C2#2144882811yescharacteristic
633C2#2145472211yescharacteristic
634C2#2145473611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#3076C2
socleC22×C61C2×C4×D4
prime radicalC2×#41×C61C23
cosocleC22×C4×C61C24
hypercenter#3076D61
hypocenterC61#22984
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21454768C2semidirect
3. #21446801C2semidirect
4. #21446595C2semidirect
5. #21446573C2semidirect
6. #21446568C2semidirect
7. #21446420C2semidirect
8. #21446420C2semidirect
11. #21445678C2semidirect
54. C61×#479C4semidirect
55. C61×#479C4semidirect
56. C61×#478C4semidirect
57. C61×#478C4semidirect
58. C61×#477C4semidirect
59. C61×#476C4semidirect
60. C61×#475C4semidirect
61. C61×#474C4semidirect
100. #14519704Q8semidirect
337. #479Dic61semidirect
338. #479Dic61semidirect
339. #478Dic61semidirect
340. #478Dic61semidirect
341. #477Dic61semidirect
342. #476Dic61semidirect
343. #475Dic61semidirect
344. #474Dic61semidirect
384. C61#22984semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214547682
#214543872
#214468012
#214465952
#214465732
#214465682
#214464202
#214460962
#214456782
#214456662
#214456592
#214456292
C61×#30762
#2298461