Finite Group Databasebeta

← #29687867

Subgroups — #29687867

Order 15616.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455609C211yesmaximal, characteristic
2#21454759C211yesmaximal, characteristic, complemented
3#21454244C211yesmaximal, characteristic
4#21446762C211yesmaximal, characteristic, complemented
5#21446595C211yesmaximal, characteristic, complemented
6#21446419C211yesmaximal, characteristic, complemented
7#21446419C211yesmaximal, characteristic, complemented
8#21446097C211yesmaximal, characteristic
9#21446097C211yesmaximal, characteristic
10#21445840C211yesmaximal, characteristic
11#21445703C211yesmaximal, characteristic, complemented
12#21445698C211yesmaximal, characteristic, complemented
13#21445663C211yesmaximal, characteristic
14#21445656C211yesmaximal, characteristic
15C61×#1771C211yesmaximal, characteristic, complemented
16C2×#14519783C2211yescharacteristic
17C2×#14519783C2211yescharacteristic
18#16840319C2211yescharacteristic
19C4×#11758222C2211yescharacteristic
20#16840070C411yescharacteristic, complemented
21#16840070C411yescharacteristic, complemented
22#16840063C411yescharacteristic, complemented
23#16840063C411yescharacteristic, complemented
24#16840058C411yescharacteristic, complemented
25#16840053C411yescharacteristic, complemented
26#16840048C411yescharacteristic, complemented
27#16840043C411yescharacteristic, complemented
28#16839890—21no—
29#16839890—21no—
30#16839890—21no—
31#16839890—21no—
32C2×#14519713C2211yescharacteristic
33C2×#14519713C2211yescharacteristic
34#16839800C2211yescharacteristic
35#16839800C2211yescharacteristic
36#16839788C2211yescharacteristic
37#16839788C2211yescharacteristic
38#16839786C2211yescharacteristic
39C2×#14519705C2211yescharacteristic
40C2×#14519705C2211yescharacteristic
41#16839759C2211yescharacteristic
42#16839759C2211yescharacteristic
43#16839719—21no—
44#16839719—21no—
45#16839717C2211yescharacteristic
46#16839717C2211yescharacteristic
47#16839717C2211yescharacteristic
48#16839717C2211yescharacteristic
49C4×#11758194C2211yescharacteristic
50#16839714—21no—
51#97×D61—21no—
52#16839527C2211yescharacteristic
53#16839527C2211yescharacteristic
54#16839522C2211yescharacteristic
55#16839522C2211yescharacteristic
56#16839522C2211yescharacteristic
57#16839522C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839522C2211yescharacteristic
60C2×C4⋊C8×C61C2211yescharacteristic
61C2×#97×C61C2211yescharacteristic
62C2×#97×C61C2211yescharacteristic
63C2×C4×C8×C61C2211yescharacteristic
64C4×C4⋊C4×C61C2211yescharacteristic
65C61×#336C2211yescharacteristic
66C61×#336C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519783—22no—
69#14519783—22no—
70#14519783—22no—
71#14519783Q814yescomplemented
72C2×#11758194C2311yescharacteristic
73C2×#11758194C2311yescharacteristic
74C2×#11758194C2311yescharacteristic
75C2×#11758194C2311yescharacteristic
76C2×#11758189—21no—
77C2×#11758189—21no—
78C2×C8×D61—21no—
79C2×C8×D61—21no—
80#14519777C2×C411yescharacteristic
81#14519777C2×C411yescharacteristic
82#14519769C2×C411yescharacteristic
83#14519769C2×C411yescharacteristic
84#14519756—21no—
85#14519756—21no—
86C4×Dic122—21no—
87C4×Dic122—21no—
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122C2×C411yescharacteristic
90C4×D244C2×C411yescharacteristic
91C4×D244C2×C411yescharacteristic
92C4×C22⋊D61—21no—
93C4×C22⋊D61—21no—
94#14519736C2311yescharacteristic
95#14519736C2311yescharacteristic
96#14519736C2311yescharacteristic
97#14519734—21no—
98C42×D61—21no—
99#14519713—22no—
100#14519713—22no—
101#14519711C2×C411yescharacteristic
102#14519711C2×C411yescharacteristic
103#14519711C2×C411yescharacteristic
104#14519711C2×C411yescharacteristic
105#14519709C2×C411yescharacteristic
106#14519709C2×C411yescharacteristic
107#14519709C2×C411yescharacteristic
108#14519709C2×C411yescharacteristic
109C4⋊#92418—21no—
110C4⋊#92418—21no—
111#14519706—21no—
112#14519706—21no—
113#14519706—21no—
114#14519706—21no—
115#14519705—22no—
116#14519705—22no—
117#14519702C2×C411yescharacteristic
118#14519702C2×C411yescharacteristic
119#14519702C2×C411yescharacteristic
120#14519702C2×C411yescharacteristic
121#14519699C2×C411yescharacteristic
122#14519699C2×C411yescharacteristic
123#14519699C2×C411yescharacteristic
124#14519699C2×C411yescharacteristic
125#14519688—21no—
126#14519687—21no—
127#14519687—21no—
128#14519687—21no—
129#14519687—21no—
130C4×#92418—21no—
131C8×Dic61—21no—
132C8×Dic61—21no—
133C8×Dic61—21no—
134C8×Dic61—21no—
135C22×C8×C61C2311yescharacteristic
136C22×C8×C61C2311yescharacteristic
137C22×C8×C61C2311yescharacteristic
138C22×C8×C61C2311yescharacteristic
139C2×C4⋊C4×C61—21no—
140C2×C42×C61C2311yescharacteristic
141C2×C42×C61C2311yescharacteristic
142C2×C42×C61C2311yescharacteristic
143C4⋊C8×C61C2×C411yescharacteristic
144C4⋊C8×C61C2×C411yescharacteristic
145C4⋊C8×C61C2×C411yescharacteristic
146C4⋊C8×C61C2×C411yescharacteristic
147#97×C61—22no—
148#97×C61D411yescharacteristic
149#97×C61D411yescharacteristic
150#97×C61D411yescharacteristic
151#97×C61D411yescharacteristic
152C4×C8×C61—22no—
153#95×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61#4111yescharacteristic
157C22×C4×C61#4111yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C2×C22⋊D61—21no—
160#11758222—21no—
161#11758222—21no—
162#11758222C2×Q812yes—
163#11758222#4112yes—
164C2×D244C22×C411yescharacteristic
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122C22×C411yescharacteristic
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C22×C411yescharacteristic
178C2×C8×C61C22×C411yescharacteristic
179C2×C8×C61C22×C411yescharacteristic
180C2×C8×C61C22×C411yescharacteristic
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C2×C8×C61C2×D411yescharacteristic
184C2×C8×C61C2×D411yescharacteristic
185C2×C8×C61C2×Q812yes—
186C2×C8×C61#4112yes—
187C4⋊C4×C61—42no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61C22×C411yescharacteristic
193C42×C61C22×C411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C22⋊Dic61—21no—
197C22⋊Dic61—21no—
198C22⋊Dic61—21no—
199#11758198—21no—
200#11758198—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C22×C411yescharacteristic
204C4⋊Dic61C22×C411yescharacteristic
205C4⋊Dic61C22×C411yescharacteristic
206C4⋊Dic61C22×C411yescharacteristic
207C4⋊Dic61C22×C411yescharacteristic
208C4⋊Dic61C22×C411yescharacteristic
209#11758196—21no—
210#11758196—21no—
211#11758196—21no—
212#11758196—21no—
213#11758196—21no—
214#11758196—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220#11758194—21no—
221#11758194—21no—
222#11758194—21no—
223#11758194—21no—
224#11758194—21no—
225#11758194—21no—
226#11758194C2×Q812yes—
227#11758194#4112yes—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232#11758189—24no—
233#11758189—24no—
234C8×D61—24no—
235C8×D61—24no—
236C23×C61C2×#4111yescharacteristic
237C22×D61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61C4×D411yescharacteristic
247C2×C4×C61C4×D411yescharacteristic
248C2×C4×C61C4×D411yescharacteristic
249C2×C4×C61C4×D411yescharacteristic
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×Q811yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C22⋊D61—22no—
257C22⋊D61—22no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265D244—21no—
266D244C4×Q812yes—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—81no—
270C4×D61—81no—
271Dic122—21no—
272Dic122—81no—
273Dic122—81no—
274Dic122C4×Q812yes—
275C8×C61—21no—
276C8×C61—21no—
277C8×C61—21no—
278C8×C61—21no—
279C8×C61—21no—
280C8×C61—21no—
281C8×C61C22⋊Q812yes—
282C8×C61C22⋊Q812yes—
283#92418—22no—
284#92418—22no—
285#92418—41no—
286#92418—41no—
287#21629—611nomaximal, Sylow
288C22×C61—21no—
289C22×C61—21no—
290C22×C61C2×C4×D411yescharacteristic
291C22×C61#52911yescharacteristic
292C22×C61#53311yescharacteristic
293C2×D61—22no—
294C2×D61—41no—
295C2×D61—41no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—42no—
301C4×C61—81no—
302C4×C61C2×C4×Q811yescharacteristic
303C4×C61C4×#4111yescharacteristic
304C4×C61C2×C22⋊Q811yescharacteristic
305C4×C61#52511yescharacteristic
306Dic61—22no—
307Dic61—41no—
308Dic61—41no—
309Dic61—41no—
310Dic61—41no—
311Dic61—41no—
312Dic61—41no—
313#3076—611no—
314#2961—611no—
315#2875—611no—
316#1986—611no—
317#1950—611no—
318#1922—611no—
319#1922—611no—
320#1866—611no—
321#1866—611no—
322#1806—611no—
323#1773—611no—
324#1772—611no—
325#1771D6111yescharacteristic, complemented
326#1765—611no—
327#1764—611no—
328C2×C61—21no—
329C2×C61—21no—
330C2×C61C4×C22⋊Q811yescharacteristic
331C2×C61#295611yescharacteristic
332C2×C61#295711yescharacteristic
333D61—42no—
334C2×#131—611no—
335C2×#131—611no—
336#533—611no—
337C4×#41—611no—
338#479—611no—
339#479—611no—
340#478—611no—
341#478—611no—
342#477—611no—
343#476—611no—
344#475—611no—
345#474—611no—
346#446—1221no—
347#446—1221no—
348#446—1221no—
349#446—1221no—
350C2×#108—611no—
351C2×#108—611no—
352#427—611no—
353#427—611no—
354#424—611no—
355#424—611no—
356#423—611no—
357C2×C4⋊C8C2×D6111yescharacteristic
358C2×#104—611no—
359C2×#104—611no—
360#416—611no—
361#416—611no—
362#405—611no—
363#405—611no—
364#405—611no—
365#405—611no—
366#405—1221no—
367#405—1221no—
368C4×M16—611no—
369C4×M16—1221no—
370C2×#97—1221no—
371C2×#97C2×D6111yescharacteristic
372C2×#97C2×D6111yescharacteristic
373C2×C4×C8C2×D6111yescharacteristic
374C4×C4⋊C4C2×D6111yescharacteristic
375#341—611no—
376#341—611no—
377#339—611no—
378#339—611no—
379#339—611no—
380#339—611no—
381#339—611no—
382#339—611no—
383#336C2×D6111yescharacteristic
384#336C2×D6111yescharacteristic
385C61#2162911yescharacteristic, complemented, Sylow
386C2×#41—611no—
387#131—614no—
388#131—1222no—
389#131—1222no—
390#131—1222no—
391C2×M16—611no—
392C2×M16—611no—
393C2×M16—611no—
394C2×M16—611no—
395C2×M16—1221no—
396C2×M16—1221no—
397C22×C8—1221no—
398C22×C8—1221no—
399C22×C8C22×D6111yescharacteristic
400C22×C8C22×D6111yescharacteristic
401C22×C8C22×D6111yescharacteristic
402C22×C8C22×D6111yescharacteristic
403C4⋊Q8—611no—
404C4⋊Q8—611no—
405#125—611no—
406#125—611no—
407C22⋊Q8—1221no—
408C22⋊Q8—1221no—
409C4×Q8—611no—
410C4×Q8—611no—
411C4×Q8—1221no—
412C4×Q8—1221no—
413C4×D4—611no—
414C4×D4—611no—
415C4×D4—1221no—
416C4×D4—1221no—
417#117—611no—
418#117—611no—
419#117—611no—
420#117—1221no—
421C2×C4⋊C4—21no—
422C2×C42—1221no—
423C2×C42C22×D6111yescharacteristic
424C2×C42C22×D6111yescharacteristic
425C2×C42C22×D6111yescharacteristic
426#108—1222no—
427#108—1222no—
428#107—611no—
429#107—611no—
430#107—611no—
431#107—611no—
432#106—611no—
433#106—611no—
434#106—611no—
435#106—611no—
436C4⋊C8—1221no—
437C4⋊C8—1221no—
438C4⋊C8—1221no—
439C4⋊C8—1221no—
440C4⋊C8—1221no—
441C4⋊C8—1221no—
442C4⋊C8C4×D6111yescharacteristic
443C4⋊C8C4×D6111yescharacteristic
444C4⋊C8C4×D6111yescharacteristic
445C4⋊C8C4×D6111yescharacteristic
446#104—1222no—
447#104—1222no—
448#103—611no—
449#103—611no—
450#103—611no—
451#103—611no—
452#102—611no—
453#102—611no—
454#102—611no—
455#102—611no—
456#97—22no—
457#97—1221no—
458#97—1221no—
459#97—1221no—
460#97—1221no—
461#97—1221no—
462#97D24411yescharacteristic
463#97D24411yescharacteristic
464#97D24411yescharacteristic
465#97D24411yescharacteristic
466C4×C8—22no—
467C4×C8—1221no—
468C4×C8—1221no—
469C4×C8—1221no—
470C4×C8—1221no—
471C4×C8—1221no—
472#95—21no—
473#41—612no—
474#41—612no—
475#41—1221no—
476#41—1221no—
477C2×Q8—611no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×D4—611no—
481C2×D4—1221no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4#1175822211yescharacteristic
488C22×C4#1175822211yescharacteristic
489C22×C4C23×D6111yescharacteristic
490M16—612no—
491M16—612no—
492M16—1224no—
493M16—1221no—
494M16—1221no—
495M16—1224no—
496M16—1221no—
497M16—1221no—
498M16—1221no—
499M16—1221no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—1224no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1224no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8C2×C4×D6111yescharacteristic
513C2×C8C2×C4×D6111yescharacteristic
514C2×C8C2×C4×D6111yescharacteristic
515C2×C8C2×C4×D6111yescharacteristic
516C2×C8C2×D24411yescharacteristic
517C2×C8C2×D24411yescharacteristic
518C2×C8C2×D24411yescharacteristic
519C2×C8C2×D24411yescharacteristic
520C2×C8#1175822412yes—
521C2×C8Q8×D6112yes—
522C4⋊C4—42no—
523C4⋊C4—611no—
524C4⋊C4—611no—
525C4⋊C4—611no—
526C4⋊C4—611no—
527C4⋊C4—611no—
528C4⋊C4—611no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C22⋊C4—1221no—
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C22⋊C4—1221no—
541C22⋊C4—1221no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—1221no—
547C42—1221no—
548C42—1221no—
549C42—1221no—
550C42—1221no—
551C42C2×C4×D6111yescharacteristic
552C42C2×C4×D6111yescharacteristic
553C42C2×D24411yescharacteristic
554C42C2×D24411yescharacteristic
555C23—1221no—
556C23C2×#1175822211yescharacteristic
557Q8—612no—
558Q8—1221no—
559Q8—4881no—
560Q8—4881no—
561D4—612no—
562D4—1222no—
563D4—1222no—
564D4—1221no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1221no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—4881no—
583C2×C4—4881no—
584C2×C4C4×D24411yescharacteristic
585C2×C4C4×D24411yescharacteristic
586C2×C4C4×D24411yescharacteristic
587C2×C4C4×D24411yescharacteristic
588C2×C4C22×C4×D6111yescharacteristic
589C2×C4C22×D24411yescharacteristic
590C2×C4C2×Q8×D6111yescharacteristic
591C2×C4#41×D6111yescharacteristic
592C2×C4#41×D6111yescharacteristic
593C2×C4C2×#1175822411yescharacteristic
594C8—21no—
595C8—21no—
596C8—21no—
597C8—21no—
598C8—21no—
599C8—21no—
600C8—1222no—
601C8—1222no—
602C8—2441no—
603C8—2441no—
604C8#1451975712yes—
605C8#1451975712yes—
606C22—21no—
607C22—21no—
608C22—1222no—
609C22—2441no—
610C22—2441no—
611C22C2×C4×D24411yescharacteristic
612C22#1684029411yescharacteristic
613C22#1684031611yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—42no—
619C4—81no—
620C4—1222no—
621C4—2441no—
622C4—2441no—
623C4—2441no—
624C4—2441no—
625C4—2441no—
626C4—2441no—
627C4C4×Q8×D6111yescharacteristic
628C4C4×#1175822411yescharacteristic
629C4C2×#1451975711yescharacteristic
630C4#1684027411yescharacteristic
631C2—21no—
632C2—21no—
633C2—2442no—
634C2C4×#1451975711yescharacteristic
635C2#2145473011yescharacteristic
636C2#2145473711yescharacteristic
subgroupG/H
centerC2×C4C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1771C2
socleC22×C61C2×C4×D4
prime radicalC2×#11758222C23
cosocleC22×C4×C61C24
hypercenter#1771D61
hypocenterC61#21629
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. #21454759C2semidirect
4. #21446762C2semidirect
5. #21446595C2semidirect
6. #21446419C2semidirect
7. #21446419C2semidirect
11. #21445703C2semidirect
12. #21445698C2semidirect
15. C61×#1771C2semidirect
20. #16840070C4semidirect
21. #16840070C4semidirect
22. #16840063C4semidirect
23. #16840063C4semidirect
24. #16840058C4semidirect
25. #16840053C4semidirect
26. #16840048C4semidirect
27. #16840043C4semidirect
71. #14519783Q8semidirect
325. #1771D61semidirect
385. C61#21629semidirect

Maximal subgroups

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

subgroupindex
#214556092
#214547592
#214542442
#214467622
#214465952
#214464192
#214460972
#214458402
#214457032
#214456982
#214456632
#214456562
C61×#17712
#2162961