Finite Group Databasebeta

← #29687865

Subgroups — #29687865

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#21455607C211yesmaximal, characteristic
2#21454759C211yesmaximal, characteristic, complemented
3#21446763C211yesmaximal, characteristic, complemented
4#21446594C211yesmaximal, characteristic, complemented
5#21446418C211yesmaximal, characteristic, complemented
6#21446418C211yesmaximal, characteristic, complemented
7#21446094C211yesmaximal, characteristic
8#21446094C211yesmaximal, characteristic
9#21445839C211yesmaximal, characteristic
10#21445704C211yesmaximal, characteristic, complemented
11#21445699C211yesmaximal, characteristic, complemented
12#21445691C211yesmaximal, characteristic, complemented
13#21445661C211yesmaximal, characteristic
14#21445654C211yesmaximal, characteristic
15C61×#2875C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840069C411yescharacteristic, complemented
18#16840069C411yescharacteristic, complemented
19#16840062C411yescharacteristic, complemented
20#16840062C411yescharacteristic, complemented
21#16840057C411yescharacteristic, complemented
22#16840052C411yescharacteristic, complemented
23#16840047C411yescharacteristic, complemented
24#16840042C411yescharacteristic, complemented
25#16839890—21no—
26#16839890—21no—
27#16839890—21no—
28#16839890—21no—
29C2×#14519712C2211yescharacteristic
30C2×#14519712C2211yescharacteristic
31#16839798C2211yescharacteristic
32#16839798C2211yescharacteristic
33#16839788C2211yescharacteristic
34#16839788C2211yescharacteristic
35#16839786C2211yescharacteristic
36C2×C4⋊#92418C2211yescharacteristic
37C2×#14519704C2211yescharacteristic
38C2×#14519704C2211yescharacteristic
39#16839758C2211yescharacteristic
40#16839758C2211yescharacteristic
41#16839717C2211yescharacteristic
42#16839717C2211yescharacteristic
43C2×#14519687C2211yescharacteristic
44C2×C8×Dic61C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46#16839526C2211yescharacteristic
47#16839526C2211yescharacteristic
48#16839523C2211yescharacteristic
49#16839523C2211yescharacteristic
50#16839523C2211yescharacteristic
51#16839523C2211yescharacteristic
52#16839521C2211yescharacteristic
53#16839521C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×#131×C61C2211yescharacteristic
57C2×#131×C61C2211yescharacteristic
58C4×#41×C61C2211yescharacteristic
59C61×#405—21no—
60C61×#405—21no—
61C61×#405C2211yescharacteristic
62C61×#405C2211yescharacteristic
63C4×M16×C61—21no—
64C4×M16×C61C2211yescharacteristic
65C2×#97×C61—21no—
66C2×#97×C61C2211yescharacteristic
67C2×#11758194C2311yescharacteristic
68C2×#11758194C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519777C2×C411yescharacteristic
72#14519777C2×C411yescharacteristic
73#14519769C2×C411yescharacteristic
74#14519769C2×C411yescharacteristic
75#14519756—21no—
76#14519756—21no—
77C4×Dic122—21no—
78C4×Dic122—21no—
79#14519736C2311yescharacteristic
80#14519736C2311yescharacteristic
81C2×C4⋊Dic61—21no—
82C2×C4×Dic61C2311yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84#14519712—22no—
85#14519712—22no—
86#14519710C2×C411yescharacteristic
87#14519710C2×C411yescharacteristic
88#14519710C2×C411yescharacteristic
89#14519710C2×C411yescharacteristic
90#14519708C2×C411yescharacteristic
91#14519708C2×C411yescharacteristic
92#14519708C2×C411yescharacteristic
93#14519708C2×C411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418—21no—
96C4⋊#92418C2×C411yescharacteristic
97C4⋊#92418C2×C411yescharacteristic
98C4⋊#92418C2×C411yescharacteristic
99C4⋊#92418C2×C411yescharacteristic
100#14519706—21no—
101#14519706—21no—
102#14519706—21no—
103#14519706—21no—
104#14519704—22no—
105#14519704—22no—
106#14519701C2×C411yescharacteristic
107#14519701C2×C411yescharacteristic
108#14519701C2×C411yescharacteristic
109#14519701C2×C411yescharacteristic
110#14519698C2×C411yescharacteristic
111#14519698C2×C411yescharacteristic
112#14519698C2×C411yescharacteristic
113#14519698C2×C411yescharacteristic
114#14519687—21no—
115#14519687—21no—
116#14519687—22no—
117C8×Dic61—21no—
118C8×Dic61—21no—
119C8×Dic61—22no—
120#14519684—21no—
121C2×#41×C61C2311yescharacteristic
122#131×C61—22no—
123#131×C61—22no—
124#131×C61—22no—
125#131×C61Q814yescomplemented
126C2×M16×C61—21no—
127C2×M16×C61—21no—
128C2×M16×C61C2311yescharacteristic
129C2×M16×C61C2311yescharacteristic
130C22×C8×C61—21no—
131C22×C8×C61—21no—
132C22×C8×C61C2311yescharacteristic
133C22×C8×C61C2311yescharacteristic
134C4×Q8×C61C2×C411yescharacteristic
135C4×Q8×C61C2×C411yescharacteristic
136C4×D4×C61—21no—
137C4×D4×C61—21no—
138C4×D4×C61C2×C411yescharacteristic
139C4×D4×C61C2×C411yescharacteristic
140#117×C61—21no—
141#117×C61C2311yescharacteristic
142C2×C42×C61—21no—
143C2×C42×C61C2311yescharacteristic
144#97×C61—21no—
145#97×C61—21no—
146#97×C61—21no—
147#97×C61D411yescharacteristic
148#97×C61D411yescharacteristic
149#97×C61D411yescharacteristic
150#97×C61D411yescharacteristic
151C4×C8×C61—21no—
152C4×C8×C61—21no—
153C4×C8×C61—21no—
154#41×C61—21no—
155#41×C61—21no—
156#41×C61C2×Q812yes—
157#41×C61#4112yes—
158C2×Q8×C61C22×C411yescharacteristic
159C2×D4×C61—21no—
160C2×D4×C61C22×C411yescharacteristic
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61#4111yescharacteristic
165C22×C4×C61#4111yescharacteristic
166C22×C4×C61C2411yescharacteristic
167C22×Dic61—21no—
168C22×Dic61—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171M16×C61—24no—
172M16×C61—21no—
173M16×C61—21no—
174M16×C61—24no—
175M16×C61C2×Q812yes—
176M16×C61#4112yes—
177C2×C8×C61—24no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—24no—
183C2×C8×C61C2×D411yescharacteristic
184C2×C8×C61C2×D411yescharacteristic
185C2×C8×C61C2×D411yescharacteristic
186C2×C8×C61C2×D411yescharacteristic
187C2×C8×C61C2×Q812yes—
188C2×C8×C61#4112yes—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61C22×C411yescharacteristic
192C4⋊C4×C61C22×C411yescharacteristic
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C42×C61—21no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61C22×C411yescharacteristic
200C42×C61C22×C411yescharacteristic
201C42×C61C2×D411yescharacteristic
202C42×C61C2×D411yescharacteristic
203C22⋊Dic61—21no—
204C22⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—42no—
208C4⋊Dic61C22×C411yescharacteristic
209C4⋊Dic61C22×C411yescharacteristic
210C4⋊Dic61C22×C411yescharacteristic
211C4⋊Dic61C22×C411yescharacteristic
212#11758196—21no—
213#11758196—21no—
214#11758196—21no—
215#11758196—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222#11758194—21no—
223#11758194—21no—
224#11758194—21no—
225#11758194—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418C22×C411yescharacteristic
233C2×#92418C22×C411yescharacteristic
234C2×#92418C22×C411yescharacteristic
235C2×#92418C22×C411yescharacteristic
236C23×C61—21no—
237C23×C61C2×#4111yescharacteristic
238Q8×C61—21no—
239Q8×C61C4×Q812yes—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—21no—
243D4×C61C4×Q812yes—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—81no—
252C2×C4×C61—81no—
253C2×C4×C61C4×D411yescharacteristic
254C2×C4×C61C4×D411yescharacteristic
255C2×C4×C61C4×D411yescharacteristic
256C2×C4×C61C4×D411yescharacteristic
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×Q811yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×Dic61—22no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273Dic122—81no—
274Dic122—81no—
275C8×C61—22no—
276C8×C61—21no—
277C8×C61—22no—
278C8×C61—21no—
279C8×C61C22⋊Q812yes—
280C8×C61C22⋊Q812yes—
281#92418—21no—
282#92418—21no—
283#92418—21no—
284#92418—21no—
285#92418—41no—
286#92418—41no—
287#21629—611nomaximal, Sylow
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—22no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61C2×C4×D411yescharacteristic
294C22×C61#52911yescharacteristic
295C22×C61#53311yescharacteristic
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—22no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61C2×C4×Q811yescharacteristic
304C4×C61C4×#4111yescharacteristic
305C4×C61C2×C22⋊Q811yescharacteristic
306C4×C61#52511yescharacteristic
307Dic61—42no—
308Dic61—41no—
309Dic61—41no—
310Dic61—41no—
311Dic61—41no—
312Dic61—81no—
313#3076—611no—
314#2961—611no—
315#2875D6111yescharacteristic
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#1771—611no—
326#1765—611no—
327#1764—611no—
328C2×C61—21no—
329C2×C61—21no—
330C2×C61—42no—
331C2×C61C4×C22⋊Q811yescharacteristic
332C2×C61#295611yescharacteristic
333C2×C61#295711yescharacteristic
334C2×#131C2×D6111yescharacteristic
335C2×#131C2×D6111yescharacteristic
336#533—611no—
337C4×#41C2×D6111yescharacteristic
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⋊C8—611no—
358C2×#104—611no—
359C2×#104—611no—
360#416—611no—
361#416—611no—
362#405—21no—
363#405—21no—
364#405—611no—
365#405—611no—
366#405C2×D6111yescharacteristic
367#405C2×D6111yescharacteristic
368C4×M16—21no—
369C4×M16C2×D6111yescharacteristic
370C2×#97—21no—
371C2×#97—611no—
372C2×#97C2×D6111yescharacteristic
373C2×C4×C8—611no—
374C4×C4⋊C4—611no—
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#336—611no—
384#336—611no—
385C61#2162911yescharacteristic, complemented, Sylow
386C2×#41C22×D6111yescharacteristic
387#131—22no—
388#131—22no—
389#131—22no—
390#131Dic12214yescomplemented
391C2×M16—21no—
392C2×M16—21no—
393C2×M16—611no—
394C2×M16—611no—
395C2×M16C22×D6111yescharacteristic
396C2×M16C22×D6111yescharacteristic
397C22×C8—21no—
398C22×C8—21no—
399C22×C8—611no—
400C22×C8—611no—
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—1221no—
410C4×Q8—1221no—
411C4×Q8C4×D6111yescharacteristic
412C4×Q8C4×D6111yescharacteristic
413C4×D4—21no—
414C4×D4—21no—
415C4×D4C4×D6111yescharacteristic
416C4×D4C4×D6111yescharacteristic
417#117—21no—
418#117—611no—
419#117—611no—
420#117C22×D6111yescharacteristic
421C2×C4⋊C4—1221no—
422C2×C42—21no—
423C2×C42—611no—
424C2×C42—611no—
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—611no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8—1221no—
441C4⋊C8—1221no—
442C4⋊C8—1221no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445C4⋊C8—1221no—
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—21no—
457#97—21no—
458#97—21no—
459#97—1222no—
460#97—1221no—
461#97—1221no—
462#97C22⋊D6111yescharacteristic
463#97C22⋊D6111yescharacteristic
464#97C22⋊D6111yescharacteristic
465#97C22⋊D6111yescharacteristic
466C4×C8—21no—
467C4×C8—21no—
468C4×C8—21no—
469C4×C8—1222no—
470C4×C8—1221no—
471C4×C8—1221no—
472#95—1221no—
473#41—21no—
474#41—21no—
475#41C2×Dic12212yes—
476#41#1175822212yes—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8C2×C4×D6111yescharacteristic
480C2×D4—21no—
481C2×D4C2×C4×D6111yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4#1175822211yescharacteristic
488C22×C4#1175822211yescharacteristic
489C22×C4C23×D6111yescharacteristic
490M16—24no—
491M16—21no—
492M16—21no—
493M16—24no—
494M16—1221no—
495M16—1221no—
496M16—1221no—
497M16—1221no—
498M16C2×Dic12212yes—
499M16#1175822212yes—
500C2×C8—24no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—24no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8C2×Dic12212yes—
517C2×C8#1175822212yes—
518C2×C8C2×C22⋊D6111yescharacteristic
519C2×C8C2×C22⋊D6111yescharacteristic
520C2×C8C2×C22⋊D6111yescharacteristic
521C2×C8C2×C22⋊D6111yescharacteristic
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—611no—
525C4⋊C4—611no—
526C4⋊C4—611no—
527C4⋊C4—611no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—2442no—
535C4⋊C4C2×C4×D6111yescharacteristic
536C4⋊C4C2×C4×D6111yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—1221no—
541C22⋊C4—1221no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—1221no—
546C42—1221no—
547C42—1221no—
548C42—1221no—
549C42—1221no—
550C42—1221no—
551C42C2×C4×D6111yescharacteristic
552C42C2×C4×D6111yescharacteristic
553C42C2×C22⋊D6111yescharacteristic
554C42C2×C22⋊D6111yescharacteristic
555C23—21no—
556C23C2×#1175822211yescharacteristic
557Q8—21no—
558Q8—4881no—
559Q8—4881no—
560Q8C4×Dic12212yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4C4×Dic12212yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—1222no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4C4×C22⋊D6111yescharacteristic
585C2×C4C4×C22⋊D6111yescharacteristic
586C2×C4C4×C22⋊D6111yescharacteristic
587C2×C4C4×C22⋊D6111yescharacteristic
588C2×C4C22×C4×D6111yescharacteristic
589C2×C4C22×C22⋊D6111yescharacteristic
590C2×C4C22×Dic12211yescharacteristic
591C2×C4C2×#1175822211yescharacteristic
592C2×C4C2×#1175822211yescharacteristic
593C2×C4C2×#1175822211yescharacteristic
594C8—22no—
595C8—21no—
596C8—22no—
597C8—21no—
598C8—1221no—
599C8—1221no—
600C8—1221no—
601C8—1221no—
602C8—2441no—
603C8—2441no—
604C8#1451975612yes—
605C8#1451975612yes—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C2×C4×C22⋊D6111yescharacteristic
612C22#1684029611yescharacteristic
613C22#1684031911yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—41no—
620C4—41no—
621C4—4881no—
622C4—2442no—
623C4—2441no—
624C4—2441no—
625C4—2441no—
626C4—2441no—
627C4C2×C4×Dic12211yescharacteristic
628C4C4×#1175822211yescharacteristic
629C4C2×#1451975611yescharacteristic
630C4#1684027311yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2C4×#1451975611yescharacteristic
635C2#2145472611yescharacteristic
636C2#2145473411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2875C2
socleC22×C61C2×C4×D4
prime radicalC2×#41×C61C23
cosocleC22×C4×C61C24
hypercenter#2875D61
hypocenterC61#21629
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #21454759C2semidirect
3. #21446763C2semidirect
4. #21446594C2semidirect
5. #21446418C2semidirect
6. #21446418C2semidirect
10. #21445704C2semidirect
11. #21445699C2semidirect
12. #21445691C2semidirect
17. #16840069C4semidirect
18. #16840069C4semidirect
19. #16840062C4semidirect
20. #16840062C4semidirect
21. #16840057C4semidirect
22. #16840052C4semidirect
23. #16840047C4semidirect
24. #16840042C4semidirect
125. #131×C61Q8semidirect
385. C61#21629semidirect
390. #131Dic122semidirect

Maximal subgroups

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

subgroupindex
#214556072
#214547592
#214467632
#214465942
#214464182
#214460942
#214458392
#214457042
#214456992
#214456912
#214456612
#214456542
C61×#28752
#2162961