Finite Group Databasebeta

← #30617345

Subgroups — #30617345

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#21462612C211yesmaximal, characteristic, complemented
2#21462390C211yesmaximal, characteristic
3#21461994C211yesmaximal, characteristic, complemented
4#21461935C211yesmaximal, characteristic, complemented
5#21461887C211yesmaximal, characteristic, complemented
6#21461884C211yesmaximal, characteristic, complemented
7#21461788C211yesmaximal, characteristic
8#21461786C211yesmaximal, characteristic
9#21461693C211yesmaximal, characteristic
10#21461626C211yesmaximal, characteristic, complemented
11#21461623C211yesmaximal, characteristic, complemented
12#21461620C211yesmaximal, characteristic, complemented
13#21461598C211yesmaximal, characteristic
14#21461594C211yesmaximal, characteristic
15#21455600C211yesmaximal, characteristic
16C2×#14519864C2211yescharacteristic
17C2×#14519864C2211yescharacteristic
18#16840814—21no—
19#16840814—21no—
20#16840813—21no—
21#16840813—21no—
22C2×#14519848C2211yescharacteristic
23C2×#14519847C2211yescharacteristic
24#16840769C2211yescharacteristic
25#16840769C2211yescharacteristic
26#16840767C2211yescharacteristic
27C2×#14519841C2211yescharacteristic
28C2×#14519841C2211yescharacteristic
29#16840727C2211yescharacteristic
30#16840727C2211yescharacteristic
31#16840727C2211yescharacteristic
32#16840727C2211yescharacteristic
33#16840726—21no—
34#16840726—21no—
35C4×#11758213C2211yescharacteristic
36C4×#11758217—21no—
37#16840721C2211yescharacteristic
38C2×#14519832—21no—
39C2×#14519831C2211yescharacteristic
40C2×C8×#6923C2211yescharacteristic
41C4×C4⋊#6923C2211yescharacteristic
42#16840638C2211yescharacteristic
43#16840638C2211yescharacteristic
44#16840632C2211yescharacteristic
45#16840632C2211yescharacteristic
46#16840632C2211yescharacteristic
47#16840632C2211yescharacteristic
48#16840632C2211yescharacteristic
49#16840632C2211yescharacteristic
50(C2×#6923)⋊C8C2211yescharacteristic
51(C2×#6923)⋊C8C2211yescharacteristic
52#16840317C2211yescharacteristic
53C4×#11758224C2211yescharacteristic
54#16840068C411yescharacteristic, complemented
55#16840068C411yescharacteristic, complemented
56#16840060C411yescharacteristic, complemented
57#16840060C411yescharacteristic, complemented
58#16840058C411yescharacteristic, complemented
59#16840052C411yescharacteristic, complemented
60#16840044C411yescharacteristic, complemented
61#16839799C2211yescharacteristic
62#16839799C2211yescharacteristic
63C4⋊C8×D61C2211yescharacteristic
64#16839755C2211yescharacteristic
65#16839755C2211yescharacteristic
66C61×#474C411yescharacteristic, complemented
67#14519864—22no—
68#14519864—22no—
69#14519864—22no—
70#14519864Q814yescomplemented
71C2×#11758213C2311yescharacteristic
72C2×#11758213C2311yescharacteristic
73C2×#11758213C2311yescharacteristic
74C2×#11758213C2311yescharacteristic
75C2×#11758217—21no—
76C2×#11758217—21no—
77C2×D61⋊C8C2311yescharacteristic
78C2×D61⋊C8C2311yescharacteristic
79C22×#92420—21no—
80C22×#92420—21no—
81C2×C4⋊#6923—21no—
82C2×C4×#6923C2311yescharacteristic
83C2×C4×#6923C2311yescharacteristic
84#14519848—22no—
85#14519847—22no—
86#14519844—21no—
87#14519844—21no—
88#14519844—21no—
89#14519844—21no—
90C4⋊#92420—21no—
91C4⋊#92420—21no—
92#14519841—22no—
93#14519841—22no—
94#14519832—21no—
95#14519832—21no—
96#14519832—21no—
97#14519832—21no—
98#14519832—21no—
99#14519832D411yescharacteristic
100#14519832D411yescharacteristic
101#14519832D411yescharacteristic
102#14519832D411yescharacteristic
103#14519831—22no—
104C4×#92420—21no—
105C4×#92420—21no—
106C4×#92420—21no—
107C4×#92420—21no—
108C4×#92420—21no—
109C8×#6923—22no—
110(C2×#6923)⋊C4—21no—
111C2×#11758224C2311yescharacteristic
112C2×C8×D61C2311yescharacteristic
113C2×C8×D61C2311yescharacteristic
114#14519777C2×C411yescharacteristic
115#14519768C2×C411yescharacteristic
116#14519768C2×C411yescharacteristic
117#14519757—21no—
118#14519753—21no—
119Q8×Dic61—21no—
120C4×Dic122C2×C411yescharacteristic
121#14519742—21no—
122#14519742C2×C411yescharacteristic
123D4×Dic61C2×C411yescharacteristic
124C4×C22⋊D61—21no—
125#14519737—21no—
126#14519735C2311yescharacteristic
127#14519735C2311yescharacteristic
128#14519735C2311yescharacteristic
129#14519733—21no—
130C2×C4×Dic61—21no—
131C42×D61C2311yescharacteristic
132#14519711C2×C411yescharacteristic
133#14519711C2×C411yescharacteristic
134#14519708C2×C411yescharacteristic
135#14519708C2×C411yescharacteristic
136#14519708C2×C411yescharacteristic
137#14519708C2×C411yescharacteristic
138C4⋊#92418C2×C411yescharacteristic
139#14519706C2×C411yescharacteristic
140#14519706C2×C411yescharacteristic
141#14519702C2×C411yescharacteristic
142#14519702C2×C411yescharacteristic
143#14519700C2×C411yescharacteristic
144#14519700C2×C411yescharacteristic
145#14519698C2×C411yescharacteristic
146#14519698C2×C411yescharacteristic
147C4⋊Q8×C61C2×C411yescharacteristic
148C4×D4×C61C2×C411yescharacteristic
149#107×C61C2×C411yescharacteristic
150#107×C61C2×C411yescharacteristic
151C4⋊C8×C61C2×C411yescharacteristic
152#102×C61C2×C411yescharacteristic
153#102×C61C2×C411yescharacteristic
154C22×#6923—21no—
155C22×#6923—21no—
156C2×Q8×C61—21no—
157C2×D4×C61C22×C411yescharacteristic
158C22×C4×C61—21no—
159C2×C22⋊D61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162#11758224—21no—
163#11758224—21no—
164#11758224C2×Q812yes—
165#11758224#4112yes—
166C2×C4×D61#4111yescharacteristic
167C2×C4×D61#4111yescharacteristic
168C2×C4×D61C2411yescharacteristic
169C2×Dic122—21no—
170C2×Dic122C22×C411yescharacteristic
171#11758217—24no—
172#11758217—24no—
173C2×#92420—24no—
174C2×#92420—21no—
175C2×#92420—21no—
176C2×#92420—24no—
177C2×#92420—21no—
178C2×#92420—21no—
179C2×#92420C2×D411yescharacteristic
180C2×#92420C2×D411yescharacteristic
181C2×#92420C2×D411yescharacteristic
182C2×#92420C2×D411yescharacteristic
183C4⋊#6923—42no—
184C4×#6923—21no—
185C4×#6923—21no—
186C4×#6923—21no—
187C4×#6923—21no—
188#11758213—21no—
189#11758213—21no—
190#11758213—21no—
191#11758213—21no—
192#11758213—21no—
193#11758213—21no—
194#11758213C2×Q812yes—
195#11758213#4112yes—
196D61⋊C8—21no—
197D61⋊C8—21no—
198D61⋊C8C2×Q812yes—
199D61⋊C8#4112yes—
200C2×C8×C61C22×C411yescharacteristic
201C2×C8×C61C22×C411yescharacteristic
202C4⋊C4×C61—21no—
203C4⋊C4×C61C22×C411yescharacteristic
204C4⋊C4×C61C22×C411yescharacteristic
205C4⋊C4×C61C22×C411yescharacteristic
206C22⋊C4×C61—21no—
207C42×C61C22×C411yescharacteristic
208C22⋊Dic61—21no—
209#11758198—21no—
210#11758198—21no—
211#11758198—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61C22×C411yescharacteristic
216C4⋊Dic61C22×C411yescharacteristic
217C4⋊Dic61C22×C411yescharacteristic
218#11758196—21no—
219#11758196—21no—
220#11758196—21no—
221#11758196—21no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—21no—
226C4×Dic61—21no—
227C4×Dic61C22×C411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C2×#92418C22×C411yescharacteristic
231C2×#92418C22×C411yescharacteristic
232C8×D61—21no—
233C8×D61—21no—
234C8×D61—21no—
235C8×D61—21no—
236C23×C61—21no—
237C22×D61C2×#4111yescharacteristic
238C2×#6923—22no—
239C2×#6923—41no—
240C2×#6923—41no—
241C2×#6923—41no—
242C2×#6923—41no—
243C2×#6923—41no—
244Q8×C61—81no—
245D4×C61—21no—
246D4×C61C4×Q812yes—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—81no—
251C2×C4×C61C4×D411yescharacteristic
252C2×C4×C61C4×D411yescharacteristic
253C2×C4×C61C23×C411yescharacteristic
254C22⋊D61—22no—
255C22⋊D61—22no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—81no—
263C2×Dic61C4×D411yescharacteristic
264C2×Dic61C4×D411yescharacteristic
265C2×Dic61C22×D411yescharacteristic
266C4×D61—41no—
267C4×D61—41no—
268C4×D61C22×Q811yescharacteristic
269C4×D61C2×#4111yescharacteristic
270C4×D61C2×#4111yescharacteristic
271C4×D61C2×#4111yescharacteristic
272Dic122—21no—
273Dic122—81no—
274Dic122C4×Q812yes—
275#92420—22no—
276#92420—21no—
277#92420—21no—
278#92420—22no—
279#92420—41no—
280#92420—41no—
281#92420C22⋊Q812yes—
282#92420C22⋊Q812yes—
283C8×C61—21no—
284C8×C61—21no—
285#92418—21no—
286#92418—21no—
287#21629—611nomaximal, Sylow
288C22×C61—22no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61C2×C4×D411yescharacteristic
292C2×D61—21no—
293C2×D61—21no—
294C2×D61#52911yescharacteristic
295C2×D61#53311yescharacteristic
296#6923—42no—
297#6923—81no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61C2×C4×Q811yescharacteristic
304C4×C61C4×#4111yescharacteristic
305Dic61—21no—
306Dic61—21no—
307Dic61—22no—
308Dic61—41no—
309Dic61—41no—
310Dic61—41no—
311Dic61C2×C22⋊Q811yescharacteristic
312Dic61#52511yescharacteristic
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#1771—611no—
326#1765—611no—
327#1764—611no—
328C2×C61—42no—
329C2×C61C4×C22⋊Q811yescharacteristic
330C2×C61#295611yescharacteristic
331C2×C61#295711yescharacteristic
332D61—21no—
333D61—21no—
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#692311yescharacteristic, complemented
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—611no—
363#405—611no—
364#405—611no—
365#405—611no—
366#405—1221no—
367#405—1221no—
368C4×M16—611no—
369C4×M16—1221no—
370C2×#97—611no—
371C2×#97—611no—
372C2×#97—1221no—
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×#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—611no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8—1221no—
402C22×C8—1221no—
403C4⋊Q8—611no—
404C4⋊Q8C2×#692311yescharacteristic
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—1221no—
415C4×D4—1221no—
416C4×D4C2×#692311yescharacteristic
417#117—611no—
418#117—611no—
419#117—611no—
420#117—1221no—
421C2×C4⋊C4—1221no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—1221no—
426#108—1222no—
427#108—1222no—
428#107—611no—
429#107—611no—
430#107C2×#692311yescharacteristic
431#107C2×#692311yescharacteristic
432#106—611no—
433#106—611no—
434#106—611no—
435#106—611no—
436C4⋊C8—611no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8—1221no—
440C4⋊C8—1221no—
441C4⋊C8—1221no—
442C4⋊C8—1221no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445C4⋊C8C2×#692311yescharacteristic
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#102C2×#692311yescharacteristic
455#102C2×#692311yescharacteristic
456#97—611no—
457#97—611no—
458#97—611no—
459#97—611no—
460#97—1221no—
461#97—1222no—
462#97—1221no—
463#97—1221no—
464#97—1221no—
465#97—1221no—
466C4×C8—1221no—
467C4×C8—1222no—
468C4×C8—1221no—
469C4×C8—1221no—
470C4×C8—1221no—
471C4×C8—1221no—
472#95—1221no—
473#41—612no—
474#41—612no—
475#41—1221no—
476#41—1221no—
477C2×Q8—21no—
478C2×Q8—611no—
479C2×Q8—1221no—
480C2×D4—1221no—
481C2×D4C22×#692311yescharacteristic
482C22×C4—21no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490M16—612no—
491M16—612no—
492M16—1224no—
493M16—1221no—
494M16—1221no—
495M16—1224no—
496M16—1221no—
497M16—1221no—
498M16—1221no—
499M16—1221no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—612no—
503C2×C8—612no—
504C2×C8—611no—
505C2×C8—611no—
506C2×C8—611no—
507C2×C8—611no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1224no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1224no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8C22×#692311yescharacteristic
521C2×C8C22×#692311yescharacteristic
522C4⋊C4—21no—
523C4⋊C4—611no—
524C4⋊C4—611no—
525C4⋊C4—611no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—2442no—
534C4⋊C4C22×#692311yescharacteristic
535C4⋊C4C22×#692311yescharacteristic
536C4⋊C4C22×#692311yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—1221no—
539C22⋊C4—1221no—
540C22⋊C4—1221no—
541C22⋊C4—1221no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—1221no—
546C42—1221no—
547C42—1221no—
548C42—1221no—
549C42—1221no—
550C42—1221no—
551C42—1221no—
552C42—1221no—
553C42—1221no—
554C42C22×#692311yescharacteristic
555C23—21no—
556C23—611no—
557Q8—81no—
558Q8—612no—
559Q8—1221no—
560Q8—4881no—
561D4—21no—
562D4—1222no—
563D4—1222no—
564D4Q8×#692312yes—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—81no—
569C2×C4—611no—
570C2×C4—611no—
571C2×C4—611no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1221no—
579C2×C4—1222no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—4881no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4D4×#692311yescharacteristic
592C2×C4D4×#692311yescharacteristic
593C2×C4C23×#692311yescharacteristic
594C8—21no—
595C8—21no—
596C8—612no—
597C8—612no—
598C8—1221no—
599C8—1221no—
600C8—1222no—
601C8—1222no—
602C8—1221no—
603C8—1221no—
604C8—2441no—
605C8—2441no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—611no—
610C22—611no—
611C22—1221no—
612C22—1221no—
613C22C2×D4×#692311yescharacteristic
614C4—21no—
615C4—21no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—611no—
620C4—611no—
621C4—1222no—
622C4—1221no—
623C4—1221no—
624C4—4881no—
625C4—2442no—
626C4—2441no—
627C4—2441no—
628C4—2441no—
629C4C2×Q8×#692311yescharacteristic
630C4#41×#692311yescharacteristic
631C2—42no—
632C2—1221no—
633C2—1221no—
634C2C22⋊Q8×#692311yescharacteristic
635C2#2146259411yescharacteristic
636C2#2146259511yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#474C4
socleC22×C61C2×C4×D4
prime radicalC2×#11758224C23
cosocleC2×C4×D61C24
hypercenter#474#6923
hypocenterC61#21629
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #21462612C2semidirect
3. #21461994C2semidirect
4. #21461935C2semidirect
5. #21461887C2semidirect
6. #21461884C2semidirect
10. #21461626C2semidirect
11. #21461623C2semidirect
12. #21461620C2semidirect
54. #16840068C4semidirect
55. #16840068C4semidirect
56. #16840060C4semidirect
57. #16840060C4semidirect
58. #16840058C4semidirect
59. #16840052C4semidirect
60. #16840044C4semidirect
66. C61×#474C4semidirect
70. #14519864Q8semidirect
345. #474#6923semidirect
385. C61#21629semidirect

Maximal subgroups

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

subgroupindex
#214626122
#214623902
#214619942
#214619352
#214618872
#214618842
#214617882
#214617862
#214616932
#214616262
#214616232
#214616202
#214615982
#214615942
#214556002
#2162961