Finite Group Databasebeta

← #29657778

Subgroups — #29657778

Order 15616.

946 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457161C211yesmaximal, characteristic, complemented
2#21456374C211yesmaximal, characteristic, complemented
3#21455616C211yesmaximal, characteristic, complemented
4#21445089C212yesmaximal, complemented
5#21445012C212yesmaximal, complemented
6#21444991C212yesmaximal, complemented
7#21444946C212yesmaximal, complemented
8#21444496C211yesmaximal, characteristic, complemented
9#21444274C211yesmaximal, characteristic, complemented
10C2×D4⋊#92418C211yesmaximal, characteristic, complemented
11C61×#3506C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840158C2211yescharacteristic, complemented
14#16840134C2211yescharacteristic, complemented
15#16840114C2211yescharacteristic, complemented
16#16840108C2211yescharacteristic, complemented
17#16840062—22no—
18#16840057C2212yescomplemented
19#16840054—21no—
20#16840052C2212yes—
21#16840042C2212yes—
22#16840001—22no—
23#16839989—22no—
24#16839972C2212yescomplemented
25#16839969—21no—
26C4×#11758200—22no—
27#16839802—21no—
28#16839786C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839767—21no—
31C2×C4×#92418C2211yescharacteristic
32#16839508C2212yescomplemented
33#16839506C2212yescomplemented
34Q8⋊#92418C2212yescomplemented
35Q8⋊#92418C2212yescomplemented
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#559—22no—
39C61×#556C2212yes—
40C61×#555—21no—
41C61×#553—22no—
42C61×#550C2212yes—
43C61×#547—21no—
44C61×#538—22no—
45C61×#535—21no—
46C61×#533C2211yescharacteristic
47C61×#532C2211yescharacteristic
48C61×#525—22no—
49C61×#520—21no—
50C2×C4×D4×C61C2211yescharacteristic
51C2×#11758202—22no—
52C2×#11758200—22no—
53C2×#11758194—21no—
54C22×#92418—21no—
55C22×#92418—21no—
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519769—21no—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63#14519736—21no—
64#14519710—22no—
65#14519710—22no—
66#14519708—22no—
67#14519708—22no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71C4⋊#92418C2312yes—
72#14519701—22no—
73#14519700—22no—
74#14519700—22no—
75#14519698—22no—
76#14519698—22no—
77#14519698—22no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C2×#41×C61—22no—
82C2×#41×C61—21no—
83C22×D4×C61D411yescharacteristic
84C23×C4×C61—21no—
85C4⋊Q8×C61D411yescharacteristic
86C4⋊Q8×C61D411yescharacteristic
87C4⋊Q8×C61C2311yescharacteristic
88#127×C61—22no—
89#127×C61C2311yescharacteristic
90#126×C61—24no—
91#125×C61—21no—
92#124×C61—24no—
93#124×C61—22no—
94#124×C61—21no—
95#123×C61—24no—
96#123×C61—24no—
97C22⋊Q8×C61—22no—
98C22⋊Q8×C61—21no—
99C22⋊Q8×C61D412yes—
100#121×C61—22no—
101#121×C61—22no—
102#121×C61—24no—
103#121×C61—21no—
104#121×C61—22no—
105#121×C61D412yes—
106#120×C61—24no—
107C4×Q8×C61—22no—
108C4×Q8×C61—21no—
109C4×Q8×C61C2312yes—
110C4×D4×C61—24no—
111C4×D4×C61—24no—
112C4×D4×C61—22no—
113C4×D4×C61—21no—
114C4×D4×C61—21no—
115C4×D4×C61—21no—
116C4×D4×C61D412yes—
117C4×D4×C61D412yes—
118C4×D4×C61C2312yes—
119#117×C61—22no—
120#117×C61—21no—
121#117×C61—22no—
122C2×C4⋊C4×C61—22no—
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C42×C61C2311yescharacteristic
126C24×C61—21no—
127#41×C61—42no—
128#41×C61—42no—
129#41×C61—82no—
130#41×C61—82no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61C2×D412yes—
134C2×D4×C61—22no—
135C2×D4×C61—24no—
136C2×D4×C61—24no—
137C2×D4×C61—21no—
138C2×D4×C61—22no—
139C2×D4×C61—22no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61C2×D412yes—
143C2×D4×C61C2×D412yes—
144C22×C4×C61—24no—
145C22×C4×C61—21no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—81no—
150C22×C4×C61—81no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C2×Dic122—21no—
155C2×Dic122—21no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61C2×D412yes—
165C4⋊C4×C61C2×D412yes—
166C4⋊C4×C61C2×D412yes—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C42×C61—24no—
176C42×C61—22no—
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61C2411yescharacteristic
181C22⋊Dic61—41no—
182#11758202—82no—
183#11758200—44no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188#11758196—41no—
189#11758196—41no—
190C4×Dic61—41no—
191#11758194—42no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—42no—
197C2×#92418—81no—
198C23×C61—22no—
199C23×C61—22no—
200C23×C61—42no—
201C23×C61—81no—
202C23×C61—81no—
203C23×C61#12011yescharacteristic
204Q8×C61—22no—
205Q8×C61—22no—
206Q8×C61—41no—
207Q8×C61—41no—
208Q8×C61—81no—
209D4×C61—22no—
210D4×C61—22no—
211D4×C61—42no—
212D4×C61—44no—
213D4×C61—41no—
214D4×C61—41no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—42no—
218D4×C61—84no—
219D4×C61—84no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—41no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—44no—
230C2×C4×C61—82no—
231C2×C4×C61—81no—
232C2×C4×C61—84no—
233C2×C4×C61#12011yescharacteristic
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#13512yes—
237C2×C4×C61#13712yes—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×Dic61—41no—
242C2×Dic61—41no—
243Dic122—42no—
244Dic122—41no—
245Dic122—41no—
246#92418—22no—
247#92418—41no—
248#92418—42no—
249#92418—42no—
250#18679—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—84no—
260C22×C61#45212yes—
261C22×C61C2×#12011yescharacteristic
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—42no—
266C4×C61—42no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61#45212yes—
270C4×C61C2×#13511yescharacteristic
271C4×C61C2×#13711yescharacteristic
272C4×C61#57611yescharacteristic
273C4×C61#57711yescharacteristic
274Dic61—81no—
275Dic61—81no—
276#3506D6111yescharacteristic
277#3249—611no—
278#3167—611no—
279#3077—611no—
280#1652—612no—
281#1641—612no—
282#1638—612no—
283#1631—612no—
284#1547—611no—
285#1493—611no—
286C2×D4⋊C8—611no—
287C2×C61—22no—
288C2×C61—82no—
289C2×C61—82no—
290C2×C61C2×#45211yescharacteristic
291C2×C61#301211yescharacteristic
292C2×C61#301811yescharacteristic
293#559—22no—
294#556C2×D6112yes—
295#555—21no—
296#553—22no—
297#550C2×D6112yes—
298#547—21no—
299#538—22no—
300#535—21no—
301#533—611no—
302#533C2×D6111yescharacteristic
303#532C2×D6111yescharacteristic
304#525—22no—
305#520—21no—
306C2×C4×D4C2×D6111yescharacteristic
307#498—611no—
308#492—611no—
309#487—611no—
310#486—611no—
311#478—1222no—
312#477—612no—
313#477—1221no—
314#476—612no—
315#474—612no—
316#467—1222no—
317#465—1222no—
318#462—612no—
319#462—1221no—
320C4×SD16—1222no—
321#428—1221no—
322#423—611no—
323C2×C4⋊C8—611no—
324#418—1221no—
325C2×C4×C8—611no—
326#332—612no—
327#331—612no—
328Q8⋊C8—612no—
329Q8⋊C8—612no—
330D4⋊C8—612no—
331D4⋊C8—612no—
332C61#1867911yescharacteristic, complemented, Sylow
333C2×#41—22no—
334C2×#41—21no—
335C22×D4C22⋊D6111yescharacteristic
336C23×C4—21no—
337C2×Q16—1222no—
338C2×SD16—1222no—
339C2×M16—1221no—
340C22×C8—1221no—
341C22×C8—1221no—
342C4⋊Q8—611no—
343C4⋊Q8—611no—
344C4⋊Q8C22⋊D6111yescharacteristic
345C4⋊Q8C22⋊D6111yescharacteristic
346C4⋊Q8C22×D6111yescharacteristic
347#127—22no—
348#127C22×D6111yescharacteristic
349#126—24no—
350#125—21no—
351#125—1221no—
352#124—24no—
353#124—22no—
354#124—21no—
355#123—24no—
356#123—24no—
357C22⋊Q8—22no—
358C22⋊Q8—21no—
359C22⋊Q8—1221no—
360C22⋊Q8—1221no—
361C22⋊Q8C22⋊D6112yes—
362#121—22no—
363#121—22no—
364#121—24no—
365#121—21no—
366#121—22no—
367#121C22⋊D6112yes—
368#120—24no—
369C4×Q8—22no—
370C4×Q8—21no—
371C4×Q8—1221no—
372C4×Q8—1221no—
373C4×Q8C22×D6112yes—
374C4×D4—24no—
375C4×D4—24no—
376C4×D4—22no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C22⋊D6112yes—
381C4×D4C22⋊D6112yes—
382C4×D4C22×D6112yes—
383#117—22no—
384#117—21no—
385#117—22no—
386#117—1221no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4C22⋊D6111yescharacteristic
389C2×C22⋊C4—21no—
390C2×C42C22×D6111yescharacteristic
391#107—1222no—
392#107—1222no—
393#106—1222no—
394#106—1222no—
395C4⋊C8—612no—
396C4⋊C8—612no—
397C4⋊C8—1221no—
398C4⋊C8—1221no—
399#103—1222no—
400#103—1222no—
401#103—1222no—
402#102—1222no—
403#102—1222no—
404#102—1222no—
405C4×C8—611no—
406C4×C8—611no—
407C4×C8—1221no—
408C24—21no—
409#41—42no—
410#41—42no—
411#41—82no—
412#41—82no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—1221no—
416C2×Q8—1221no—
417C2×Q8C2×C22⋊D6112yes—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4C2×C22⋊D6112yes—
427C2×D4C2×C22⋊D6112yes—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438Q16—4882no—
439SD16—2444no—
440M16—2442no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—4881no—
446C2×C8—2442no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—2441no—
460C4⋊C4—2441no—
461C4⋊C4C2×C22⋊D6112yes—
462C4⋊C4C2×C22⋊D6112yes—
463C4⋊C4C2×C22⋊D6112yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—2441no—
473C42—24no—
474C42—22no—
475C42—2441no—
476C42C2×C22⋊D6111yescharacteristic
477C42C2×C22⋊D6111yescharacteristic
478C42C2×C22⋊D6111yescharacteristic
479C42C23×D6111yescharacteristic
480C23—22no—
481C23—22no—
482C23—42no—
483C23—81no—
484C23—81no—
485C23#1451974611yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—41no—
489Q8—41no—
490Q8—81no—
491Q8—2442no—
492Q8—2441no—
493Q8—2441no—
494D4—22no—
495D4—22no—
496D4—42no—
497D4—44no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—42no—
503D4—84no—
504D4—84no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—84no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4#1451974611yescharacteristic
521C2×C4#1451974611yescharacteristic
522C2×C4#1451974611yescharacteristic
523C2×C4#1451979712yes—
524C2×C4#1451980712yes—
525C2×C4C22×C22⋊D6111yescharacteristic
526C2×C4C22×C22⋊D6111yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C8—1222no—
529C8—2441no—
530C8—2442no—
531C8—2442no—
532C22—21no—
533C22—21no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—84no—
541C22#1683992412yes—
542C22C2×#1451974611yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—4881no—
551C4—4881no—
552C4#1683992412yes—
553C4C2×#1451979711yescharacteristic
554C4C2×#1451980711yescharacteristic
555C4#1684056511yescharacteristic
556C4#1684057311yescharacteristic
557C2—22no—
558C2—82no—
559C2—82no—
560C2C2×#1683992411yescharacteristic
561C2#2145512311yescharacteristic
562C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3506C2
socleC22×C61C2×#120
hypercenter#3506D61
hypocenterC61#18679
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21457161C2semidirect
2. #21456374C2semidirect
3. #21455616C2semidirect
4. #21445089C2semidirect
5. #21445012C2semidirect
6. #21444991C2semidirect
7. #21444946C2semidirect
8. #21444496C2semidirect
9. #21444274C2semidirect
10. C2×D4⋊#92418C2semidirect
13. #16840158C22semidirect
14. #16840134C22semidirect
15. #16840114C22semidirect
16. #16840108C22semidirect
18. #16840057C22semidirect
24. #16839972C22semidirect
32. #16839508C22semidirect
33. #16839506C22semidirect
34. Q8⋊#92418C22semidirect
35. Q8⋊#92418C22semidirect
332. C61#18679semidirect

Maximal subgroups

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

subgroupindex
#214571612
#214563742
#214556162
#214450892
#214450122
#214449912
#214449462
#214444962
#214442742
C2×D4⋊#924182
C61×#35062
#1867961