Finite Group Databasebeta

← #29657800

Subgroups — #29657800

Order 15616.

946 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457135C211yesmaximal, characteristic, complemented
2#21456401C211yesmaximal, characteristic, complemented
3#21455616C211yesmaximal, characteristic, complemented
4#21445054C212yesmaximal, complemented
5#21445047C212yesmaximal, complemented
6#21444970C212yesmaximal, complemented
7#21444964C212yesmaximal, complemented
8#21444509C211yesmaximal, characteristic, complemented
9#21444270C211yesmaximal, characteristic, complemented
10C2×D4⋊#92418C211yesmaximal, characteristic, complemented
11C61×#3506C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840165C2211yescharacteristic, complemented
14#16840141C2211yescharacteristic, complemented
15#16840114C2211yescharacteristic, complemented
16#16840112C2211yescharacteristic, complemented
17#16840062—22no—
18#16840052C2212yes—
19#16840052C2212yescomplemented
20#16840044—21no—
21#16840042C2212yes—
22#16840001—22no—
23#16839989—22no—
24#16839967C2212yescomplemented
25#16839959—21no—
26C4×#11758202—22no—
27#16839802—21no—
28#16839786C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839767—21no—
31C2×C4×#92418C2211yescharacteristic
32#16839511C2212yescomplemented
33#16839508C2212yescomplemented
34D4⋊#92418C2212yes—
35D4⋊#92418C2212yes—
36D4⋊#92418C2212yescomplemented
37D4⋊#92418C2212yescomplemented
38C61×#559—22no—
39C61×#556—21no—
40C61×#555C2212yes—
41C61×#553—22no—
42C61×#550—21no—
43C61×#547C2212yes—
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×C61C2311yescharacteristic
87C4⋊Q8×C61C2311yescharacteristic
88#127×C61—22no—
89#127×C61D411yescharacteristic
90#126×C61—24no—
91#125×C61—21no—
92#124×C61—22no—
93#124×C61—24no—
94#124×C61—21no—
95#123×C61—24no—
96#123×C61—24no—
97C22⋊Q8×C61—21no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61D412yes—
100#121×C61—24no—
101#121×C61—22no—
102#121×C61—22no—
103#121×C61—22no—
104#121×C61—21no—
105#121×C61D412yes—
106#120×C61—24no—
107C4×Q8×C61—22no—
108C4×Q8×C61—21no—
109C4×Q8×C61D412yes—
110C4×D4×C61—24no—
111C4×D4×C61—22no—
112C4×D4×C61—24no—
113C4×D4×C61—21no—
114C4×D4×C61—21no—
115C4×D4×C61—21no—
116C4×D4×C61D412yes—
117C4×D4×C61C2312yes—
118C4×D4×C61C2312yes—
119#117×C61—21no—
120#117×C61—22no—
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—21no—
139C2×D4×C61—21no—
140C2×D4×C61—22no—
141C2×D4×C61—22no—
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—22no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
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—44no—
183#11758200—82no—
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—42no—
205Q8×C61—41no—
206Q8×C61—41no—
207Q8×C61—42no—
208D4×C61—22no—
209D4×C61—22no—
210D4×C61—22no—
211D4×C61—22no—
212D4×C61—44no—
213D4×C61—42no—
214D4×C61—42no—
215D4×C61—81no—
216D4×C61—84no—
217D4×C61—84no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—82no—
230C2×C4×C61—81no—
231C2×C4×C61—84no—
232C2×C4×C61#12011yescharacteristic
233C2×C4×C61#12011yescharacteristic
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#13512yes—
236C2×C4×C61#13712yes—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×Dic61—41no—
241C2×Dic61—41no—
242Dic122—42no—
243Dic122—41no—
244Dic122—41no—
245#92418—22no—
246#92418—41no—
247#92418—42no—
248#92418—42no—
249#18681—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—84no—
259C22×C61#45212yes—
260C22×C61C2×#12011yescharacteristic
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—42no—
265C4×C61—42no—
266C4×C61—42no—
267C4×C61—42no—
268C4×C61#45212yes—
269C4×C61C2×#13511yescharacteristic
270C4×C61C2×#13711yescharacteristic
271C4×C61#57611yescharacteristic
272C4×C61#57711yescharacteristic
273Dic61—81no—
274Dic61—81no—
275#3506D6111yescharacteristic
276#3246—611no—
277#3170—611no—
278#3077—611no—
279#1647—612no—
280#1646—612no—
281#1635—612no—
282#1634—612no—
283#1550—611no—
284#1492—611no—
285C2×D4⋊C8—611no—
286C2×C61—22no—
287C2×C61—82no—
288C2×C61—82no—
289C2×C61C2×#45211yescharacteristic
290C2×C61#301211yescharacteristic
291C2×C61#301811yescharacteristic
292#559—22no—
293#556—21no—
294#555C2×D6112yes—
295#553—22no—
296#550—21no—
297#547C2×D6112yes—
298#538—22no—
299#535—21no—
300#533—611no—
301#533C2×D6111yescharacteristic
302#532C2×D6111yescharacteristic
303#525—22no—
304#520—21no—
305C2×C4×D4C2×D6111yescharacteristic
306#500—611no—
307#494—611no—
308#487—611no—
309#487—611no—
310#478—1222no—
311#476—612no—
312#476—612no—
313#475—1221no—
314#474—612no—
315#467—1222no—
316#465—1222no—
317#461—612no—
318#460—1221no—
319C4×Q16—1222no—
320#428—1221no—
321#423—611no—
322C2×C4⋊C8—611no—
323#418—1221no—
324C2×C4×C8—611no—
325#333—612no—
326#332—612no—
327D4⋊C8—612no—
328D4⋊C8—612no—
329D4⋊C8—612no—
330D4⋊C8—612no—
331C61#1868111yescharacteristic, complemented, Sylow
332C2×#41—22no—
333C2×#41—21no—
334C22×D4C22⋊D6111yescharacteristic
335C23×C4—21no—
336C2×Q16—1222no—
337C2×SD16—1222no—
338C2×M16—1221no—
339C22×C8—1221no—
340C22×C8—1221no—
341C4⋊Q8—611no—
342C4⋊Q8—611no—
343C4⋊Q8C22⋊D6111yescharacteristic
344C4⋊Q8C22×D6111yescharacteristic
345C4⋊Q8C22×D6111yescharacteristic
346#127—22no—
347#127C22⋊D6111yescharacteristic
348#126—24no—
349#125—21no—
350#125—1221no—
351#124—22no—
352#124—24no—
353#124—21no—
354#123—24no—
355#123—24no—
356C22⋊Q8—21no—
357C22⋊Q8—22no—
358C22⋊Q8—1221no—
359C22⋊Q8—1221no—
360C22⋊Q8C22⋊D6112yes—
361#121—24no—
362#121—22no—
363#121—22no—
364#121—22no—
365#121—21no—
366#121C22⋊D6112yes—
367#120—24no—
368C4×Q8—22no—
369C4×Q8—21no—
370C4×Q8—1221no—
371C4×Q8—1221no—
372C4×Q8C22⋊D6112yes—
373C4×D4—24no—
374C4×D4—22no—
375C4×D4—24no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4C22⋊D6112yes—
380C4×D4C22×D6112yes—
381C4×D4C22×D6112yes—
382#117—21no—
383#117—22no—
384#117—22no—
385#117—1221no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4C22⋊D6111yescharacteristic
388C2×C22⋊C4—21no—
389C2×C42C22×D6111yescharacteristic
390#107—1222no—
391#107—1222no—
392#106—1222no—
393#106—1222no—
394C4⋊C8—612no—
395C4⋊C8—612no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398#103—1222no—
399#103—1222no—
400#103—1222no—
401#102—1222no—
402#102—1222no—
403#102—1222no—
404C4×C8—611no—
405C4×C8—611no—
406C4×C8—1221no—
407C24—21no—
408#41—42no—
409#41—42no—
410#41—82no—
411#41—82no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—1221no—
415C2×Q8—1221no—
416C2×Q8C2×C22⋊D6112yes—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4C2×C22⋊D6112yes—
426C2×D4C2×C22⋊D6112yes—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437Q16—2444no—
438SD16—4882no—
439M16—2442no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—4881no—
445C2×C8—2442no—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—2441no—
459C4⋊C4—2441no—
460C4⋊C4C2×C22⋊D6112yes—
461C4⋊C4C2×C22⋊D6112yes—
462C4⋊C4C2×C22⋊D6112yes—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—2441no—
472C42—24no—
473C42—22no—
474C42—2441no—
475C42C2×C22⋊D6111yescharacteristic
476C42C2×C22⋊D6111yescharacteristic
477C42C2×C22⋊D6111yescharacteristic
478C42C23×D6111yescharacteristic
479C23—22no—
480C23—22no—
481C23—42no—
482C23—81no—
483C23—81no—
484C23#1451974611yescharacteristic
485Q8—42no—
486Q8—41no—
487Q8—41no—
488Q8—42no—
489Q8—2442no—
490Q8—2441no—
491Q8—2441no—
492D4—22no—
493D4—22no—
494D4—22no—
495D4—22no—
496D4—44no—
497D4—42no—
498D4—42no—
499D4—81no—
500D4—84no—
501D4—84no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—84no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4#1451974611yescharacteristic
519C2×C4#1451974611yescharacteristic
520C2×C4#1451974611yescharacteristic
521C2×C4#1451979712yes—
522C2×C4#1451980712yes—
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4C22×C22⋊D6111yescharacteristic
525C2×C4C22×C22⋊D6111yescharacteristic
526C8—1222no—
527C8—2441no—
528C8—2442no—
529C8—2442no—
530C22—21no—
531C22—21no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—84no—
539C22#1683992412yes—
540C22C2×#1451974611yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—4881no—
549C4—4881no—
550C4#1683992412yes—
551C4C2×#1451979711yescharacteristic
552C4C2×#1451980711yescharacteristic
553C4#1684056511yescharacteristic
554C4#1684057311yescharacteristic
555C2—22no—
556C2—82no—
557C2—82no—
558C2C2×#1683992411yescharacteristic
559C2#2145512311yescharacteristic
560C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3506C2
socleC22×C61C2×#120
hypercenter#3506D61
hypocenterC61#18681
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. #21457135C2semidirect
2. #21456401C2semidirect
3. #21455616C2semidirect
4. #21445054C2semidirect
5. #21445047C2semidirect
6. #21444970C2semidirect
7. #21444964C2semidirect
8. #21444509C2semidirect
9. #21444270C2semidirect
10. C2×D4⋊#92418C2semidirect
13. #16840165C22semidirect
14. #16840141C22semidirect
15. #16840114C22semidirect
16. #16840112C22semidirect
19. #16840052C22semidirect
24. #16839967C22semidirect
32. #16839511C22semidirect
33. #16839508C22semidirect
36. D4⋊#92418C22semidirect
37. D4⋊#92418C22semidirect
331. C61#18681semidirect

Maximal subgroups

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

subgroupindex
#214571352
#214564012
#214556162
#214450542
#214450472
#214449702
#214449642
#214445092
#214442702
C2×D4⋊#924182
C61×#35062
#1868161