Finite Group Databasebeta

← #29670271

Subgroups — #29670271

Order 15616.

564 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458729C211yesmaximal, characteristic
2#21458694C211yesmaximal, characteristic
3#21458495C211yesmaximal, characteristic, complemented
4#21445495C211yesmaximal, characteristic, complemented
5#21445327C211yesmaximal, characteristic, complemented
6#21445270C211yesmaximal, characteristic
7#21445221C211yesmaximal, characteristic, complemented
8#21445220C211yesmaximal, characteristic, complemented
9#21445033C211yesmaximal, characteristic
10#21444975C211yesmaximal, characteristic, complemented
11#21444958C211yesmaximal, characteristic, complemented
12#21444953C211yesmaximal, characteristic
13#21444670C211yesmaximal, characteristic, complemented
14#21444631C211yesmaximal, characteristic
15C61×#3338C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17#16840490C2211yescharacteristic
18#16840172—21no—
19#16840165C2211yescharacteristic
20#16840161—21no—
21#16840158C2211yescharacteristic
22#16840141C2211yescharacteristic
23#16840113C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840111—21no—
26#16840066—21no—
27#16840066—21no—
28#16840062—21no—
29#16840057C2211yescharacteristic
30#16840056—21no—
31#16840054C2211yescharacteristic
32#16840054C2211yescharacteristic
33#16840052C2211yescharacteristic
34#16840042—21no—
35#16840042C2211yescharacteristic
36#16839972C2211yescharacteristic
37#16839970—21no—
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40C8×Dic122C2211yescharacteristic
41#16839862—21no—
42#16839854—21no—
43C4×#11758200—21no—
44C4×#11758200C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839509C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#554C2211yescharacteristic
57C61×#554C2211yescharacteristic
58C61×#500C2211yescharacteristic
59C61×#484—21no—
60C61×#484—21no—
61C61×#477C2211yescharacteristic
62C61×#477C2211yescharacteristic
63C61×#467—21no—
64C4×Q16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#107×C61—21no—
67C2×#11758202—21no—
68C2×#11758202—41no—
69C2×#11758200D411yescharacteristic
70#14519777—21no—
71#14519777C2311yescharacteristic
72#14519777C2311yescharacteristic
73#14519776—21no—
74#14519776—41no—
75#14519775—21no—
76#14519769—21no—
77#14519767—41no—
78Q8×Dic61—21no—
79C4×Dic122—21no—
80C4×Dic122—21no—
81C4×Dic122C2311yescharacteristic
82#14519742—21no—
83#14519710—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708D411yescharacteristic
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519702—21no—
100#14519702—21no—
101#14519701—21no—
102#14519701—41no—
103#14519700—21no—
104#14519700—21no—
105#14519700—21no—
106#14519700D411yescharacteristic
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110#14519688—21no—
111C4×#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×Q16×C61D411yescharacteristic
116C22×C8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119#125×C61—21no—
120#125×C61—21no—
121C22⋊Q8×C61—21no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—41no—
124C22⋊Q8×C61—41no—
125C4×Q8×C61C2311yescharacteristic
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61—21no—
132#107×C61—21no—
133#107×C61—21no—
134#107×C61—21no—
135#107×C61—21no—
136#107×C61D411yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#103×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—41no—
151C22×C4×C61—41no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—41no—
155C2×Dic122C2×D411yescharacteristic
156Q16×C61—41no—
157C2×C8×C61—21no—
158C2×C8×C61—42no—
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2×D411yescharacteristic
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—41no—
168C4⋊C4×C61—41no—
169C4⋊C4×C61—41no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊C4×C61—41no—
176C22⋊C4×C61—41no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61C2411yescharacteristic
180#11758202—42no—
181#11758202—82no—
182#11758200Q1614yescomplemented
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—41no—
188C4⋊Dic61C2×D411yescharacteristic
189#11758196—21no—
190#11758196—41no—
191#11758196—41no—
192#11758196—41no—
193#11758196—41no—
194C4×Dic61—21no—
195C4×Dic61—41no—
196C4×Dic61—41no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418C2×D411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203C23×C61—21no—
204Q8×C61—21no—
205Q8×C61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—81no—
209Q8×C61—81no—
210D4×C61—21no—
211D4×C61C2×Q1612yes—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—161no—
222C2×C4×C61—161no—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61H(5,𝔽2)11yescharacteristic
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230Dic122—21no—
231Dic122—41no—
232Dic122—41no—
233Dic122—81no—
234Dic122—161no—
235Dic122C2×Q1612yes—
236C8×C61—21no—
237C8×C61—21no—
238C8×C61—41no—
239#92418—21no—
240#92418—41no—
241#92418—41no—
242#92418—41no—
243#92418—41no—
244#92418C2×Q1612yes—
245#19816—611nomaximal, Sylow
246C22×C61—22no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61D4211yescharacteristic
250C4×C61—21no—
251C4×C61—41no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61—41no—
255C4×C61—41no—
256C4×C61—81no—
257C4×C61—81no—
258C4×C61C22×Q1611yescharacteristic
259C4×C61#57511yescharacteristic
260C4×C61#57811yescharacteristic
261C4×C61#57811yescharacteristic
262Dic61—22no—
263Dic61—41no—
264Dic61—81no—
265Dic61—81no—
266#3385—611no—
267#3381—611no—
268#3363—611no—
269#3338D6111yescharacteristic
270#1723—611no—
271#1692—611no—
272#1683—611no—
273#1676—611no—
274#1676—611no—
275#1644—611no—
276#1636—611no—
277#1633—611no—
278#1632—611no—
279#1584—611no—
280SD16⋊C8—611no—
281C2×C61—42no—
282C2×C61#326911yescharacteristic
283C2×C61D4×Q1611yescharacteristic
284C2×C61#328811yescharacteristic
285Q82—611no—
286#557—611no—
287#554C2×D6111yescharacteristic
288#554C2×D6111yescharacteristic
289#501—1221no—
290#500—611no—
291#500C2×D6111yescharacteristic
292#499—1221no—
293#498—611no—
294#494—611no—
295#487—611no—
296#487—611no—
297#487—1221no—
298#484—21no—
299#484—21no—
300#479—1221no—
301#479—1221no—
302#478—1221no—
303#477—611no—
304#477—611no—
305#477—611no—
306#477—1221no—
307#477C2×D6111yescharacteristic
308#477C2×D6111yescharacteristic
309#476—611no—
310#474—611no—
311#474—1221no—
312#467—21no—
313#462—611no—
314#462—1221no—
315#446—611no—
316C8×Q8—611no—
317C8×Q8—611no—
318#440—1221no—
319C4×Q16—1221no—
320C4×Q16C2×D6111yescharacteristic
321C4×SD16—611no—
322C4×SD16—1221no—
323C8×D4C2×D6111yescharacteristic
324C2×#107—21no—
325#335—611no—
326#335—611no—
327#333—611no—
328#333—611no—
329#333—611no—
330#332—611no—
331Q8⋊C8—611no—
332Q8⋊C8—611no—
333Q8⋊C8—611no—
334D4⋊C8—611no—
335#322—611no—
336C61#1981611yescharacteristic, complemented, Sylow
337C2×Q16—1221no—
338C2×Q16—2441no—
339C2×Q16C22⋊D6111yescharacteristic
340C2×SD16—611no—
341C22×C8—21no—
342C4⋊Q8—611no—
343C4⋊Q8—611no—
344C4⋊Q8—1221no—
345C4⋊Q8—1221no—
346C4⋊Q8—1221no—
347C4⋊Q8—2441no—
348C4⋊Q8C22×D6111yescharacteristic
349C4⋊Q8C22×D6111yescharacteristic
350#125—21no—
351#125—21no—
352#125—1221no—
353#125—2441no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—41no—
357C22⋊Q8—41no—
358C4×Q8—611no—
359C4×Q8—1221no—
360C4×Q8—1221no—
361C4×Q8—1221no—
362C4×Q8—1221no—
363C4×Q8C22×D6111yescharacteristic
364C4×Q8C22×D6111yescharacteristic
365C4×D4—21no—
366C4×D4—21no—
367C4×D4C22×D6111yescharacteristic
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370#107—21no—
371#107—21no—
372#107—21no—
373#107—21no—
374#107—1221no—
375#107—1221no—
376#107—1221no—
377#107—1221no—
378#107—1221no—
379#107—1221no—
380#107C22⋊D6111yescharacteristic
381#106—611no—
382#106—1221no—
383#106—1221no—
384#106—1221no—
385C4⋊C8—611no—
386C4⋊C8—611no—
387C4⋊C8—611no—
388C4⋊C8—1221no—
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391C4⋊C8C22×D6111yescharacteristic
392#103—21no—
393#103—21no—
394#103—611no—
395#103—1221no—
396#103—1221no—
397#103—1221no—
398#103—1221no—
399#103—1221no—
400#103—1221no—
401#103—2441no—
402#103C22⋊D6111yescharacteristic
403#103C22⋊D6111yescharacteristic
404#102—611no—
405#102—1221no—
406#102—1221no—
407C22⋊C8—21no—
408#97—1221no—
409C4×C8—611no—
410C4×C8—611no—
411C4×C8—1221no—
412C4×C8—1221no—
413C4×C8C22×D6111yescharacteristic
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—611no—
417C2×Q8—1221no—
418C2×Q8—1221no—
419C2×Q8—2441no—
420C2×Q8C2×C22⋊D6111yescharacteristic
421C2×Q8C2×C22⋊D6111yescharacteristic
422C2×D4D4×D6111yescharacteristic
423C22×C4—21no—
424C22×C4—41no—
425C22×C4—41no—
426Q16—41no—
427Q16—4882no—
428Q16—2442no—
429SD16—614no—
430C2×C8—21no—
431C2×C8—42no—
432C2×C8—611no—
433C2×C8—611no—
434C2×C8—1221no—
435C2×C8—1221no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8C2×C22⋊D6111yescharacteristic
439C2×C8C2×C22⋊D6111yescharacteristic
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—611no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—2441no—
457C4⋊C4—2441no—
458C4⋊C4—2441no—
459C4⋊C4—2441no—
460C4⋊C4—2441no—
461C4⋊C4D4×D6111yescharacteristic
462C4⋊C4C2×C22⋊D6111yescharacteristic
463C4⋊C4C2×C22⋊D6111yescharacteristic
464C22⋊C4—21no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C42—21no—
468C42—21no—
469C42—1221no—
470C42—2441no—
471C42—2441no—
472C42C23×D6111yescharacteristic
473C23—21no—
474Q8—21no—
475Q8—21no—
476Q8—21no—
477Q8—21no—
478Q8—81no—
479Q8—81no—
480Q8—612no—
481Q8—1221no—
482Q8—9761no—
483Q8—4881no—
484Q8—2441no—
485Q8—2441no—
486D4—21no—
487D4Q16×D6112yes—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—161no—
498C2×C4—161no—
499C2×C4—1221no—
500C2×C4—1221no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4C2×D4×D6111yescharacteristic
504C2×C4C22×C22⋊D6111yescharacteristic
505C2×C4#1451982211yescharacteristic
506C8—21no—
507C8—21no—
508C8—41no—
509C8—612no—
510C8—1221no—
511C8—2441no—
512C8—2441no—
513C8—2441no—
514C8—2441no—
515C22—22no—
516C22—41no—
517C22—41no—
518C22D4×C22⋊D6111yescharacteristic
519C4—21no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—81no—
526C4—81no—
527C4—1222no—
528C4—4881no—
529C4—4881no—
530C4—2441no—
531C4C2×Q16×D6111yescharacteristic
532C4#1684056011yescharacteristic
533C4#1684057511yescharacteristic
534C4#1684057711yescharacteristic
535C2—42no—
536C2#2145737011yescharacteristic
537C2Q16×C22⋊D6111yescharacteristic
538C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3338C2
socleC22×C61D42
prime radicalC2×D4×C61C2×D4
hypercenter#3338D61
hypocenterC61#19816
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3338

Complements

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

NH ≅ G/Ntype
3. #21458495C2semidirect
4. #21445495C2semidirect
5. #21445327C2semidirect
7. #21445221C2semidirect
8. #21445220C2semidirect
10. #21444975C2semidirect
11. #21444958C2semidirect
13. #21444670C2semidirect
182. #11758200Q16semidirect
336. C61#19816semidirect

Maximal subgroups

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

subgroupindex
#214587292
#214586942
#214584952
#214454952
#214453272
#214452702
#214452212
#214452202
#214450332
#214449752
#214449582
#214449532
#214446702
#214446312
C61×#33382
#1981661