Finite Group Databasebeta

← #29671006

Subgroups — #29671006

Order 15616.

572 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458684C211yesmaximal, characteristic
2#21458428C211yesmaximal, characteristic, complemented
3#21458207C211yesmaximal, characteristic, complemented
4#21445499C211yesmaximal, characteristic, complemented
5#21445340C211yesmaximal, characteristic, complemented
6#21445274C211yesmaximal, characteristic, complemented
7#21445223C211yesmaximal, characteristic, complemented
8#21445192C211yesmaximal, characteristic, complemented
9#21445024C211yesmaximal, characteristic, complemented
10#21444991C211yesmaximal, characteristic, complemented
11#21444953C211yesmaximal, characteristic, complemented
12#21444951C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic
14#21444636C211yesmaximal, characteristic, complemented
15C61×#3360C211yesmaximal, characteristic
16#16840469C2211yescharacteristic
17#16840469C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840165C2211yescharacteristic
20#16840163—21no—
21#16840158C2211yescharacteristic
22#16840141C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840103—21no—
25#16840097—21no—
26#16840095—21no—
27#16840090—21no—
28#16840060—21no—
29#16840060—21no—
30#16840057C2211yescharacteristic
31#16840057C2211yescharacteristic
32#16840054C2211yescharacteristic
33#16840052C2211yescharacteristic
34#16840052C2211yescharacteristic
35#16840044C2211yescharacteristic
36#16840018—21no—
37#16840001—21no—
38C8×Dic122C2211yescharacteristic
39C4×#11758202C2211yescharacteristic
40D8×Dic61—21no—
41#16839840C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839805—21no—
44C2×#14519710—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#556C2211yescharacteristic
57C61×#554C2211yescharacteristic
58C61×#488—21no—
59C61×#487C2211yescharacteristic
60C61×#479—21no—
61C61×#476C2211yescharacteristic
62C61×#463—21no—
63C61×#461C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758202—41no—
68C2×#11758202D411yescharacteristic
69C2×#11758194—21no—
70C22×#92418—21no—
71#14519777C2311yescharacteristic
72#14519777C2311yescharacteristic
73#14519768—21no—
74#14519768—21no—
75#14519756—21no—
76#14519756—21no—
77#14519754—41no—
78#14519754—41no—
79C4×Dic122C2311yescharacteristic
80D4×Dic61—21no—
81D4×Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84#14519711—21no—
85#14519711—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710D411yescharacteristic
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519706—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701D411yescharacteristic
104#14519700—21no—
105#14519700—21no—
106#14519700—41no—
107#14519700D411yescharacteristic
108#14519698—21no—
109#14519698—21no—
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C2×SD16×C61—21no—
116C2×D8×C61D411yescharacteristic
117C4⋊Q8×C61C2311yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119#126×C61—41no—
120#125×C61—21no—
121#124×C61—21no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C4×Q8×C61—21no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61C2311yescharacteristic
129C4×D4×C61C2311yescharacteristic
130#117×C61—21no—
131C2×C4⋊C4×C61—21no—
132#107×C61D411yescharacteristic
133#106×C61—21no—
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61—21no—
138#103×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61C2×D411yescharacteristic
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—41no—
152C22×Dic61—41no—
153C22×Dic61—41no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122C2×D411yescharacteristic
157SD16×C61—42no—
158D8×C61—22no—
159C2×C8×C61—21no—
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61C2×D411yescharacteristic
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×C61C2×D411yescharacteristic
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D411yescharacteristic
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—41no—
176C22⋊C4×C61—41no—
177C22⋊C4×C61—41no—
178C42×C61—21no—
179C42×C61—41no—
180C42×C61C2411yescharacteristic
181C22⋊Dic61—41no—
182C22⋊Dic61—41no—
183#11758202—22no—
184#11758202—82no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—41no—
190C4⋊Dic61—41no—
191C4⋊Dic61C2×D411yescharacteristic
192#11758196—21no—
193#11758196—41no—
194#11758196—41no—
195C4×Dic61—21no—
196#11758194—42no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—42no—
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C23×C61—21no—
205C23×C61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—41no—
209Q8×C61—41no—
210Q8×C61—81no—
211D4×C61—21no—
212D4×C61—21no—
213D4×C61—21no—
214D4×C61#13712yes—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—161no—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61H(5,𝔽2)11yescharacteristic
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—161no—
234C2×Dic61—161no—
235Dic122—21no—
236Dic122—21no—
237Dic122—81no—
238Dic122—81no—
239C8×C61—21no—
240C8×C61—21no—
241C8×C61—41no—
242#92418—21no—
243#92418—41no—
244#92418—41no—
245#92418—41no—
246#92418—41no—
247#92418#13512yes—
248#19865—611nomaximal, Sylow
249C22×C61—22no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—41no—
255C22×C61D4211yescharacteristic
256C4×C61—21no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—81no—
262C4×C61—81no—
263C4×C61C2×#13511yescharacteristic
264C4×C61C2×#13711yescharacteristic
265C4×C61#57811yescharacteristic
266C4×C61#57811yescharacteristic
267Dic61—41no—
268Dic61—41no—
269Dic61—81no—
270Dic61—81no—
271#3380—611no—
272#3360D6111yescharacteristic
273#3356—611no—
274#3338—611no—
275#1723—611no—
276#1695—611no—
277#1684—611no—
278#1676—611no—
279#1671—611no—
280#1643—611no—
281#1638—611no—
282#1632—611no—
283#1632—611no—
284#1581—611no—
285#1579—611no—
286C2×C61—42no—
287C2×C61—81no—
288C2×C61#327011yescharacteristic
289C2×C61#328311yescharacteristic
290C2×C61#328811yescharacteristic
291#556C2×D6111yescharacteristic
292#554—611no—
293#554—611no—
294#554C2×D6111yescharacteristic
295#500—611no—
296#500—611no—
297#499—1221no—
298#498—611no—
299#494—611no—
300#488—21no—
301#487—611no—
302#487C2×D6111yescharacteristic
303#485—1221no—
304#484—1221no—
305#484—1221no—
306#483—1221no—
307#479—21no—
308#478—1221no—
309#478—1221no—
310#477—611no—
311#477—611no—
312#477—611no—
313#476—611no—
314#476—611no—
315#476C2×D6111yescharacteristic
316#475—611no—
317#470—1221no—
318#467—1221no—
319#463—21no—
320#461C2×D6111yescharacteristic
321C8×Q8—611no—
322C8×Q8C2×D6111yescharacteristic
323C4×Q16—611no—
324C4×SD16—21no—
325C4×D8—1221no—
326C4×D8C2×D6111yescharacteristic
327#436—611no—
328C8×D4—611no—
329#428—1221no—
330C2×#107—1221no—
331#335—611no—
332#334—611no—
333#333—611no—
334#333—611no—
335#333—611no—
336#332—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339D4⋊C8—611no—
340D4⋊C8—611no—
341#322—611no—
342C61#1986511yescharacteristic, complemented, Sylow
343C2×Q16—611no—
344C2×Q16—2441no—
345C2×SD16—21no—
346C2×D8C22⋊D6111yescharacteristic
347C2×M16—1221no—
348C22×C8—1221no—
349C4⋊Q8—611no—
350C4⋊Q8—611no—
351C4⋊Q8C22×D6111yescharacteristic
352C4⋊Q8C22×D6111yescharacteristic
353#126—41no—
354#125—21no—
355#125—1221no—
356#125—1221no—
357#124—21no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—41no—
361C22⋊Q8—1221no—
362C22⋊Q8—1221no—
363C22⋊Q8—2441no—
364C22⋊Q8—2441no—
365C4×Q8—21no—
366C4×Q8—611no—
367C4×Q8C22×D6111yescharacteristic
368C4×D4—21no—
369C4×D4—1221no—
370C4×D4—1221no—
371C4×D4C22×D6111yescharacteristic
372C4×D4C22×D6111yescharacteristic
373#117—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377#107—611no—
378#107—1221no—
379#107—1221no—
380#107—1221no—
381#107—1221no—
382#107—1221no—
383#107—1221no—
384#107—1221no—
385#107C22⋊D6111yescharacteristic
386#106—21no—
387#106—1221no—
388#106—1221no—
389#106—1221no—
390#106—1221no—
391#106—1221no—
392C4⋊C8—21no—
393C4⋊C8—611no—
394C4⋊C8—611no—
395C4⋊C8—611no—
396C4⋊C8—1221no—
397C4⋊C8C22×D6111yescharacteristic
398#103—21no—
399#103—21no—
400#103—21no—
401#103—611no—
402#103—611no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—2441no—
408#102—21no—
409#102—1221no—
410#102—1221no—
411#102C22⋊D6111yescharacteristic
412#102C22⋊D6111yescharacteristic
413C22⋊C8—1221no—
414C22⋊C8—1221no—
415C4×C8—21no—
416C4×C8—611no—
417C4×C8—611no—
418C4×C8—1221no—
419C4×C8C22×D6111yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—611no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8D4×D6111yescharacteristic
426C2×D4C2×C22⋊D6111yescharacteristic
427C2×D4C2×C22⋊D6111yescharacteristic
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—41no—
431C22×C4—2441no—
432C22×C4—2441no—
433Q16—1222no—
434Q16—4882no—
435SD16—42no—
436D8—22no—
437M16—2442no—
438C2×C8—21no—
439C2×C8—611no—
440C2×C8—611no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—2442no—
446C2×C8C2×C22⋊D6111yescharacteristic
447C2×C8C2×C22⋊D6111yescharacteristic
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—611no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C4⋊C4—2441no—
465C4⋊C4—2441no—
466C4⋊C4D4×D6111yescharacteristic
467C4⋊C4C2×C22⋊D6111yescharacteristic
468C4⋊C4C2×C22⋊D6111yescharacteristic
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—2441no—
475C22⋊C4—2441no—
476C42—21no—
477C42—41no—
478C42—1221no—
479C42C23×D6111yescharacteristic
480C23—21no—
481C23—21no—
482Q8—21no—
483Q8—21no—
484Q8—41no—
485Q8—41no—
486Q8—81no—
487Q8—1221no—
488Q8—1221no—
489Q8—4881no—
490Q8—4881no—
491D4—21no—
492D4—21no—
493D4—21no—
494D4#1451980712yes—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—161no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—9761no—
509C2×C4—9761no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4C2×D4×D6111yescharacteristic
513C2×C4C22×C22⋊D6111yescharacteristic
514C2×C4#1451982211yescharacteristic
515C8—21no—
516C8—21no—
517C8—41no—
518C8—612no—
519C8—1221no—
520C8—2441no—
521C8—2441no—
522C8—2441no—
523C8—2441no—
524C22—22no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22D4×C22⋊D6111yescharacteristic
531C4—21no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—81no—
537C4—81no—
538C4—4881no—
539C4—4881no—
540C4—2441no—
541C4—2441no—
542C4#135×D6111yescharacteristic
543C4C2×#1451980711yescharacteristic
544C4#1684057611yescharacteristic
545C4#1684057711yescharacteristic
546C2—42no—
547C2—81no—
548C2#2145738511yescharacteristic
549C2#2145753511yescharacteristic
550C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3360C2
socleC22×C61D42
prime radicalC2×D8×C61D4
hypercenter#3360D61
hypocenterC61#19865
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3360

Complements

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

NH ≅ G/Ntype
2. #21458428C2semidirect
3. #21458207C2semidirect
4. #21445499C2semidirect
5. #21445340C2semidirect
6. #21445274C2semidirect
7. #21445223C2semidirect
8. #21445192C2semidirect
9. #21445024C2semidirect
10. #21444991C2semidirect
11. #21444953C2semidirect
12. #21444951C2semidirect
14. #21444636C2semidirect
342. C61#19865semidirect

Maximal subgroups

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

subgroupindex
#214586842
#214584282
#214582072
#214454992
#214453402
#214452742
#214452232
#214451922
#214450242
#214449912
#214449532
#214449512
#214446492
#214446362
C61×#33602
#1986561