Finite Group Databasebeta

← #29677323

Subgroups — #29677323

Order 15616.

570 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#21458729C211yesmaximal, characteristic, complemented
2#21458727C211yesmaximal, characteristic, complemented
3#21458439C211yesmaximal, characteristic
4#21445436C211yesmaximal, characteristic, complemented
5#21445345C211yesmaximal, characteristic, complemented
6#21445344C211yesmaximal, characteristic, complemented
7#21445342C211yesmaximal, characteristic, complemented
8#21445340C211yesmaximal, characteristic, complemented
9#21444964C211yesmaximal, characteristic
10#21444958C211yesmaximal, characteristic, complemented
11#21444953C211yesmaximal, characteristic, complemented
12#21444951C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14#21444707C211yesmaximal, characteristic, complemented
15C61×#3357C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17#16840469C2211yescharacteristic
18#16840172—21no—
19#16840169—21no—
20#16840167C2211yescharacteristic
21#16840165C2211yescharacteristic
22#16840165C2211yescharacteristic
23#16840141C2211yescharacteristic
24#16840141C2211yescharacteristic
25#16840103—21no—
26#16840073—21no—
27#16840071—21no—
28#16840066—21no—
29#16840057C2211yescharacteristic
30#16840054C2211yescharacteristic
31#16840054C2211yescharacteristic
32#16840052C2211yescharacteristic
33#16840052C2211yescharacteristic
34#16840052C2211yescharacteristic
35#16840042—21no—
36#16840040—21no—
37#16840006—21no—
38#16839893C2211yescharacteristic
39#16839892C2211yescharacteristic
40#16839863—21no—
41#16839862—21no—
42C4×#11758200C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839805—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839510C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52#16839492C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#555C2211yescharacteristic
57C61×#554C2211yescharacteristic
58C61×#500C2211yescharacteristic
59C61×#485—21no—
60C61×#480—21no—
61C61×#477C2211yescharacteristic
62C61×#476C2211yescharacteristic
63C61×#468—21no—
64C4×SD16×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#428—21no—
67C2×#11758202—41no—
68C2×#11758202—41no—
69C2×#11758200D411yescharacteristic
70C2×#11758194—21no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519776—21no—
75#14519776—41no—
76#14519768—21no—
77#14519756—21no—
78#14519754—41no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122C2311yescharacteristic
82D4×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#14519710—21no—
92#14519710—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708D411yescharacteristic
97C4⋊#92418—21no—
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519706—21no—
102#14519701—21no—
103#14519700—21no—
104#14519700—21no—
105#14519700D411yescharacteristic
106#14519698—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111#14519688—21no—
112#14519687—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×SD16×C61D411yescharacteristic
116C2×M16×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119#125×C61—21no—
120#124×C61—41no—
121C22⋊Q8×C61—21no—
122C22⋊Q8×C61—41no—
123#121×C61—21no—
124#121×C61—21no—
125C4×Q8×C61C2311yescharacteristic
126C4×D4×C61—21no—
127C4×D4×C61C2311yescharacteristic
128C4×D4×C61C2311yescharacteristic
129#117×C61—21no—
130#117×C61—21no—
131C2×C4⋊C4×C61—21no—
132#107×C61—21no—
133#107×C61—21no—
134#107×C61—21no—
135#106×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—41no—
148C2×D4×C61C2×D411yescharacteristic
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—41no—
153C22×Dic61—41no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—41no—
157C2×Dic122C2×D411yescharacteristic
158SD16×C61—22no—
159M16×C61—42no—
160C2×C8×C61—21no—
161C2×C8×C61C2×D411yescharacteristic
162C2×C8×C61C2×D411yescharacteristic
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
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—
182#11758202—82no—
183#11758202—82no—
184#11758200—22no—
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—
196C4×Dic61—41no—
197#11758194—42no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C23×C61—21no—
205C23×C61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—81no—
209Q8×C61—81no—
210D4×C61—21no—
211D4×C61—21no—
212D4×C61—21no—
213D4×C61—21no—
214D4×C61—161no—
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—
234Dic122—21no—
235Dic122—21no—
236Dic122—81no—
237Dic122—81no—
238Dic122—161no—
239C8×C61—21no—
240C8×C61—41no—
241C8×C61#13712yes—
242#92418—22no—
243#92418—21no—
244#92418—41no—
245#92418—41no—
246#92418—41no—
247#92418#13712yes—
248#20324—611nomaximal, Sylow
249C22×C61—41no—
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×#13711yescharacteristic
264C4×C61C2×#13711yescharacteristic
265C4×C61#57811yescharacteristic
266C4×C61#57811yescharacteristic
267Dic61—41no—
268Dic61—41no—
269Dic61—81no—
270Dic61—81no—
271#3385—611no—
272#3385—611no—
273#3357—611no—
274#3357D6111yescharacteristic
275#1713—611no—
276#1696—611no—
277#1696—611no—
278#1695—611no—
279#1695—611no—
280#1634—611no—
281#1633—611no—
282#1632—611no—
283#1632—611no—
284#1590—611no—
285#1590—611no—
286C2×C61—81no—
287C2×C61—81no—
288C2×C61#327611yescharacteristic
289C2×C61#327611yescharacteristic
290C2×C61#329211yescharacteristic
291Q82—611no—
292#555C2×D6111yescharacteristic
293#554—611no—
294#554C2×D6111yescharacteristic
295#501—1221no—
296#501—1221no—
297#500—611no—
298#500—611no—
299#500—611no—
300#500C2×D6111yescharacteristic
301#494—611no—
302#494—611no—
303#485—21no—
304#485—1221no—
305#480—21no—
306#480—1221no—
307#479—1221no—
308#479—1221no—
309#477—611no—
310#477—611no—
311#477—611no—
312#477C2×D6111yescharacteristic
313#476—611no—
314#476—611no—
315#476—611no—
316#476C2×D6111yescharacteristic
317#474—1221no—
318#474—1221no—
319#468—21no—
320#468—1221no—
321#446—611no—
322#446—611no—
323#440—1221no—
324#440—1221no—
325C4×SD16—611no—
326C4×SD16C2×D6111yescharacteristic
327#436—611no—
328#436C2×D6111yescharacteristic
329#428—21no—
330#428—1221no—
331#334—611no—
332#334—611no—
333#333—611no—
334#333—611no—
335#333—611no—
336#333—611no—
337Q8⋊C8—611no—
338Q8⋊C8—611no—
339D4⋊C8—611no—
340D4⋊C8—611no—
341C82—611no—
342C61#2032411yescharacteristic, complemented, Sylow
343C2×Q16—2441no—
344C2×Q16—2441no—
345C2×SD16—611no—
346C2×SD16C22⋊D6111yescharacteristic
347C2×M16—21no—
348C2×M16—1221no—
349C4⋊Q8—611no—
350C4⋊Q8—611no—
351C4⋊Q8—1221no—
352C4⋊Q8—1221no—
353C4⋊Q8—2441no—
354C4⋊Q8C22×D6111yescharacteristic
355C4⋊Q8C22×D6111yescharacteristic
356#125—21no—
357#125—1221no—
358#124—41no—
359C22⋊Q8—21no—
360C22⋊Q8—41no—
361C22⋊Q8—1221no—
362C22⋊Q8—2441no—
363#121—21no—
364#121—21no—
365C4×Q8—611no—
366C4×Q8—1221no—
367C4×Q8—1221no—
368C4×Q8C22×D6111yescharacteristic
369C4×D4—21no—
370C4×D4—1221no—
371C4×D4C22×D6111yescharacteristic
372C4×D4C22×D6111yescharacteristic
373#117—21no—
374#117—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—1221no—
377#107—21no—
378#107—21no—
379#107—21no—
380#107—1221no—
381#107—1221no—
382#107—1221no—
383#107—1221no—
384#107—1221no—
385#107—1221no—
386#107—1221no—
387#107—1221no—
388#107—1221no—
389#106—21no—
390#106—611no—
391#106—1221no—
392#106—1221no—
393#106—1221no—
394#106C22⋊D6111yescharacteristic
395C4⋊C8—611no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—1221no—
399C4⋊C8—1221no—
400C4⋊C8C22×D6111yescharacteristic
401#103—21no—
402#103—611no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103C22⋊D6111yescharacteristic
407#102—21no—
408#102—611no—
409#102—1221no—
410#102—1221no—
411#102—1221no—
412#102C22⋊D6111yescharacteristic
413C22⋊C8—21no—
414C22⋊C8—1221no—
415#97—1221no—
416#97—1221no—
417C4×C8—611no—
418C4×C8—611no—
419C4×C8C22×D6111yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—611no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—2441no—
426C2×Q8C2×C22⋊D6111yescharacteristic
427C2×D4—41no—
428C2×D4D4×D6111yescharacteristic
429C2×D4C2×C22⋊D6111yescharacteristic
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—41no—
433C22×C4—2441no—
434Q16—4882no—
435Q16—4882no—
436SD16—22no—
437SD16—1222no—
438M16—42no—
439M16—2442no—
440C2×C8—21no—
441C2×C8—611no—
442C2×C8—611no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8C2×C22⋊D6111yescharacteristic
448C2×C8C2×C22⋊D6111yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
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—
475C42—21no—
476C42—41no—
477C42—1221no—
478C42—2441no—
479C42C23×D6111yescharacteristic
480C23—21no—
481C23—21no—
482Q8—21no—
483Q8—21no—
484Q8—81no—
485Q8—81no—
486Q8—1221no—
487Q8—1221no—
488Q8—9761no—
489Q8—4881no—
490Q8—4881no—
491D4—21no—
492D4—21no—
493D4—21no—
494D4—21no—
495D4—161no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—161no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—9761no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4C2×D4×D6111yescharacteristic
513C2×C4C22×C22⋊D6111yescharacteristic
514C2×C4#1451982211yescharacteristic
515C8—21no—
516C8—41no—
517C8—612no—
518C8—1222no—
519C8—1221no—
520C8—2441no—
521C8—2441no—
522C8—2441no—
523C8#1451980712yes—
524C22—41no—
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#137×D6111yescharacteristic
543C4C2×#1451980711yescharacteristic
544C4#1684057711yescharacteristic
545C4#1684057811yescharacteristic
546C2—81no—
547C2—81no—
548C2#2145745911yescharacteristic
549C2#2145746111yescharacteristic
550C2#2145761611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3357C2
socleC22×C61D42
prime radicalC61×#555C22
hypercenter#3357D61
hypocenterC61#20324
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3357

Complements

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

NH ≅ G/Ntype
1. #21458729C2semidirect
2. #21458727C2semidirect
4. #21445436C2semidirect
5. #21445345C2semidirect
6. #21445344C2semidirect
7. #21445342C2semidirect
8. #21445340C2semidirect
10. #21444958C2semidirect
11. #21444953C2semidirect
12. #21444951C2semidirect
13. #21444710C2semidirect
14. #21444707C2semidirect
342. C61#20324semidirect

Maximal subgroups

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

subgroupindex
#214587292
#214587272
#214584392
#214454362
#214453452
#214453442
#214453422
#214453402
#214449642
#214449582
#214449532
#214449512
#214447102
#214447072
C61×#33572
#2032461