Finite Group Databasebeta

← #29662884

Subgroups — #29662884

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456360C211yesmaximal, characteristic
2#21455860C211yesmaximal, characteristic
3#462×D61C211yesmaximal, characteristic
4Q8×#11758189C211yesmaximal, characteristic
5#21445338C211yesmaximal, characteristic, complemented
6#21445336C211yesmaximal, characteristic, complemented
7#21445328C211yesmaximal, characteristic, complemented
8#21445322C211yesmaximal, characteristic, complemented
9#21445213C211yesmaximal, characteristic, complemented
10#21445211C211yesmaximal, characteristic, complemented
11#21445203C211yesmaximal, characteristic, complemented
12#21444568C211yesmaximal, characteristic
13#21444506C211yesmaximal, characteristic
14#21444502C211yesmaximal, characteristic
15C61×#1672C211yesmaximal, characteristic, complemented
16C2×Q16×D61—21no—
17#16840317C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4×Q8×D61C2211yescharacteristic
20#16840169—21no—
21#16840169—21no—
22#16840167C2211yescharacteristic
23#16840165C2211yescharacteristic
24#16840158C2211yescharacteristic
25#16840099—21no—
26#16840054C2211yescharacteristic
27#16839973C2211yescharacteristic
28#16839973C2211yescharacteristic
29#16839972C2211yescharacteristic
30#16839972C2211yescharacteristic
31#16839971—21no—
32#16839970—21no—
33#16839969C2211yescharacteristic
34#16839919—21no—
35#16839892—21no—
36#16839892C2211yescharacteristic
37#16839889—21no—
38#16839889C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40#16839856—21no—
41#16839854—21no—
42#16839814C2211yescharacteristic
43#16839801—21no—
44#16839801—21no—
45#16839784—21no—
46#16839784C2211yescharacteristic
47C4⋊C8×D61C2211yescharacteristic
48#103×D61—21no—
49C4×#11758189—21no—
50C4×#11758189C2211yescharacteristic
51#16839515C2211yescharacteristic
52#16839514C2211yescharacteristic
53#16839512C2211yescharacteristic
54#16839511C2211yescharacteristic
55#16839509C2211yescharacteristic
56#16839509C2211yescharacteristic
57#16839508C2211yescharacteristic
58#16839507C2211yescharacteristic
59#16839507C2211yescharacteristic
60C61×#498C2211yescharacteristic
61C61×#477C2211yescharacteristic
62C61×#462C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic
65C61×#333C2211yescharacteristic
66C61×#332C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×Q8×D61D411yescharacteristic
69C2×Dic244—21no—
70C2×Dic244—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758202—21no—
74Q16×D61—44no—
75C2×#11758189—21no—
76C2×#11758189D411yescharacteristic
77C2×#11758189D411yescharacteristic
78C2×C8×D61—21no—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519776—21no—
82#14519776—21no—
83#14519775—21no—
84#14519768—21no—
85#14519757—21no—
86#14519753—21no—
87Q8×Dic61—21no—
88Q8×Dic61D411yescharacteristic
89C4×Dic122—21no—
90C4×Dic122C2311yescharacteristic
91#14519742—21no—
92#14519742—21no—
93#14519742D411yescharacteristic
94#14519735—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61D411yescharacteristic
99C42×D61—21no—
100C42×D61C2311yescharacteristic
101#14519711—21no—
102#14519711—21no—
103#14519710—21no—
104#14519710—21no—
105#14519710—21no—
106#14519708—21no—
107#14519708—21no—
108C4⋊#92418—21no—
109C4⋊#92418C2311yescharacteristic
110C4⋊#92418C2311yescharacteristic
111C4⋊#92418C2311yescharacteristic
112#14519706—21no—
113#14519706—21no—
114#14519706—21no—
115#14519706—21no—
116#14519702—21no—
117#14519702—21no—
118#14519701—21no—
119#14519700—21no—
120#14519700—21no—
121#14519700—21no—
122#14519689—21no—
123#14519687—21no—
124#14519687—21no—
125#14519687D411yescharacteristic
126#14519687D411yescharacteristic
127C4×#92418—21no—
128C4×#92418C2311yescharacteristic
129C8×Dic61—21no—
130C2×Q16×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132C4⋊Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134#107×C61—21no—
135#106×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#103×C61—21no—
142#103×C61—21no—
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148Q8×D61—22no—
149Q8×D61—41no—
150Q8×D61—42no—
151Q8×D61—42no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—41no—
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61#4111yescharacteristic
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—41no—
162C2×Dic122C2×D411yescharacteristic
163Q16×C61—42no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C2×D411yescharacteristic
175C42×C61—21no—
176C42×C61C2411yescharacteristic
177#11758202—42no—
178#11758202—42no—
179#11758202—42no—
180#11758198—41no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—41no—
187C4⋊Dic61C2×D411yescharacteristic
188#11758196—21no—
189#11758196—21no—
190#11758196—21no—
191#11758196—41no—
192#11758196—41no—
193#11758196—41no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—41no—
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206Dic244—42no—
207Dic244—42no—
208#11758189—22no—
209#11758189—42no—
210#11758189Q1614yescomplemented
211C8×D61—42no—
212C22×D61#12111yescharacteristic
213Q8×C61—21no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—81no—
217Q8×C61AGL(1,ℤ/8)12yes—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61#12111yescharacteristic
230C2×Dic61#12111yescharacteristic
231C2×Dic61#12111yescharacteristic
232C4×D61—21no—
233C4×D61—21no—
234C4×D61—41no—
235C4×D61—42no—
236C4×D61—81no—
237C4×D61—82no—
238C4×D61C2×Q1612yes—
239C4×D61#13712yes—
240Dic122—21no—
241Dic122—21no—
242Dic122—42no—
243Dic122—41no—
244Dic122—41no—
245Dic122—41no—
246Dic122—41no—
247Dic122—82no—
248C8×C61—21no—
249C8×C61—41no—
250C8×C61—41no—
251C8×C61—41no—
252C8×C61C2×Q1612yescomplemented
253#92418—21no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418C2×Q1612yescomplemented
258#19172—611nomaximal, Sylow
259C22×C61C2×#12111yescharacteristic
260C2×D61—21no—
261C2×D61—21no—
262C2×D61#46212yes—
263C4×C61—22no—
264C4×C61—21no—
265C4×C61—41no—
266C4×C61—81no—
267C4×C61—81no—
268C4×C61C22×Q1611yescharacteristic
269C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
270C4×C61C2×#13711yescharacteristic
271C4×C61#57511yescharacteristic
272Dic61—21no—
273Dic61—21no—
274Dic61—41no—
275Dic61—41no—
276Dic61—81no—
277Dic61—81no—
278Dic61#46212yes—
279#3165—611no—
280#3104—611no—
281C2×#462—611no—
282Q8×M16—611no—
283#1694—611no—
284#1694—611no—
285#1692—611no—
286#1692—611no—
287#1674—611no—
288#1674—611no—
289#1672—611no—
290#1672D6111yescharacteristic, complemented
291#1565—611no—
292#1550—611no—
293#1549—611no—
294C2×C61C2×#46211yescharacteristic
295C2×C61#305511yescharacteristic
296C2×C61#306911yescharacteristic
297D61—22no—
298C22×Q16—1221no—
299#533—611no—
300C2×C4⋊Q8—611no—
301C2×C4×Q8—611no—
302#501—1221no—
303#501—1221no—
304#500—611no—
305#500—611no—
306#498—611no—
307#498C2×D6111yescharacteristic
308#484—1221no—
309#477—611no—
310#477C2×D6111yescharacteristic
311#462—611no—
312#462—611no—
313#462—611no—
314#462—611no—
315#462—611no—
316#462—1221no—
317#462—1221no—
318#462C2×D6111yescharacteristic
319#451—1221no—
320#446—611no—
321#446—611no—
322#446—1221no—
323#446—1221no—
324C8×Q8—611no—
325C8×Q8C2×D6111yescharacteristic
326C4×Q16—1221no—
327C4×Q16—1221no—
328#431—611no—
329#428—1221no—
330#428—1221no—
331#423—611no—
332#423—1221no—
333C2×C4⋊C8—611no—
334C2×#103—1221no—
335C4×M16—611no—
336C4×M16—1221no—
337#335—611no—
338#335—611no—
339#334—611no—
340#334C2×D6111yescharacteristic
341#333—611no—
342#333C2×D6111yescharacteristic
343#332—611no—
344#332—611no—
345#332—611no—
346#332—611no—
347#332—611no—
348#332C2×D6111yescharacteristic
349C61#1917211yescharacteristic, complemented, Sylow
350C22×Q8—611no—
351C22×Q8—1221no—
352C2×Q16—21no—
353C2×Q16—1221no—
354C2×Q16—1221no—
355C2×Q16—1221no—
356C2×Q16—1221no—
357C2×Q16—1221no—
358C2×Q16—2444no—
359C2×M16—611no—
360C2×M16—611no—
361C2×M16—1221no—
362C22×C8—1221no—
363C4⋊Q8—611no—
364C4⋊Q8—611no—
365C4⋊Q8—1221no—
366C4⋊Q8—1221no—
367C4⋊Q8—1221no—
368C4⋊Q8C22×D6111yescharacteristic
369C4⋊Q8C22×D6111yescharacteristic
370#125—1221no—
371C22⋊Q8—1221no—
372C22⋊Q8—1221no—
373C4×Q8—611no—
374C4×Q8—611no—
375C4×Q8—611no—
376C4×Q8—1221no—
377C4×Q8—1221no—
378C4×Q8—1221no—
379C4×Q8—1221no—
380C4×Q8C22×D6111yescharacteristic
381#117—1221no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C42—611no—
387C2×C42—1221no—
388#107—21no—
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#107—1221no—
394#106—21no—
395#106—21no—
396#106—1221no—
397#106—1221no—
398C4⋊C8—21no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405C4⋊C8—1221no—
406C4⋊C8—1221no—
407C4⋊C8C22×D6111yescharacteristic
408C4⋊C8C22×D6111yescharacteristic
409C4⋊C8C22×D6111yescharacteristic
410#103—21no—
411#103—21no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418C22⋊C8—1221no—
419#97—611no—
420#97—611no—
421#97—1221no—
422#97—1221no—
423C4×C8—21no—
424C4×C8—611no—
425C4×C8—1221no—
426C4×C8—1221no—
427C4×C8C22×D6111yescharacteristic
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—611no—
431C2×Q8—1221no—
432C2×Q8—1222no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—2441no—
436C2×Q8—2442no—
437C2×Q8—2442no—
438C2×Q8—2441no—
439C2×Q8D4×D6111yescharacteristic
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—2441no—
446Q16—42no—
447Q16—2442no—
448Q16—2442no—
449Q16—2442no—
450Q16—2442no—
451Q16—2442no—
452M16—614no—
453M16—1222no—
454M16—2442no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—611no—
459C2×C8—611no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—2442no—
464C2×C8D4×D6111yescharacteristic
465C2×C8D4×D6111yescharacteristic
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—611no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—2441no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4—2441no—
484C4⋊C4D4×D6111yescharacteristic
485C22⋊C4—2441no—
486C42—21no—
487C42—611no—
488C42—611no—
489C42—611no—
490C42—1221no—
491C42—1221no—
492C42—1221no—
493C42—2441no—
494C42C23×D6111yescharacteristic
495C23—611no—
496Q8—21no—
497Q8—41no—
498Q8—41no—
499Q8—81no—
500Q8—1221no—
501Q8—1221no—
502Q8—4882no—
503Q8—2442no—
504Q8—2441no—
505Q8—2441no—
506Q8—2441no—
507Q8—2441no—
508Q8#1451980412yes—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—611no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—611no—
517C2×C4—611no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—4881no—
523C2×C4—4882no—
524C2×C4—2441no—
525C2×C4—2442no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4C2×D4×D6111yescharacteristic
530C2×C4#41×D6111yescharacteristic
531C8—21no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—612no—
536C8—1221no—
537C8—2441no—
538C8—2441no—
539C8—2441no—
540C8Q16×D6112yescomplemented
541C22—612no—
542C22—1221no—
543C22—1221no—
544C22#121×D6111yescharacteristic
545C4—22no—
546C4—21no—
547C4—41no—
548C4—81no—
549C4—81no—
550C4—612no—
551C4—1221no—
552C4—1221no—
553C4—4881no—
554C4—4881no—
555C4—2441no—
556C4—2441no—
557C4C2×Q16×D6111yescharacteristic
558C4C2×#1451980411yescharacteristic
559C4#137×D6111yescharacteristic
560C4#1684055911yescharacteristic
561C2—1222no—
562C2#462×D6111yescharacteristic
563C2#2145543111yescharacteristic
564C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1672C2
socleC22×C61C2×#121
prime radicalC22×D61#121
hypercenter#1672D61
hypocenterC61#19172
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1672

Complements

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

NH ≅ G/Ntype
5. #21445338C2semidirect
6. #21445336C2semidirect
7. #21445328C2semidirect
8. #21445322C2semidirect
9. #21445213C2semidirect
10. #21445211C2semidirect
11. #21445203C2semidirect
15. C61×#1672C2semidirect
210. #11758189Q16semidirect
252. C8×C61C2×Q16semidirect
257. #92418C2×Q16semidirect
290. #1672D61semidirect
349. C61#19172semidirect
540. C8Q16×D61semidirect

Maximal subgroups

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

subgroupindex
#214563602
#214558602
#462×D612
Q8×#117581892
#214453382
#214453362
#214453282
#214453222
#214452132
#214452112
#214452032
#214445682
#214445062
#214445022
C61×#16722
#1917261