Finite Group Databasebeta

← #27156520

Subgroups — #27156520

Order 12416.

1412 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261571C211yesmaximal, characteristic, complemented
2C2×#15690857C211yesmaximal, characteristic, complemented
3#20261551C214yesmaximal, complemented
4#20261549C214yesmaximal, complemented
5#20261545C214yesmaximal, complemented
6#20261542C214yesmaximal, complemented
7#20261533C211yesmaximal, characteristic, complemented
8#20261531C214yesmaximal, complemented
9C2×#15690843C211yesmaximal, characteristic, complemented
10C2×#15690840C211yesmaximal, characteristic, complemented
11#137×D97C214yesmaximal, complemented
12#20261501C211yesmaximal, characteristic, complemented
13#575×C97C211yesmaximal, characteristic, complemented
14#15690858C2212yescomplemented
15#15690857—22no—
16#15690857—22no—
17#15690857—21no—
18#15690857C2214yescomplemented
19#15690857C2214yescomplemented
20#15690857C2212yescomplemented
21#15690856—21no—
22#15690856C2212yescomplemented
23#15690855—22no—
24#15690855C2212yescomplemented
25C2×#13095511—22no—
26C2×#13095511—21no—
27C2×#13095511C2211yescharacteristic
28C2×#13095511C2211yescharacteristic
29C2×#13095510—22no—
30C2×#13095510C2212yescomplemented
31#41×D97—22no—
32#41×D97C2214yes—
33C22×Dic194—21no—
34C22×Dic194C2211yescharacteristic, complemented
35C2×Q8×D97C2212yescomplemented
36#15690843C2214yescomplemented
37#15690843C2214yescomplemented
38#15690842C2218yescomplemented
39#15690841C2212yescomplemented
40#15690840C2214yescomplemented
41#15690840C2214yescomplemented
42#15690840C2212yescomplemented
43#15690839C2218yescomplemented
44#15690839C2218yescomplemented
45#15690836C2212yescomplemented
46#15690835C2212yescomplemented
47#15690834C2218yescomplemented
48#15690831C2218yescomplemented
49#15690830C2214yescomplemented
50#15690829C2218yescomplemented
51C2×Dic388C2212yescomplemented
52C2×#13095501C2212yescomplemented
53Q16×D97C2218yescomplemented
54C2×#13095500C2212yescomplemented
55C2×#13095499C2212yescomplemented
56SD16×D97C2218yescomplemented
57#15690818C2214yescomplemented
58#15690816C2214yescomplemented
59C2×#13095495C2211yescharacteristic, complemented
60M16×D97C2214yescomplemented
61C2×#41×C97C2211yescharacteristic
62C2×#41×C97C2211yescharacteristic, complemented
63#137×C97C2212yescomplemented
64#137×C97C2212yescomplemented
65AGL(1,ℤ/8)×C97C2212yescomplemented
66AGL(1,ℤ/8)×C97C2212yescomplemented
67#135×C97C2214yescomplemented
68C2×M16×C97C2211yescharacteristic, complemented
69#13095512—22no—
70#13095512C2314yes—
71#13095511—21no—
72#13095511—21no—
73#13095511—24no—
74#13095511—22no—
75#13095511—24no—
76#13095511—24no—
77#13095511—24no—
78#13095511—22no—
79#13095511D412yes—
80#13095511D412yes—
81#13095511C2312yes—
82#13095511C2312yes—
83#13095511C2312yes—
84#13095511C2312yes—
85#13095510—24no—
86#13095510—28no—
87#13095510—28no—
88#13095510—24no—
89#13095510—24no—
90#13095510—22no—
91#13095510C2314yes—
92#13095510C2318yes—
93C2×Dic194—21no—
94C2×Dic194—21no—
95C2×Dic194—21no—
96C2×Dic194—22no—
97C2×Dic194—24no—
98C2×Dic194—24no—
99C2×Dic194—22no—
100C2×Dic194D411yescharacteristic
101C2×Dic194C2312yes—
102C2×Dic194C2312yes—
103C2×Dic194C2311yescharacteristic
104C2×Dic194C2312yes—
105Q8×D97—24no—
106Q8×D97—22no—
107Q8×D97C2318yes—
108Q8×D97C2314yes—
109C2×D388—22no—
110C2×D388D411yescharacteristic
111C2×D388C2311yescharacteristic
112C2×C22⋊D97—21no—
113C2×C22⋊D97—22no—
114C2×C22⋊D97—24no—
115C2×C22⋊D97D412yes—
116D4×D97—24no—
117D4×D97—24no—
118D4×D97—22no—
119D4×D97C2314yes—
120C22×Dic97—21no—
121C22×Dic97—22no—
122C2×C4×D97—22no—
123C2×C4×D97—21no—
124C2×C4×D97—24no—
125C2×C4×D97C2312yes—
126Dic388C2318yes—
127#13095501C2314yes—
128#13095501C2318yes—
129#13095500C2318yes—
130#13095499C2318yes—
131#13095499C2314yes—
132#13095498C2314yes—
133#13095496C2314yes—
134#13095495C2312yes—
135#13095495C2312yes—
136#13095494C2318yes—
137C2×#11715895C2312yes—
138C8×D97C2318yes—
139#41×C97—21no—
140#41×C97—21no—
141#41×C97—21no—
142#41×C97—21no—
143#41×C97C2312yes—
144#41×C97C2312yes—
145#41×C97C2312yes—
146#41×C97C2312yes—
147C2×Q8×C97C2311yescharacteristic
148C2×Q8×C97C2311yescharacteristic
149C2×D4×C97—21no—
150C2×D4×C97—21no—
151C2×D4×C97C2311yescharacteristic
152C2×D4×C97C2311yescharacteristic
153C22×C4×C97—21no—
154C22×C4×C97—21no—
155C22×C4×C97C2311yescharacteristic
156Q16×C97C2314yes—
157SD16×C97C2314yes—
158SD16×C97C2314yes—
159D8×C97C2314yes—
160M16×C97C2312yes—
161M16×C97C2312yes—
162C2×C8×C97C2312yes—
163C23×C97—21no—
164C23×C97—21no—
165C23×C97C2×D411yescharacteristic
166C22×D97—21no—
167C22×D97—24no—
168C22×D97C2×D412yes—
169Q8×C97—21no—
170Q8×C97—21no—
171Q8×C97C2412yes—
172Q8×C97C2412yes—
173D4×C97—22no—
174D4×C97—22no—
175D4×C97—22no—
176D4×C97—22no—
177D4×C97—21no—
178D4×C97—21no—
179D4×C97C2412yes—
180D4×C97C2412yes—
181C2×C4×C97—21no—
182C2×C4×C97—21no—
183C2×C4×C97—22no—
184C2×C4×C97—22no—
185C2×C4×C97—22no—
186C2×C4×C97—22no—
187C2×C4×C97C2×D411yescharacteristic
188C2×C4×C97C2×D411yescharacteristic
189C2×C4×C97C2×D411yescharacteristic
190C2×C4×C97C2411yescharacteristic
191C2×C4×C97C2411yescharacteristic
192C2×C4×C97C2411yescharacteristic
193C22⋊D97—22no—
194C22⋊D97—22no—
195C22⋊D97—24no—
196C22⋊D97—28no—
197C22⋊D97—24no—
198C22⋊D97C2×D414yes—
199C22⋊D97C2×D414yes—
200C2×Dic97—22no—
201C2×Dic97—21no—
202C2×Dic97—22no—
203C2×Dic97—22no—
204C2×Dic97—24no—
205C2×Dic97—28no—
206C2×Dic97C2×D412yes—
207D388—24no—
208D388—21no—
209D388C2×D412yes—
210D388C2×D412yes—
211D388C2412yes—
212C4×D97—24no—
213C4×D97—28no—
214C4×D97—22no—
215C4×D97—22no—
216C4×D97—24no—
217C4×D97C2×D414yes—
218C4×D97C2414yes—
219Dic194—22no—
220Dic194—22no—
221Dic194—22no—
222Dic194—21no—
223Dic194—28no—
224Dic194—24no—
225Dic194C2×D412yes—
226Dic194C2×D412yes—
227Dic194C2414yes—
228Dic194C2412yes—
229C8×C97C2414yes—
230#11715895C2414yes—
231C22×C97—22no—
232C22×C97—22no—
233C22×C97—41no—
234C22×C97—42no—
235C22×C97—81no—
236C22×C97C22×D411yescharacteristic
237C22×C97C22×D411yescharacteristic
238C22×C97C22×D411yescharacteristic
239C2×D97—22no—
240C2×D97—42no—
241C2×D97—84no—
242C2×D97—81no—
243C2×D97C22×D414yes—
244C4×C97—22no—
245C4×C97—22no—
246C4×C97C22×D411yescharacteristic
247C4×C97C22×D411yescharacteristic
248C4×C97C22×D411yescharacteristic
249C4×C97C2511yescharacteristic
250Dic97—24no—
251Dic97—22no—
252Dic97C22×D414yes—
253C2×C97—21no—
254C2×C97—21no—
255C2×C97—21no—
256C2×C97—42no—
257C2×C97—42no—
258C2×C97C23×D411yescharacteristic
259D97—24no—
260D97—42no—
261#3586—971nomaximal, Sylow
262C97#358611yescharacteristic, complemented, Sylow
263#585—971no—
264C2×2−1+4—971no—
265#578—974no—
266#578—974no—
267#577—974no—
268#577—974no—
269#575—971no—
270#575—974no—
271#575D9711yescharacteristic, complemented
272C2×#137—971no—
273C2×#137—974no—
274C2×#137—971no—
275#568—971no—
2762−1+4—974no—
2772−1+4—974no—
2782−1+4—972no—
2792−1+4—972no—
2802−1+4—1941no—
2812−1+4—1942no—
2822−1+4—1942no—
283H(5,𝔽2)—972no—
284H(5,𝔽2)—972no—
285H(5,𝔽2)—1942no—
286H(5,𝔽2)—1941no—
287C2×#41—971no—
288C2×#41—974no—
289C2×#41—972no—
290C2×#41—971no—
291C2×#41—1941no—
292C2×#41—1942no—
293C2×#41—1942no—
294C2×#41—1942no—
295C2×#41C2×D9711yescharacteristic
296C2×#41C2×D9711yescharacteristic, complemented
297C22×Q8—972no—
298C22×Q8—971no—
299C22×Q8—1941no—
300#137—974no—
301#137—974no—
302#137—972no—
303#137—972no—
304#137—978no—
305#137—978no—
306#137—978no—
307#137—974no—
308#137—974no—
309#137C2×D9712yescomplemented
310#137C2×D9712yescomplemented
311AGL(1,ℤ/8)—972no—
312AGL(1,ℤ/8)—972no—
313AGL(1,ℤ/8)—978no—
314AGL(1,ℤ/8)C2×D9712yescomplemented
315AGL(1,ℤ/8)C2×D9712yescomplemented
316#135—974no—
317#135—978no—
318#135—978no—
319#135C2×D9714yescomplemented
320C2×Q16—972no—
321C2×Q16—978no—
322C2×Q16—972no—
323C2×SD16—972no—
324C2×SD16—978no—
325C2×SD16—972no—
326#131—974no—
327#131—974no—
328C2×M16—971no—
329C2×M16—974no—
330C2×M16C2×D9711yescharacteristic, complemented
331#41—21no—
332#41—21no—
333#41—21no—
334#41—21no—
335#41—972no—
336#41—972no—
337#41—972no—
338#41—972no—
339#41—974no—
340#41—978no—
341#41—974no—
342#41—972no—
343#41—972no—
344#41—1944no—
345#41—1944no—
346#41—1942no—
347#41—1944no—
348#41—1944no—
349#41—1942no—
350#41—1944no—
351#41—1944no—
352#41—1944no—
353#41—1948no—
354#41—1948no—
355#41—1942no—
356#41—1942no—
357#41—1941no—
358#41—1941no—
359#41C22×D9712yes—
360#41C22×D9712yes—
361#41C22×D9712yes—
362#41C22×D9712yes—
363C2×Q8—971no—
364C2×Q8—978no—
365C2×Q8—974no—
366C2×Q8—972no—
367C2×Q8—972no—
368C2×Q8—971no—
369C2×Q8—972no—
370C2×Q8—1942no—
371C2×Q8—1944no—
372C2×Q8—1944no—
373C2×Q8—1942no—
374C2×Q8—1944no—
375C2×Q8—1942no—
376C2×Q8—1941no—
377C2×Q8—1941no—
378C2×Q8—1941no—
379C2×Q8C22×D9711yescharacteristic
380C2×Q8C22×D9711yescharacteristic
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—972no—
384C2×D4—974no—
385C2×D4—971no—
386C2×D4—971no—
387C2×D4—1942no—
388C2×D4—1944no—
389C2×D4—1942no—
390C2×D4—1944no—
391C2×D4—1944no—
392C2×D4—1942no—
393C2×D4—1941no—
394C2×D4C22×D9711yescharacteristic
395C2×D4C22×D9711yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—972no—
399C22×C4—1944no—
400C22×C4—1942no—
401C22×C4—1942no—
402C22×C4—1941no—
403C22×C4—1941no—
404C22×C4C22×D9711yescharacteristic
405Q16—978no—
406Q16—974no—
407Q16—978no—
408Q16C22×D9714yes—
409SD16—974no—
410SD16—978no—
411SD16—974no—
412SD16—978no—
413SD16C22×D9714yes—
414SD16C22×D9714yes—
415D8—974no—
416D8C22×D9714yes—
417M16—972no—
418M16—972no—
419M16—978no—
420M16C22×D9712yes—
421M16C22×D9712yes—
422C2×C8—972no—
423C2×C8—978no—
424C2×C8C22×D9712yes—
425C23—21no—
426C23—21no—
427C23—972no—
428C23—1944no—
429C23—1941no—
430C23D4×D9711yescharacteristic
431Q8—21no—
432Q8—21no—
433Q8—972no—
434Q8—972no—
435Q8—974no—
436Q8—972no—
437Q8—1948no—
438Q8—1944no—
439Q8—1942no—
440Q8—1942no—
441Q8—1942no—
442Q8—1941no—
443Q8C23×D9712yes—
444Q8C23×D9712yes—
445D4—22no—
446D4—22no—
447D4—22no—
448D4—22no—
449D4—21no—
450D4—21no—
451D4—974no—
452D4—974no—
453D4—972no—
454D4—972no—
455D4—972no—
456D4—1944no—
457D4—1944no—
458D4—1944no—
459D4—1948no—
460D4—1942no—
461D4—1942no—
462D4—1941no—
463D4C23×D9712yes—
464D4C23×D9712yes—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—972no—
472C2×C4—974no—
473C2×C4—974no—
474C2×C4—1944no—
475C2×C4—1944no—
476C2×C4—1948no—
477C2×C4—1944no—
478C2×C4—1948no—
479C2×C4—1942no—
480C2×C4—1941no—
481C2×C4—1942no—
482C2×C4—1942no—
483C2×C4—1942no—
484C2×C4—1942no—
485C2×C4D4×D9711yescharacteristic
486C2×C4D4×D9711yescharacteristic
487C2×C4D4×D9711yescharacteristic
488C2×C4C23×D9711yescharacteristic
489C2×C4C23×D9711yescharacteristic
490C2×C4C23×D9711yescharacteristic
491C8—974no—
492C8C23×D9714yes—
493C22—22no—
494C22—22no—
495C22—41no—
496C22—42no—
497C22—81no—
498C22—7761no—
499C22—7764no—
500C22—974no—
501C22—3882no—
502C22—1942no—
503C22C2×D4×D9711yescharacteristic
504C22C2×D4×D9711yescharacteristic
505C22C2×D4×D9711yescharacteristic
506C4—22no—
507C4—22no—
508C4—974no—
509C4—1942no—
510C4—1944no—
511C4C2×D4×D9711yescharacteristic
512C4C2×D4×D9711yescharacteristic
513C4C2×D4×D9711yescharacteristic
514C4C24×D9711yescharacteristic
515C2—21no—
516C2—21no—
517C2—21no—
518C2—42no—
519C2—42no—
520C2—3882no—
521C2—1944no—
522C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC2C22×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
Fitting#575×C97C2
socleC2×C97C23×D4
hypercenter#575D97
hypocenterC97#3586
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20261571C2semidirect
2. C2×#15690857C2semidirect
3. #20261551C2semidirect
4. #20261549C2semidirect
5. #20261545C2semidirect
6. #20261542C2semidirect
7. #20261533C2semidirect
8. #20261531C2semidirect
9. C2×#15690843C2semidirect
10. C2×#15690840C2semidirect
11. #137×D97C2semidirect
12. #20261501C2semidirect
13. #575×C97C2semidirect
14. #15690858C22semidirect
18. #15690857C22semidirect
19. #15690857C22semidirect
20. #15690857C22semidirect
22. #15690856C22semidirect
24. #15690855C22semidirect
30. C2×#13095510C22semidirect
34. C22×Dic194C22semidirect
35. C2×Q8×D97C22semidirect
36. #15690843C22semidirect
37. #15690843C22semidirect
38. #15690842C22semidirect
39. #15690841C22semidirect
40. #15690840C22semidirect
41. #15690840C22semidirect
42. #15690840C22semidirect
43. #15690839C22semidirect
44. #15690839C22semidirect
45. #15690836C22semidirect
46. #15690835C22semidirect
47. #15690834C22semidirect
48. #15690831C22semidirect
49. #15690830C22semidirect
50. #15690829C22semidirect
51. C2×Dic388C22semidirect
52. C2×#13095501C22semidirect
53. Q16×D97C22semidirect
54. C2×#13095500C22semidirect
55. C2×#13095499C22semidirect
56. SD16×D97C22semidirect
57. #15690818C22semidirect
58. #15690816C22semidirect
59. C2×#13095495C22semidirect
60. M16×D97C22semidirect
62. C2×#41×C97C22semidirect
63. #137×C97C22semidirect
64. #137×C97C22semidirect
65. AGL(1,ℤ/8)×C97C22semidirect
66. AGL(1,ℤ/8)×C97C22semidirect
67. #135×C97C22semidirect
68. C2×M16×C97C22semidirect
262. C97#3586semidirect
271. #575D97semidirect
296. C2×#41C2×D97semidirect
309. #137C2×D97semidirect
310. #137C2×D97semidirect
314. AGL(1,ℤ/8)C2×D97semidirect
315. AGL(1,ℤ/8)C2×D97semidirect
319. #135C2×D97semidirect
330. C2×M16C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615712
C2×#156908572
#202615512
#202615492
#202615452
#202615422
#202615332
#202615312
C2×#156908432
C2×#156908402
#137×D972
#202615012
#575×C972
#358697