Finite Group Databasebeta

← #27156506

Subgroups — #27156506

Order 12416.

1424 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261572C212yesmaximal, complemented
2#20261548C213yesmaximal, complemented
3#20261544C216yesmaximal, complemented
4#20261543C212yesmaximal, complemented
5#20261540C211yesmaximal, characteristic, complemented
6#20261537C213yesmaximal, complemented
7#20261534C216yesmaximal, complemented
8C2×#15690833C213yesmaximal, complemented
9#135×D97C213yesmaximal, complemented
10#131×D97C211yesmaximal, characteristic, complemented
11#578×C97C211yesmaximal, characteristic, complemented
12#15690858—22no—
13#15690858C2216yescomplemented
14#15690856—26no—
15#15690856C2212yescomplemented
16#15690856C2216yescomplemented
17C2×#13095512—22no—
18C2×#13095512C2216yescomplemented
19#41×D97—26no—
20#41×D97C2211yescharacteristic
21#41×D97C2216yescomplemented
22#15690842C22112yescomplemented
23#15690841C2216yescomplemented
24#15690838C2216yescomplemented
25#15690837C22112yescomplemented
26#15690835C2216yescomplemented
27#15690833C2213yescomplemented
28#15690833C2216yescomplemented
29#15690833C2216yescomplemented
30#15690833C2216yescomplemented
31#15690832C2213yescomplemented
32#15690831C2216yescomplemented
33#15690830C2216yescomplemented
34#15690829C2211yescharacteristic, complemented
35Q16×D97C2213yescomplemented
36C2×#13095498C2216yescomplemented
37SD16×D97C2216yescomplemented
38C2×D776C2213yescomplemented
39D8×D97C2213yescomplemented
40#15690818C2213yescomplemented
41#15690817C2211yescharacteristic, complemented
42#15690816C2213yescomplemented
43M16×D97C2213yescomplemented
44C2×C8×D97C2213yescomplemented
452−1+4×C97C2212yescomplemented
46#137×C97C2216yescomplemented
47#135×C97C2213yescomplemented
48C2×Q16×C97C2213yescomplemented
49#131×C97C2211yescharacteristic, complemented
50#13095512—22no—
51#13095512—26no—
52#13095512—26no—
53#13095512—26no—
54#13095512D411yescharacteristic
55#13095512C2316yes—
56#13095512C23112yes—
57#13095511—212no—
58#13095511—26no—
59#13095511—26no—
60#13095511C2313yes—
61#13095511C2316yes—
62#13095510—26no—
63#13095510D413yes—
64#13095510C2312yes—
65Q8×D97—26no—
66Q8×D97D411yescharacteristic
67Q8×D97C2316yes—
68C2×D388—26no—
69C2×D388—26no—
70C2×D388C2316yes—
71D4×D97—26no—
72D4×D97—26no—
73D4×D97—212no—
74D4×D97D413yes—
75D4×D97C2316yes—
76C2×C4×D97—26no—
77C2×C4×D97—26no—
78C2×C4×D97C2313yes—
79Dic388C2311yescharacteristic
80#13095501C2316yes—
81#13095500C2316yes—
82#13095499C2312yes—
83#13095498C2316yes—
84#13095498C23112yes—
85D776C2313yes—
86D776C2316yes—
87#13095496C2316yes—
88#13095495C2313yes—
89#13095494C2316yes—
90C2×#11715895C2313yes—
91C8×D97C2311yescharacteristic
92C8×D97C2313yes—
93C8×D97C2313yes—
94C8×D97C2313yes—
95#41×C97—26no—
96#41×C97C2316yes—
97#41×C97C2311yescharacteristic
98C2×Q8×C97—22no—
99C2×Q8×C97C2316yes—
100Q16×C97C2313yes—
101Q16×C97C2316yes—
102SD16×C97C2316yes—
103D8×C97C2311yescharacteristic
104M16×C97C2313yes—
105C2×C8×C97C2313yes—
106C22×D97—26no—
107C22×D97—26no—
108C22×D97C2×D413yes—
109Q8×C97—26no—
110Q8×C97C2×D411yescharacteristic
111Q8×C97C2416yes—
112D4×C97—26no—
113D4×C97C2×D413yes—
114D4×C97C2412yes—
115C2×C4×C97—26no—
116C2×C4×C97—26no—
117C2×C4×C97C2413yes—
118C22⋊D97—26no—
119C22⋊D97—22no—
120C22⋊D97—26no—
121C22⋊D97C2×D416yes—
122C2×Dic97—22no—
123C2×Dic97C2×D413yes—
124D388—212no—
125D388—26no—
126D388—26no—
127D388C2×D413yes—
128D388C2416yes—
129C4×D97—26no—
130C4×D97—26no—
131C4×D97—26no—
132C4×D97—212no—
133C4×D97C2×D413yes—
134C4×D97C2×D413yes—
135C4×D97C2411yescharacteristic
136C4×D97C2413yes—
137Dic194—26no—
138Dic194C2×D413yes—
139Dic194C2412yes—
140C8×C97C2411yescharacteristic
141C8×C97C2413yes—
142#11715895C2411yescharacteristic
143#11715895C2413yes—
144C22×C97—22no—
145C22×C97C22×D413yes—
146C2×D97—26no—
147C2×D97—43no—
148C2×D97—86no—
149C2×D97—86no—
150C2×D97C22×D413yes—
151C2×D97C22×D411yescharacteristic
152C4×C97—26no—
153C4×C97C22×D413yes—
154C4×C97C2511yescharacteristic
155Dic97—22no—
156Dic97C22×D413yes—
157Dic97C22×D411yescharacteristic
158C2×C97—23no—
159C2×C97—42no—
160C2×C97C23×D411yescharacteristic
161D97—21no—
162D97—23no—
163D97—46no—
164#3584—971nomaximal, Sylow
165C97#358411yescharacteristic, complemented, Sylow
166#585—972no—
167#578—973no—
168#578D9711yescharacteristic, complemented
169#577—972no—
170#577—976no—
171#576—973no—
172#576—971no—
173#575—976no—
174C2×#135—973no—
175C2×#135—973no—
176C2×#131—971no—
1772−1+4—976no—
1782−1+4—1942no—
1792−1+4C2×D9712yescomplemented
180H(5,𝔽2)—976no—
181H(5,𝔽2)—972no—
182H(5,𝔽2)—1946no—
183C2×#41—971no—
184C2×#41—976no—
185C2×#41—976no—
186C2×#41—1942no—
187C2×#41—1946no—
188#137—976no—
189#137—9712no—
190#137C2×D9716yescomplemented
191AGL(1,ℤ/8)—976no—
192AGL(1,ℤ/8)—9712no—
193AGL(1,ℤ/8)—976no—
194#135—976no—
195#135—973no—
196#135—976no—
197#135—976no—
198#135—976no—
199#135—976no—
200#135—971no—
201#135—973no—
202#135C2×D9713yescomplemented
203C2×Q16—973no—
204C2×Q16C2×D9713yescomplemented
205C2×SD16—976no—
206C2×SD16—976no—
207C2×D8—973no—
208C2×D8—973no—
209#131—971no—
210#131—973no—
211#131—973no—
212#131C2×D9711yescharacteristic, complemented
213C2×M16—973no—
214C22×C8—973no—
215#41—26no—
216#41—971no—
217#41—973no—
218#41—973no—
219#41—976no—
220#41—9712no—
221#41—972no—
222#41—976no—
223#41—1946no—
224#41—19412no—
225#41—1946no—
226#41—1946no—
227#41—1946no—
228#41—1946no—
229#41—1946no—
230#41—1942no—
231#41C22×D9716yes—
232#41C22×D9711yescharacteristic
233C2×Q8—22no—
234C2×Q8—971no—
235C2×Q8—976no—
236C2×Q8—1946no—
237C2×Q8C22×D9716yes—
238C2×D4—973no—
239C2×D4—976no—
240C2×D4—976no—
241C2×D4—1946no—
242C2×D4—1946no—
243C2×D4—19412no—
244C2×D4—1946no—
245C2×D4—1946no—
246C22×C4—973no—
247C22×C4—1946no—
248C22×C4—1946no—
249Q16—976no—
250Q16—971no—
251Q16C22×D9713yes—
252Q16C22×D9716yes—
253SD16—976no—
254SD16—9712no—
255SD16—972no—
256SD16—976no—
257SD16C22×D9716yes—
258D8—976no—
259D8—973no—
260D8—976no—
261D8C22×D9711yescharacteristic
262M16—973no—
263M16—976no—
264M16C22×D9713yes—
265C2×C8—973no—
266C2×C8—971no—
267C2×C8—973no—
268C2×C8—973no—
269C2×C8—973no—
270C2×C8C22×D9713yes—
271C23—973no—
272C23—1946no—
273C23—1946no—
274Q8—26no—
275Q8—973no—
276Q8—972no—
277Q8—1946no—
278Q8D4×D9711yescharacteristic
279Q8C23×D9716yes—
280D4—26no—
281D4—976no—
282D4—973no—
283D4—976no—
284D4—1946no—
285D4—19412no—
286D4—1946no—
287D4—1942no—
288D4—1946no—
289D4—1946no—
290D4D4×D9713yes—
291D4C23×D9712yes—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—973no—
295C2×C4—973no—
296C2×C4—973no—
297C2×C4—971no—
298C2×C4—973no—
299C2×C4—1946no—
300C2×C4—19412no—
301C2×C4—1946no—
302C2×C4—1942no—
303C2×C4—1946no—
304C2×C4C23×D9713yes—
305C8—971no—
306C8—973no—
307C8C23×D9711yescharacteristic
308C8C23×D9713yes—
309C22—22no—
310C22—7766no—
311C22—7766no—
312C22—971no—
313C22—973no—
314C22—3883no—
315C22—1946no—
316C22C2×D4×D9713yes—
317C4—26no—
318C4—971no—
319C4—973no—
320C4—1942no—
321C4C2×D4×D9713yes—
322C4C24×D9711yescharacteristic
323C2—23no—
324C2—42no—
325C2—3886no—
326C2—1943no—
327C2—1941no—
328C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC2C22×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
Fitting#578×C97C2
socleC2×C97C23×D4
hypercenter#578D97
hypocenterC97#3584
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ #41 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #20261572C2semidirect
2. #20261548C2semidirect
3. #20261544C2semidirect
4. #20261543C2semidirect
5. #20261540C2semidirect
6. #20261537C2semidirect
7. #20261534C2semidirect
8. C2×#15690833C2semidirect
9. #135×D97C2semidirect
10. #131×D97C2semidirect
11. #578×C97C2semidirect
13. #15690858C22semidirect
15. #15690856C22semidirect
16. #15690856C22semidirect
18. C2×#13095512C22semidirect
21. #41×D97C22semidirect
22. #15690842C22semidirect
23. #15690841C22semidirect
24. #15690838C22semidirect
25. #15690837C22semidirect
26. #15690835C22semidirect
27. #15690833C22semidirect
28. #15690833C22semidirect
29. #15690833C22semidirect
30. #15690833C22semidirect
31. #15690832C22semidirect
32. #15690831C22semidirect
33. #15690830C22semidirect
34. #15690829C22semidirect
35. Q16×D97C22semidirect
36. C2×#13095498C22semidirect
37. SD16×D97C22semidirect
38. C2×D776C22semidirect
39. D8×D97C22semidirect
40. #15690818C22semidirect
41. #15690817C22semidirect
42. #15690816C22semidirect
43. M16×D97C22semidirect
44. C2×C8×D97C22semidirect
45. 2−1+4×C97C22semidirect
46. #137×C97C22semidirect
47. #135×C97C22semidirect
48. C2×Q16×C97C22semidirect
49. #131×C97C22semidirect
165. C97#3584semidirect
168. #578D97semidirect
179. 2−1+4C2×D97semidirect
190. #137C2×D97semidirect
202. #135C2×D97semidirect
204. C2×Q16C2×D97semidirect
212. #131C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615722
#202615482
#202615442
#202615432
#202615402
#202615372
#202615342
C2×#156908332
#135×D972
#131×D972
#578×C972
#358497