Finite Group Databasebeta

← #27151555

Subgroups — #27151555

Order 12416.

384 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#125×D97C211yesmaximal, characteristic, complemented
2C4×#13095512C211yesmaximal, characteristic
3#20261044C211yesmaximal, characteristic, complemented
4#20261042C211yesmaximal, characteristic, complemented
5#20261040C211yesmaximal, characteristic, complemented
6#20261038C211yesmaximal, characteristic, complemented
7#20261036C211yesmaximal, characteristic, complemented
8#20261034C211yesmaximal, characteristic, complemented
9#20261032C211yesmaximal, characteristic, complemented
10#20260776C211yesmaximal, characteristic, complemented
11#20260774C211yesmaximal, characteristic, complemented
12#20260757C211yesmaximal, characteristic, complemented
13#20260735C211yesmaximal, characteristic
14#20260735C211yesmaximal, characteristic
15#479×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095494C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19#15690802C2211yescharacteristic
20#15690801C2211yescharacteristic
21#15690801C2211yescharacteristic
22#15690800—21no—
23#15690800—21no—
24Q8×Dic97C2211yescharacteristic
25C4×D388—21no—
26C4×D388C2211yescharacteristic
27#15690772—21no—
28#15690772C2211yescharacteristic
29#15690768—21no—
30#15690768C2211yescharacteristic
31C4⋊C4×D97—21no—
32C4⋊C4×D97—21no—
33C4⋊C4×D97C2211yescharacteristic
34C4⋊C4×D97C2211yescharacteristic
35C42×D97—21no—
36C42×D97C2211yescharacteristic
37#15690744C2211yescharacteristic
38#15690743C2211yescharacteristic
39#15690743C2211yescharacteristic
40#15690742C2211yescharacteristic
41#15690741C2211yescharacteristic
42#15690741C2211yescharacteristic
43C4⋊#11715895C2211yescharacteristic
44#15690739C2211yescharacteristic
45#15690739C2211yescharacteristic
46#15690734C2211yescharacteristic
47#15690734C2211yescharacteristic
48#15690732C2211yescharacteristic
49#15690732C2211yescharacteristic
50#15690730C2211yescharacteristic
51#15690730C2211yescharacteristic
52#125×C97C2211yescharacteristic
53C4×Q8×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic
55#106×C97C2211yescharacteristic
56C4⋊C8×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic
59#13095512—22no—
60#13095512Q814yes—
61C2×D388—21no—
62C2×D388C2311yescharacteristic
63C2×C4×D97—21no—
64C2×C4×D97—21no—
65C2×C4×D97—21no—
66C2×C4×D97—21no—
67C2×C4×D97—21no—
68C2×C4×D97D411yescharacteristic
69C2×C4×D97D411yescharacteristic
70C2×C4×D97C2311yescharacteristic
71#13095494—22no—
72#13095494—22no—
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C4⋊Dic97—21no—
76C4⋊Dic97—21no—
77C4⋊Dic97—21no—
78C4⋊Dic97C2311yescharacteristic
79C4⋊Dic97C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86#13095490—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90C4×Dic97—21no—
91C4×Dic97—21no—
92C4×Dic97D411yescharacteristic
93C4×Dic97D411yescharacteristic
94C4×Dic97C2311yescharacteristic
95C2×Q8×C97C2311yescharacteristic
96C2×C8×C97C2311yescharacteristic
97C2×C8×C97C2311yescharacteristic
98C4⋊C4×C97—21no—
99C4⋊C4×C97—21no—
100C4⋊C4×C97—21no—
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C42×C97—21no—
105C42×C97C2311yescharacteristic
106C22×D97—21no—
107C22×D97C2×D411yescharacteristic
108Q8×C97—21no—
109Q8×C97C2×Q812yes—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2×D411yescharacteristic
116C2×C4×C97C2411yescharacteristic
117C2×Dic97—21no—
118C2×Dic97—21no—
119C2×Dic97—21no—
120C2×Dic97C2×D411yescharacteristic
121C2×Dic97C2×D411yescharacteristic
122C2×Dic97C2×D411yescharacteristic
123D388—24no—
124D388—21no—
125D388C2×Q812yes—
126C4×D97—24no—
127C4×D97—41no—
128C4×D97—41no—
129C4×D97—42no—
130C4×D97—42no—
131C4×D97—42no—
132C4×D97—42no—
133C4×D97C2×Q812yes—
134C4×D97#4112yes—
135C8×C97—21no—
136C8×C97—21no—
137#11715895—21no—
138#11715895—21no—
139C22×C97C22×D411yescharacteristic
140C2×D97—21no—
141C2×D97—21no—
142C2×D97—22no—
143C2×D97—41no—
144C2×D97—41no—
145C2×D97C22⋊Q812yes—
146C4×C97—22no—
147C4×C97—21no—
148C4×C97—21no—
149C4×C97—41no—
150C4×C97—41no—
151C4×C97—41no—
152C4×C97C22×Q811yescharacteristic
153C4×C97C2×#4111yescharacteristic
154Dic97—21no—
155Dic97—21no—
156Dic97—41no—
157Dic97—41no—
158Dic97—41no—
159Dic97C22⋊Q812yes—
160C2×C97C2×C22⋊Q811yescharacteristic
161C2×C97#57511yescharacteristic
162C2×C97#57511yescharacteristic
163D97—22no—
164D97—42no—
165#3080—971nomaximal, Sylow
166C97#308011yescharacteristic, complemented, Sylow
167C2×#125—971no—
168C4×#41—971no—
169#479—971no—
170#479—971no—
171#479—971no—
172#479D9711yescharacteristic, complemented
173#478—971no—
174#478—971no—
175#478—971no—
176#478—971no—
177#428—971no—
178#428—971no—
179#423—971no—
180#417—971no—
181#417—971no—
182C2×#41—971no—
183C2×M16—971no—
184C2×M16—971no—
185#125—971no—
186#125—971no—
187#125—971no—
188#125—1941no—
189#125—1941no—
190#125C2×D9711yescharacteristic
191C4×Q8—971no—
192C4×Q8C2×D9711yescharacteristic
193C4×D4—971no—
194C4×D4—971no—
195C4×D4—1941no—
196C4×D4—1941no—
197#117—971no—
198#117—1941no—
199C2×C4⋊C4—971no—
200C2×C4⋊C4—971no—
201C2×C4⋊C4—1941no—
202C2×C4⋊C4—1941no—
203C2×C42—971no—
204C2×C42—1941no—
205#107—971no—
206#107—971no—
207#107—971no—
208#107C2×D9711yescharacteristic
209#106—971no—
210#106—971no—
211#106—971no—
212#106C2×D9711yescharacteristic
213C4⋊C8—971no—
214C4⋊C8—971no—
215C4⋊C8—971no—
216C4⋊C8C2×D9711yescharacteristic
217#103—971no—
218#103—971no—
219#103C2×D9711yescharacteristic
220#103C2×D9711yescharacteristic
221#102—971no—
222#102—971no—
223#102—971no—
224#102—971no—
225#41—974no—
226#41—1942no—
227C2×Q8C22×D9711yescharacteristic
228C2×D4—971no—
229C2×D4—1941no—
230C22×C4—971no—
231C22×C4—971no—
232C22×C4—971no—
233C22×C4—1941no—
234C22×C4—1941no—
235C22×C4—1941no—
236C22×C4—1941no—
237C22×C4—1941no—
238M16—1942no—
239M16—1942no—
240C2×C8—971no—
241C2×C8—971no—
242C2×C8C22×D9711yescharacteristic
243C2×C8C22×D9711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—971no—
248C4⋊C4—971no—
249C4⋊C4—971no—
250C4⋊C4—1941no—
251C4⋊C4—1941no—
252C4⋊C4—1941no—
253C4⋊C4—1941no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4—1941no—
257C4⋊C4—1941no—
258C4⋊C4—1941no—
259C4⋊C4C22×D9711yescharacteristic
260C4⋊C4C22×D9711yescharacteristic
261C4⋊C4C22×D9711yescharacteristic
262C22⋊C4—1941no—
263C22⋊C4—1941no—
264C22⋊C4—1941no—
265C42—21no—
266C42—971no—
267C42—971no—
268C42—971no—
269C42—1941no—
270C42—1941no—
271C42C22×D9711yescharacteristic
272C23—971no—
273C23—1941no—
274Q8—21no—
275Q8Q8×D9712yes—
276D4—972no—
277D4—1944no—
278D4—1941no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—971no—
284C2×C4—972no—
285C2×C4—972no—
286C2×C4—971no—
287C2×C4—971no—
288C2×C4—3882no—
289C2×C4—3881no—
290C2×C4—3881no—
291C2×C4—3882no—
292C2×C4—3882no—
293C2×C4—3882no—
294C2×C4—1944no—
295C2×C4—1941no—
296C2×C4—1941no—
297C2×C4—1941no—
298C2×C4D4×D9711yescharacteristic
299C2×C4D4×D9711yescharacteristic
300C2×C4C23×D9711yescharacteristic
301C8—21no—
302C8—21no—
303C8—1941no—
304C8—1941no—
305C22—972no—
306C22—3881no—
307C22—3881no—
308C22—1942no—
309C22—1941no—
310C22—1941no—
311C22C2×D4×D9711yescharacteristic
312C4—22no—
313C4—21no—
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—972no—
319C4—3881no—
320C4—3881no—
321C4—3881no—
322C4—1941no—
323C4—1941no—
324C4C2×Q8×D9711yescharacteristic
325C4#41×D9711yescharacteristic
326C2—3882no—
327C2—1942no—
328C2C22⋊Q8×D9711yescharacteristic
329C2#2026153211yescharacteristic
330C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#479×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095512C22
hypercenter#479D97
hypocenterC97#3080
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #125×D97C2semidirect
3. #20261044C2semidirect
4. #20261042C2semidirect
5. #20261040C2semidirect
6. #20261038C2semidirect
7. #20261036C2semidirect
8. #20261034C2semidirect
9. #20261032C2semidirect
10. #20260776C2semidirect
11. #20260774C2semidirect
12. #20260757C2semidirect
15. #479×C97C2semidirect
166. C97#3080semidirect
172. #479D97semidirect

Maximal subgroups

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

subgroupindex
#125×D972
C4×#130955122
#202610442
#202610422
#202610402
#202610382
#202610362
#202610342
#202610322
#202607762
#202607742
#202607572
#202607352
#479×C972
#308097