Finite Group Databasebeta

← #27151522

Subgroups — #27151522

Order 12416.

388 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#20261288C211yesmaximal, characteristic, complemented
2#20261203C211yesmaximal, characteristic
3#20261045C211yesmaximal, characteristic, complemented
4#20261041C211yesmaximal, characteristic, complemented
5#20261038C211yesmaximal, characteristic, complemented
6#20261034C211yesmaximal, characteristic, complemented
7#20261029C211yesmaximal, characteristic, complemented
8#20261024C211yesmaximal, characteristic, complemented
9#20261019C211yesmaximal, characteristic, complemented
10#20261014C211yesmaximal, characteristic, complemented
11#20260771C211yesmaximal, characteristic, complemented
12C2×#15690739C211yesmaximal, characteristic, complemented
13#20260732C211yesmaximal, characteristic
14#20260729C211yesmaximal, characteristic
15#427×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×#11715895C2211yescharacteristic
18#15690809C2211yescharacteristic
19#15690809C2211yescharacteristic
20#15690801C2211yescharacteristic
21#15690801C2211yescharacteristic
22#15690787—21no—
23#15690787—21no—
24C4×Dic194—21no—
25C4×Dic194—21no—
26#15690775C2211yescharacteristic
27#15690775C2211yescharacteristic
28#15690772C2211yescharacteristic
29#15690772C2211yescharacteristic
30#15690770—21no—
31#15690770—21no—
32#15690769C2211yescharacteristic
33#15690767—21no—
34C2×C4×Dic97C2211yescharacteristic
35C42×D97—21no—
36#15690743C2211yescharacteristic
37#15690743C2211yescharacteristic
38#15690741C2211yescharacteristic
39#15690741C2211yescharacteristic
40#15690739C2211yescharacteristic
41#15690739C2211yescharacteristic
42#15690739C2211yescharacteristic
43#15690739C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690735C2211yescharacteristic
46#15690733C2211yescharacteristic
47#15690733C2211yescharacteristic
48#15690732C2211yescharacteristic
49#15690732C2211yescharacteristic
50#15690730C2211yescharacteristic
51#15690730C2211yescharacteristic
52C22×C8×C97C2211yescharacteristic
53#117×C97C2211yescharacteristic
54#117×C97C2211yescharacteristic
55#107×C97C2211yescharacteristic
56#107×C97C2211yescharacteristic
57#106×C97C2211yescharacteristic
58#106×C97C2211yescharacteristic
59#13095511—22no—
60#13095511Q814yescomplemented
61C2×Dic194—21no—
62C2×Dic194—21no—
63C2×Dic194C2311yescharacteristic
64C2×D388C2311yescharacteristic
65C2×C22⋊D97—21no—
66C22×Dic97D411yescharacteristic
67C22×Dic97D411yescharacteristic
68C2×C4×D97—21no—
69C2×C4×D97—21no—
70C2×C4×D97—21no—
71C2×#11715895—22no—
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C4⋊Dic97—21no—
75C4⋊Dic97—21no—
76C4⋊Dic97C2311yescharacteristic
77C4⋊Dic97C2311yescharacteristic
78#13095490—21no—
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84C22⋊Dic97—21no—
85#13095488—21no—
86#13095488—21no—
87C4×Dic97—21no—
88C4×Dic97—21no—
89C4×Dic97—21no—
90C4×Dic97D411yescharacteristic
91C4×Dic97D411yescharacteristic
92C4×Dic97C2311yescharacteristic
93C4×Dic97C2311yescharacteristic
94C22×C4×C97C2311yescharacteristic
95C2×C8×C97—22no—
96C2×C8×C97C2311yescharacteristic
97C2×C8×C97C2311yescharacteristic
98C4⋊C4×C97C2311yescharacteristic
99C4⋊C4×C97C2311yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C22⋊C4×C97—21no—
103C22⋊C4×C97—21no—
104C42×C97—21no—
105C42×C97—21no—
106C23×C97C2×D411yescharacteristic
107C22×D97—21no—
108C2×C4×C97—21no—
109C2×C4×C97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97C2×D411yescharacteristic
113C2×C4×C97C2×Q812yes—
114C2×C4×C97#4112yes—
115C2×C4×C97C2411yescharacteristic
116C22⋊D97—24no—
117C2×Dic97—21no—
118C2×Dic97—22no—
119C2×Dic97—21no—
120C2×Dic97—21no—
121C2×Dic97—41no—
122C2×Dic97C2×D411yescharacteristic
123C2×Dic97C2×D411yescharacteristic
124C2×Dic97C2×D411yescharacteristic
125C2×Dic97C2×D411yescharacteristic
126D388—21no—
127D388C2×Q812yes—
128C4×D97—24no—
129C4×D97—42no—
130C4×D97—42no—
131Dic194—21no—
132Dic194—42no—
133Dic194—42no—
134Dic194C2×Q812yes—
135C8×C97—21no—
136C8×C97—21no—
137#11715895—21no—
138#11715895—21no—
139C22×C97—21no—
140C22×C97—21no—
141C22×C97C22⋊Q812yes—
142C22×C97C22×D411yescharacteristic
143C2×D97—22no—
144C2×D97—41no—
145C2×D97—41no—
146C4×C97—41no—
147C4×C97—41no—
148C4×C97—41no—
149C4×C97—41no—
150C4×C97C22⋊Q812yes—
151C4×C97C22×Q811yescharacteristic
152C4×C97C2×#4111yescharacteristic
153Dic97—22no—
154Dic97—21no—
155Dic97—21no—
156Dic97—21no—
157Dic97—41no—
158Dic97—41no—
159Dic97#13512yes—
160C2×C97—22no—
161C2×C97C2×C22⋊Q811yescharacteristic
162C2×C97C2×#13511yescharacteristic
163C2×C97#57511yescharacteristic
164D97—42no—
165#3076—971nomaximal, Sylow
166C97#307611yescharacteristic, complemented, Sylow
167#533—971no—
168C4×#41—971no—
169#479—971no—
170#479—971no—
171#478—971no—
172#478—971no—
173#477—971no—
174#476—971no—
175#475—971no—
176#474—971no—
177#427—971no—
178#427D9711yescharacteristic, complemented
179C2×C4⋊C8—971no—
180#416—971no—
181#416—971no—
182C2×#41—971no—
183C22×C8—971no—
184C22×C8C2×D9711yescharacteristic
185C4⋊Q8—971no—
186C4⋊Q8—971no—
187#125—971no—
188#125—971no—
189C22⋊Q8—1941no—
190C22⋊Q8—1941no—
191C4×Q8—971no—
192C4×Q8—971no—
193C4×Q8—1941no—
194C4×Q8—1941no—
195C4×D4—971no—
196C4×D4—971no—
197C4×D4—1941no—
198C4×D4—1941no—
199#117—971no—
200#117—1941no—
201#117C2×D9711yescharacteristic
202#117C2×D9711yescharacteristic
203C2×C42—971no—
204C2×C42—1941no—
205#107—971no—
206#107—971no—
207#107C2×D9711yescharacteristic
208#107C2×D9711yescharacteristic
209#106—971no—
210#106—971no—
211#106C2×D9711yescharacteristic
212#106C2×D9711yescharacteristic
213C4⋊C8—971no—
214C4⋊C8—971no—
215C4⋊C8—971no—
216C4⋊C8—971no—
217#103—971no—
218#103—971no—
219#103—971no—
220#103—971no—
221#102—971no—
222#102—971no—
223#102—971no—
224#102—971no—
225#41—974no—
226#41—1942no—
227C2×Q8—971no—
228C2×Q8—1941no—
229C2×Q8—1941no—
230C2×D4—971no—
231C2×D4—1941no—
232C22×C4—971no—
233C22×C4—971no—
234C22×C4—1941no—
235C22×C4—1941no—
236C22×C4—1941no—
237C22×C4C22×D9711yescharacteristic
238C2×C8—22no—
239C2×C8—971no—
240C2×C8—971no—
241C2×C8—1942no—
242C2×C8C22×D9711yescharacteristic
243C2×C8C22×D9711yescharacteristic
244C4⋊C4—971no—
245C4⋊C4—971no—
246C4⋊C4—1941no—
247C4⋊C4—1941no—
248C4⋊C4—1941no—
249C4⋊C4—1941no—
250C4⋊C4—1941no—
251C4⋊C4—1941no—
252C4⋊C4—1941no—
253C4⋊C4—1941no—
254C4⋊C4C22×D9711yescharacteristic
255C4⋊C4C22×D9711yescharacteristic
256C4⋊C4C22×D9711yescharacteristic
257C4⋊C4C22×D9711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—1941no—
261C22⋊C4—1941no—
262C22⋊C4—1941no—
263C42—21no—
264C42—21no—
265C42—971no—
266C42—971no—
267C42—971no—
268C42—971no—
269C42—1941no—
270C42—1941no—
271C42—1941no—
272C23—1941no—
273C23D4×D9711yescharacteristic
274Q8—972no—
275Q8—3882no—
276Q8—3882no—
277Q8—1941no—
278D4—972no—
279D4—1944no—
280D4—1941no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—971no—
286C2×C4—971no—
287C2×C4—971no—
288C2×C4—971no—
289C2×C4—3882no—
290C2×C4—3882no—
291C2×C4—3881no—
292C2×C4—1942no—
293C2×C4—1941no—
294C2×C4—1941no—
295C2×C4—1941no—
296C2×C4—1944no—
297C2×C4D4×D9711yescharacteristic
298C2×C4Q8×D9712yes—
299C2×C4#1309551112yes—
300C2×C4C23×D9711yescharacteristic
301C8—21no—
302C8—21no—
303C8—1941no—
304C8—1941no—
305C22—21no—
306C22—21no—
307C22—3881no—
308C22—3881no—
309C22—1942no—
310C22#1569078812yes—
311C22C2×D4×D9711yescharacteristic
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—972no—
317C4—3881no—
318C4—3881no—
319C4—1942no—
320C4—1941no—
321C4—1941no—
322C4—1941no—
323C4#1569078812yes—
324C4C2×Q8×D9711yescharacteristic
325C4C2×#1309551111yescharacteristic
326C2—22no—
327C2—3882no—
328C2C2×#1569078811yescharacteristic
329C2#135×D9711yescharacteristic
330C2#2026153211yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#427×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095511C22
hypercenter#427D97
hypocenterC97#3076
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #20261288C2semidirect
3. #20261045C2semidirect
4. #20261041C2semidirect
5. #20261038C2semidirect
6. #20261034C2semidirect
7. #20261029C2semidirect
8. #20261024C2semidirect
9. #20261019C2semidirect
10. #20261014C2semidirect
11. #20260771C2semidirect
12. C2×#15690739C2semidirect
15. #427×C97C2semidirect
60. #13095511Q8semidirect
166. C97#3076semidirect
178. #427D97semidirect

Maximal subgroups

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

subgroupindex
#202612882
#202612032
#202610452
#202610412
#202610382
#202610342
#202610292
#202610242
#202610192
#202610142
#202607712
C2×#156907392
#202607322
#202607292
#427×C972
#307697