Finite Group Databasebeta

← #27151521

Subgroups — #27151521

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#20261291C211yesmaximal, characteristic, complemented
2C4×#13095510C211yesmaximal, characteristic
3#20261043C211yesmaximal, characteristic, complemented
4#20261039C211yesmaximal, characteristic, complemented
5#20261035C211yesmaximal, characteristic, complemented
6#20261029C211yesmaximal, characteristic, complemented
7#20261023C211yesmaximal, characteristic, complemented
8#20261019C211yesmaximal, characteristic, complemented
9#20261013C211yesmaximal, characteristic, complemented
10#20260772C211yesmaximal, characteristic, complemented
11#20260770C211yesmaximal, characteristic, complemented
12C4⋊C8×D97C211yesmaximal, characteristic, complemented
13#20260728C211yesmaximal, characteristic
14#20260728C211yesmaximal, characteristic
15#478×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C2×C8×D97C2211yescharacteristic
18C2×C8×D97C2211yescharacteristic
19#15690809C2211yescharacteristic
20#15690809C2211yescharacteristic
21#15690802C2211yescharacteristic
22#15690788—21no—
23#15690788—21no—
24C4×Dic194C2211yescharacteristic
25#15690775—21no—
26#15690775—21no—
27#15690775C2211yescharacteristic
28D4×Dic97C2211yescharacteristic
29C4×C22⋊D97—21no—
30#15690770—21no—
31#15690768C2211yescharacteristic
32#15690768C2211yescharacteristic
33#15690768C2211yescharacteristic
34#15690766—21no—
35C2×C4×Dic97—21no—
36C42×D97C2211yescharacteristic
37#15690744C2211yescharacteristic
38#15690743C2211yescharacteristic
39#15690743C2211yescharacteristic
40#15690742C2211yescharacteristic
41#15690741C2211yescharacteristic
42#15690741C2211yescharacteristic
43C4⋊#11715895C2211yescharacteristic
44#15690739C2211yescharacteristic
45#15690739C2211yescharacteristic
46#15690735C2211yescharacteristic
47#15690735C2211yescharacteristic
48#15690733C2211yescharacteristic
49#15690733C2211yescharacteristic
50#15690731C2211yescharacteristic
51#15690731C2211yescharacteristic
52#125×C97C2211yescharacteristic
53C4×D4×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic
55#106×C97C2211yescharacteristic
56C4⋊C8×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59#13095510—22no—
60#13095510Q814yescomplemented
61C2×Dic194—21no—
62C2×Dic194—21no—
63C2×Dic194C2311yescharacteristic
64C2×C22⋊D97—21no—
65C22×Dic97—21no—
66C22×Dic97—21no—
67C2×C4×D97D411yescharacteristic
68C2×C4×D97D411yescharacteristic
69C2×C4×D97C2311yescharacteristic
70C2×#11715895C2311yescharacteristic
71C2×#11715895C2311yescharacteristic
72C8×D97—22no—
73C8×D97—22no—
74C4⋊Dic97C2311yescharacteristic
75C4⋊Dic97C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83C22⋊Dic97—21no—
84#13095488—21no—
85#13095488—21no—
86#13095488—21no—
87C4×Dic97—21no—
88C4×Dic97—21no—
89C4×Dic97—21no—
90C4×Dic97—21no—
91C4×Dic97—21no—
92C4×Dic97D411yescharacteristic
93C4×Dic97D411yescharacteristic
94C4×Dic97C2311yescharacteristic
95C2×D4×C97C2311yescharacteristic
96C22×C4×C97—21no—
97C2×C8×C97C2311yescharacteristic
98C2×C8×C97C2311yescharacteristic
99C4⋊C4×C97—21no—
100C4⋊C4×C97—21no—
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C22⋊C4×C97—21no—
105C42×C97C2311yescharacteristic
106C23×C97—21no—
107C22×D97C2×D411yescharacteristic
108D4×C97—21no—
109D4×C97C2×Q812yes—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—42no—
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2×D411yescharacteristic
116C2×C4×C97C2411yescharacteristic
117C22⋊D97—24no—
118C2×Dic97—21no—
119C2×Dic97—24no—
120C2×Dic97—21no—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—42no—
124C2×Dic97C2×D411yescharacteristic
125C2×Dic97C2×D411yescharacteristic
126C2×Dic97C2×D411yescharacteristic
127C4×D97—22no—
128C4×D97—41no—
129C4×D97C2×Q812yes—
130C4×D97#4112yes—
131Dic194—21no—
132Dic194—42no—
133Dic194—42no—
134Dic194C2×Q812yes—
135C8×C97—21no—
136C8×C97—21no—
137#11715895—21no—
138#11715895—21no—
139C22×C97—22no—
140C22×C97—41no—
141C22×C97—41no—
142C22×C97C22×D411yescharacteristic
143C2×D97—21no—
144C2×D97—21no—
145C2×D97C22⋊Q812yes—
146C4×C97—21no—
147C4×C97—21no—
148C4×C97—41no—
149C4×C97—41no—
150C4×C97—41no—
151C4×C97C22×Q811yescharacteristic
152C4×C97C2×#4111yescharacteristic
153Dic97—22no—
154Dic97—21no—
155Dic97—41no—
156Dic97—41no—
157Dic97—41no—
158Dic97C22⋊Q812yes—
159Dic97#13512yes—
160C2×C97—42no—
161C2×C97C2×C22⋊Q811yescharacteristic
162C2×C97C2×#13511yescharacteristic
163C2×C97#57511yescharacteristic
164D97—22no—
165#3076—971nomaximal, Sylow
166C97#307611yescharacteristic, complemented, Sylow
167#533—971no—
168C4×#41—971no—
169#479—971no—
170#479—971no—
171#478—971no—
172#478D9711yescharacteristic, complemented
173#477—971no—
174#476—971no—
175#475—971no—
176#474—971no—
177#427—971no—
178#427—971no—
179C2×C4⋊C8—971no—
180#416—971no—
181#416—971no—
182C2×#41—971no—
183C22×C8—971no—
184C22×C8—971no—
185C4⋊Q8—971no—
186C4⋊Q8—971no—
187#125—971no—
188#125C2×D9711yescharacteristic
189C22⋊Q8—1941no—
190C22⋊Q8—1941no—
191C4×Q8—971no—
192C4×Q8—971no—
193C4×Q8—1941no—
194C4×Q8—1941no—
195C4×D4—971no—
196C4×D4—1941no—
197C4×D4—1941no—
198C4×D4C2×D9711yescharacteristic
199#117—971no—
200#117—971no—
201#117—971no—
202#117—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#103—971no—
220#103—971no—
221#102—971no—
222#102—971no—
223#102C2×D9711yescharacteristic
224#102C2×D9711yescharacteristic
225#41—974no—
226#41—1942no—
227C2×Q8—971no—
228C2×Q8—1941no—
229C2×Q8—1941no—
230C2×D4—1941no—
231C2×D4C22×D9711yescharacteristic
232C22×C4—21no—
233C22×C4—971no—
234C22×C4—971no—
235C22×C4—971no—
236C22×C4—1941no—
237C22×C4—1941no—
238C2×C8—971no—
239C2×C8—971no—
240C2×C8—1942no—
241C2×C8—1942no—
242C2×C8C22×D9711yescharacteristic
243C2×C8C22×D9711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—971no—
247C4⋊C4—971no—
248C4⋊C4—971no—
249C4⋊C4—1941no—
250C4⋊C4—1941no—
251C4⋊C4—1941no—
252C4⋊C4—1941no—
253C4⋊C4—1941no—
254C4⋊C4—1941no—
255C4⋊C4C22×D9711yescharacteristic
256C4⋊C4C22×D9711yescharacteristic
257C4⋊C4C22×D9711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—1941no—
260C22⋊C4—1941no—
261C22⋊C4—1941no—
262C22⋊C4—1941no—
263C42—971no—
264C42—971no—
265C42—971no—
266C42—1941no—
267C42—1941no—
268C42—1941no—
269C42—1941no—
270C42—1941no—
271C42C22×D9711yescharacteristic
272C23—21no—
273C23—971no—
274Q8—972no—
275Q8—3882no—
276Q8—3882no—
277Q8—1941no—
278D4—21no—
279D4—1944no—
280D4Q8×D9712yes—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—971no—
286C2×C4—971no—
287C2×C4—972no—
288C2×C4—972no—
289C2×C4—971no—
290C2×C4—3881no—
291C2×C4—3882no—
292C2×C4—1942no—
293C2×C4—1944no—
294C2×C4—1941no—
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—22no—
306C22—41no—
307C22—41no—
308C22—972no—
309C22—1941no—
310C22—1941no—
311C22C2×D4×D9711yescharacteristic
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—972no—
318C4—972no—
319C4—3881no—
320C4—3881no—
321C4—3881no—
322C4—1942no—
323C4—1941no—
324C4C2×Q8×D9711yescharacteristic
325C4#41×D9711yescharacteristic
326C2—42no—
327C2—1942no—
328C2C22⋊Q8×D9711yescharacteristic
329C2#135×D9711yescharacteristic
330C2#2026153111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#478×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095510C22
hypercenter#478D97
hypocenterC97#3076
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #20261291C2semidirect
3. #20261043C2semidirect
4. #20261039C2semidirect
5. #20261035C2semidirect
6. #20261029C2semidirect
7. #20261023C2semidirect
8. #20261019C2semidirect
9. #20261013C2semidirect
10. #20260772C2semidirect
11. #20260770C2semidirect
12. C4⋊C8×D97C2semidirect
15. #478×C97C2semidirect
60. #13095510Q8semidirect
166. C97#3076semidirect
172. #478D97semidirect

Maximal subgroups

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

subgroupindex
#202612912
C4×#130955102
#202610432
#202610392
#202610352
#202610292
#202610232
#202610192
#202610132
#202607722
#202607702
C4⋊C8×D972
#202607282
#478×C972
#307697