Finite Group Databasebeta

← #27151520

Subgroups — #27151520

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#41×Dic97C211yesmaximal, characteristic
3#20261044C211yesmaximal, characteristic, complemented
4#20261040C211yesmaximal, characteristic, complemented
5#20261037C211yesmaximal, characteristic, complemented
6#20261033C211yesmaximal, characteristic, complemented
7#20261028C211yesmaximal, characteristic, complemented
8#20261023C211yesmaximal, characteristic, complemented
9#20261018C211yesmaximal, characteristic, complemented
10#20261013C211yesmaximal, characteristic, complemented
11#20260773C211yesmaximal, characteristic, complemented
12#20260771C211yesmaximal, characteristic, complemented
13C2×#15690739C211yesmaximal, characteristic, complemented
14#20260731C211yesmaximal, characteristic
15#416×C97C211yesmaximal, characteristic
16C22×#11715895C2211yescharacteristic
17#15690809C2211yescharacteristic
18#15690809C2211yescharacteristic
19#15690801C2211yescharacteristic
20#15690801C2211yescharacteristic
21#15690787—21no—
22#15690787—21no—
23Q8×Dic97C2211yescharacteristic
24Q8×Dic97C2211yescharacteristic
25C4×Dic194—21no—
26C4×Dic194—21no—
27D4×Dic97—21no—
28D4×Dic97—21no—
29D4×Dic97C2211yescharacteristic
30D4×Dic97C2211yescharacteristic
31#15690769—21no—
32#15690769C2211yescharacteristic
33#15690769C2211yescharacteristic
34C2×C4×Dic97—21no—
35C2×C4×Dic97C2211yescharacteristic
36#15690744C2211yescharacteristic
37#15690744C2211yescharacteristic
38#15690743C2211yescharacteristic
39#15690743C2211yescharacteristic
40#15690742C2211yescharacteristic
41#15690742C2211yescharacteristic
42#15690741C2211yescharacteristic
43#15690741C2211yescharacteristic
44#15690739C2211yescharacteristic
45#15690739C2211yescharacteristic
46#15690739C2211yescharacteristic
47#15690739C2211yescharacteristic
48#15690734C2211yescharacteristic
49#15690734C2211yescharacteristic
50#15690731C2211yescharacteristic
51#15690731C2211yescharacteristic
52C2×#41×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic
54#117×C97C2211yescharacteristic
55#103×C97C2211yescharacteristic
56#103×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194—21no—
61C22×Dic97—21no—
62C22×Dic97—21no—
63C22×Dic97D411yescharacteristic
64C22×Dic97D411yescharacteristic
65C2×#11715895—22no—
66C2×#11715895C2311yescharacteristic
67C2×#11715895C2311yescharacteristic
68C4⋊Dic97—21no—
69C4⋊Dic97—21no—
70C4⋊Dic97—21no—
71C4⋊Dic97—21no—
72C4⋊Dic97C2311yescharacteristic
73C4⋊Dic97C2311yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75C4⋊Dic97C2311yescharacteristic
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80C22⋊Dic97—21no—
81C22⋊Dic97—21no—
82C22⋊Dic97—21no—
83C22⋊Dic97—21no—
84C4×Dic97—21no—
85C4×Dic97—21no—
86C4×Dic97—21no—
87C4×Dic97—21no—
88C4×Dic97D411yescharacteristic
89C4×Dic97D411yescharacteristic
90C4×Dic97C2311yescharacteristic
91C4×Dic97C2311yescharacteristic
92#41×C97—22no—
93#41×C97Q814yescomplemented
94C2×Q8×C97C2311yescharacteristic
95C2×D4×C97—21no—
96C2×D4×C97C2311yescharacteristic
97C22×C4×C97—21no—
98C22×C4×C97C2311yescharacteristic
99C2×C8×C97—22no—
100C2×C8×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C22⋊C4×C97—21no—
105C42×C97—21no—
106C23×C97—21no—
107C23×C97C2×D411yescharacteristic
108Q8×C97—21no—
109Q8×C97C2×Q812yes—
110D4×C97—24no—
111D4×C97—21no—
112D4×C97C2×Q812yes—
113C2×C4×C97—21no—
114C2×C4×C97—24no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97C2×D411yescharacteristic
118C2×C4×C97C2×Q812yes—
119C2×C4×C97#4112yes—
120C2×C4×C97C2411yescharacteristic
121C2×Dic97—22no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—41no—
127C2×Dic97—42no—
128C2×Dic97—42no—
129C2×Dic97C2×D411yescharacteristic
130C2×Dic97C2×D411yescharacteristic
131C2×Dic97C2×D411yescharacteristic
132C2×Dic97C2×D411yescharacteristic
133Dic194—42no—
134Dic194—42no—
135C8×C97—21no—
136C8×C97—21no—
137#11715895—21no—
138#11715895—21no—
139C22×C97—21no—
140C22×C97—21no—
141C22×C97—22no—
142C22×C97—41no—
143C22×C97—41no—
144C22×C97C22⋊Q812yes—
145C22×C97C22×D411yescharacteristic
146C4×C97—22no—
147C4×C97—41no—
148C4×C97—41no—
149C4×C97C22⋊Q812yes—
150C4×C97C22×Q811yescharacteristic
151C4×C97C2×#4111yescharacteristic
152Dic97—21no—
153Dic97—21no—
154Dic97—21no—
155Dic97—41no—
156Dic97—41no—
157Dic97—41no—
158Dic97—41no—
159Dic97#13512yes—
160C2×C97—22no—
161C2×C97—42no—
162C2×C97C2×C22⋊Q811yescharacteristic
163C2×C97C2×#13511yescharacteristic
164C2×C97#57511yescharacteristic
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#427—971no—
179C2×C4⋊C8—971no—
180#416—971no—
181#416D9711yescharacteristic
182C2×#41C2×D9711yescharacteristic
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—971no—
201#117—1941no—
202#117C2×D9711yescharacteristic
203C2×C42—971no—
204C2×C42—1941no—
205#107—971no—
206#107—971no—
207#107—971no—
208#107—971no—
209#106—971no—
210#106—971no—
211#106—971no—
212#106—971no—
213C4⋊C8—971no—
214C4⋊C8—971no—
215C4⋊C8—971no—
216C4⋊C8—971no—
217#103—971no—
218#103—971no—
219#103C2×D9711yescharacteristic
220#103C2×D9711yescharacteristic
221#102—971no—
222#102—971no—
223#102C2×D9711yescharacteristic
224#102C2×D9711yescharacteristic
225#41—22no—
226#41Dic19414yescomplemented
227C2×Q8—1941no—
228C2×Q8—1941no—
229C2×Q8C22×D9711yescharacteristic
230C2×D4—21no—
231C2×D4C22×D9711yescharacteristic
232C22×C4—21no—
233C22×C4—971no—
234C22×C4—971no—
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—971no—
247C4⋊C4—971no—
248C4⋊C4—1941no—
249C4⋊C4—1941no—
250C4⋊C4—1941no—
251C4⋊C4—1941no—
252C4⋊C4—1941no—
253C4⋊C4—1941no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4C22×D9711yescharacteristic
257C4⋊C4C22×D9711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—1941no—
260C22⋊C4—1941no—
261C22⋊C4—1941no—
262C22⋊C4—1941no—
263C42—21no—
264C42—971no—
265C42—971no—
266C42—971no—
267C42—971no—
268C42—1941no—
269C42—1941no—
270C42—1941no—
271C42—1941no—
272C23—21no—
273C23D4×D9711yescharacteristic
274Q8—21no—
275Q8—3882no—
276Q8—3882no—
277Q8C2×Dic19412yes—
278D4—24no—
279D4—21no—
280D4C2×Dic19412yes—
281C2×C4—21no—
282C2×C4—24no—
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—1941no—
297C2×C4D4×D9711yescharacteristic
298C2×C4C2×Dic19412yes—
299C2×C4#1309551012yes—
300C2×C4C23×D9711yescharacteristic
301C8—21no—
302C8—21no—
303C8—1941no—
304C8—1941no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—41no—
309C22—41no—
310C22#1569078712yes—
311C22C2×D4×D9711yescharacteristic
312C4—22no—
313C4—41no—
314C4—41no—
315C4—972no—
316C4—3881no—
317C4—3881no—
318C4—3881no—
319C4—3881no—
320C4—1941no—
321C4—1941no—
322C4—1941no—
323C4#1569078712yes—
324C4C22×Dic19411yescharacteristic
325C4C2×#1309551011yescharacteristic
326C2—22no—
327C2—42no—
328C2C2×#1569078711yescharacteristic
329C2#135×D9711yescharacteristic
330C2#2026153211yescharacteristic
subgroupG/H
centerC2×C4D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#416×C97C2
socleC22×C97C22×D4
prime radicalC2×#41×C97C22
hypercenter#416D97
hypocenterC97#3076
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20261288C2semidirect
3. #20261044C2semidirect
4. #20261040C2semidirect
5. #20261037C2semidirect
6. #20261033C2semidirect
7. #20261028C2semidirect
8. #20261023C2semidirect
9. #20261018C2semidirect
10. #20261013C2semidirect
11. #20260773C2semidirect
12. #20260771C2semidirect
13. C2×#15690739C2semidirect
93. #41×C97Q8semidirect
166. C97#3076semidirect
226. #41Dic194semidirect

Maximal subgroups

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

subgroupindex
#202612882
#41×Dic972
#202610442
#202610402
#202610372
#202610332
#202610282
#202610232
#202610182
#202610132
#202607732
#202607712
C2×#156907392
#202607312
#416×C972
#307697