Finite Group Databasebeta

← #27151539

Subgroups — #27151539

Order 12416.

376 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261291C211yesmaximal, characteristic
2C4×Q8×D97C211yesmaximal, characteristic
3#20261043C211yesmaximal, characteristic, complemented
4#20261042C211yesmaximal, characteristic, complemented
5#20261039C211yesmaximal, characteristic, complemented
6#20261029C211yesmaximal, characteristic, complemented
7#20261028C211yesmaximal, characteristic, complemented
8#20261019C211yesmaximal, characteristic, complemented
9#20261018C211yesmaximal, characteristic, complemented
10#20260776C211yesmaximal, characteristic
11#20260774C211yesmaximal, characteristic
12#20260757C211yesmaximal, characteristic
13#20260742C211yesmaximal, characteristic
14#20260742C211yesmaximal, characteristic
15#479×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2211yescharacteristic
17C2×#13095494C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19#15690809C2211yescharacteristic
20#15690809C2211yescharacteristic
21#15690802C2211yescharacteristic
22#15690788—21no—
23#15690788—21no—
24Q8×Dic97C2211yescharacteristic
25C4×Dic194—21no—
26C4×Dic194C2211yescharacteristic
27#15690775—21no—
28#15690775—21no—
29#15690775—21no—
30#15690775C2211yescharacteristic
31#15690768C2211yescharacteristic
32#15690768C2211yescharacteristic
33C4⋊C4×D97—21no—
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#15690735C2211yescharacteristic
47#15690735C2211yescharacteristic
48#15690734C2211yescharacteristic
49#15690734C2211yescharacteristic
50#15690733C2211yescharacteristic
51#15690733C2211yescharacteristic
52#125×C97C2211yescharacteristic
53C4×Q8×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic
55#106×C97C2211yescharacteristic
56C4⋊C8×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194—21no—
61C2×Dic194—21no—
62C2×Dic194C2311yescharacteristic
63Q8×D97—22no—
64Q8×D97Q814yescomplemented
65C2×C4×D97—21no—
66C2×C4×D97—21no—
67C2×C4×D97D411yescharacteristic
68C2×C4×D97D411yescharacteristic
69C2×C4×D97C2311yescharacteristic
70#13095494—22no—
71#13095494—22no—
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C4⋊Dic97—21no—
75C4⋊Dic97C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77C4⋊Dic97C2311yescharacteristic
78#13095490—21no—
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490—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×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×D97C2×D411yescharacteristic
107Q8×C97—21no—
108Q8×C97C2×Q812yes—
109C2×C4×C97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97C2×D411yescharacteristic
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2411yescharacteristic
116C2×Dic97—21no—
117C2×Dic97—21no—
118C2×Dic97—21no—
119C2×Dic97—21no—
120C2×Dic97C2×D411yescharacteristic
121C2×Dic97C2×D411yescharacteristic
122C2×Dic97C2×D411yescharacteristic
123C4×D97—24no—
124C4×D97—22no—
125C4×D97—41no—
126C4×D97—42no—
127C4×D97C2×Q812yes—
128C4×D97#4112yes—
129Dic194—21no—
130Dic194—24no—
131Dic194—42no—
132Dic194—42no—
133Dic194C2×Q812yes—
134C8×C97—21no—
135C8×C97—21no—
136#11715895—21no—
137#11715895—21no—
138C22×C97C22×D411yescharacteristic
139C2×D97—21no—
140C2×D97—21no—
141C2×D97C22⋊Q812yes—
142C4×C97—22no—
143C4×C97—21no—
144C4×C97—21no—
145C4×C97—41no—
146C4×C97—41no—
147C4×C97—41no—
148C4×C97C22×Q811yescharacteristic
149C4×C97C2×#4111yescharacteristic
150Dic97—22no—
151Dic97—21no—
152Dic97—41no—
153Dic97—41no—
154Dic97—41no—
155Dic97C22⋊Q812yes—
156Dic97#13712yes—
157C2×C97C2×C22⋊Q811yescharacteristic
158C2×C97C2×#13711yescharacteristic
159C2×C97#57511yescharacteristic
160D97—22no—
161#3078—971nomaximal, Sylow
162C97#307811yescharacteristic, complemented, Sylow
163#533—971no—
164C2×C4×Q8—971no—
165#479—971no—
166#479—971no—
167#479—971no—
168#479D9711yescharacteristic, complemented
169#477—971no—
170#477—971no—
171#475—971no—
172#475—971no—
173#428—971no—
174#428—971no—
175#423—971no—
176#419—971no—
177#419—971no—
178C22×Q8—971no—
179C2×M16—971no—
180C2×M16—971no—
181C4⋊Q8—971no—
182C4⋊Q8—971no—
183#125—971no—
184#125C2×D9711yescharacteristic
185C22⋊Q8—1941no—
186C22⋊Q8—1941no—
187C4×Q8—971no—
188C4×Q8—971no—
189C4×Q8—971no—
190C4×Q8—1941no—
191C4×Q8—1941no—
192C4×Q8—1941no—
193C4×Q8—1941no—
194C4×Q8C2×D9711yescharacteristic
195#117—971no—
196#117—971no—
197C2×C4⋊C4—971no—
198C2×C4⋊C4—1941no—
199C2×C42—971no—
200C2×C42—1941no—
201#107—971no—
202#107—971no—
203#107—971no—
204#107C2×D9711yescharacteristic
205#106—971no—
206#106—971no—
207#106—971no—
208#106C2×D9711yescharacteristic
209C4⋊C8—971no—
210C4⋊C8—971no—
211C4⋊C8—971no—
212C4⋊C8C2×D9711yescharacteristic
213#103—971no—
214#103—971no—
215#103—971no—
216#103—971no—
217#103—971no—
218#103—971no—
219#103C2×D9711yescharacteristic
220#103C2×D9711yescharacteristic
221C2×Q8—974no—
222C2×Q8—971no—
223C2×Q8—1941no—
224C2×Q8—1941no—
225C2×Q8—1941no—
226C2×Q8—1942no—
227C2×Q8C22×D9711yescharacteristic
228C22×C4—971no—
229C22×C4—971no—
230C22×C4—971no—
231C22×C4—1941no—
232C22×C4—1941no—
233M16—1942no—
234M16—1942no—
235C2×C8—971no—
236C2×C8—971no—
237C2×C8C22×D9711yescharacteristic
238C2×C8C22×D9711yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—971no—
243C4⋊C4—971no—
244C4⋊C4—971no—
245C4⋊C4—1941no—
246C4⋊C4—1941no—
247C4⋊C4—1941no—
248C4⋊C4—1941no—
249C4⋊C4—1941no—
250C4⋊C4—1941no—
251C4⋊C4—1941no—
252C4⋊C4—1941no—
253C4⋊C4C22×D9711yescharacteristic
254C4⋊C4C22×D9711yescharacteristic
255C4⋊C4C22×D9711yescharacteristic
256C22⋊C4—1941no—
257C22⋊C4—1941no—
258C42—21no—
259C42—971no—
260C42—971no—
261C42—971no—
262C42—1941no—
263C42—1941no—
264C42—1941no—
265C42—1941no—
266C42—1941no—
267C42C22×D9711yescharacteristic
268C23—971no—
269Q8—21no—
270Q8—972no—
271Q8—3882no—
272Q8—3882no—
273Q8—1944no—
274Q8—1941no—
275Q8Q8×D9712yes—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—971no—
281C2×C4—971no—
282C2×C4—972no—
283C2×C4—972no—
284C2×C4—971no—
285C2×C4—3881no—
286C2×C4—3882no—
287C2×C4—1942no—
288C2×C4—1944no—
289C2×C4—1941no—
290C2×C4—1941no—
291C2×C4—1941no—
292C2×C4—1941no—
293C2×C4D4×D9711yescharacteristic
294C2×C4D4×D9711yescharacteristic
295C2×C4C23×D9711yescharacteristic
296C8—21no—
297C8—21no—
298C8—1941no—
299C8—1941no—
300C22—972no—
301C22—1941no—
302C22—1941no—
303C22C2×D4×D9711yescharacteristic
304C4—22no—
305C4—21no—
306C4—21no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—972no—
311C4—972no—
312C4—3881no—
313C4—3881no—
314C4—3881no—
315C4—1942no—
316C4—1941no—
317C4C2×Q8×D9711yescharacteristic
318C4#41×D9711yescharacteristic
319C2—1942no—
320C2C22⋊Q8×D9711yescharacteristic
321C2#137×D9711yescharacteristic
322C2#2026153211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#479×C97C2
socleC22×C97C22×D4
prime radicalC22×D97C2×D4
hypercenter#479D97
hypocenterC97#3078
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
3. #20261043C2semidirect
4. #20261042C2semidirect
5. #20261039C2semidirect
6. #20261029C2semidirect
7. #20261028C2semidirect
8. #20261019C2semidirect
9. #20261018C2semidirect
15. #479×C97C2semidirect
64. Q8×D97Q8semidirect
162. C97#3078semidirect
168. #479D97semidirect

Maximal subgroups

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

subgroupindex
#202612912
C4×Q8×D972
#202610432
#202610422
#202610392
#202610292
#202610282
#202610192
#202610182
#202607762
#202607742
#202607572
#202607422
#479×C972
#307897