Finite Group Databasebeta

← #27150399

Subgroups — #27150399

Order 12416.

392 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261203C211yesmaximal, characteristic
2#20260796C211yesmaximal, characteristic, complemented
3#20260795C211yesmaximal, characteristic, complemented
4#20260795C211yesmaximal, characteristic, complemented
5#20260790C211yesmaximal, characteristic, complemented
6#20260789C211yesmaximal, characteristic, complemented
7#20260786C211yesmaximal, characteristic, complemented
8#20260785C211yesmaximal, characteristic, complemented
9#20260757C211yesmaximal, characteristic, complemented
10C4⋊C8×D97C211yesmaximal, characteristic, complemented
11#20260700C211yesmaximal, characteristic
12#20260700C211yesmaximal, characteristic
13#20260692C211yesmaximal, characteristic, complemented
14#20260689C211yesmaximal, characteristic, complemented
15#433×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×#13095494C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic
21#15690775C411yescharacteristic
22#15690775C411yescharacteristic
23#15690772C411yescharacteristic
24#15690772C411yescharacteristic
25#15690770C411yescharacteristic
26#15690770C411yescharacteristic
27#15690770C411yescharacteristic
28#15690770C411yescharacteristic
29#15690767C2211yescharacteristic
30#15690767C2211yescharacteristic
31C2×C4×Dic97C2211yescharacteristic
32C42×D97C2211yescharacteristic
33C42×D97C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35C4⋊#11715895C2211yescharacteristic
36#15690739C2211yescharacteristic
37#15690739C2211yescharacteristic
38#15690739C2211yescharacteristic
39#15690739C2211yescharacteristic
40C22⋊#11715895C2211yescharacteristic
41C22⋊#11715895C2211yescharacteristic
42#15690722C2211yescharacteristic
43#15690722C2211yescharacteristic
44#15690722C2211yescharacteristic
45#15690722C2211yescharacteristic
46#15690721C2211yescharacteristic
47#15690720C2211yescharacteristic
48#15690720C2211yescharacteristic
49C4×#11715895C2211yescharacteristic
50C8×Dic97C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52#117×C97C2211yescharacteristic
53C4⋊C8×C97C2211yescharacteristic
54C4⋊C8×C97C2211yescharacteristic
55C22⋊C8×C97C2211yescharacteristic
56C22⋊C8×C97C2211yescharacteristic
57#97×C97C2211yescharacteristic
58C4×C8×C97C2211yescharacteristic
59#13095511—22no—
60#13095511—22no—
61C2×Dic194C2×C411yescharacteristic
62C2×D388C2×C411yescharacteristic
63C2×C22⋊D97C2×C411yescharacteristic
64C2×C22⋊D97C2×C411yescharacteristic
65C22×Dic97C2×C411yescharacteristic
66C22×Dic97C2×C411yescharacteristic
67C2×C4×D97C2×C411yescharacteristic
68C2×C4×D97C2×C411yescharacteristic
69C2×C4×D97C2×C411yescharacteristic
70C2×C4×D97C2×C411yescharacteristic
71C2×C4×D97C2311yescharacteristic
72C2×C4×D97C2311yescharacteristic
73#13095494—22no—
74#13095494—22no—
75C2×#11715895C2311yescharacteristic
76C2×#11715895C2311yescharacteristic
77C2×#11715895C2311yescharacteristic
78C2×#11715895C2311yescharacteristic
79C8×D97—22no—
80C8×D97—22no—
81#13095490C2×C411yescharacteristic
82#13095490C2×C411yescharacteristic
83#13095490C2×C411yescharacteristic
84#13095490C2×C411yescharacteristic
85#13095488C2×C411yescharacteristic
86#13095488C2×C411yescharacteristic
87#13095488C2×C411yescharacteristic
88#13095488C2×C411yescharacteristic
89C4×Dic97C2×C411yescharacteristic
90C4×Dic97C2×C411yescharacteristic
91C4×Dic97C2×C411yescharacteristic
92C4×Dic97C2×C411yescharacteristic
93C4×Dic97C2×C411yescharacteristic
94C4×Dic97C2×C411yescharacteristic
95C4×Dic97C2311yescharacteristic
96C4×Dic97C2311yescharacteristic
97C22×C4×C97C2311yescharacteristic
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C2×C8×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C4⋊C4×C97C2×C411yescharacteristic
103C4⋊C4×C97C2×C411yescharacteristic
104C22⋊C4×C97C2×C411yescharacteristic
105C22⋊C4×C97C2×C411yescharacteristic
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97C22×C411yescharacteristic
109C22×D97C22×C411yescharacteristic
110C22×D97C22×C411yescharacteristic
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97C22×C411yescharacteristic
114C2×C4×C97C22×C411yescharacteristic
115C2×C4×C97C22×C411yescharacteristic
116C2×C4×C97C22×C411yescharacteristic
117C2×C4×C97C22×C411yescharacteristic
118C2×C4×C97C2411yescharacteristic
119C22⋊D97—22no—
120C22⋊D97—22no—
121C2×Dic97—22no—
122C2×Dic97—41no—
123C2×Dic97C22×C411yescharacteristic
124C2×Dic97C22×C411yescharacteristic
125C2×Dic97C22×C411yescharacteristic
126C2×Dic97C22×C411yescharacteristic
127C2×Dic97C22×C411yescharacteristic
128C2×Dic97C22×C411yescharacteristic
129D388—22no—
130C4×D97—21no—
131C4×D97—21no—
132C4×D97—22no—
133C4×D97—22no—
134C4×D97—41no—
135C4×D97—41no—
136C4×D97#4112yes—
137C4×D97#4112yes—
138Dic194—22no—
139C8×C97—21no—
140C8×C97—21no—
141C8×C97—21no—
142C8×C97—21no—
143#11715895—21no—
144#11715895—21no—
145#11715895—21no—
146#11715895—21no—
147C22×C97—21no—
148C22×C97—21no—
149C22×C97—21no—
150C22×C97C23×C411yescharacteristic
151C2×D97—21no—
152C2×D97—21no—
153C2×D97—21no—
154C2×D97—21no—
155C2×D97—21no—
156C2×D97#11712yes—
157C4×C97—21no—
158C4×C97—21no—
159C4×C97—21no—
160C4×C97—21no—
161C4×C97—21no—
162C4×C97C2×#4111yescharacteristic
163C4×C97C2×#4111yescharacteristic
164Dic97—21no—
165Dic97—21no—
166Dic97—21no—
167Dic97—21no—
168Dic97#11712yes—
169Dic97#13112yes—
170C2×C97—41no—
171C2×C97C2×#11711yescharacteristic
172C2×C97C2×#13111yescharacteristic
173C2×C97#56811yescharacteristic
174D97—22no—
175D97—41no—
176#2925—971nomaximal, Sylow
177C97#292511yescharacteristic, complemented, Sylow
178C4×#41—971no—
179#433—971no—
180#433—971no—
181#433—971no—
182#433D9711yescharacteristic, complemented
183#432—971no—
184#432—971no—
185#431—971no—
186#431—971no—
187#423—971no—
188C2×C4⋊C8—971no—
189#408—971no—
190#408—971no—
191#405—971no—
192#405—971no—
193C2×#41—971no—
194C2×M16—971no—
195C2×M16—971no—
196C22×C8—971no—
197C22×C8—971no—
198C4×Q8—971no—
199C4×Q8—971no—
200C4×D4—971no—
201C4×D4—971no—
202C4×D4—971no—
203C4×D4—971no—
204C4×D4—971no—
205C4×D4—971no—
206#117—971no—
207#117—971no—
208#117C2×D9711yescharacteristic
209C2×C42—971no—
210C2×C42—971no—
211C2×C42—971no—
212C4⋊C8—971no—
213C4⋊C8—971no—
214C4⋊C8—971no—
215C4⋊C8—971no—
216C4⋊C8—971no—
217C4⋊C8—971no—
218C4⋊C8C2×D9711yescharacteristic
219C4⋊C8C2×D9711yescharacteristic
220C22⋊C8—971no—
221C22⋊C8—971no—
222C22⋊C8—971no—
223C22⋊C8—971no—
224C22⋊C8—971no—
225C22⋊C8—971no—
226C22⋊C8C2×D9711yescharacteristic
227C22⋊C8C2×D9711yescharacteristic
228#97—971no—
229#97—971no—
230#97—971no—
231#97C2×D9711yescharacteristic
232C4×C8—971no—
233C4×C8—971no—
234C4×C8—971no—
235C4×C8C2×D9711yescharacteristic
236#41—1942no—
237#41—1942no—
238C2×Q8—971no—
239C2×D4—971no—
240C2×D4—971no—
241C2×D4—971no—
242C22×C4—971no—
243C22×C4—971no—
244C22×C4—971no—
245C22×C4—971no—
246C22×C4—971no—
247C22×C4—971no—
248C22×C4—971no—
249C22×C4—971no—
250C22×C4C22×D9711yescharacteristic
251M16—1942no—
252M16—1942no—
253C2×C8—971no—
254C2×C8—971no—
255C2×C8—971no—
256C2×C8—971no—
257C2×C8—1942no—
258C2×C8—1942no—
259C2×C8C22×D9711yescharacteristic
260C2×C8C22×D9711yescharacteristic
261C2×C8C22×D9711yescharacteristic
262C2×C8C22×D9711yescharacteristic
263C4⋊C4—971no—
264C4⋊C4—971no—
265C4⋊C4—971no—
266C4⋊C4—971no—
267C4⋊C4C4×D9711yescharacteristic
268C4⋊C4C4×D9711yescharacteristic
269C22⋊C4—971no—
270C22⋊C4—971no—
271C22⋊C4—971no—
272C22⋊C4—971no—
273C22⋊C4C4×D9711yescharacteristic
274C22⋊C4C4×D9711yescharacteristic
275C42—971no—
276C42—971no—
277C42—971no—
278C42—971no—
279C42—971no—
280C42—971no—
281C42—971no—
282C42—971no—
283C42C22×D9711yescharacteristic
284C42C22×D9711yescharacteristic
285C23—971no—
286C23—971no—
287C23C2×C4×D9711yescharacteristic
288Q8—1942no—
289D4—1942no—
290D4—1942no—
291D4—1942no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—971no—
295C2×C4—971no—
296C2×C4—972no—
297C2×C4—972no—
298C2×C4—971no—
299C2×C4—971no—
300C2×C4—971no—
301C2×C4—971no—
302C2×C4—3881no—
303C2×C4—3881no—
304C2×C4—3881no—
305C2×C4—1942no—
306C2×C4—1942no—
307C2×C4—1942no—
308C2×C4—1941no—
309C2×C4—1941no—
310C2×C4C2×C4×D9711yescharacteristic
311C2×C4C2×C4×D9711yescharacteristic
312C2×C4C2×C4×D9711yescharacteristic
313C2×C4C2×C4×D9711yescharacteristic
314C2×C4C2×C4×D9711yescharacteristic
315C2×C4C23×D9711yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8—21no—
320C8—1941no—
321C8—1941no—
322C8—1941no—
323C8—1941no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—972no—
328C22—1941no—
329C22—1941no—
330C22—1941no—
331C22—1941no—
332C22—1941no—
333C22C22×C4×D9711yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—972no—
340C4—972no—
341C4—1941no—
342C4—1941no—
343C4—1941no—
344C4—1941no—
345C4#41×D9711yescharacteristic
346C4#41×D9711yescharacteristic
347C2—41no—
348C2—3881no—
349C2—1942no—
350C2#117×D9711yescharacteristic
351C2#131×D9711yescharacteristic
352C2#2026150011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#433×C97C2
prime radicalC2×#13095511C22
cosocleC2×C4×C97C24
hypercenter#433D97
hypocenterC97#2925
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #20260796C2semidirect
3. #20260795C2semidirect
4. #20260795C2semidirect
5. #20260790C2semidirect
6. #20260789C2semidirect
7. #20260786C2semidirect
8. #20260785C2semidirect
9. #20260757C2semidirect
10. C4⋊C8×D97C2semidirect
13. #20260692C2semidirect
14. #20260689C2semidirect
15. #433×C97C2semidirect
177. C97#2925semidirect
182. #433D97semidirect

Maximal subgroups

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

subgroupindex
#202612032
#202607962
#202607952
#202607902
#202607892
#202607862
#202607852
#202607572
C4⋊C8×D972
#202607002
#202606922
#202606892
#433×C972
#292597