Finite Group Databasebeta

← #27150809

Subgroups — #27150809

Order 12416.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261281C211yesmaximal, characteristic
2#20260811C211yesmaximal, characteristic, complemented
3#20260810C211yesmaximal, characteristic, complemented
4#20260809C211yesmaximal, characteristic, complemented
5#20260806C211yesmaximal, characteristic, complemented
6#20260804C211yesmaximal, characteristic, complemented
7#20260803C211yesmaximal, characteristic, complemented
8C8×C22⋊D97C211yesmaximal, characteristic, complemented
9#20260798C211yesmaximal, characteristic, complemented
10#20260789C211yesmaximal, characteristic, complemented
11#20260785C211yesmaximal, characteristic, complemented
12#20260702C211yesmaximal, characteristic, complemented
13#20260700C211yesmaximal, characteristic, complemented
14#20260700C211yesmaximal, characteristic, complemented
15#408×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×#13095495C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19C2×#13095494C2211yescharacteristic
20C22×#11715895C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic
22C2×C8×D97C2211yescharacteristic
23#15690808C411yescharacteristic
24#15690806C411yescharacteristic
25#15690799C411yescharacteristic
26#15690797C411yescharacteristic
27#15690782C411yescharacteristic
28#15690782C411yescharacteristic
29#15690782C411yescharacteristic
30#15690782C411yescharacteristic
31C4×C22⋊D97C2211yescharacteristic
32C4×C22⋊D97C2211yescharacteristic
33C4×C22⋊D97C2211yescharacteristic
34C4×C22⋊D97C2211yescharacteristic
35C2×C4×Dic97C2211yescharacteristic
36#15690739C2211yescharacteristic
37#15690739C2211yescharacteristic
38#15690739C2211yescharacteristic
39#15690739C2211yescharacteristic
40C22⋊#11715895C2211yescharacteristic
41C22⋊#11715895C2211yescharacteristic
42C22⋊#11715895C2211yescharacteristic
43C22⋊#11715895C2211yescharacteristic
44#15690722C2211yescharacteristic
45#15690722C2211yescharacteristic
46#15690722C2211yescharacteristic
47#15690722C2211yescharacteristic
48#15690720C2211yescharacteristic
49#15690720C2211yescharacteristic
50C8×Dic97C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C2×#41×C97C2211yescharacteristic
53C2×M16×C97C2211yescharacteristic
54C22×C8×C97C2211yescharacteristic
55C22⋊C8×C97C2211yescharacteristic
56C22⋊C8×C97C2211yescharacteristic
57C22⋊C8×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic
59#13095511—41no—
60#13095511—41no—
61C2×Dic194C2×C411yescharacteristic
62C2×D388C2×C411yescharacteristic
63C2×C22⋊D97C2×C411yescharacteristic
64C2×C22⋊D97C2×C411yescharacteristic
65C2×C22⋊D97C2×C411yescharacteristic
66C2×C22⋊D97C2×C411yescharacteristic
67C2×C22⋊D97C2×C411yescharacteristic
68C2×C22⋊D97C2×C411yescharacteristic
69C22×Dic97C2×C411yescharacteristic
70C22×Dic97C2×C411yescharacteristic
71C2×C4×D97C2311yescharacteristic
72C2×C4×D97C2311yescharacteristic
73#13095495—41no—
74#13095494—41no—
75#13095494—41no—
76C2×#11715895—41no—
77C2×#11715895C2311yescharacteristic
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C2×#11715895C2311yescharacteristic
81C8×D97—41no—
82C8×D97—41no—
83#13095490C2×C411yescharacteristic
84#13095490C2×C411yescharacteristic
85#13095490C2×C411yescharacteristic
86#13095490C2×C411yescharacteristic
87C22⋊Dic97C2×C411yescharacteristic
88C22⋊Dic97C2×C411yescharacteristic
89C22⋊Dic97C2×C411yescharacteristic
90C22⋊Dic97C2×C411yescharacteristic
91#13095488C2×C411yescharacteristic
92#13095488C2×C411yescharacteristic
93#13095488C2×C411yescharacteristic
94#13095488C2×C411yescharacteristic
95C4×Dic97C2×C411yescharacteristic
96C4×Dic97C2×C411yescharacteristic
97C4×Dic97C2311yescharacteristic
98C4×Dic97C2311yescharacteristic
99#41×C97—41no—
100#41×C97—41no—
101C2×Q8×C97C2×C411yescharacteristic
102C2×D4×C97C2×C411yescharacteristic
103C2×D4×C97C2×C411yescharacteristic
104C2×D4×C97C2×C411yescharacteristic
105C22×C4×C97C2311yescharacteristic
106C22×C4×C97C2311yescharacteristic
107C22×C4×C97C2311yescharacteristic
108M16×C97—41no—
109C2×C8×C97—41no—
110C2×C8×C97C2311yescharacteristic
111C2×C8×C97C2311yescharacteristic
112C2×C8×C97C2311yescharacteristic
113C2×C8×C97C2311yescharacteristic
114C23×C97C22×C411yescharacteristic
115C23×C97C22×C411yescharacteristic
116C23×C97C22×C411yescharacteristic
117C22×D97C22×C411yescharacteristic
118C22×D97C22×C411yescharacteristic
119Q8×C97—41no—
120D4×C97—41no—
121D4×C97—41no—
122D4×C97—41no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97C22×C411yescharacteristic
130C2×C4×C97C22×C411yescharacteristic
131C2×C4×C97C22×C411yescharacteristic
132C2×C4×C97C2411yescharacteristic
133C22⋊D97—22no—
134C22⋊D97—22no—
135C22⋊D97—41no—
136C22⋊D97—41no—
137C22⋊D97—41no—
138C22⋊D97—41no—
139C2×Dic97—22no—
140C2×Dic97—41no—
141C2×Dic97C22×C411yescharacteristic
142C2×Dic97C22×C411yescharacteristic
143C2×Dic97C22×C411yescharacteristic
144C2×Dic97C22×C411yescharacteristic
145C2×Dic97C22×C411yescharacteristic
146C2×Dic97C22×C411yescharacteristic
147D388—41no—
148C4×D97—21no—
149C4×D97—21no—
150C4×D97—21no—
151C4×D97—21no—
152Dic194—41no—
153C8×C97—21no—
154C8×C97—21no—
155C8×C97—21no—
156C8×C97—21no—
157#11715895—21no—
158#11715895—21no—
159#11715895—21no—
160#11715895—21no—
161C22×C97—21no—
162C22×C97—21no—
163C22×C97—21no—
164C22×C97—21no—
165C22×C97—21no—
166C22×C97—21no—
167C22×C97—21no—
168C22×C97—21no—
169C22×C97—21no—
170C22×C97C23×C411yescharacteristic
171C2×D97—21no—
172C2×D97—21no—
173C2×D97—21no—
174C2×D97—21no—
175C2×D97—21no—
176C2×D97—21no—
177C4×C97—21no—
178C4×C97—21no—
179C4×C97—21no—
180C4×C97H(5,𝔽2)11yescharacteristic
181C4×C97H(5,𝔽2)11yescharacteristic
182Dic97—21no—
183Dic97—21no—
184Dic97—21no—
185Dic97—21no—
186Dic97—21no—
187Dic97#13112yes—
188C2×C97—41no—
189C2×C97—41no—
190C2×C97—41no—
191C2×C97#51811yescharacteristic
192C2×C97C2×#13111yescharacteristic
193C2×C97#56811yescharacteristic
194D97—41no—
195D97—41no—
196#2981—971nomaximal, Sylow
197C97#298111yescharacteristic, complemented, Sylow
198#532—971no—
199#436—971no—
200#436—971no—
201#435—971no—
202#435—971no—
203#435—971no—
204#435—971no—
205C8×D4—971no—
206C8×D4—971no—
207#432—971no—
208#431—971no—
209#408—971no—
210#408—971no—
211#408—971no—
212#408D9711yescharacteristic, complemented
213C2×#41—971no—
214C2×#41C2×D9711yescharacteristic
215C2×M16—971no—
216C2×M16—971no—
217C2×M16—971no—
218C2×M16C2×D9711yescharacteristic
219C22×C8—971no—
220C22×C8—971no—
221C22×C8—971no—
222C22×C8C2×D9711yescharacteristic
223C4⋊Q8—971no—
224#127—971no—
225#124—971no—
226#124—971no—
227#121—971no—
228#121—971no—
229#121—971no—
230#121—971no—
231C4×D4—971no—
232C4×D4—971no—
233C4×D4—971no—
234C4×D4—971no—
235C2×C42—971no—
236C4⋊C8—971no—
237C4⋊C8—971no—
238C4⋊C8—971no—
239C4⋊C8—971no—
240C22⋊C8—971no—
241C22⋊C8—971no—
242C22⋊C8—971no—
243C22⋊C8—971no—
244C22⋊C8—971no—
245C22⋊C8—971no—
246C22⋊C8—971no—
247C22⋊C8—971no—
248C22⋊C8C2×D9711yescharacteristic
249C22⋊C8C2×D9711yescharacteristic
250C22⋊C8C2×D9711yescharacteristic
251C22⋊C8C2×D9711yescharacteristic
252#97—971no—
253#97—971no—
254C4×C8—971no—
255C4×C8—971no—
256#41—41no—
257#41—41no—
258#41—3881no—
259#41—3881no—
260C2×Q8—971no—
261C2×Q8C4×D9711yescharacteristic
262C2×D4—971no—
263C2×D4—971no—
264C2×D4—971no—
265C2×D4—971no—
266C2×D4—971no—
267C2×D4—971no—
268C2×D4—971no—
269C2×D4C4×D9711yescharacteristic
270C2×D4C4×D9711yescharacteristic
271C2×D4C4×D9711yescharacteristic
272C22×C4—971no—
273C22×C4—971no—
274C22×C4—971no—
275C22×C4—971no—
276C22×C4C22×D9711yescharacteristic
277C22×C4C22×D9711yescharacteristic
278C22×C4C22×D9711yescharacteristic
279M16—41no—
280M16—3881no—
281M16—3881no—
282M16—3881no—
283C2×C8—41no—
284C2×C8—971no—
285C2×C8—971no—
286C2×C8—971no—
287C2×C8—971no—
288C2×C8—3881no—
289C2×C8—3881no—
290C2×C8—3881no—
291C2×C8C22×D9711yescharacteristic
292C2×C8C22×D9711yescharacteristic
293C2×C8C22×D9711yescharacteristic
294C2×C8C22×D9711yescharacteristic
295C4⋊C4—971no—
296C4⋊C4—971no—
297C4⋊C4—971no—
298C4⋊C4—971no—
299C22⋊C4—971no—
300C22⋊C4—971no—
301C22⋊C4—971no—
302C22⋊C4—971no—
303C22⋊C4—971no—
304C22⋊C4—971no—
305C22⋊C4—971no—
306C22⋊C4—971no—
307C42—971no—
308C42—971no—
309C42—971no—
310C42—971no—
311C23—971no—
312C23—971no—
313C23C2×C4×D9711yescharacteristic
314C23C2×C4×D9711yescharacteristic
315C23C2×C4×D9711yescharacteristic
316Q8—41no—
317Q8—3881no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—3881no—
322D4—3881no—
323D4—3881no—
324D4—3881no—
325D4—3881no—
326D4—1942no—
327D4—1942no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—971no—
335C2×C4—971no—
336C2×C4—971no—
337C2×C4—971no—
338C2×C4—971no—
339C2×C4—971no—
340C2×C4—3881no—
341C2×C4—1942no—
342C2×C4—1941no—
343C2×C4—1941no—
344C2×C4—1941no—
345C2×C4—1941no—
346C2×C4C2×C4×D9711yescharacteristic
347C2×C4C2×C4×D9711yescharacteristic
348C2×C4C2×C4×D9711yescharacteristic
349C2×C4C23×D9711yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—1941no—
355C8—1941no—
356C8—1941no—
357C8—1941no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—1941no—
368C22—1941no—
369C22—1941no—
370C22—1941no—
371C22—1941no—
372C22—1941no—
373C22C22×C4×D9711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—972no—
378C4—1941no—
379C4—1941no—
380C4—1941no—
381C4—1941no—
382C4—1941no—
383C4#1569085511yescharacteristic
384C4#1569085511yescharacteristic
385C2—41no—
386C2—41no—
387C2—41no—
388C2—3881no—
389C2—3881no—
390C2#2026120811yescharacteristic
391C2#131×D9711yescharacteristic
392C2#2026150011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#408×C97C2
prime radical#20261281C2
cosocleC2×C4×C97C24
hypercenter#408D97
hypocenterC97#2981
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #20260811C2semidirect
3. #20260810C2semidirect
4. #20260809C2semidirect
5. #20260806C2semidirect
6. #20260804C2semidirect
7. #20260803C2semidirect
8. C8×C22⋊D97C2semidirect
9. #20260798C2semidirect
10. #20260789C2semidirect
11. #20260785C2semidirect
12. #20260702C2semidirect
13. #20260700C2semidirect
14. #20260700C2semidirect
15. #408×C97C2semidirect
197. C97#2981semidirect
212. #408D97semidirect

Maximal subgroups

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

subgroupindex
#202612812
#202608112
#202608102
#202608092
#202608062
#202608042
#202608032
C8×C22⋊D972
#202607982
#202607892
#202607852
#202607022
#202607002
#408×C972
#298197