Finite Group Databasebeta

← #27150392

Subgroups — #27150392

Order 12416.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#117×D97C211yesmaximal, characteristic
2#20260796C211yesmaximal, characteristic, complemented
3#20260793C211yesmaximal, characteristic, complemented
4#20260793C211yesmaximal, characteristic, complemented
5#20260791C211yesmaximal, characteristic, complemented
6#20260790C211yesmaximal, characteristic, complemented
7#20260787C211yesmaximal, characteristic, complemented
8#20260786C211yesmaximal, characteristic, complemented
9#20260760C211yesmaximal, characteristic, complemented
10#20260760C211yesmaximal, characteristic, complemented
11#20260697C211yesmaximal, characteristic
12C22⋊C8×D97C211yesmaximal, characteristic
13#20260692C211yesmaximal, characteristic, complemented
14#20260689C211yesmaximal, characteristic, complemented
15#433×C97C211yesmaximal, characteristic, complemented
16C22×C4×D97C2211yescharacteristic
17C2×#13095494C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic
21#15690769C2211yescharacteristic
22#15690768C411yescharacteristic
23#15690768C411yescharacteristic
24#15690767C2211yescharacteristic
25#15690767C2211yescharacteristic
26#15690766C411yescharacteristic
27#15690766C411yescharacteristic
28C4⋊C4×D97C411yescharacteristic
29C4⋊C4×D97C411yescharacteristic
30C22⋊C4×D97C411yescharacteristic
31C22⋊C4×D97C411yescharacteristic
32C42×D97C2211yescharacteristic
33C42×D97C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic, complemented
35C4⋊#11715895C2211yescharacteristic, complemented
36#15690739C2211yescharacteristic, complemented
37#15690739C2211yescharacteristic, complemented
38#15690739C2211yescharacteristic, complemented
39#15690739C2211yescharacteristic, complemented
40C22⋊#11715895C2211yescharacteristic
41C22⋊#11715895C2211yescharacteristic
42#15690722C2211yescharacteristic
43#15690722C2211yescharacteristic
44#15690722C2211yescharacteristic
45#15690722C2211yescharacteristic
46#15690721C2211yescharacteristic, complemented
47#15690720C2211yescharacteristic, complemented
48#15690720C2211yescharacteristic, complemented
49C4×#11715895C2211yescharacteristic, complemented
50C8×Dic97C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52#117×C97C2211yescharacteristic
53C4⋊C8×C97C2211yescharacteristic, complemented
54C4⋊C8×C97C2211yescharacteristic, complemented
55C22⋊C8×C97C2211yescharacteristic
56C22⋊C8×C97C2211yescharacteristic
57#97×C97C2211yescharacteristic, complemented
58C4×C8×C97C2211yescharacteristic, complemented
59C23×D97C2×C411yescharacteristic
60C22×Dic97C2×C411yescharacteristic
61C2×C4×D97—22no—
62C2×C4×D97—22no—
63C2×C4×D97C2×C411yescharacteristic
64C2×C4×D97C2×C411yescharacteristic
65C2×C4×D97C2×C411yescharacteristic
66C2×C4×D97C2×C411yescharacteristic
67C2×C4×D97C2×C411yescharacteristic
68C2×C4×D97C2×C411yescharacteristic
69C2×C4×D97C2311yescharacteristic
70C2×C4×D97C2311yescharacteristic
71#13095494—22no—
72#13095494—22no—
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C2×#11715895C2311yescharacteristic
77C8×D97—22no—
78C8×D97—22no—
79C4⋊Dic97C2×C411yescharacteristic
80C4⋊Dic97C2×C411yescharacteristic
81#13095490C2×C411yescharacteristic
82#13095490C2×C411yescharacteristic
83#13095490C2×C411yescharacteristic
84#13095490C2×C411yescharacteristic
85C22⋊Dic97C2×C411yescharacteristic
86C22⋊Dic97C2×C411yescharacteristic
87#13095488C2×C411yescharacteristic
88#13095488C2×C411yescharacteristic
89#13095488C2×C411yescharacteristic
90#13095488C2×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×D97—22no—
110C22×D97—22no—
111C22×D97—41no—
112C22×D97C22×C411yescharacteristic
113C22×D97C22×C411yescharacteristic
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97C22×C411yescharacteristic
117C2×C4×C97C22×C411yescharacteristic
118C2×C4×C97C22×C411yescharacteristic
119C2×C4×C97C22×C411yescharacteristic
120C2×C4×C97C22×C411yescharacteristic
121C2×C4×C97C2411yescharacteristic
122C2×Dic97—22no—
123C2×Dic97C22×C411yescharacteristic
124C2×Dic97C22×C411yescharacteristic
125C2×Dic97C22×C411yescharacteristic
126C2×Dic97C22×C411yescharacteristic
127C2×Dic97C22×C411yescharacteristic
128C2×Dic97C22×C411yescharacteristic
129C4×D97—22no—
130C4×D97—22no—
131C4×D97—22no—
132C4×D97—21no—
133C4×D97—21no—
134C4×D97—22no—
135C4×D97—41no—
136C4×D97—41no—
137C4×D97#4112yes—
138C4×D97#4112yes—
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—44no—
156C2×D97#11712yes—
157C2×D97#13112yes—
158C4×C97—21no—
159C4×C97—21no—
160C4×C97—21no—
161C4×C97—21no—
162C4×C97—21no—
163C4×C97C2×#4111yescharacteristic
164C4×C97C2×#4111yescharacteristic
165Dic97—21no—
166Dic97—21no—
167Dic97—21no—
168Dic97—21no—
169Dic97—21no—
170Dic97#11712yes—
171C2×C97—41no—
172C2×C97C2×#11711yescharacteristic
173C2×C97C2×#13111yescharacteristic
174C2×C97#56811yescharacteristic
175D97—22no—
176D97—22no—
177#2924—971nomaximal, Sylow
178C97#292411yescharacteristic, complemented, Sylow
179C2×#117—971no—
180#433—971no—
181#433—971no—
182#433—971no—
183#433D9711yescharacteristic, complemented
184#432—971no—
185#432—971no—
186#431—971no—
187#431—971no—
188#424—971no—
189#424—971no—
190#407—971no—
191C2×C22⋊C8—971no—
192#405—971no—
193#405—971no—
194C23×C4—971no—
195C2×M16—971no—
196C2×M16—971no—
197C22×C8—971no—
198C22×C8—971no—
199#117—971no—
200#117—971no—
201#117—971no—
202#117—971no—
203#117—971no—
204#117—971no—
205#117—971no—
206#117C2×D9711yescharacteristic
207C2×C4⋊C4—971no—
208C2×C4⋊C4—971no—
209C2×C22⋊C4—971no—
210C2×C22⋊C4—971no—
211C2×C42—971no—
212C2×C42—971no—
213C4⋊C8—971no—
214C4⋊C8—971no—
215C4⋊C8—971no—
216C4⋊C8—971no—
217C4⋊C8—971no—
218C4⋊C8—971no—
219C4⋊C8C2×D9711yescharacteristic, complemented
220C4⋊C8C2×D9711yescharacteristic, complemented
221C22⋊C8—971no—
222C22⋊C8—971no—
223C22⋊C8—971no—
224C22⋊C8—971no—
225C22⋊C8—971no—
226C22⋊C8—971no—
227C22⋊C8C2×D9711yescharacteristic
228C22⋊C8C2×D9711yescharacteristic
229#97—971no—
230#97—971no—
231#97—971no—
232#97C2×D9711yescharacteristic, complemented
233C4×C8—971no—
234C4×C8—971no—
235C4×C8—971no—
236C4×C8C2×D9711yescharacteristic, complemented
237C24—971no—
238C22×C4—971no—
239C22×C4—971no—
240C22×C4—971no—
241C22×C4—971no—
242C22×C4—971no—
243C22×C4—971no—
244C22×C4—971no—
245C22×C4—971no—
246C22×C4—971no—
247C22×C4—1942no—
248C22×C4—1942no—
249C22×C4C22×D9711yescharacteristic
250M16—1942no—
251M16—1942no—
252C2×C8—971no—
253C2×C8—971no—
254C2×C8—971no—
255C2×C8—971no—
256C2×C8—1942no—
257C2×C8—1942no—
258C2×C8C22×D9711yescharacteristic
259C2×C8C22×D9711yescharacteristic
260C2×C8C22×D9711yescharacteristic
261C2×C8C22×D9711yescharacteristic
262C4⋊C4—971no—
263C4⋊C4—971no—
264C4⋊C4—971no—
265C4⋊C4—971no—
266C4⋊C4—971no—
267C4⋊C4—971no—
268C4⋊C4C4×D9711yescharacteristic
269C4⋊C4C4×D9711yescharacteristic
270C22⋊C4—971no—
271C22⋊C4—971no—
272C22⋊C4—971no—
273C22⋊C4—971no—
274C22⋊C4—971no—
275C22⋊C4—971no—
276C22⋊C4C4×D9711yescharacteristic
277C22⋊C4C4×D9711yescharacteristic
278C42—971no—
279C42—971no—
280C42—971no—
281C42—971no—
282C42—971no—
283C42—971no—
284C42C22×D9711yescharacteristic
285C42C22×D9711yescharacteristic
286C23—971no—
287C23—971no—
288C23—3881no—
289C23—1942no—
290C23—1942no—
291C23C2×C4×D9711yescharacteristic
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—971no—
295C2×C4—972no—
296C2×C4—972no—
297C2×C4—971no—
298C2×C4—971no—
299C2×C4—971no—
300C2×C4—971no—
301C2×C4—971no—
302C2×C4—3881no—
303C2×C4—3881no—
304C2×C4—1942no—
305C2×C4—1942no—
306C2×C4—1942no—
307C2×C4—1942no—
308C2×C4—1942no—
309C2×C4—1941no—
310C2×C4—1941no—
311C2×C4C2×C4×D9711yescharacteristic
312C2×C4C2×C4×D9711yescharacteristic
313C2×C4C2×C4×D9711yescharacteristic
314C2×C4C2×C4×D9711yescharacteristic
315C2×C4C2×C4×D9711yescharacteristic
316C2×C4C23×D9711yescharacteristic
317C8—21no—
318C8—21no—
319C8—21no—
320C8—21no—
321C8—1941no—
322C8—1941no—
323C8—1941no—
324C8—1941no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—972no—
329C22—972no—
330C22—3884no—
331C22—1941no—
332C22—1941no—
333C22—1941no—
334C22—1941no—
335C22C22×C4×D9711yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—972no—
342C4—1941no—
343C4—1941no—
344C4—1941no—
345C4—1941no—
346C4—1941no—
347C4#41×D9711yescharacteristic
348C4#41×D9711yescharacteristic
349C2—41no—
350C2—1942no—
351C2—1942no—
352C2#117×D9711yescharacteristic
353C2#131×D9711yescharacteristic
354C2#2026150011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D97
commutator, socleC22×C97C23×C4
Fitting#433×C97C2
prime radicalC23×D97C2×C4
cosocleC2×C4×C97C24
hypercenter#433D97
hypocenterC97#2924
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. #20260793C2semidirect
4. #20260793C2semidirect
5. #20260791C2semidirect
6. #20260790C2semidirect
7. #20260787C2semidirect
8. #20260786C2semidirect
9. #20260760C2semidirect
10. #20260760C2semidirect
13. #20260692C2semidirect
14. #20260689C2semidirect
15. #433×C97C2semidirect
34. C4⋊#11715895C22semidirect
35. C4⋊#11715895C22semidirect
36. #15690739C22semidirect
37. #15690739C22semidirect
38. #15690739C22semidirect
39. #15690739C22semidirect
46. #15690721C22semidirect
47. #15690720C22semidirect
48. #15690720C22semidirect
49. C4×#11715895C22semidirect
50. C8×Dic97C22semidirect
51. C8×Dic97C22semidirect
53. C4⋊C8×C97C22semidirect
54. C4⋊C8×C97C22semidirect
57. #97×C97C22semidirect
58. C4×C8×C97C22semidirect
178. C97#2924semidirect
183. #433D97semidirect
219. C4⋊C8C2×D97semidirect
220. C4⋊C8C2×D97semidirect
232. #97C2×D97semidirect
236. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#117×D972
#202607962
#202607932
#202607912
#202607902
#202607872
#202607862
#202607602
#202606972
C22⋊C8×D972
#202606922
#202606892
#433×C972
#292497