Finite Group Databasebeta

← #27151648

Subgroups — #27151648

Order 12416.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261245C211yesmaximal, characteristic
2#117×D97C211yesmaximal, characteristic, complemented
3#20261077C211yesmaximal, characteristic, complemented
4#20261076C211yesmaximal, characteristic, complemented
5#20261072C211yesmaximal, characteristic, complemented
6#20261067C211yesmaximal, characteristic, complemented
7#20261056C211yesmaximal, characteristic, complemented
8#20261052C211yesmaximal, characteristic, complemented
9#20261047C211yesmaximal, characteristic, complemented
10#20260772C211yesmaximal, characteristic, complemented
11#20260770C211yesmaximal, characteristic, complemented
12#20260728C211yesmaximal, characteristic, complemented
13#20260728C211yesmaximal, characteristic, complemented
14C22⋊C8×D97C211yesmaximal, characteristic, complemented
15#482×C97C211yesmaximal, characteristic, complemented
16C2×#13095510C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18C2×C8×D97C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20#15690792—21no—
21#15690790C2211yescharacteristic
22#15690789C2211yescharacteristic
23#15690781C2211yescharacteristic
24#15690778—21no—
25D4×Dic97—21no—
26#15690770—21no—
27#15690769C2211yescharacteristic
28#15690768C2211yescharacteristic
29#15690768C2211yescharacteristic
30#15690768C2211yescharacteristic
31#15690767—21no—
32#15690766—21no—
33C4⋊C4×D97C2211yescharacteristic
34C4⋊C4×D97C2211yescharacteristic
35C22⋊C4×D97—21no—
36C42×D97—21no—
37#15690744C2211yescharacteristic, complemented
38#15690743C2211yescharacteristic, complemented
39#15690743C2211yescharacteristic, complemented
40#15690742C2211yescharacteristic, complemented
41#15690741C2211yescharacteristic, complemented
42#15690741C2211yescharacteristic, complemented
43#15690735C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690733C2211yescharacteristic, complemented
46#15690733C2211yescharacteristic, complemented
47#15690731C2211yescharacteristic, complemented
48#15690731C2211yescharacteristic, complemented
49C22⋊#11715895C2211yescharacteristic
50#15690722C2211yescharacteristic
51#15690722C2211yescharacteristic
52#121×C97C2211yescharacteristic
53#117×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic, complemented
55#106×C97C2211yescharacteristic, complemented
56#102×C97C2211yescharacteristic, complemented
57#102×C97C2211yescharacteristic, complemented
58C22⋊C8×C97C2211yescharacteristic
59C23×D97D411yescharacteristic
60#13095510—22no—
61#13095510—22no—
62C2×Dic194C2311yescharacteristic
63C2×C22⋊D97—21no—
64C2×C22⋊D97—21no—
65C22×Dic97—21no—
66C22×Dic97D411yescharacteristic
67C2×C4×D97—22no—
68C2×C4×D97—22no—
69C2×C4×D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97D411yescharacteristic
72C2×C4×D97D411yescharacteristic
73C2×C4×D97C2311yescharacteristic
74C2×C4×D97C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C2×#11715895C2311yescharacteristic
77C8×D97—22no—
78C8×D97—22no—
79C4⋊Dic97—21no—
80C4⋊Dic97C2311yescharacteristic
81C4⋊Dic97C2311yescharacteristic
82C4⋊Dic97C2311yescharacteristic
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86C22⋊Dic97—21no—
87C22⋊Dic97—21no—
88C22⋊Dic97—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93C4×Dic97—21no—
94C4×Dic97—21no—
95C4×Dic97—21no—
96C4×Dic97—21no—
97C2×D4×C97—21no—
98C2×D4×C97C2311yescharacteristic
99C22×C4×C97C2311yescharacteristic
100C2×C8×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C4⋊C4×C97C2311yescharacteristic
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C42×C97—21no—
108C23×C97—21no—
109C23×C97C2×D411yescharacteristic
110C22×D97—22no—
111C22×D97—22no—
112C22×D97—41no—
113C22×D97C2×D411yescharacteristic
114C22×D97C2×D411yescharacteristic
115D4×C97—21no—
116D4×C97—21no—
117D4×C97—42no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2411yescharacteristic
125C22⋊D97—42no—
126C22⋊D97—42no—
127C2×Dic97—22no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—42no—
133C2×Dic97C2×D411yescharacteristic
134C2×Dic97C2×D411yescharacteristic
135C4×D97—22no—
136C4×D97—21no—
137C4×D97—21no—
138C4×D97—22no—
139C4×D97—42no—
140C4×D97—42no—
141C4×D97#4112yes—
142C4×D97#4112yes—
143Dic194—21no—
144Dic194—21no—
145C8×C97—21no—
146C8×C97—21no—
147#11715895—21no—
148#11715895—21no—
149C22×C97—21no—
150C22×C97—21no—
151C22×C97—21no—
152C22×C97—41no—
153C22×C97—41no—
154C22×C97—41no—
155C22×C97C22×D411yescharacteristic
156C2×D97—21no—
157C2×D97—21no—
158C2×D97—21no—
159C2×D97—21no—
160C2×D97—44no—
161C2×D97#12312yes—
162C2×D97#13512yes—
163C4×C97—21no—
164C4×C97—41no—
165C4×C97—41no—
166C4×C97—41no—
167C4×C97C2×#4111yescharacteristic
168C4×C97C2×#4111yescharacteristic
169Dic97—21no—
170Dic97—41no—
171Dic97—41no—
172Dic97—41no—
173Dic97—41no—
174Dic97#12312yes—
175C2×C97—41no—
176C2×C97—81no—
177C2×C97C2×#12311yescharacteristic
178C2×C97C2×#13511yescharacteristic
179C2×C97#57511yescharacteristic
180D97—22no—
181D97—22no—
182#3091—971nomaximal, Sylow
183C97#309111yescharacteristic, complemented, Sylow
184#525—971no—
185C2×#117—971no—
186#485—971no—
187#485—971no—
188#484—971no—
189#483—971no—
190#482—971no—
191#482D9711yescharacteristic, complemented
192#481—971no—
193#480—971no—
194#427—971no—
195#427—971no—
196#416—971no—
197#416—971no—
198C2×C22⋊C8—971no—
199C2×#41—971no—
200C23×C4—971no—
201C22×C8—971no—
202C22×C8—971no—
203#123—1941no—
204C22⋊Q8—971no—
205C22⋊Q8—971no—
206#121—971no—
207#121C2×D9711yescharacteristic
208#120—1941no—
209C4×D4—1941no—
210C4×D4—1941no—
211#117—971no—
212#117—971no—
213#117—971no—
214#117—971no—
215#117—1941no—
216#117—1941no—
217#117C2×D9711yescharacteristic
218C2×C4⋊C4—971no—
219C2×C4⋊C4—971no—
220C2×C22⋊C4—1941no—
221C2×C42—1941no—
222#107—971no—
223#107—971no—
224#107—971no—
225#107C2×D9711yescharacteristic, complemented
226#106—971no—
227#106—971no—
228#106—971no—
229#106C2×D9711yescharacteristic, complemented
230#103—971no—
231#103—971no—
232#103—971no—
233#103—971no—
234#102—971no—
235#102—971no—
236#102C2×D9711yescharacteristic, complemented
237#102C2×D9711yescharacteristic, complemented
238C22⋊C8—971no—
239C22⋊C8—971no—
240C22⋊C8—971no—
241C22⋊C8C2×D9711yescharacteristic
242C24—971no—
243#41—1942no—
244#41—1942no—
245C2×Q8—971no—
246C2×D4—21no—
247C2×D4—1941no—
248C2×D4—1941no—
249C2×D4C22×D9711yescharacteristic
250C22×C4—971no—
251C22×C4—971no—
252C22×C4—971no—
253C22×C4—971no—
254C22×C4—971no—
255C22×C4—1942no—
256C22×C4—1942no—
257C22×C4—1941no—
258C22×C4—1941no—
259C22×C4—1941no—
260C22×C4C22×D9711yescharacteristic
261C2×C8—971no—
262C2×C8—971no—
263C2×C8—1942no—
264C2×C8—1942no—
265C2×C8C22×D9711yescharacteristic
266C2×C8C22×D9711yescharacteristic
267C4⋊C4—971no—
268C4⋊C4—971no—
269C4⋊C4—971no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4—1941no—
274C4⋊C4C22×D9711yescharacteristic
275C4⋊C4C22×D9711yescharacteristic
276C4⋊C4C22×D9711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4—1941no—
284C22⋊C4—1941no—
285C22⋊C4—1941no—
286C42—21no—
287C42—1941no—
288C42—1941no—
289C42—1941no—
290C42—1941no—
291C23—21no—
292C23—971no—
293C23—971no—
294C23—3881no—
295C23—1942no—
296C23—1942no—
297C23D4×D9711yescharacteristic
298Q8—1941no—
299Q8—1941no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—3882no—
304D4—3882no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—971no—
311C2×C4—972no—
312C2×C4—972no—
313C2×C4—971no—
314C2×C4—3882no—
315C2×C4—3882no—
316C2×C4—3882no—
317C2×C4—1942no—
318C2×C4—1942no—
319C2×C4—1942no—
320C2×C4—1941no—
321C2×C4—1941no—
322C2×C4—1941no—
323C2×C4—1941no—
324C2×C4—1941no—
325C2×C4—1941no—
326C2×C4D4×D9711yescharacteristic
327C2×C4C23×D9711yescharacteristic
328C8—21no—
329C8—21no—
330C8—1941no—
331C8—1941no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—972no—
339C22—972no—
340C22—3884no—
341C22—1941no—
342C22—1941no—
343C22—1941no—
344C22—1941no—
345C22C2×D4×D9711yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—972no—
351C4—3881no—
352C4—3881no—
353C4—3881no—
354C4—3881no—
355C4—1941no—
356C4#41×D9711yescharacteristic
357C4#41×D9711yescharacteristic
358C2—41no—
359C2—81no—
360C2—1942no—
361C2—1942no—
362C2#123×D9711yescharacteristic
363C2#135×D9711yescharacteristic
364C2#2026153111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#482×C97C2
socleC22×C97C22×D4
prime radical#20261245C2
hypercenter#482D97
hypocenterC97#3091
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #117×D97C2semidirect
3. #20261077C2semidirect
4. #20261076C2semidirect
5. #20261072C2semidirect
6. #20261067C2semidirect
7. #20261056C2semidirect
8. #20261052C2semidirect
9. #20261047C2semidirect
10. #20260772C2semidirect
11. #20260770C2semidirect
12. #20260728C2semidirect
13. #20260728C2semidirect
14. C22⋊C8×D97C2semidirect
15. #482×C97C2semidirect
37. #15690744C22semidirect
38. #15690743C22semidirect
39. #15690743C22semidirect
40. #15690742C22semidirect
41. #15690741C22semidirect
42. #15690741C22semidirect
43. #15690735C22semidirect
44. #15690735C22semidirect
45. #15690733C22semidirect
46. #15690733C22semidirect
47. #15690731C22semidirect
48. #15690731C22semidirect
54. #107×C97C22semidirect
55. #106×C97C22semidirect
56. #102×C97C22semidirect
57. #102×C97C22semidirect
183. C97#3091semidirect
191. #482D97semidirect
225. #107C2×D97semidirect
229. #106C2×D97semidirect
236. #102C2×D97semidirect
237. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612452
#117×D972
#202610772
#202610762
#202610722
#202610672
#202610562
#202610522
#202610472
#202607722
#202607702
#202607282
C22⋊C8×D972
#482×C972
#309197