Finite Group Databasebeta

← #38752659

Subgroups — #38752659

Order 25216.

410 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D197C211yesmaximal, characteristic
2#117×D197C211yesmaximal, characteristic
3#27161421C211yesmaximal, characteristic, complemented
4#27161420C211yesmaximal, characteristic, complemented
5#27161417C211yesmaximal, characteristic, complemented
6#27161416C211yesmaximal, characteristic, complemented
7#27161413C211yesmaximal, characteristic, complemented
8#27161411C211yesmaximal, characteristic, complemented
9#27161408C211yesmaximal, characteristic, complemented
10#27161120C211yesmaximal, characteristic, complemented
11#27161118C211yesmaximal, characteristic, complemented
12#27161086C211yesmaximal, characteristic, complemented
13#27161086C211yesmaximal, characteristic, complemented
14#27161041C211yesmaximal, characteristic
15#485×C197C211yesmaximal, characteristic, complemented
16C2×Q8×D197C2211yescharacteristic
17C22×C4×D197C2211yescharacteristic
18C2×#15691341C2211yescharacteristic
19C2×#15691341C2211yescharacteristic
20#20263306C2211yescharacteristic
21#20263305C2211yescharacteristic
22#20263304—21no—
23#20263303—21no—
24#20263302C2211yescharacteristic
25#20263285C2211yescharacteristic
26#20263284C2211yescharacteristic
27#20263284C2211yescharacteristic
28#20263283—21no—
29#20263282—21no—
30C4⋊C4×D197—21no—
31C4⋊C4×D197C2211yescharacteristic
32C4⋊C4×D197C2211yescharacteristic
33C4⋊C4×D197C2211yescharacteristic
34C22⋊C4×D197—21no—
35C22⋊C4×D197—21no—
36C42×D197—21no—
37#20263260C2211yescharacteristic, complemented
38#20263259C2211yescharacteristic, complemented
39#20263259C2211yescharacteristic, complemented
40#20263258C2211yescharacteristic, complemented
41#20263257C2211yescharacteristic, complemented
42#20263257C2211yescharacteristic, complemented
43#20263251C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263250C2211yescharacteristic, complemented
46#20263250C2211yescharacteristic, complemented
47#20263249C2211yescharacteristic, complemented
48#20263249C2211yescharacteristic, complemented
49C22⋊#13096802C2211yescharacteristic
50#20263238C2211yescharacteristic
51#20263238C2211yescharacteristic
52C22⋊Q8×C197C2211yescharacteristic
53#117×C197C2211yescharacteristic
54#107×C197C2211yescharacteristic, complemented
55#106×C197C2211yescharacteristic, complemented
56#103×C197C2211yescharacteristic, complemented
57#103×C197C2211yescharacteristic, complemented
58C22⋊C8×C197C2211yescharacteristic
59C23×D197D411yescharacteristic
60C2×Dic394—21no—
61C2×Dic394C2311yescharacteristic
62Q8×D197—22no—
63Q8×D197—22no—
64C22×Dic197D411yescharacteristic
65C2×C4×D197—22no—
66C2×C4×D197—22no—
67C2×C4×D197—21no—
68C2×C4×D197—21no—
69C2×C4×D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197D411yescharacteristic
72C2×C4×D197D411yescharacteristic
73C2×C4×D197C2311yescharacteristic
74C2×C4×D197C2311yescharacteristic
75#15691341—22no—
76#15691341—22no—
77C2×#13096802C2311yescharacteristic
78C2×#13096802C2311yescharacteristic
79C4⋊Dic197—21no—
80C4⋊Dic197C2311yescharacteristic
81C4⋊Dic197C2311yescharacteristic
82C4⋊Dic197C2311yescharacteristic
83#15691337—21no—
84#15691337—21no—
85#15691337—21no—
86#15691337—21no—
87#15691337—21no—
88C22⋊Dic197—21no—
89C22⋊Dic197—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94C4×Dic197—21no—
95C4×Dic197—21no—
96C4×Dic197—21no—
97C2×Q8×C197C2311yescharacteristic
98C22×C4×C197C2311yescharacteristic
99C2×C8×C197C2311yescharacteristic
100C2×C8×C197C2311yescharacteristic
101C4⋊C4×C197—21no—
102C4⋊C4×C197C2311yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C42×C197—21no—
108C22×D197—22no—
109C22×D197—22no—
110C22×D197—41no—
111C22×D197C2×D411yescharacteristic
112C22×D197C2×D411yescharacteristic
113Dic394—21no—
114Dic394—21no—
115Dic394—42no—
116C2×Dic197—22no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197C2×D411yescharacteristic
122C2×Dic197C2×D411yescharacteristic
123C4×D197—22no—
124C4×D197—21no—
125C4×D197—21no—
126C4×D197—22no—
127C4×D197—42no—
128C4×D197—42no—
129C4×D197—42no—
130C4×D197—42no—
131C4×D197#4112yes—
132C4×D197#4112yes—
133#13096802—21no—
134#13096802—21no—
135C23×C197C2×D411yescharacteristic
136Q8×C197—21no—
137Q8×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2411yescharacteristic
146C8×C197—21no—
147C8×C197—21no—
148C22×C197—21no—
149C22×C197—21no—
150C22×C197—21no—
151C22×C197C22×D411yescharacteristic
152C2×D197—21no—
153C2×D197—21no—
154C2×D197—21no—
155C2×D197—21no—
156C2×D197—44no—
157C2×D197#12312yes—
158C2×D197#13712yes—
159C4×C197—21no—
160C4×C197—41no—
161C4×C197—41no—
162C4×C197—41no—
163C4×C197—41no—
164C4×C197C2×#4111yescharacteristic
165C4×C197C2×#4111yescharacteristic
166Dic197—21no—
167Dic197—41no—
168Dic197—41no—
169Dic197—41no—
170Dic197—41no—
171Dic197#12312yes—
172C2×C197—41no—
173C2×C197C2×#12311yescharacteristic
174C2×C197C2×#13711yescharacteristic
175C2×C197#57511yescharacteristic
176D197—22no—
177D197—22no—
178C197#309511yescharacteristic, complemented, Sylow
179#3095—1971nomaximal, Sylow
180C2×C22⋊Q8—1971no—
181C2×#117—1971no—
182#485—1971no—
183#485—1971no—
184#485—1971no—
185#485D19711yescharacteristic, complemented
186#484—1971no—
187#484—1971no—
188#483—1971no—
189#483—1971no—
190#428—1971no—
191#428—1971no—
192#419—1971no—
193#419—1971no—
194#407—1971no—
195C22×Q8—1971no—
196C23×C4—1971no—
197C2×M16—1971no—
198C2×M16—1971no—
199C22⋊Q8—3941no—
200C22⋊Q8—3941no—
201C22⋊Q8—1971no—
202C22⋊Q8—1971no—
203C22⋊Q8—1971no—
204C22⋊Q8C2×D19711yescharacteristic
205#117—3941no—
206#117—3941no—
207#117—1971no—
208#117—1971no—
209#117—1971no—
210#117C2×D19711yescharacteristic
211C2×C4⋊C4—3941no—
212C2×C4⋊C4—1971no—
213C2×C4⋊C4—1971no—
214C2×C4⋊C4—1971no—
215C2×C22⋊C4—3941no—
216C2×C22⋊C4—3941no—
217C2×C42—3941no—
218#107—1971no—
219#107—1971no—
220#107—1971no—
221#107C2×D19711yescharacteristic, complemented
222#106—1971no—
223#106—1971no—
224#106—1971no—
225#106C2×D19711yescharacteristic, complemented
226#103—1971no—
227#103—1971no—
228#103—1971no—
229#103—1971no—
230#103—1971no—
231#103—1971no—
232#103C2×D19711yescharacteristic, complemented
233#103C2×D19711yescharacteristic, complemented
234C22⋊C8—1971no—
235C22⋊C8—1971no—
236C22⋊C8—1971no—
237C22⋊C8C2×D19711yescharacteristic
238C24—1971no—
239C2×Q8—3941no—
240C2×Q8—3942no—
241C2×Q8—3942no—
242C2×Q8—1971no—
243C2×Q8C22×D19711yescharacteristic
244C22×C4—3941no—
245C22×C4—3941no—
246C22×C4—3941no—
247C22×C4—3941no—
248C22×C4—3942no—
249C22×C4—3942no—
250C22×C4—1971no—
251C22×C4—1971no—
252C22×C4—1971no—
253C22×C4—1971no—
254C22×C4—1971no—
255C22×C4C22×D19711yescharacteristic
256M16—3942no—
257M16—3942no—
258C2×C8—1971no—
259C2×C8—1971no—
260C2×C8C22×D19711yescharacteristic
261C2×C8C22×D19711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—3941no—
264C4⋊C4—3941no—
265C4⋊C4—3941no—
266C4⋊C4—3941no—
267C4⋊C4—3941no—
268C4⋊C4—3941no—
269C4⋊C4—1971no—
270C4⋊C4—1971no—
271C4⋊C4—1971no—
272C4⋊C4C22×D19711yescharacteristic
273C4⋊C4C22×D19711yescharacteristic
274C4⋊C4C22×D19711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3941no—
278C22⋊C4—3941no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C42—21no—
284C42—3941no—
285C42—3941no—
286C42—3941no—
287C23—7881no—
288C23—3942no—
289C23—3942no—
290C23—1971no—
291C23—1971no—
292C23D4×D19711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—7882no—
296Q8—3941no—
297Q8—3941no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—7882no—
305C2×C4—7882no—
306C2×C4—7882no—
307C2×C4—7882no—
308C2×C4—3942no—
309C2×C4—3942no—
310C2×C4—3941no—
311C2×C4—3941no—
312C2×C4—3941no—
313C2×C4—3941no—
314C2×C4—3942no—
315C2×C4—3941no—
316C2×C4—3941no—
317C2×C4—1971no—
318C2×C4—1971no—
319C2×C4—1972no—
320C2×C4—1972no—
321C2×C4D4×D19711yescharacteristic
322C2×C4C23×D19711yescharacteristic
323C8—21no—
324C8—21no—
325C8—3941no—
326C8—3941no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—7884no—
331C22—3941no—
332C22—3941no—
333C22—3941no—
334C22—3941no—
335C22—1972no—
336C22—1972no—
337C22C2×D4×D19711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—7881no—
344C4—7881no—
345C4—7881no—
346C4—7881no—
347C4—3941no—
348C4—1972no—
349C4#41×D19711yescharacteristic
350C4#41×D19711yescharacteristic
351C2—41no—
352C2—3942no—
353C2—3942no—
354C2#123×D19711yescharacteristic
355C2#137×D19711yescharacteristic
356C2#2716187611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#485×C197C2
socleC22×C197C22×D4
prime radicalC23×D197D4
hypercenter#485D197
hypocenterC197#3095
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. #27161421C2semidirect
4. #27161420C2semidirect
5. #27161417C2semidirect
6. #27161416C2semidirect
7. #27161413C2semidirect
8. #27161411C2semidirect
9. #27161408C2semidirect
10. #27161120C2semidirect
11. #27161118C2semidirect
12. #27161086C2semidirect
13. #27161086C2semidirect
15. #485×C197C2semidirect
37. #20263260C22semidirect
38. #20263259C22semidirect
39. #20263259C22semidirect
40. #20263258C22semidirect
41. #20263257C22semidirect
42. #20263257C22semidirect
43. #20263251C22semidirect
44. #20263251C22semidirect
45. #20263250C22semidirect
46. #20263250C22semidirect
47. #20263249C22semidirect
48. #20263249C22semidirect
54. #107×C197C22semidirect
55. #106×C197C22semidirect
56. #103×C197C22semidirect
57. #103×C197C22semidirect
178. C197#3095semidirect
185. #485D197semidirect
221. #107C2×D197semidirect
225. #106C2×D197semidirect
232. #103C2×D197semidirect
233. #103C2×D197semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1972
#117×D1972
#271614212
#271614202
#271614172
#271614162
#271614132
#271614112
#271614082
#271611202
#271611182
#271610862
#271610412
#485×C1972
#3095197