Finite Group Databasebeta

← #38754763

Subgroups — #38754763

Order 25216.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161769C211yesmaximal, characteristic
2#27161447C211yesmaximal, characteristic, complemented
3#27161417C211yesmaximal, characteristic, complemented
4#27161407C211yesmaximal, characteristic, complemented
5#27161386C211yesmaximal, characteristic, complemented
6#27161335C211yesmaximal, characteristic, complemented
7#27161318C211yesmaximal, characteristic, complemented
8#27161286C211yesmaximal, characteristic, complemented
9#27161235C211yesmaximal, characteristic, complemented
10#27161235C211yesmaximal, characteristic, complemented
11#27161184C211yesmaximal, characteristic, complemented
12#27161151C211yesmaximal, characteristic
13#27161088C211yesmaximal, characteristic, complemented
14C2×#20263250C211yesmaximal, characteristic, complemented
15D4×Q8×C197C211yesmaximal, characteristic
16C2×#15691348C2211yescharacteristic
17C2×#15691348C2211yescharacteristic
18C2×#15691348C2211yescharacteristic
19C2×#15691342C2211yescharacteristic
20C22×#13096802C2211yescharacteristic
21#20263319—21no—
22#20263318C2211yescharacteristic
23#20263316—21no—
24#20263307—21no—
25#20263305C2211yescharacteristic
26#20263305C2211yescharacteristic
27C4×Dic394C2211yescharacteristic
28#20263285C2211yescharacteristic
29#20263282—21no—
30C2×C4⋊Dic197C2211yescharacteristic
31#20263259C2211yescharacteristic
32#20263257C2211yescharacteristic
33#20263257C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35#20263250C2211yescharacteristic
36#20263250C2211yescharacteristic
37#20263250C2211yescharacteristic
38#20263250C2211yescharacteristic
39#20263250C2211yescharacteristic
40#20263250C2211yescharacteristic
41#20263250C2211yescharacteristic
42#20263250C2211yescharacteristic
43#20263250C2211yescharacteristic
44#20263249C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic
47#20263237C2211yescharacteristic
48C22×Q8×C197C2211yescharacteristic
49C22×Q8×C197C2211yescharacteristic
50C4⋊Q8×C197—21no—
51C4⋊Q8×C197C2211yescharacteristic
52C22⋊Q8×C197—21no—
53C22⋊Q8×C197—21no—
54C22⋊Q8×C197C2211yescharacteristic
55C22⋊Q8×C197C2211yescharacteristic
56C4×Q8×C197C2211yescharacteristic
57C4×D4×C197—21no—
58C4×D4×C197C2211yescharacteristic
59C2×Dic394C2311yescharacteristic
60C22×Dic197—21no—
61#15691348—22no—
62#15691348—41no—
63#15691348—41no—
64#15691342—41no—
65C2×#13096802—41no—
66C2×#13096802C2311yescharacteristic
67C2×#13096802C2311yescharacteristic
68C2×#13096802C2311yescharacteristic
69C2×#13096802C2311yescharacteristic
70C4⋊Dic197—21no—
71C4⋊Dic197C2311yescharacteristic
72C4⋊Dic197C2311yescharacteristic
73C4⋊Dic197C2311yescharacteristic
74#15691337—21no—
75#15691337—21no—
76#15691337—21no—
77#15691337—21no—
78#15691337—21no—
79C22⋊Dic197—21no—
80C22⋊Dic197—21no—
81C22⋊Dic197—21no—
82C4×Dic197—21no—
83C4×Dic197—21no—
84C2×Q8×C197—21no—
85C2×Q8×C197—21no—
86C2×Q8×C197—22no—
87C2×Q8×C197—22no—
88C2×Q8×C197—41no—
89C2×Q8×C197—41no—
90C2×Q8×C197C2311yescharacteristic
91C2×Q8×C197C2311yescharacteristic
92C2×Q8×C197C2311yescharacteristic
93C2×D4×C197D411yescharacteristic
94C22×C4×C197—21no—
95C22×C4×C197—21no—
96C22×C4×C197C2311yescharacteristic
97C22×C4×C197C2311yescharacteristic
98C4⋊C4×C197—21no—
99C4⋊C4×C197—21no—
100C4⋊C4×C197—21no—
101C4⋊C4×C197—21no—
102C4⋊C4×C197—21no—
103C4⋊C4×C197D411yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C22⋊C4×C197D411yescharacteristic
108C22⋊C4×C197D411yescharacteristic
109C42×C197—21no—
110C42×C197C2311yescharacteristic
111Dic394—21no—
112Dic394—21no—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—42no—
118#13096802—21no—
119#13096802—21no—
120#13096802—21no—
121#13096802—21no—
122C23×C197C2×D411yescharacteristic
123C23×C197C2×D411yescharacteristic
124Q8×C197—21no—
125Q8×C197—21no—
126Q8×C197—21no—
127Q8×C197—21no—
128Q8×C197—21no—
129Q8×C197—42no—
130Q8×C197—42no—
131Q8×C197#4112yes—
132D4×C197—22no—
133C2×C4×C197—21no—
134C2×C4×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—42no—
142C2×C4×C197—42no—
143C2×C4×C197C2×D411yescharacteristic
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2×D411yescharacteristic
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2411yescharacteristic
148C22×C197—21no—
149C22×C197—21no—
150C22×C197—21no—
151C22×C197—21no—
152C22×C197—21no—
153C22×C197#13712yes—
154C22×C197C22×D411yescharacteristic
155C4×C197—21no—
156C4×C197—21no—
157C4×C197—22no—
158C4×C197—21no—
159C4×C197—21no—
160C4×C197—41no—
161C4×C197—41no—
162C4×C197—41no—
163C4×C197C2×#4111yescharacteristic
164C4×C197H(5,𝔽2)11yescharacteristic
165Dic197—41no—
166Dic197—41no—
167Dic197—41no—
168Dic197—41no—
169C2×C197—22no—
170C2×C197—41no—
171C2×C197#54611yescharacteristic
172C2×C197C2×#13711yescharacteristic
173C2×C197#57811yescharacteristic
174C197#331211yescharacteristic, complemented, Sylow
175#3312—1971nomaximal, Sylow
176#552—1971no—
177D4×Q8D19711yescharacteristic
178#491—1971no—
179#485—1971no—
180#483—1971no—
181#479—1971no—
182#470—1971no—
183#467—1971no—
184#462—1971no—
185#451—1971no—
186#451—1971no—
187#441—1971no—
188#435—1971no—
189#419—1971no—
190C2×#103—1971no—
191C22×Q8C2×D19711yescharacteristic
192C22×Q8C2×D19711yescharacteristic
193C2×Q16—1971no—
194C2×Q16—1971no—
195C2×Q16—1971no—
196C2×M16—1971no—
197C22×C8—1971no—
198C4⋊Q8—21no—
199C4⋊Q8C2×D19711yescharacteristic
200#126—3941no—
201#125—3941no—
202#125—1971no—
203#123—3941no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—1971no—
207C22⋊Q8—1971no—
208C22⋊Q8C2×D19711yescharacteristic
209C22⋊Q8C2×D19711yescharacteristic
210C4×Q8—1971no—
211C4×Q8C2×D19711yescharacteristic
212C4×D4—21no—
213C4×D4C2×D19711yescharacteristic
214#117—3941no—
215#117—1971no—
216C2×C4⋊C4—1971no—
217#107—1971no—
218#106—1971no—
219#106—1971no—
220C4⋊C8—1971no—
221#103—1971no—
222#103—1971no—
223#103—1971no—
224#103—1971no—
225#103—1971no—
226#103—1971no—
227#103—1971no—
228#103—1971no—
229#103—1971no—
230#103—1971no—
231C22⋊C8—1971no—
232C22⋊C8—1971no—
233#97—1971no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—1971no—
241C2×Q8C22×D19711yescharacteristic
242C2×Q8C22×D19711yescharacteristic
243C2×Q8C22×D19711yescharacteristic
244C2×D4C22⋊D19711yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3941no—
248C22×C4C22×D19711yescharacteristic
249C22×C4C22×D19711yescharacteristic
250Q16—7881no—
251Q16—7881no—
252Q16—3942no—
253M16—7881no—
254C2×C8—7881no—
255C2×C8—1971no—
256C2×C8—1971no—
257C2×C8—1971no—
258C2×C8—1971no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—3941no—
265C4⋊C4—3941no—
266C4⋊C4—3941no—
267C4⋊C4—3941no—
268C4⋊C4—3941no—
269C4⋊C4—3941no—
270C4⋊C4—1971no—
271C4⋊C4—1971no—
272C4⋊C4—1971no—
273C4⋊C4C22⋊D19711yescharacteristic
274C4⋊C4C22×D19711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3941no—
278C22⋊C4—3941no—
279C22⋊C4—3941no—
280C22⋊C4C22⋊D19711yescharacteristic
281C22⋊C4C22⋊D19711yescharacteristic
282C42—21no—
283C42—3941no—
284C42—3941no—
285C42C22×D19711yescharacteristic
286C23C2×C22⋊D19711yescharacteristic
287C23C2×C22⋊D19711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—42no—
294Q8—42no—
295Q8—3941no—
296Q8—3941no—
297Q8#1569135712yes—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—7882no—
310C2×C4—3941no—
311C2×C4—3941no—
312C2×C4—3941no—
313C2×C4—3941no—
314C2×C4C2×C22⋊D19711yescharacteristic
315C2×C4C2×C22⋊D19711yescharacteristic
316C2×C4C2×C22⋊D19711yescharacteristic
317C2×C4C2×C22⋊D19711yescharacteristic
318C2×C4C23×D19711yescharacteristic
319C8—3941no—
320C8—3941no—
321C8—3941no—
322C8—3941no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#2026335712yes—
329C22C22×C22⋊D19711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—7881no—
339C4—7881no—
340C4—7881no—
341C4—7881no—
342C4C2×#1569135711yescharacteristic
343C4#2026337111yescharacteristic
344C2—22no—
345C2—41no—
346C2#2716172311yescharacteristic
347C2C2×#2026335711yescharacteristic
348C2#2716189411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingD4×Q8×C197C2
socleC22×C197C22×D4
prime radicalC2×D4×C197D4
hypercenterD4×Q8D197
hypocenterC197#3312
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
2. #27161447C2semidirect
3. #27161417C2semidirect
4. #27161407C2semidirect
5. #27161386C2semidirect
6. #27161335C2semidirect
7. #27161318C2semidirect
8. #27161286C2semidirect
9. #27161235C2semidirect
10. #27161235C2semidirect
11. #27161184C2semidirect
13. #27161088C2semidirect
14. C2×#20263250C2semidirect
174. C197#3312semidirect

Maximal subgroups

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

subgroupindex
#271617692
#271614472
#271614172
#271614072
#271613862
#271613352
#271613182
#271612862
#271612352
#271611842
#271611512
#271610882
C2×#202632502
D4×Q8×C1972
#3312197