Finite Group Databasebeta

← #38754757

Subgroups — #38754757

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
1D4×Dic394C211yesmaximal, characteristic
2#27161449C211yesmaximal, characteristic, complemented
3#27161420C211yesmaximal, characteristic, complemented
4#27161409C211yesmaximal, characteristic, complemented
5#27161383C211yesmaximal, characteristic, complemented
6#27161335C211yesmaximal, characteristic, complemented
7#27161318C211yesmaximal, characteristic, complemented
8#27161289C211yesmaximal, characteristic, complemented
9#27161234C211yesmaximal, characteristic, complemented
10#27161234C211yesmaximal, characteristic, complemented
11#27161184C211yesmaximal, characteristic, complemented
12#27161151C211yesmaximal, characteristic
13#27161087C211yesmaximal, characteristic, complemented
14C2×#20263249C211yesmaximal, characteristic, complemented
15#552×C197C211yesmaximal, characteristic
16C22×Dic394C2211yescharacteristic
17C22×Dic394C2211yescharacteristic
18C2×#15691348C2211yescharacteristic
19C2×#15691348C2211yescharacteristic
20C2×#15691348C2211yescharacteristic
21C2×#15691342C2211yescharacteristic
22C22×#13096802C2211yescharacteristic
23#20263326C2211yescharacteristic
24#20263325—21no—
25#20263305C2211yescharacteristic
26#20263305C2211yescharacteristic
27#20263303—21no—
28#20263303—21no—
29C4×Dic394C2211yescharacteristic
30D4×Dic197—21no—
31#20263259C2211yescharacteristic
32#20263257C2211yescharacteristic
33#20263257C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35#20263250C2211yescharacteristic
36#20263249C2211yescharacteristic
37#20263249C2211yescharacteristic
38#20263249C2211yescharacteristic
39#20263249C2211yescharacteristic
40#20263249C2211yescharacteristic
41#20263249C2211yescharacteristic
42#20263249C2211yescharacteristic
43#20263249C2211yescharacteristic
44#20263249C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic
47#20263237C2211yescharacteristic
48#126×C197—21no—
49#125×C197—21no—
50#125×C197C2211yescharacteristic
51#123×C197—21no—
52C22⋊Q8×C197C2211yescharacteristic
53C22⋊Q8×C197C2211yescharacteristic
54C4×Q8×C197C2211yescharacteristic
55C4×D4×C197C2211yescharacteristic
56#117×C197—21no—
57#117×C197C2211yescharacteristic
58C2×C4⋊C4×C197C2211yescharacteristic
59C2×Dic394—22no—
60C2×Dic394—22no—
61C2×Dic394—21no—
62C2×Dic394—21no—
63C2×Dic394—41no—
64C2×Dic394—41no—
65C2×Dic394C2311yescharacteristic
66C2×Dic394C2311yescharacteristic
67C2×Dic394C2311yescharacteristic
68C22×Dic197—21no—
69C22×Dic197—21no—
70#15691348—22no—
71#15691348—41no—
72#15691348—41no—
73#15691342—41no—
74C2×#13096802—41no—
75C2×#13096802C2311yescharacteristic
76C2×#13096802C2311yescharacteristic
77C2×#13096802C2311yescharacteristic
78C2×#13096802C2311yescharacteristic
79C4⋊Dic197—21no—
80C4⋊Dic197—21no—
81C4⋊Dic197C2311yescharacteristic
82#15691337—21no—
83#15691337—21no—
84#15691337—21no—
85C22⋊Dic197—21no—
86C22⋊Dic197—21no—
87C4×Dic197—21no—
88C2×Q8×C197C2311yescharacteristic
89C2×D4×C197D411yescharacteristic
90C22×C4×C197—21no—
91C22×C4×C197C2311yescharacteristic
92C22×C4×C197C2311yescharacteristic
93C4⋊C4×C197—21no—
94C4⋊C4×C197—21no—
95C4⋊C4×C197—21no—
96C4⋊C4×C197—21no—
97C4⋊C4×C197—21no—
98C4⋊C4×C197—21no—
99C4⋊C4×C197D411yescharacteristic
100C4⋊C4×C197C2311yescharacteristic
101C4⋊C4×C197C2311yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C22⋊C4×C197—21no—
104C22⋊C4×C197—21no—
105C22⋊C4×C197—21no—
106C22⋊C4×C197D411yescharacteristic
107C22⋊C4×C197D411yescharacteristic
108C42×C197—21no—
109C42×C197—21no—
110C42×C197C2311yescharacteristic
111Dic394—21no—
112Dic394—21no—
113Dic394—21no—
114Dic394—21no—
115Dic394—21no—
116Dic394—42no—
117Dic394—42no—
118Dic394#4112yes—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—42no—
124C2×Dic197—42no—
125#13096802—21no—
126#13096802—21no—
127#13096802—21no—
128#13096802—21no—
129C23×C197C2×D411yescharacteristic
130C23×C197C2×D411yescharacteristic
131Q8×C197—21no—
132Q8×C197—21no—
133D4×C197—22no—
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—21no—
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—21no—
158C4×C197—21no—
159C4×C197—41no—
160C4×C197—41no—
161C4×C197—41no—
162C4×C197—41no—
163C4×C197C2×#4111yescharacteristic
164C4×C197H(5,𝔽2)11yescharacteristic
165Dic197—22no—
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#552D19711yescharacteristic
177D4×Q8—1971no—
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×Q8—1971no—
192C22×Q8—1971no—
193C2×Q16—1971no—
194C2×Q16—1971no—
195C2×Q16—1971no—
196C2×M16—1971no—
197C22×C8—1971no—
198C4⋊Q8—3941no—
199C4⋊Q8—1971no—
200#126—21no—
201#125—21no—
202#125C2×D19711yescharacteristic
203#123—21no—
204C22⋊Q8—3941no—
205C22⋊Q8—3941no—
206C22⋊Q8—1971no—
207C22⋊Q8—1971no—
208C22⋊Q8C2×D19711yescharacteristic
209C22⋊Q8C2×D19711yescharacteristic
210C4×Q8—1971no—
211C4×Q8C2×D19711yescharacteristic
212C4×D4—3941no—
213C4×D4C2×D19711yescharacteristic
214#117—21no—
215#117C2×D19711yescharacteristic
216C2×C4⋊C4C2×D19711yescharacteristic
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—7881no—
235C2×Q8—7881no—
236C2×Q8—3942no—
237C2×Q8—3942no—
238C2×Q8—3941no—
239C2×Q8—3941no—
240C2×Q8—1971no—
241C2×Q8—1971no—
242C2×Q8—1971no—
243C2×Q8C22×D19711yescharacteristic
244C2×D4C22⋊D19711yescharacteristic
245C22×C4—21no—
246C22×C4—3941no—
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—21no—
265C4⋊C4—3941no—
266C4⋊C4—3941no—
267C4⋊C4—3941no—
268C4⋊C4—3941no—
269C4⋊C4—3941no—
270C4⋊C4—1971no—
271C4⋊C4C22⋊D19711yescharacteristic
272C4⋊C4C22×D19711yescharacteristic
273C4⋊C4C22×D19711yescharacteristic
274C4⋊C4C22×D19711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—3941no—
279C22⋊C4—3941no—
280C22⋊C4C22⋊D19711yescharacteristic
281C22⋊C4C22⋊D19711yescharacteristic
282C42—21no—
283C42—21no—
284C42—3941no—
285C42C22×D19711yescharacteristic
286C23C2×C22⋊D19711yescharacteristic
287C23C2×C22⋊D19711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—7882no—
291Q8—7882no—
292Q8—3941no—
293Q8—3941no—
294Q8—3941no—
295Q8—3941no—
296Q8—3941no—
297Q8—1972no—
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—7882no—
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#2026335612yes—
329C22C22×C22⋊D19711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—7881no—
339C4—7881no—
340C4—7881no—
341C4—3942no—
342C4#41×D19711yescharacteristic
343C4#2026337211yescharacteristic
344C2—22no—
345C2—41no—
346C2#2716171711yescharacteristic
347C2C2×#2026335611yescharacteristic
348C2#2716189411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#552×C197C2
socleC22×C197C22×D4
prime radicalC2×D4×C197D4
hypercenter#552D197
hypocenterC197#3312
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #27161449C2semidirect
3. #27161420C2semidirect
4. #27161409C2semidirect
5. #27161383C2semidirect
6. #27161335C2semidirect
7. #27161318C2semidirect
8. #27161289C2semidirect
9. #27161234C2semidirect
10. #27161234C2semidirect
11. #27161184C2semidirect
13. #27161087C2semidirect
14. C2×#20263249C2semidirect
174. C197#3312semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3942
#271614492
#271614202
#271614092
#271613832
#271613352
#271613182
#271612892
#271612342
#271611842
#271611512
#271610872
C2×#202632492
#552×C1972
#3312197