Finite Group Databasebeta

← #38754759

Subgroups — #38754759

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#27161764C211yesmaximal, characteristic
2Q8×C22⋊D197C211yesmaximal, characteristic
3#27161446C211yesmaximal, characteristic, complemented
4#27161419C211yesmaximal, characteristic, complemented
5#27161409C211yesmaximal, characteristic, complemented
6#27161388C211yesmaximal, characteristic, complemented
7#27161336C211yesmaximal, characteristic, complemented
8#27161320C211yesmaximal, characteristic, complemented
9#27161288C211yesmaximal, characteristic, complemented
10#27161236C211yesmaximal, characteristic, complemented
11#27161235C211yesmaximal, characteristic, complemented
12#27161183C211yesmaximal, characteristic, complemented
13#27161150C211yesmaximal, characteristic
14#27161086C211yesmaximal, characteristic, complemented
15C2×#103×C197C211yesmaximal, characteristic, complemented
16C2×Q8×D197C2211yescharacteristic
17C2×Dic788C2211yescharacteristic
18C2×#15691348C2211yescharacteristic
19C2×#15691348C2211yescharacteristic
20C2×#15691341C2211yescharacteristic
21#20263324—21no—
22#20263324C2211yescharacteristic
23#20263321—21no—
24#20263317C2211yescharacteristic
25#20263316—21no—
26#20263309—21no—
27#20263306C2211yescharacteristic
28#20263305—21no—
29#20263305C2211yescharacteristic
30#20263305C2211yescharacteristic
31#20263302—21no—
32#20263302C2211yescharacteristic
33Q8×Dic197C2211yescharacteristic
34#20263291C2211yescharacteristic
35C4×C22⋊D197—21no—
36C4×C22⋊D197C2211yescharacteristic
37#20263285—21no—
38#20263284C2211yescharacteristic
39#20263260C2211yescharacteristic
40#20263257C2211yescharacteristic
41#20263257C2211yescharacteristic
42#20263255C2211yescharacteristic
43#20263251C2211yescharacteristic
44#20263251C2211yescharacteristic
45#20263250C2211yescharacteristic
46#20263250C2211yescharacteristic
47#20263249C2211yescharacteristic
48#20263249C2211yescharacteristic
49C22⋊#13096802C2211yescharacteristic
50#20263238C2211yescharacteristic
51#20263236C2211yescharacteristic
52C22×Q8×C197C2211yescharacteristic
53C22×C8×C197C2211yescharacteristic
54C2×C4⋊C4×C197C2211yescharacteristic
55#103×C197C2211yescharacteristic
56#103×C197C2211yescharacteristic
57#103×C197C2211yescharacteristic
58#103×C197C2211yescharacteristic
59C2×Dic394—21no—
60C2×Dic394C2311yescharacteristic
61C2×Dic394C2311yescharacteristic
62Q8×D197—22no—
63Q8×D197—41no—
64C2×C22⋊D197D411yescharacteristic
65C2×C4×D197—21no—
66C2×C4×D197C2311yescharacteristic
67Dic788—41no—
68#15691348—22no—
69#15691348—41no—
70#15691341—41no—
71C2×#13096802C2311yescharacteristic
72C2×#13096802C2311yescharacteristic
73C4⋊Dic197—21no—
74C4⋊Dic197—21no—
75C4⋊Dic197—21no—
76C4⋊Dic197C2311yescharacteristic
77C4⋊Dic197C2311yescharacteristic
78#15691337—21no—
79#15691337—21no—
80#15691337—21no—
81#15691337—21no—
82#15691337—21no—
83#15691337—21no—
84#15691337—21no—
85#15691337D411yescharacteristic
86C22⋊Dic197—21no—
87C22⋊Dic197—21no—
88C22⋊Dic197D411yescharacteristic
89#15691335—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335D411yescharacteristic
93C4×Dic197—21no—
94C4×Dic197—21no—
95C4×Dic197—21no—
96C4×Dic197C2311yescharacteristic
97C2×Q8×C197—21no—
98C2×Q8×C197—22no—
99C2×Q8×C197—41no—
100C2×Q8×C197C2311yescharacteristic
101C2×Q8×C197C2311yescharacteristic
102C22×C4×C197—21no—
103C22×C4×C197—21no—
104C22×C4×C197C2311yescharacteristic
105C2×C8×C197—41no—
106C2×C8×C197C2311yescharacteristic
107C2×C8×C197C2311yescharacteristic
108C4⋊C4×C197—21no—
109C4⋊C4×C197C2311yescharacteristic
110C4⋊C4×C197C2311yescharacteristic
111C22×D197C2×D411yescharacteristic
112Dic394—21no—
113Dic394—21no—
114Dic394—21no—
115Dic394—21no—
116Dic394—42no—
117C22⋊D197—22no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197C2×D411yescharacteristic
123C2×Dic197C2×D411yescharacteristic
124C2×Dic197C2×D411yescharacteristic
125C4×D197—21no—
126C4×D197—21no—
127C4×D197—42no—
128#13096802—21no—
129#13096802—21no—
130C23×C197C2×D411yescharacteristic
131Q8×C197—21no—
132Q8×C197—21no—
133Q8×C197—21no—
134Q8×C197—42no—
135Q8×C197#4112yes—
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×C197—42no—
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2411yescharacteristic
146C8×C197—21no—
147C8×C197—21no—
148C22×C197—21no—
149C22×C197—21no—
150C22×C197#13712yes—
151C22×C197C22×D411yescharacteristic
152C2×D197—21no—
153C2×D197—21no—
154C2×D197—21no—
155C4×C197—21no—
156C4×C197—22no—
157C4×C197—41no—
158C4×C197—41no—
159C4×C197—41no—
160C4×C197C2×#4111yescharacteristic
161C4×C197H(5,𝔽2)11yescharacteristic
162Dic197—21no—
163Dic197—21no—
164Dic197—21no—
165Dic197—41no—
166Dic197—41no—
167Dic197—41no—
168Dic197—41no—
169C2×C197—22no—
170C2×C197#54611yescharacteristic
171C2×C197C2×#13711yescharacteristic
172C2×C197#57811yescharacteristic
173D197—41no—
174C197#331211yescharacteristic, complemented, Sylow
175#3312—1971nomaximal, Sylow
176#552—1971no—
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×#103D19711yescharacteristic, complemented
191C22×Q8—1971no—
192C22×Q8C2×D19711yescharacteristic
193C2×Q16—1971no—
194C2×Q16—1971no—
195C2×Q16—1971no—
196C2×M16—1971no—
197C22×C8C2×D19711yescharacteristic
198C4⋊Q8—3941no—
199C4⋊Q8—1971no—
200#126—3941no—
201#125—3941no—
202#125—1971no—
203#123—3941no—
204C22⋊Q8—3941no—
205C22⋊Q8—3941no—
206C22⋊Q8—1971no—
207C22⋊Q8—1971no—
208C22⋊Q8—1971no—
209C22⋊Q8—1971no—
210C4×Q8—1971no—
211C4×Q8—1971no—
212C4×D4—3941no—
213C4×D4—1971no—
214#117—3941no—
215#117—1971no—
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#103C2×D19711yescharacteristic
228#103C2×D19711yescharacteristic
229#103C2×D19711yescharacteristic
230#103C2×D19711yescharacteristic
231C22⋊C8—1971no—
232C22⋊C8—1971no—
233#97—1971no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—7881no—
238C2×Q8—3941no—
239C2×Q8—3942no—
240C2×Q8—1971no—
241C2×Q8—1971no—
242C2×Q8C22×D19711yescharacteristic
243C2×Q8C22×D19711yescharacteristic
244C2×D4—1971no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3941no—
248C22×C4—1971no—
249C22×C4C22×D19711yescharacteristic
250Q16—7881no—
251Q16—7881no—
252Q16—3942no—
253M16—7881no—
254C2×C8—41no—
255C2×C8—1971no—
256C2×C8—1971no—
257C2×C8C22×D19711yescharacteristic
258C2×C8C22×D19711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—3941no—
261C4⋊C4—3941no—
262C4⋊C4—3941no—
263C4⋊C4—3941no—
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—3941no—
276C22⋊C4—3941no—
277C22⋊C4—3941no—
278C22⋊C4—3941no—
279C22⋊C4—3941no—
280C22⋊C4—1971no—
281C22⋊C4—1971no—
282C42—3941no—
283C42—3941no—
284C42—3941no—
285C42—1971no—
286C23—1971no—
287C23D4×D19711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—42no—
292Q8—7882no—
293Q8—3941no—
294Q8—3941no—
295Q8—3941no—
296Q8—3941no—
297Q8#1569135812yes—
298D4—3942no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—7882no—
308C2×C4—3941no—
309C2×C4—3941no—
310C2×C4—3941no—
311C2×C4—3941no—
312C2×C4—3941no—
313C2×C4—3941no—
314C2×C4—1971no—
315C2×C4—1971no—
316C2×C4—1971no—
317C2×C4D4×D19711yescharacteristic
318C2×C4C23×D19711yescharacteristic
319C8—21no—
320C8—21no—
321C8—3941no—
322C8—3941no—
323C22—21no—
324C22—21no—
325C22—3941no—
326C22—3941no—
327C22—3941no—
328C22#2026335512yes—
329C22C2×D4×D19711yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—7881no—
336C4—7881no—
337C4—7881no—
338C4—7881no—
339C4—3941no—
340C4—3941no—
341C4—3941no—
342C4C2×#1569135811yescharacteristic
343C4#2026337111yescharacteristic
344C2—22no—
345C2—7881no—
346C2#2716171911yescharacteristic
347C2C2×#2026335511yescharacteristic
348C2#2716189211yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#103×C197C2
socleC22×C197C22×D4
prime radicalC2×C22⋊D197D4
hypercenterC2×#103D197
hypocenterC197#3312
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #27161446C2semidirect
4. #27161419C2semidirect
5. #27161409C2semidirect
6. #27161388C2semidirect
7. #27161336C2semidirect
8. #27161320C2semidirect
9. #27161288C2semidirect
10. #27161236C2semidirect
11. #27161235C2semidirect
12. #27161183C2semidirect
14. #27161086C2semidirect
15. C2×#103×C197C2semidirect
174. C197#3312semidirect
190. C2×#103D197semidirect

Maximal subgroups

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

subgroupindex
#271617642
Q8×C22⋊D1972
#271614462
#271614192
#271614092
#271613882
#271613362
#271613202
#271612882
#271612362
#271612352
#271611832
#271611502
#271610862
C2×#103×C1972
#3312197