Finite Group Databasebeta

← #38754584

Subgroups — #38754584

Order 25216.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161784C211yesmaximal, characteristic
2Q8×C22⋊D197C211yesmaximal, characteristic
3#27161428C211yesmaximal, characteristic, complemented
4#27161414C211yesmaximal, characteristic, complemented
5#27161413C211yesmaximal, characteristic, complemented
6#27161372C211yesmaximal, characteristic, complemented
7#27161320C211yesmaximal, characteristic, complemented
8#27161319C211yesmaximal, characteristic, complemented
9#27161288C211yesmaximal, characteristic, complemented
10#27161236C211yesmaximal, characteristic, complemented
11#27161235C211yesmaximal, characteristic, complemented
12#27161171C211yesmaximal, characteristic, complemented
13C8×C22⋊D197C211yesmaximal, characteristic
14#103×D197C211yesmaximal, characteristic, complemented
15C2×#103×C197C211yesmaximal, characteristic, complemented
16C2×Q8×D197C2211yescharacteristic
17C2×Dic788C2211yescharacteristic
18C2×#15691348C2211yescharacteristic
19C2×#15691348C2211yescharacteristic
20C2×C8×D197C2211yescharacteristic
21#20263325C2211yescharacteristic
22#20263324—21no—
23#20263324C2211yescharacteristic
24#20263316—21no—
25#20263306C2211yescharacteristic
26#20263305—21no—
27#20263305—21no—
28#20263305C2211yescharacteristic
29#20263305C2211yescharacteristic
30#20263304—21no—
31#20263302—21no—
32#20263302C2211yescharacteristic
33Q8×Dic197C2211yescharacteristic
34#20263291C2211yescharacteristic
35C4×C22⋊D197—21no—
36C4×C22⋊D197—21no—
37C4×C22⋊D197C2211yescharacteristic
38C4⋊C4×D197C2211yescharacteristic
39#20263260C2211yescharacteristic
40#20263259C2211yescharacteristic
41#20263259C2211yescharacteristic
42#20263255C2211yescharacteristic
43#20263251C2211yescharacteristic
44#20263251C2211yescharacteristic
45#20263250C2211yescharacteristic
46#20263250C2211yescharacteristic
47#20263249C2211yescharacteristic
48#20263249C2211yescharacteristic
49C22⋊#13096802C2211yescharacteristic
50#20263238C2211yescharacteristic
51C8×Dic197C2211yescharacteristic
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×Dic394—21no—
61C2×Dic394C2311yescharacteristic
62C2×Dic394C2311yescharacteristic
63Q8×D197—22no—
64Q8×D197—41no—
65C2×C22⋊D197D411yescharacteristic
66C2×C4×D197—21no—
67C2×C4×D197—21no—
68C2×C4×D197C2311yescharacteristic
69Dic788—41no—
70#15691348—22no—
71#15691348—41no—
72C2×#13096802C2311yescharacteristic
73C2×#13096802C2311yescharacteristic
74C8×D197—41no—
75C4⋊Dic197—21no—
76C4⋊Dic197—21no—
77C4⋊Dic197C2311yescharacteristic
78C4⋊Dic197C2311yescharacteristic
79#15691337—21no—
80#15691337—21no—
81#15691337—21no—
82#15691337—21no—
83#15691337—21no—
84#15691337—21no—
85#15691337—21no—
86#15691337—21no—
87#15691337D411yescharacteristic
88C22⋊Dic197—21no—
89C22⋊Dic197—21no—
90C22⋊Dic197D411yescharacteristic
91#15691335—21no—
92#15691335—21no—
93#15691335D411yescharacteristic
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—
117Dic394—81no—
118C22⋊D197Q1614yescomplemented
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197C2×D411yescharacteristic
124C2×Dic197C2×D411yescharacteristic
125C2×Dic197C2×D411yescharacteristic
126C4×D197—21no—
127C4×D197—21no—
128C4×D197—42no—
129C4×D197—81no—
130#13096802—21no—
131#13096802—21no—
132C23×C197C2×D411yescharacteristic
133Q8×C197—21no—
134Q8×C197—21no—
135Q8×C197—21no—
136Q8×C197—42no—
137Q8×C197#4112yes—
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×C197—42no—
145C2×C4×C197—81no—
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2411yescharacteristic
148C8×C197—21no—
149C8×C197—21no—
150C22×C197—21no—
151C22×C197—21no—
152C22×C197C2×Q1612yes—
153C22×C197C22×D411yescharacteristic
154C2×D197—21no—
155C2×D197—21no—
156C2×D197C2×Q1612yes—
157C4×C197—21no—
158C4×C197—22no—
159C4×C197—41no—
160C4×C197—41no—
161C4×C197—41no—
162C4×C197C2×#4111yescharacteristic
163C4×C197H(5,𝔽2)11yescharacteristic
164Dic197—21no—
165Dic197—21no—
166Dic197—41no—
167Dic197—41no—
168Dic197—41no—
169Dic197—41no—
170Dic197C2×Q1612yes—
171C2×C197—22no—
172C2×C197#54611yescharacteristic
173C2×C197C22×Q1611yescharacteristic
174C2×C197#57711yescharacteristic
175D197—22no—
176C197#329911yescharacteristic, complemented, Sylow
177#3299—1971nomaximal, Sylow
178#554—1971no—
179D4×Q8—1971no—
180#487—1971no—
181#484—1971no—
182#484—1971no—
183#477—1971no—
184#467—1971no—
185#467—1971no—
186#462—1971no—
187#451—1971no—
188#451—1971no—
189C4×Q16—1971no—
190C8×D4—1971no—
191C2×#103—1971no—
192C2×#103D19711yescharacteristic, complemented
193C22×Q8—1971no—
194C22×Q8C2×D19711yescharacteristic
195C2×Q16—1971no—
196C2×Q16—1971no—
197C2×Q16—1971no—
198C22×C8—1971no—
199C22×C8C2×D19711yescharacteristic
200C4⋊Q8—3941no—
201C4⋊Q8—1971no—
202C4⋊Q8—1971no—
203#125—3941no—
204C22⋊Q8—3941no—
205C22⋊Q8—3941no—
206C22⋊Q8—3941no—
207C22⋊Q8—3941no—
208C22⋊Q8—1971no—
209C22⋊Q8—1971no—
210C22⋊Q8—1971no—
211C22⋊Q8—1971no—
212C4×Q8—1971no—
213C4×Q8—1971no—
214C4×D4—3941no—
215C4×D4—3941no—
216C4×D4—1971no—
217C2×C4⋊C4—1971no—
218C2×C4⋊C4C2×D19711yescharacteristic
219#107—1971no—
220#107—1971no—
221#107—1971no—
222C4⋊C8—1971no—
223#103—1971no—
224#103—1971no—
225#103—1971no—
226#103—1971no—
227#103—1971no—
228#103—1971no—
229#103C2×D19711yescharacteristic
230#103C2×D19711yescharacteristic
231#103C2×D19711yescharacteristic
232#103C2×D19711yescharacteristic
233C22⋊C8—1971no—
234C22⋊C8—1971no—
235C4×C8—1971no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—7881no—
240C2×Q8—3941no—
241C2×Q8—3941no—
242C2×Q8—3942no—
243C2×Q8—1971no—
244C2×Q8—1971no—
245C2×Q8C22×D19711yescharacteristic
246C2×Q8C22×D19711yescharacteristic
247C2×D4—1971no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—3941no—
251C22×C4—3941no—
252C22×C4—1971no—
253C22×C4C22×D19711yescharacteristic
254Q16—7881no—
255Q16—7881no—
256Q16—3942no—
257C2×C8—41no—
258C2×C8—7881no—
259C2×C8—1971no—
260C2×C8—1971no—
261C2×C8C22×D19711yescharacteristic
262C2×C8C22×D19711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—3941no—
265C4⋊C4—3941no—
266C4⋊C4—3941no—
267C4⋊C4—3941no—
268C4⋊C4—3941no—
269C4⋊C4—3941no—
270C4⋊C4—3941no—
271C4⋊C4—3941no—
272C4⋊C4—3941no—
273C4⋊C4—3941no—
274C4⋊C4—1971no—
275C4⋊C4—1971no—
276C4⋊C4—1971no—
277C4⋊C4C22×D19711yescharacteristic
278C4⋊C4C22×D19711yescharacteristic
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—1971no—
284C22⋊C4—1971no—
285C42—3941no—
286C42—3941no—
287C42—1971no—
288C23—1971no—
289C23D4×D19711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—42no—
294Q8—7882no—
295Q8—15761no—
296Q8—3941no—
297Q8—3941no—
298Q8—3941no—
299Q8—3941no—
300Q8#1569135812yes—
301D4—1974no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—7882no—
311C2×C4—15761no—
312C2×C4—3941no—
313C2×C4—3941no—
314C2×C4—3941no—
315C2×C4—3941no—
316C2×C4—3941no—
317C2×C4—3941no—
318C2×C4—1971no—
319C2×C4—1971no—
320C2×C4—1971no—
321C2×C4D4×D19711yescharacteristic
322C2×C4C23×D19711yescharacteristic
323C8—21no—
324C8—21no—
325C8—3941no—
326C8—3941no—
327C22—21no—
328C22—21no—
329C22—3941no—
330C22—3941no—
331C22—1972no—
332C22Q16×D19712yes—
333C22C2×D4×D19711yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—7881no—
340C4—7881no—
341C4—7881no—
342C4—7881no—
343C4—3941no—
344C4—3941no—
345C4—1972no—
346C4C2×#1569135811yescharacteristic
347C4#2026337111yescharacteristic
348C2—22no—
349C2—3942no—
350C2#2716171911yescharacteristic
351C2C2×Q16×D19711yescharacteristic
352C2#2716188511yescharacteristic
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#3299
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. #27161428C2semidirect
4. #27161414C2semidirect
5. #27161413C2semidirect
6. #27161372C2semidirect
7. #27161320C2semidirect
8. #27161319C2semidirect
9. #27161288C2semidirect
10. #27161236C2semidirect
11. #27161235C2semidirect
12. #27161171C2semidirect
14. #103×D197C2semidirect
15. C2×#103×C197C2semidirect
118. C22⋊D197Q16semidirect
176. C197#3299semidirect
192. C2×#103D197semidirect

Maximal subgroups

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

subgroupindex
#271617842
Q8×C22⋊D1972
#271614282
#271614142
#271614132
#271613722
#271613202
#271613192
#271612882
#271612362
#271612352
#271611712
C8×C22⋊D1972
#103×D1972
C2×#103×C1972
#3299197