Finite Group Databasebeta

← #38751554

Subgroups — #38751554

Order 25216.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263358C211yesmaximal, characteristic
2#27161561C211yesmaximal, characteristic
3#27161555C211yesmaximal, characteristic, complemented
4#27161183C211yesmaximal, characteristic, complemented
5#27161182C211yesmaximal, characteristic, complemented
6#27161181C211yesmaximal, characteristic, complemented
7Q16×Dic197C211yesmaximal, characteristic, complemented
8C4×#15691348C211yesmaximal, characteristic, complemented
9C4×#15691345C211yesmaximal, characteristic, complemented
10#27161166C211yesmaximal, characteristic, complemented
11#27161118C211yesmaximal, characteristic, complemented
12#27161074C211yesmaximal, characteristic
13#103×D197C211yesmaximal, characteristic, complemented
14#27161036C211yesmaximal, characteristic, complemented
15#441×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×Q8×D197C2211yescharacteristic
18#20263358C418yescomplemented
19C2×#15691348C2211yescharacteristic
20C2×#15691347C2211yescharacteristic
21C2×#15691345C2211yescharacteristic
22C2×#15691341C2211yescharacteristic
23C2×C8×D197C2211yescharacteristic
24Q8×Dic197C2211yescharacteristic
25Q8×Dic197C2211yescharacteristic
26C4×Dic394—21no—
27C4×Dic394C2211yescharacteristic
28#20263291—21no—
29#20263291C2211yescharacteristic
30C4×D788—21no—
31C4×D788C2211yescharacteristic
32#20263288—21no—
33#20263288C2211yescharacteristic
34#20263284—21no—
35#20263284C2211yescharacteristic
36#20263283—21no—
37#20263283—21no—
38#20263283C2211yescharacteristic
39C4⋊C4×D197—21no—
40C4⋊C4×D197C2211yescharacteristic
41#20263260C2211yescharacteristic
42#20263259C2211yescharacteristic
43#20263257C2211yescharacteristic
44#20263251C2211yescharacteristic
45#20263250C2211yescharacteristic
46#20263250C2211yescharacteristic
47#20263249C2211yescharacteristic
48#20263248C2211yescharacteristic
49#20263246C2211yescharacteristic
50#20263236C2211yescharacteristic
51C4×#13096802C2211yescharacteristic
52C8×Dic197C2211yescharacteristic
53C2×Q16×C197C2211yescharacteristic
54C4×Q8×C197C2211yescharacteristic
55C4×Q8×C197C2211yescharacteristic
56#106×C197C2211yescharacteristic
57#103×C197C2211yescharacteristic
58#103×C197C2211yescharacteristic
59#97×C197C2211yescharacteristic
60#15691359—22no—
61#15691359C2×C414yes—
62C2×Dic394—21no—
63C2×Dic394C2311yescharacteristic
64Q8×D197—22no—
65Q8×D197C2×C414yescomplemented
66C2×D788—21no—
67C2×D788C2311yescharacteristic
68C2×C4×D197—21no—
69C2×C4×D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197—21no—
73C2×C4×D197C2311yescharacteristic
74#15691348C2×C414yescomplemented
75#15691347C2×C414yescomplemented
76#15691345C2×C414yescomplemented
77#15691341C2×C414yescomplemented
78C2×#13096802C2311yescharacteristic
79C2×#13096802C2311yescharacteristic
80C8×D197—22no—
81C4⋊Dic197—21no—
82C4⋊Dic197—21no—
83C4⋊Dic197C2311yescharacteristic
84C4⋊Dic197C2311yescharacteristic
85#15691337—21no—
86#15691337—21no—
87#15691337—21no—
88#15691337D411yescharacteristic
89#15691337D411yescharacteristic
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335D411yescharacteristic
94#15691335D411yescharacteristic
95C4×Dic197—21no—
96C4×Dic197—21no—
97C4×Dic197—21no—
98C4×Dic197—21no—
99C4×Dic197—21no—
100C4×Dic197C2311yescharacteristic
101C2×Q8×C197C2311yescharacteristic
102C2×Q8×C197C2311yescharacteristic
103Q16×C197C2×C414yescomplemented
104C2×C8×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C4⋊C4×C197—21no—
107C4⋊C4×C197—21no—
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C42×C197—21no—
111C42×C197—21no—
112C42×C197C2311yescharacteristic
113C22×D197—21no—
114C22×D197C2×D411yescharacteristic
115Dic394—21no—
116Dic394—24no—
117Dic394C22×C412yes—
118D788—24no—
119D788—21no—
120D788C22×C412yes—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—21no—
124C2×Dic197C2×D411yescharacteristic
125C2×Dic197C2×D411yescharacteristic
126C2×Dic197C2×D411yescharacteristic
127C4×D197—24no—
128C4×D197—24no—
129C4×D197—81no—
130C4×D197—81no—
131C4×D197—81no—
132C4×D197C22×C412yes—
133C4×D197#4112yes—
134#13096802—21no—
135#13096802C22×C412yes—
136Q8×C197—21no—
137Q8×C197—21no—
138Q8×C197C22×C412yes—
139Q8×C197C22×C412yes—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2×D411yescharacteristic
146C2×C4×C197C2411yescharacteristic
147C8×C197—21no—
148C8×C197C22×C412yes—
149C22×C197C22×D411yescharacteristic
150C2×D197—21no—
151C2×D197—21no—
152C2×D197—22no—
153C2×D197—41no—
154C2×D197—41no—
155C2×D197C4×D412yes—
156C4×C197—22no—
157C4×C197—22no—
158C4×C197—21no—
159C4×C197—21no—
160C4×C197—41no—
161C4×C197—41no—
162C4×C197C23×C411yescharacteristic
163C4×C197C2×#4111yescharacteristic
164Dic197—22no—
165Dic197—21no—
166Dic197—21no—
167Dic197—41no—
168Dic197—41no—
169Dic197C4×D412yes—
170C2×C197C2×C4×D411yescharacteristic
171C2×C197#57711yescharacteristic
172C2×C197#57811yescharacteristic
173D197—22no—
174D197—42no—
175C197#294711yescharacteristic, complemented, Sylow
176#2947—1971nomaximal, Sylow
177C2×#137—1971no—
178#520—1971no—
179#519—1971no—
180#441—1971no—
181#441D19711yescharacteristic, complemented
182#440—1971no—
183#440—1971no—
184C4×Q16—1971no—
185C4×Q16—1971no—
186C4×SD16—1971no—
187C4×SD16—1971no—
188#428—1971no—
189#416—1971no—
190C2×#103—1971no—
191#405—1971no—
192C2×#41—1971no—
193C22×Q8—1971no—
194#137—1978no—
195C2×Q16—1971no—
196C2×Q16C2×D19711yescharacteristic
197C2×SD16—1971no—
198C2×SD16—1971no—
199C2×M16—1971no—
200C22×C8—1971no—
201C4×Q8—3941no—
202C4×Q8—3941no—
203C4×Q8—1971no—
204C4×Q8—1971no—
205C4×Q8—1971no—
206C4×Q8—1971no—
207C4×Q8C2×D19711yescharacteristic
208C4×Q8C2×D19711yescharacteristic
209C4×D4—3941no—
210C4×D4—3941no—
211C4×D4—1971no—
212C4×D4—1971no—
213#117—3941no—
214#117—3941no—
215#117—3941no—
216#117—1971no—
217#117—1971no—
218C2×C4⋊C4—3941no—
219C2×C4⋊C4—1971no—
220#107—1971no—
221#107—1971no—
222#106—1971no—
223#106C2×D19711yescharacteristic
224#103—1971no—
225#103—1971no—
226#103—1971no—
227#103—1971no—
228#103C2×D19711yescharacteristic
229#103C2×D19711yescharacteristic
230#102—1971no—
231#102—1971no—
232#97—1971no—
233#97C2×D19711yescharacteristic
234C4×C8—1971no—
235C4×C8—1971no—
236#41—3942no—
237#41—1974no—
238C2×Q8—3941no—
239C2×Q8—3942no—
240C2×Q8—1974no—
241C2×Q8—1971no—
242C2×Q8C22×D19711yescharacteristic
243C2×Q8C22×D19711yescharacteristic
244C2×D4—3941no—
245C2×D4—1971no—
246C22×C4—3941no—
247C22×C4—3941no—
248C22×C4—3941no—
249C22×C4—3941no—
250C22×C4—3941no—
251C22×C4—1971no—
252Q16—1974no—
253Q16C4×D19714yescomplemented
254SD16—1974no—
255SD16—1974no—
256M16—1974no—
257C2×C8—3942no—
258C2×C8—1971no—
259C2×C8—1971no—
260C2×C8C22×D19711yescharacteristic
261C2×C8C22×D19711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
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⋊C4—1971no—
273C4⋊C4C22×D19711yescharacteristic
274C4⋊C4C22×D19711yescharacteristic
275C22⋊C4—3941no—
276C22⋊C4—3941no—
277C22⋊C4—3941no—
278C22⋊C4—1971no—
279C22⋊C4—1971no—
280C42—21no—
281C42—21no—
282C42—3941no—
283C42—3941no—
284C42—3941no—
285C42—3941no—
286C42—3941no—
287C42—1971no—
288C42C22×D19711yescharacteristic
289C23—3941no—
290C23—1971no—
291Q8—21no—
292Q8—21no—
293Q8—3944no—
294Q8—3941no—
295Q8—1972no—
296Q8C2×C4×D19712yes—
297Q8C2×C4×D19712yes—
298D4—3944no—
299D4—3941no—
300D4—1972no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—15761no—
306C2×C4—15761no—
307C2×C4—15761no—
308C2×C4—3944no—
309C2×C4—3944no—
310C2×C4—3941no—
311C2×C4—3941no—
312C2×C4—3941no—
313C2×C4—1971no—
314C2×C4—1971no—
315C2×C4—1971no—
316C2×C4—1972no—
317C2×C4—1972no—
318C2×C4D4×D19711yescharacteristic
319C2×C4D4×D19711yescharacteristic
320C2×C4C23×D19711yescharacteristic
321C8—21no—
322C8—3941no—
323C8—1972no—
324C8C2×C4×D19712yes—
325C22—7881no—
326C22—7881no—
327C22—3942no—
328C22—3941no—
329C22—3941no—
330C22—1972no—
331C22C2×D4×D19711yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—7881no—
339C4—7881no—
340C4—3941no—
341C4—3941no—
342C4—3942no—
343C4—1972no—
344C4C22×C4×D19711yescharacteristic
345C4#41×D19711yescharacteristic
346C2—7882no—
347C2—3942no—
348C2C4×D4×D19711yescharacteristic
349C2#2716188811yescharacteristic
350C2#2716189511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutatorC4×C197C23×C4
FrattiniC2×C4C23×D197
Fitting#441×C197C2
socleC22×C197C22×D4
prime radicalC2×#15691359C22
cosocleC2×C4×C197C24
hypercenter#441D197
hypocenterC197#2947
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
3. #27161555C2semidirect
4. #27161183C2semidirect
5. #27161182C2semidirect
6. #27161181C2semidirect
7. Q16×Dic197C2semidirect
8. C4×#15691348C2semidirect
9. C4×#15691345C2semidirect
10. #27161166C2semidirect
11. #27161118C2semidirect
13. #103×D197C2semidirect
14. #27161036C2semidirect
15. #441×C197C2semidirect
18. #20263358C4semidirect
65. Q8×D197C2×C4semidirect
74. #15691348C2×C4semidirect
75. #15691347C2×C4semidirect
76. #15691345C2×C4semidirect
77. #15691341C2×C4semidirect
103. Q16×C197C2×C4semidirect
175. C197#2947semidirect
181. #441D197semidirect
253. Q16C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633582
#271615612
#271615552
#271611832
#271611822
#271611812
Q16×Dic1972
C4×#156913482
C4×#156913452
#271611662
#271611182
#271610742
#103×D1972
#271610362
#441×C1972
#2947197