Finite Group Databasebeta

← #38751528

Subgroups — #38751528

Order 25216.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263358C211yesmaximal, characteristic
2C4×#15691359C211yesmaximal, characteristic
3C4×Q8×D197C211yesmaximal, characteristic, complemented
4#27161184C211yesmaximal, characteristic, complemented
5#27161181C211yesmaximal, characteristic, complemented
6#27161177C211yesmaximal, characteristic, complemented
7Q16×Dic197C211yesmaximal, characteristic, complemented
8#27161171C211yesmaximal, characteristic, complemented
9#27161166C211yesmaximal, characteristic, complemented
10#27161164C211yesmaximal, characteristic, complemented
11#27161118C211yesmaximal, characteristic, complemented
12#27161086C211yesmaximal, characteristic, complemented
13#27161079C211yesmaximal, characteristic
14#27161031C211yesmaximal, characteristic, complemented
15#441×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×Q8×D197C2211yescharacteristic
18#20263358C418yescomplemented
19C2×#15691348C2211yescharacteristic
20C2×#15691347C2211yescharacteristic
21C2×#15691345C2211yescharacteristic
22C2×#15691341C2211yescharacteristic
23C2×#15691341C2211yescharacteristic
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
36C4⋊C4×D197—21no—
37C4⋊C4×D197C2211yescharacteristic
38C42×D197—21no—
39C42×D197—21no—
40C42×D197C2211yescharacteristic
41#20263260C2211yescharacteristic
42#20263259C2211yescharacteristic
43#20263257C2211yescharacteristic
44#20263251C2211yescharacteristic
45#20263250C2211yescharacteristic
46#20263250C2211yescharacteristic
47#20263249C2211yescharacteristic
48#20263248C2211yescharacteristic
49#20263246C2211yescharacteristic
50#20263237C2211yescharacteristic
51C8×Dic197C2211yescharacteristic
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×D197D411yescharacteristic
74C2×C4×D197D411yescharacteristic
75C2×C4×D197C2311yescharacteristic
76#15691348C2×C414yescomplemented
77#15691347C2×C414yescomplemented
78#15691345C2×C414yescomplemented
79#15691341—22no—
80#15691341C2×C414yescomplemented
81C2×#13096802C2311yescharacteristic
82C2×#13096802C2311yescharacteristic
83C4⋊Dic197—21no—
84C4⋊Dic197—21no—
85C4⋊Dic197C2311yescharacteristic
86C4⋊Dic197C2311yescharacteristic
87#15691337—21no—
88#15691337—21no—
89#15691337—21no—
90#15691335—21no—
91#15691335—21no—
92#15691335—21no—
93C4×Dic197—21no—
94C4×Dic197—21no—
95C4×Dic197—21no—
96C4×Dic197—21no—
97C4×Dic197—21no—
98C4×Dic197D411yescharacteristic
99C4×Dic197D411yescharacteristic
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—22no—
130C4×D197—41no—
131C4×D197—42no—
132C4×D197—42no—
133C4×D197—42no—
134C4×D197C22×C412yes—
135C4×D197#4112yes—
136#13096802—21no—
137#13096802C22×C412yes—
138Q8×C197—21no—
139Q8×C197—21no—
140Q8×C197C22×C412yes—
141Q8×C197C22×C412yes—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2411yescharacteristic
149C8×C197—21no—
150C8×C197C22×C412yes—
151C22×C197C22×D411yescharacteristic
152C2×D197—21no—
153C2×D197—21no—
154C2×D197—22no—
155C2×D197—41no—
156C2×D197—41no—
157C2×D197C4×D412yes—
158C4×C197—22no—
159C4×C197—22no—
160C4×C197—21no—
161C4×C197—21no—
162C4×C197—41no—
163C4×C197—41no—
164C4×C197C23×C411yescharacteristic
165C4×C197C2×#4111yescharacteristic
166Dic197—22no—
167Dic197—21no—
168Dic197—41no—
169Dic197—41no—
170Dic197C4×D412yes—
171Dic197#13712yescomplemented
172C2×C197C2×C4×D411yescharacteristic
173C2×C197C2×#13711yescharacteristic
174C2×C197#57511yescharacteristic
175D197—22no—
176D197—42no—
177C197C4×#13711yescharacteristic, complemented, Sylow
178C4×#137—1971nomaximal, Sylow
179C2×#137—1971no—
180C4×#41—1971no—
181C2×C4×Q8—1971no—
182#441—1971no—
183#441D19711yescharacteristic, complemented
184#440—1971no—
185#440—1971no—
186C4×Q16—1971no—
187C4×Q16—1971no—
188C4×SD16—1971no—
189C4×SD16—1971no—
190#428—1971no—
191#419—1971no—
192#417—1971no—
193C4×M16—1971no—
194C2×#41—1971no—
195C22×Q8—1971no—
196#137—1978no—
197C2×Q16—1971no—
198C2×Q16C2×D19711yescharacteristic
199C2×SD16—1971no—
200C2×SD16—1971no—
201C2×M16—1971no—
202C2×M16—1971no—
203C4×Q8—3941no—
204C4×Q8—3941no—
205C4×Q8—1971no—
206C4×Q8—1971no—
207C4×Q8—1971no—
208C4×Q8—1971no—
209C4×Q8C2×D19711yescharacteristic
210C4×Q8C2×D19711yescharacteristic
211C4×D4—3941no—
212C4×D4—3941no—
213C4×D4—1971no—
214C4×D4—1971no—
215#117—3941no—
216#117—1971no—
217C2×C4⋊C4—3941no—
218C2×C4⋊C4—1971no—
219C2×C42—3941no—
220C2×C42—3941no—
221C2×C42—1971no—
222#107—1971no—
223#107—1971no—
224#106—1971no—
225#106C2×D19711yescharacteristic
226#103—1971no—
227#103—1971no—
228#103—1971no—
229#103—1971no—
230#103C2×D19711yescharacteristic
231#103C2×D19711yescharacteristic
232#102—1971no—
233#102—1971no—
234#97—1971no—
235#97C2×D19711yescharacteristic
236C4×C8—1971no—
237C4×C8—1971no—
238#41—3942no—
239#41—1974no—
240C2×Q8—3941no—
241C2×Q8—3942no—
242C2×Q8—1974no—
243C2×Q8—1971no—
244C2×Q8C22×D19711yescharacteristic
245C2×Q8C22×D19711yescharacteristic
246C2×D4—3941no—
247C2×D4—1971no—
248C22×C4—3941no—
249C22×C4—3941no—
250C22×C4—3941no—
251C22×C4—3941no—
252C22×C4—3941no—
253C22×C4—1971no—
254C22×C4—1971no—
255C22×C4—1971no—
256Q16—1974no—
257Q16C4×D19714yescomplemented
258SD16—1974no—
259SD16—1974no—
260M16—3942no—
261M16—1974no—
262C2×C8—1971no—
263C2×C8—1971no—
264C2×C8C22×D19711yescharacteristic
265C2×C8C22×D19711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—3941no—
269C4⋊C4—3941no—
270C4⋊C4—3941no—
271C4⋊C4—3941no—
272C4⋊C4—3941no—
273C4⋊C4—1971no—
274C4⋊C4—1971no—
275C4⋊C4C22×D19711yescharacteristic
276C4⋊C4C22×D19711yescharacteristic
277C22⋊C4—3941no—
278C22⋊C4—3941no—
279C22⋊C4—3941no—
280C42—21no—
281C42—21no—
282C42—3941no—
283C42—3941no—
284C42—3941no—
285C42—3941no—
286C42—3941no—
287C42—1971no—
288C42—1971no—
289C42—1971no—
290C42C22×D19711yescharacteristic
291C23—3941no—
292C23—1971no—
293Q8—21no—
294Q8—21no—
295Q8—3944no—
296Q8—3941no—
297Q8—1972no—
298Q8C2×C4×D19712yes—
299Q8C2×C4×D19712yes—
300D4—3944no—
301D4—3941no—
302D4—1972no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—7882no—
308C2×C4—7881no—
309C2×C4—7882no—
310C2×C4—7882no—
311C2×C4—3944no—
312C2×C4—3944no—
313C2×C4—3942no—
314C2×C4—3941no—
315C2×C4—3941no—
316C2×C4—3941no—
317C2×C4—1971no—
318C2×C4—1971no—
319C2×C4—1971no—
320C2×C4—1972no—
321C2×C4—1972no—
322C2×C4D4×D19711yescharacteristic
323C2×C4D4×D19711yescharacteristic
324C2×C4C23×D19711yescharacteristic
325C8—21no—
326C8—3941no—
327C8—1972no—
328C8C2×C4×D19712yes—
329C22—7881no—
330C22—7881no—
331C22—3942no—
332C22—3941no—
333C22—3941no—
334C22—1972no—
335C22C2×D4×D19711yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—7881no—
343C4—7881no—
344C4—3941no—
345C4—3942no—
346C4—1972no—
347C4—1972no—
348C4C22×C4×D19711yescharacteristic
349C4#41×D19711yescharacteristic
350C2—7882no—
351C2—3942no—
352C2C4×D4×D19711yescharacteristic
353C2#137×D19711yescharacteristic
354C2#2716187811yescharacteristic
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
hypocenterC197C4×#137
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. C4×Q8×D197C2semidirect
4. #27161184C2semidirect
5. #27161181C2semidirect
6. #27161177C2semidirect
7. Q16×Dic197C2semidirect
8. #27161171C2semidirect
9. #27161166C2semidirect
10. #27161164C2semidirect
11. #27161118C2semidirect
12. #27161086C2semidirect
14. #27161031C2semidirect
15. #441×C197C2semidirect
18. #20263358C4semidirect
65. Q8×D197C2×C4semidirect
76. #15691348C2×C4semidirect
77. #15691347C2×C4semidirect
78. #15691345C2×C4semidirect
80. #15691341C2×C4semidirect
103. Q16×C197C2×C4semidirect
171. Dic197#137semidirect
177. C197C4×#137semidirect
183. #441D197semidirect
257. Q16C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633582
C4×#156913592
C4×Q8×D1972
#271611842
#271611812
#271611772
Q16×Dic1972
#271611712
#271611662
#271611642
#271611182
#271610862
#271610792
#271610312
#441×C1972
C4×#137197