Finite Group Databasebeta

← #38751523

Subgroups — #38751523

Order 25216.

548 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263350C211yesmaximal, characteristic
2C4×#15691357C211yesmaximal, characteristic, complemented
3C4×D4×D197C211yesmaximal, characteristic, complemented
4#27161189C211yesmaximal, characteristic, complemented
5#27161181C211yesmaximal, characteristic, complemented
6#27161179C211yesmaximal, characteristic, complemented
7#27161170C211yesmaximal, characteristic, complemented
8#27161166C211yesmaximal, characteristic, complemented
9D8×Dic197C211yesmaximal, characteristic, complemented
10#27161159C211yesmaximal, characteristic, complemented
11#27161118C211yesmaximal, characteristic, complemented
12#27161082C211yesmaximal, characteristic, complemented
13#27161077C211yesmaximal, characteristic, complemented
14#27161031C211yesmaximal, characteristic, complemented
15#442×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic
18C22×C4×D197—21no—
19#20263350C418yescomplemented
20C2×#15691347C2211yescharacteristic
21C2×#15691346C2211yescharacteristic
22C2×#15691343C2211yescharacteristic
23C2×#15691341C2211yescharacteristic
24C2×#15691341C2211yescharacteristic
25C4×Dic394C2211yescharacteristic, complemented
26#20263291C2211yescharacteristic, complemented
27C4×D788C2211yescharacteristic
28D4×Dic197C2211yescharacteristic
29D4×Dic197C2211yescharacteristic, complemented
30#20263288C2211yescharacteristic
31C4×C22⋊D197—21no—
32C4×C22⋊D197—21no—
33#20263286—21no—
34#20263286—21no—
35#20263284C2211yescharacteristic
36#20263282—21no—
37C4⋊C4×D197C2211yescharacteristic
38C22⋊C4×D197—21no—
39C2×C4×Dic197—21no—
40C42×D197C2211yescharacteristic
41#20263260C2211yescharacteristic, complemented
42#20263259C2211yescharacteristic, complemented
43#20263257C2211yescharacteristic, complemented
44#20263251C2211yescharacteristic, complemented
45#20263249C2211yescharacteristic, complemented
46#20263248C2211yescharacteristic
47#20263247C2211yescharacteristic
48#20263247C2211yescharacteristic, complemented
49#20263246C2211yescharacteristic
50#20263237C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52C8×Dic197C2211yescharacteristic, complemented
53C2×D8×C197C2211yescharacteristic
54C4×D4×C197C2211yescharacteristic
55C4×D4×C197C2211yescharacteristic, complemented
56#106×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic
58#102×C197C2211yescharacteristic, complemented
59#97×C197C2211yescharacteristic, complemented
60C23×D197—21no—
61#15691357—22no—
62#15691357C2×C414yescomplemented
63C2×Dic394C2311yescharacteristic
64C2×D788C2311yescharacteristic
65C2×C22⋊D197—21no—
66C2×C22⋊D197—21no—
67D4×D197—22no—
68D4×D197C2×C414yescomplemented
69C22×Dic197—21no—
70C22×Dic197—21no—
71C22×Dic197—21no—
72C2×C4×D197—21no—
73C2×C4×D197—21no—
74C2×C4×D197—21no—
75C2×C4×D197—44no—
76C2×C4×D197D411yescharacteristic
77C2×C4×D197D411yescharacteristic
78C2×C4×D197C2311yescharacteristic
79#15691347C2×C414yescomplemented
80#15691346C2×C414yescomplemented
81#15691343C2×C414yescomplemented
82#15691341—22no—
83#15691341C2×C414yescomplemented
84C2×#13096802C2311yescharacteristic
85C2×#13096802C2311yescharacteristic
86C4⋊Dic197C2311yescharacteristic
87C4⋊Dic197C2311yescharacteristic
88#15691337—21no—
89#15691337—21no—
90#15691337—21no—
91C22⋊Dic197—21no—
92C22⋊Dic197—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96C4×Dic197—21no—
97C4×Dic197—21no—
98C4×Dic197—21no—
99C4×Dic197D411yescharacteristic
100C4×Dic197D411yescharacteristic
101C4×Dic197C2311yescharacteristic
102C2×D4×C197C2311yescharacteristic
103C2×D4×C197C2311yescharacteristic
104C22×C4×C197—21no—
105C22×C4×C197—21no—
106D8×C197C2×C414yescomplemented
107C2×C8×C197C2311yescharacteristic
108C2×C8×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C4⋊C4×C197C2311yescharacteristic
111C22⋊C4×C197—21no—
112C22⋊C4×C197—21no—
113C42×C197C2311yescharacteristic
114C22×D197—21no—
115C22×D197—24no—
116C22×D197—42no—
117C22×D197—42no—
118C22×D197C2×D411yescharacteristic
119Dic394—21no—
120Dic394C22×C412yescomplemented
121D788—21no—
122D788C22×C412yes—
123C22⋊D197—24no—
124C22⋊D197—24no—
125C2×Dic197—21no—
126C2×Dic197—24no—
127C2×Dic197—21no—
128C2×Dic197—21no—
129C2×Dic197—42no—
130C2×Dic197—42no—
131C2×Dic197C2×D411yescharacteristic
132C2×Dic197C2×D411yescharacteristic
133C2×Dic197C2×D411yescharacteristic
134C4×D197—22no—
135C4×D197—41no—
136C4×D197—42no—
137C4×D197—42no—
138C4×D197—42no—
139C4×D197C22×C412yes—
140C4×D197#4112yes—
141#13096802—21no—
142#13096802C22×C412yescomplemented
143C23×C197—21no—
144C23×C197—21no—
145D4×C197—21no—
146D4×C197—21no—
147D4×C197C22×C412yes—
148D4×C197C22×C412yescomplemented
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—42no—
152C2×C4×C197—42no—
153C2×C4×C197C2×D411yescharacteristic
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2411yescharacteristic
156C8×C197—21no—
157C8×C197C22×C412yescomplemented
158C22×C197—22no—
159C22×C197—22no—
160C22×C197—41no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197—41no—
164C22×C197C22×D411yescharacteristic
165C2×D197—21no—
166C2×D197—21no—
167C2×D197—22no—
168C2×D197—48no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197C4×D412yes—
172C4×C197—21no—
173C4×C197—21no—
174C4×C197—41no—
175C4×C197—41no—
176C4×C197C23×C411yescharacteristic
177C4×C197C2×#4111yescharacteristic
178Dic197—22no—
179Dic197—21no—
180Dic197—41no—
181Dic197—41no—
182Dic197C4×D412yes—
183Dic197AGL(1,ℤ/8)12yescomplemented
184C2×C197—42no—
185C2×C197—42no—
186C2×C197C2×C4×D411yescharacteristic
187C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C197#57511yescharacteristic
189D197—22no—
190D197—42no—
191C197C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
192C4×AGL(1,ℤ/8)—1971nomaximal, Sylow
193C2×AGL(1,ℤ/8)—1971no—
194C4×#41—1971no—
195C2×C4×D4—1971no—
196#442—1971no—
197#442D19711yescharacteristic, complemented
198#440—1971no—
199#440—1971no—
200C4×SD16—1971no—
201C4×SD16—1971no—
202C4×D8—1971no—
203C4×D8—1971no—
204#428—1971no—
205#418—1971no—
206#417—1971no—
207C4×M16—1971no—
208C2×#41—1971no—
209C22×D4—1971no—
210C23×C4—3941no—
211AGL(1,ℤ/8)—1978no—
212C2×SD16—1971no—
213C2×SD16—1971no—
214C2×D8—1971no—
215C2×D8C2×D19711yescharacteristic
216C2×M16—1971no—
217C2×M16—1971no—
218C4×Q8—1971no—
219C4×Q8—1971no—
220C4×D4—3941no—
221C4×D4—3941no—
222C4×D4—3941no—
223C4×D4—3941no—
224C4×D4—1971no—
225C4×D4—1971no—
226C4×D4—1971no—
227C4×D4—1971no—
228C4×D4C2×D19711yescharacteristic
229C4×D4C2×D19711yescharacteristic, complemented
230#117—3941no—
231#117—1971no—
232C2×C4⋊C4—1971no—
233C2×C22⋊C4—3941no—
234C2×C42—3941no—
235C2×C42—1971no—
236#107—1971no—
237#107—1971no—
238#106—1971no—
239#106C2×D19711yescharacteristic, complemented
240#103—1971no—
241#103—1971no—
242#102—1971no—
243#102—1971no—
244#102—1971no—
245#102—1971no—
246#102C2×D19711yescharacteristic
247#102C2×D19711yescharacteristic, complemented
248#97—1971no—
249#97C2×D19711yescharacteristic, complemented
250C4×C8—1971no—
251C4×C8—1971no—
252C24—3941no—
253#41—3942no—
254#41—1974no—
255C2×Q8—1971no—
256C2×D4—3941no—
257C2×D4—3942no—
258C2×D4—3941no—
259C2×D4—1974no—
260C2×D4—1971no—
261C2×D4C22×D19711yescharacteristic
262C2×D4C22×D19711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—7884no—
266C22×C4—3941no—
267C22×C4—3941no—
268C22×C4—3941no—
269C22×C4—3941no—
270C22×C4—3941no—
271C22×C4—3941no—
272C22×C4—1971no—
273C22×C4—1971no—
274C22×C4—1971no—
275SD16—1974no—
276SD16—1974no—
277D8—1974no—
278D8C4×D19714yescomplemented
279M16—3942no—
280M16—1974no—
281C2×C8—1971no—
282C2×C8—1971no—
283C2×C8C22×D19711yescharacteristic
284C2×C8C22×D19711yescharacteristic
285C4⋊C4—3941no—
286C4⋊C4—3941no—
287C4⋊C4—3941no—
288C4⋊C4—1971no—
289C4⋊C4—1971no—
290C4⋊C4C22×D19711yescharacteristic
291C4⋊C4C22×D19711yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—3941no—
295C22⋊C4—3941no—
296C22⋊C4—3941no—
297C22⋊C4—3941no—
298C22⋊C4—3941no—
299C42—3941no—
300C42—3941no—
301C42—3941no—
302C42—1971no—
303C42—1971no—
304C42—1971no—
305C42C22×D19711yescharacteristic
306C23—21no—
307C23—21no—
308C23—7882no—
309C23—7882no—
310C23—3944no—
311C23—3941no—
312C23—1971no—
313Q8—3941no—
314Q8—1972no—
315D4—21no—
316D4—21no—
317D4—3944no—
318D4—3944no—
319D4—3941no—
320D4—1972no—
321D4C2×C4×D19712yes—
322D4C2×C4×D19712yescomplemented
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—7882no—
328C2×C4—7881no—
329C2×C4—7882no—
330C2×C4—7882no—
331C2×C4—7882no—
332C2×C4—7882no—
333C2×C4—3942no—
334C2×C4—3941no—
335C2×C4—3941no—
336C2×C4—3941no—
337C2×C4—3944no—
338C2×C4—1971no—
339C2×C4—1971no—
340C2×C4—1971no—
341C2×C4—1972no—
342C2×C4—1972no—
343C2×C4D4×D19711yescharacteristic
344C2×C4D4×D19711yescharacteristic
345C2×C4C23×D19711yescharacteristic
346C8—21no—
347C8—3941no—
348C8—1972no—
349C8C2×C4×D19712yescomplemented
350C22—22no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—7888no—
357C22—7881no—
358C22—7881no—
359C22—3942no—
360C22—3941no—
361C22—3941no—
362C22—1972no—
363C22C2×D4×D19711yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—7881no—
369C4—7881no—
370C4—3941no—
371C4—3942no—
372C4—1972no—
373C4—1972no—
374C4C22×C4×D19711yescharacteristic
375C4#41×D19711yescharacteristic
376C2—42no—
377C2—42no—
378C2—7882no—
379C2—3942no—
380C2C4×D4×D19711yescharacteristic
381C2AGL(1,ℤ/8)×D19711yescharacteristic
382C2#2716187511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutatorC4×C197C23×C4
FrattiniC2×C4C23×D197
Fitting#442×C197C2
socleC22×C197C22×D4
prime radicalC2×#20263350C2
cosocleC2×C4×C197C24
hypercenter#442D197
hypocenterC197C4×AGL(1,ℤ/8)
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. C4×#15691357C2semidirect
3. C4×D4×D197C2semidirect
4. #27161189C2semidirect
5. #27161181C2semidirect
6. #27161179C2semidirect
7. #27161170C2semidirect
8. #27161166C2semidirect
9. D8×Dic197C2semidirect
10. #27161159C2semidirect
11. #27161118C2semidirect
12. #27161082C2semidirect
13. #27161077C2semidirect
14. #27161031C2semidirect
15. #442×C197C2semidirect
19. #20263350C4semidirect
25. C4×Dic394C22semidirect
26. #20263291C22semidirect
29. D4×Dic197C22semidirect
41. #20263260C22semidirect
42. #20263259C22semidirect
43. #20263257C22semidirect
44. #20263251C22semidirect
45. #20263249C22semidirect
48. #20263247C22semidirect
50. #20263237C22semidirect
51. C8×Dic197C22semidirect
52. C8×Dic197C22semidirect
55. C4×D4×C197C22semidirect
56. #106×C197C22semidirect
58. #102×C197C22semidirect
59. #97×C197C22semidirect
62. #15691357C2×C4semidirect
68. D4×D197C2×C4semidirect
79. #15691347C2×C4semidirect
80. #15691346C2×C4semidirect
81. #15691343C2×C4semidirect
83. #15691341C2×C4semidirect
106. D8×C197C2×C4semidirect
120. Dic394C22×C4semidirect
142. #13096802C22×C4semidirect
148. D4×C197C22×C4semidirect
157. C8×C197C22×C4semidirect
183. Dic197AGL(1,ℤ/8)semidirect
191. C197C4×AGL(1,ℤ/8)semidirect
197. #442D197semidirect
229. C4×D4C2×D197semidirect
239. #106C2×D197semidirect
247. #102C2×D197semidirect
249. #97C2×D197semidirect
278. D8C4×D197semidirect
322. D4C2×C4×D197semidirect
349. C8C2×C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
C4×#156913572
C4×D4×D1972
#271611892
#271611812
#271611792
#271611702
#271611662
D8×Dic1972
#271611592
#271611182
#271610822
#271610772
#271610312
#442×C1972
C4×AGL(1,ℤ/8)197