Finite Group Databasebeta

← #38751539

Subgroups — #38751539

Order 25216.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263353C211yesmaximal, characteristic
2#27161561C211yesmaximal, characteristic, complemented
3#27161551C211yesmaximal, characteristic, complemented
4#27161190C211yesmaximal, characteristic, complemented
5#27161189C211yesmaximal, characteristic, complemented
6#27161180C211yesmaximal, characteristic, complemented
7#27161177C211yesmaximal, characteristic, complemented
8#27161164C211yesmaximal, characteristic, complemented
9C4×D1576C211yesmaximal, characteristic, complemented
10#27161159C211yesmaximal, characteristic, complemented
11#27161118C211yesmaximal, characteristic, complemented
12#27161074C211yesmaximal, characteristic, complemented
13#102×D197C211yesmaximal, characteristic, complemented
14#27161033C211yesmaximal, characteristic, complemented
15C4×SD16×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×D4×D197C2211yescharacteristic
18#20263353C418yescomplemented
19C2×#15691345C2211yescharacteristic
20C2×D1576C2211yescharacteristic
21C2×#15691343C2211yescharacteristic
22C2×#15691341C2211yescharacteristic
23C2×C8×D197C2211yescharacteristic
24Q8×Dic197C2211yescharacteristic, complemented
25C4×D788—21no—
26C4×D788C2211yescharacteristic
27C4×D788C2211yescharacteristic, complemented
28D4×Dic197C2211yescharacteristic
29#20263288—21no—
30#20263288C2211yescharacteristic
31#20263288C2211yescharacteristic, complemented
32C4×C22⋊D197—21no—
33#20263286—21no—
34#20263284C2211yescharacteristic
35#20263283—21no—
36#20263283C2211yescharacteristic
37C4⋊C4×D197—21no—
38C4⋊C4×D197C2211yescharacteristic
39C2×#15691335—21no—
40C22⋊C4×D197—21no—
41#20263260C2211yescharacteristic, complemented
42#20263259C2211yescharacteristic, complemented
43#20263257C2211yescharacteristic, complemented
44#20263250C2211yescharacteristic, complemented
45#20263248C2211yescharacteristic
46#20263248C2211yescharacteristic, complemented
47#20263247C2211yescharacteristic
48#20263246C2211yescharacteristic
49#20263246C2211yescharacteristic, complemented
50#20263237C2211yescharacteristic, complemented
51#20263236C2211yescharacteristic, complemented
52C8×Dic197C2211yescharacteristic, complemented
53C2×SD16×C197C2211yescharacteristic
54C4×Q8×C197C2211yescharacteristic, complemented
55C4×D4×C197C2211yescharacteristic
56#106×C197C2211yescharacteristic, complemented
57#103×C197C2211yescharacteristic, complemented
58#102×C197C2211yescharacteristic
59C4×C8×C197C2211yescharacteristic, complemented
60C23×D197—21no—
61#15691359—22no—
62#15691359C2×C414yescomplemented
63C2×D788—21no—
64C2×D788C2311yescharacteristic
65C2×D788C2311yescharacteristic
66C2×C22⋊D197—21no—
67D4×D197—22no—
68D4×D197C2×C414yescomplemented
69C22×Dic197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197—21no—
73C2×C4×D197—21no—
74C2×C4×D197—21no—
75C2×C4×D197—21no—
76C2×C4×D197C2311yescharacteristic
77#15691345C2×C414yescomplemented
78D1576C2×C414yescomplemented
79#15691343C2×C414yescomplemented
80#15691341C2×C414yescomplemented
81C2×#13096802C2311yescharacteristic
82C2×#13096802C2311yescharacteristic
83C8×D197—22no—
84C4⋊Dic197—21no—
85C4⋊Dic197C2311yescharacteristic
86C4⋊Dic197C2311yescharacteristic
87#15691337—21no—
88#15691337D411yescharacteristic
89#15691337D411yescharacteristic
90C22⋊Dic197—21no—
91#15691335—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96#15691335D411yescharacteristic
97#15691335D411yescharacteristic
98C4×Dic197—21no—
99C4×Dic197—21no—
100C4×Dic197C2311yescharacteristic
101C2×Q8×C197C2311yescharacteristic
102C2×D4×C197C2311yescharacteristic
103C22×C4×C197—21no—
104SD16×C197C2×C414yescomplemented
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197—21no—
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C22⋊C4×C197—21no—
111C42×C197—21no—
112C42×C197C2311yescharacteristic
113C22×D197—21no—
114C22×D197—21no—
115C22×D197—24no—
116C22×D197—81no—
117C22×D197—81no—
118C22×D197C2×D411yescharacteristic
119D788—24no—
120D788—21no—
121D788—21no—
122D788C22×C412yes—
123D788C22×C412yes—
124C22⋊D197—24no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C2×Dic197—81no—
128C2×Dic197C2×D411yescharacteristic
129C2×Dic197C2×D411yescharacteristic
130C2×Dic197C2×D411yescharacteristic
131C4×D197—24no—
132C4×D197—81no—
133C4×D197—81no—
134C4×D197—81no—
135C4×D197—81no—
136C4×D197—81no—
137C4×D197C22×C412yes—
138C4×D197#4112yes—
139#13096802—21no—
140#13096802C22×C412yes—
141C23×C197—21no—
142Q8×C197—21no—
143Q8×C197C22×C412yes—
144D4×C197—21no—
145D4×C197C22×C412yes—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—81no—
150C2×C4×C197C2×D411yescharacteristic
151C2×C4×C197C2×D411yescharacteristic
152C2×C4×C197C2411yescharacteristic
153C8×C197—21no—
154C8×C197C22×C412yes—
155C22×C197—22no—
156C22×C197—41no—
157C22×C197—41no—
158C22×C197C22×D411yescharacteristic
159C2×D197—21no—
160C2×D197—21no—
161C2×D197—22no—
162C2×D197—22no—
163C2×D197—41no—
164C2×D197—41no—
165C2×D197—41no—
166C2×D197—41no—
167C2×D197—84no—
168C2×D197C4×D412yes—
169C4×C197—22no—
170C4×C197—21no—
171C4×C197—21no—
172C4×C197—41no—
173C4×C197—41no—
174C4×C197C23×C411yescharacteristic
175C4×C197C2×#4111yescharacteristic
176Dic197—21no—
177Dic197—21no—
178Dic197—41no—
179Dic197—41no—
180Dic197C4×D412yes—
181C2×C197—42no—
182C2×C197C2×C4×D411yescharacteristic
183C2×C197#57611yescharacteristic
184C2×C197#57711yescharacteristic
185D197—22no—
186D197—42no—
187D197—42no—
188C197#294611yescharacteristic, complemented, Sylow
189#2946—1971nomaximal, Sylow
190C2×AGL(1,ℤ/8)—1971no—
191#520—1971no—
192#518—1971no—
193#442—1971no—
194#442—1971no—
195#440—1971no—
196#440—1971no—
197C4×SD16—1971no—
198C4×SD16D19711yescharacteristic, complemented
199C4×D8—1971no—
200C4×D8—1971no—
201#428—1971no—
202#416—1971no—
203C2×#102—1971no—
204#405—1971no—
205C2×#41—1971no—
206C22×D4—1971no—
207AGL(1,ℤ/8)—1978no—
208C2×SD16—1971no—
209C2×SD16C2×D19711yescharacteristic
210C2×D8—1971no—
211C2×D8—1971no—
212C2×M16—1971no—
213C22×C8—1971no—
214C4×Q8—1971no—
215C4×Q8C2×D19711yescharacteristic, complemented
216C4×D4—3941no—
217C4×D4—3941no—
218C4×D4—3941no—
219C4×D4—3941no—
220C4×D4—1971no—
221C4×D4—1971no—
222C4×D4—1971no—
223C4×D4—1971no—
224C4×D4—1971no—
225C4×D4C2×D19711yescharacteristic
226#117—3941no—
227#117—1971no—
228#117—1971no—
229C2×C4⋊C4—3941no—
230C2×C4⋊C4—1971no—
231C2×C22⋊C4—3941no—
232C2×C22⋊C4—3941no—
233#107—1971no—
234#107—1971no—
235#106—1971no—
236#106C2×D19711yescharacteristic, complemented
237#103—1971no—
238#103C2×D19711yescharacteristic, complemented
239#102—1971no—
240#102—1971no—
241#102—1971no—
242#102—1971no—
243#102—1971no—
244#102C2×D19711yescharacteristic
245#97—1971no—
246#97—1971no—
247C4×C8—1971no—
248C4×C8C2×D19711yescharacteristic, complemented
249C24—3941no—
250#41—3942no—
251#41—1974no—
252C2×Q8C22×D19711yescharacteristic
253C2×D4—3941no—
254C2×D4—3942no—
255C2×D4—3941no—
256C2×D4—1974no—
257C2×D4—1971no—
258C2×D4—1971no—
259C2×D4C22×D19711yescharacteristic
260C22×C4—21no—
261C22×C4—3941no—
262C22×C4—3941no—
263C22×C4—3941no—
264C22×C4—3941no—
265C22×C4—3941no—
266C22×C4—3941no—
267C22×C4—3941no—
268C22×C4—1971no—
269SD16—1974no—
270SD16C4×D19714yescomplemented
271D8—1974no—
272D8—1974no—
273M16—1974no—
274C2×C8—3942no—
275C2×C8—1971no—
276C2×C8—1971no—
277C2×C8C22×D19711yescharacteristic
278C2×C8C22×D19711yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—3941no—
281C4⋊C4—3941no—
282C4⋊C4—1971no—
283C4⋊C4—1971no—
284C4⋊C4—1971no—
285C4⋊C4—1971no—
286C4⋊C4C22×D19711yescharacteristic
287C4⋊C4C22×D19711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—3941no—
290C22⋊C4—3941no—
291C22⋊C4—3941no—
292C22⋊C4—3941no—
293C22⋊C4—3941no—
294C22⋊C4—3941no—
295C22⋊C4—1971no—
296C22⋊C4—1971no—
297C42—21no—
298C42—3941no—
299C42—3941no—
300C42—1971no—
301C42C22×D19711yescharacteristic
302C23—21no—
303C23—15761no—
304C23—15761no—
305C23—3944no—
306C23—3941no—
307C23—3941no—
308C23—1971no—
309Q8—21no—
310Q8C2×C4×D19712yes—
311D4—21no—
312D4—3944no—
313D4—3944no—
314D4—3941no—
315D4—3941no—
316D4—1972no—
317D4—1972no—
318D4C2×C4×D19712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—15761no—
324C2×C4—15761no—
325C2×C4—15761no—
326C2×C4—15761no—
327C2×C4—15761no—
328C2×C4—15761no—
329C2×C4—3944no—
330C2×C4—3941no—
331C2×C4—3941no—
332C2×C4—1971no—
333C2×C4—1971no—
334C2×C4—1971no—
335C2×C4—1972no—
336C2×C4—1972no—
337C2×C4D4×D19711yescharacteristic
338C2×C4D4×D19711yescharacteristic
339C2×C4C23×D19711yescharacteristic
340C8—21no—
341C8—3941no—
342C8—1972no—
343C8C2×C4×D19712yes—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—7881no—
348C22—7881no—
349C22—7881no—
350C22—7881no—
351C22—15764no—
352C22—3942no—
353C22—3942no—
354C22—3941no—
355C22—3941no—
356C22—1972no—
357C22C2×D4×D19711yescharacteristic
358C4—21no—
359C4—21no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—7881no—
364C4—7881no—
365C4—3941no—
366C4—3941no—
367C4—1972no—
368C4C22×C4×D19711yescharacteristic
369C4#41×D19711yescharacteristic
370C2—42no—
371C2—7882no—
372C2—7882no—
373C2—3942no—
374C2C4×D4×D19711yescharacteristic
375C2#2716188111yescharacteristic
376C2#2716188511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutatorC4×C197C23×C4
FrattiniC2×C4C23×D197
FittingC4×SD16×C197C2
socleC22×C197C22×D4
prime radicalC2×#20263353C2
cosocleC2×C4×C197C24
hypercenterC4×SD16D197
hypocenterC197#2946
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #27161561C2semidirect
3. #27161551C2semidirect
4. #27161190C2semidirect
5. #27161189C2semidirect
6. #27161180C2semidirect
7. #27161177C2semidirect
8. #27161164C2semidirect
9. C4×D1576C2semidirect
10. #27161159C2semidirect
11. #27161118C2semidirect
12. #27161074C2semidirect
13. #102×D197C2semidirect
14. #27161033C2semidirect
15. C4×SD16×C197C2semidirect
18. #20263353C4semidirect
24. Q8×Dic197C22semidirect
27. C4×D788C22semidirect
31. #20263288C22semidirect
41. #20263260C22semidirect
42. #20263259C22semidirect
43. #20263257C22semidirect
44. #20263250C22semidirect
46. #20263248C22semidirect
49. #20263246C22semidirect
50. #20263237C22semidirect
51. #20263236C22semidirect
52. C8×Dic197C22semidirect
54. C4×Q8×C197C22semidirect
56. #106×C197C22semidirect
57. #103×C197C22semidirect
59. C4×C8×C197C22semidirect
62. #15691359C2×C4semidirect
68. D4×D197C2×C4semidirect
77. #15691345C2×C4semidirect
78. D1576C2×C4semidirect
79. #15691343C2×C4semidirect
80. #15691341C2×C4semidirect
104. SD16×C197C2×C4semidirect
188. C197#2946semidirect
198. C4×SD16D197semidirect
215. C4×Q8C2×D197semidirect
236. #106C2×D197semidirect
238. #103C2×D197semidirect
248. C4×C8C2×D197semidirect
270. SD16C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633532
#271615612
#271615512
#271611902
#271611892
#271611802
#271611772
#271611642
C4×D15762
#271611592
#271611182
#271610742
#102×D1972
#271610332
C4×SD16×C1972
#2946197