Finite Group Databasebeta

← #38754253

Subgroups — #38754253

Order 25216.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263351C211yesmaximal, characteristic, complemented
2C2×#20263346C211yesmaximal, characteristic, complemented
3#27161725C211yesmaximal, characteristic, complemented
4#27161466C211yesmaximal, characteristic, complemented
5#27161323C211yesmaximal, characteristic, complemented
6#27161313C211yesmaximal, characteristic, complemented
7#27161301C211yesmaximal, characteristic, complemented
8#27161271C211yesmaximal, characteristic, complemented
9#27161241C211yesmaximal, characteristic, complemented
10#27161224C211yesmaximal, characteristic, complemented
11#27161222C211yesmaximal, characteristic, complemented
12#27161212C211yesmaximal, characteristic, complemented
13#27161177C211yesmaximal, characteristic, complemented
14#27161151C211yesmaximal, characteristic, complemented
15#550×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×D788C2211yescharacteristic
18#20263351—24no—
19#20263346—24no—
20C2×#15691348C2211yescharacteristic, complemented
21C2×#15691346C2211yescharacteristic, complemented
22C2×#15691346C2211yescharacteristic, complemented
23C2×#15691345C2211yescharacteristic
24C2×#15691345C2211yescharacteristic
25C2×#15691343C2211yescharacteristic
26C2×#15691343C2211yescharacteristic
27C2×#15691343C2211yescharacteristic
28C2×#15691343C2211yescharacteristic
29C2×#15691342C2211yescharacteristic
30C22×#13096802C2211yescharacteristic, complemented
31#20263314C2211yescharacteristic
32#20263311—21no—
33#20263305C2211yescharacteristic
34#20263300C2211yescharacteristic
35#20263299—21no—
36#20263294—21no—
37C4×D788C2211yescharacteristic
38C22⋊C4×D197—21no—
39#20263259C2211yescharacteristic, complemented
40C4⋊#13096802C2211yescharacteristic, complemented
41#20263250C2211yescharacteristic, complemented
42#20263249C2211yescharacteristic, complemented
43#20263247C2211yescharacteristic, complemented
44#20263247C2211yescharacteristic, complemented
45#20263246C2211yescharacteristic
46#20263246C2211yescharacteristic
47C22⋊#13096802C2211yescharacteristic
48C22⋊#13096802C2211yescharacteristic, complemented
49#20263237C2211yescharacteristic, complemented
50C2×#41×C197C2211yescharacteristic
51C2×#41×C197C2211yescharacteristic, complemented
52#127×C197—21no—
53#127×C197C2211yescharacteristic, complemented
54#121×C197—21no—
55#121×C197—21no—
56#121×C197C2211yescharacteristic
57#121×C197C2211yescharacteristic, complemented
58C4×Q8×C197C2211yescharacteristic, complemented
59C4×D4×C197—21no—
60C4×D4×C197C2211yescharacteristic
61C23×D197—21no—
62#15691358—22no—
63#15691358—41no—
64C2×Dic394C2311yescharacteristic
65C2×D788—22no—
66C2×D788—21no—
67C2×D788—41no—
68C2×D788C2311yescharacteristic
69C2×D788C2311yescharacteristic
70C2×C22⋊D197—21no—
71C2×C22⋊D197—21no—
72C2×C4×D197—21no—
73C2×C4×D197—21no—
74#15691348—22no—
75#15691346—22no—
76#15691346—41no—
77#15691345—22no—
78#15691345D414yescomplemented
79#15691343—22no—
80#15691343—22no—
81#15691343—22no—
82#15691343—41no—
83#15691342—22no—
84C2×#13096802—22no—
85C2×#13096802C2311yescharacteristic
86C2×#13096802C2311yescharacteristic
87C2×#13096802C2311yescharacteristic
88C2×#13096802C2311yescharacteristic
89C4⋊Dic197C2311yescharacteristic
90#15691337—21no—
91C22⋊Dic197—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96#41×C197—22no—
97#41×C197—22no—
98#41×C197—41no—
99#41×C197—41no—
100C2×Q8×C197C2311yescharacteristic
101C2×D4×C197—21no—
102C2×D4×C197—21no—
103C2×D4×C197—21no—
104C2×D4×C197—21no—
105C2×D4×C197—21no—
106C2×D4×C197—21no—
107C2×D4×C197D411yescharacteristic
108C2×D4×C197C2311yescharacteristic
109C2×D4×C197C2311yescharacteristic
110C22×C4×C197—21no—
111C22×C4×C197—21no—
112C22×C4×C197C2311yescharacteristic
113C22×C4×C197C2311yescharacteristic
114C4⋊C4×C197—21no—
115C4⋊C4×C197D411yescharacteristic
116C4⋊C4×C197C2311yescharacteristic
117C22⋊C4×C197—21no—
118C22⋊C4×C197—21no—
119C22⋊C4×C197D411yescharacteristic
120C22⋊C4×C197D411yescharacteristic
121C42×C197—21no—
122C42×C197C2311yescharacteristic
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—42no—
126C22×D197—42no—
127C22×D197—81no—
128Dic394—21no—
129Dic394—21no—
130D788—21no—
131D788—21no—
132D788—21no—
133D788—42no—
134D788C2×D412yes—
135C22⋊D197—42no—
136C22⋊D197—81no—
137C2×Dic197—21no—
138C2×Dic197—21no—
139C4×D197—42no—
140C4×D197—81no—
141#13096802—21no—
142#13096802—21no—
143#13096802—21no—
144#13096802C2×D412yes—
145C23×C197—21no—
146C23×C197—21no—
147C23×C197C2×D411yescharacteristic
148C23×C197C2×D411yescharacteristic
149Q8×C197—21no—
150Q8×C197C2×D412yes—
151D4×C197—21no—
152D4×C197—21no—
153D4×C197—21no—
154D4×C197—21no—
155D4×C197—41no—
156D4×C197—42no—
157D4×C197—42no—
158D4×C197—42no—
159D4×C197—81no—
160D4×C197—81no—
161D4×C197—81no—
162C2×C4×C197—21no—
163C2×C4×C197—21no—
164C2×C4×C197—21no—
165C2×C4×C197—21no—
166C2×C4×C197—21no—
167C2×C4×C197—21no—
168C2×C4×C197—42no—
169C2×C4×C197—42no—
170C2×C4×C197C2×D411yescharacteristic
171C2×C4×C197C2×D411yescharacteristic
172C2×C4×C197C2×D411yescharacteristic
173C2×C4×C197C2×D411yescharacteristic
174C2×C4×C197C2411yescharacteristic
175C22×C197—21no—
176C22×C197—21no—
177C22×C197—21no—
178C22×C197—21no—
179C22×C197—21no—
180C22×C197—21no—
181C22×C197—41no—
182C22×C197—41no—
183C22×C197—41no—
184C22×C197—41no—
185C22×C197—41no—
186C22×C197—41no—
187C22×C197C22×D411yescharacteristic
188C2×D197—22no—
189C2×D197—41no—
190C2×D197—41no—
191C2×D197—41no—
192C2×D197—41no—
193C2×D197—41no—
194C2×D197—84no—
195C4×C197—21no—
196C4×C197—21no—
197C4×C197—22no—
198C4×C197—21no—
199C4×C197—21no—
200C4×C197—41no—
201C4×C197C22×D411yescharacteristic
202C4×C197H(5,𝔽2)11yescharacteristic
203Dic197—41no—
204Dic197—41no—
205C2×C197—41no—
206C2×C197—41no—
207C2×C197—81no—
208C2×C197—81no—
209C2×C197D4211yescharacteristic
210C2×C197#57511yescharacteristic
211C2×C197#57611yescharacteristic
212D197—42no—
213D197—81no—
214C197#326811yescharacteristic, complemented, Sylow
215#3268—1971nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1971no—
217C2×#135—1971no—
218#550D19711yescharacteristic, complemented
219#546—1971no—
220#496—1971no—
221#468—1971no—
222#466—1971no—
223#464—1971no—
224#459—1971no—
225#452—1971no—
226#449—1971no—
227#449—1971no—
228#447—1971no—
229#440—1971no—
230#435—1971no—
231C2×#41—1971no—
232C2×#41C2×D19711yescharacteristic
233C2×#41C2×D19711yescharacteristic, complemented
234C22×D4—1971no—
235AGL(1,ℤ/8)—3944no—
236#135—3944no—
237C2×Q16—1971no—
238C2×SD16—1971no—
239C2×SD16—1971no—
240C2×SD16—1971no—
241C2×SD16—1971no—
242C2×D8—1971no—
243C2×D8—1971no—
244C2×D8—1971no—
245C2×D8—1971no—
246C2×M16—1971no—
247C22×C8—1971no—
248#127—21no—
249#127C2×D19711yescharacteristic, complemented
250#124—1971no—
251#123—3941no—
252C22⋊Q8—1971no—
253#121—21no—
254#121—21no—
255#121—3941no—
256#121—1971no—
257#121C2×D19711yescharacteristic
258#121C2×D19711yescharacteristic, complemented
259#120—3941no—
260C4×Q8C2×D19711yescharacteristic, complemented
261C4×D4—21no—
262C4×D4—1971no—
263C4×D4C2×D19711yescharacteristic
264C2×C22⋊C4—3941no—
265#107—1971no—
266C4⋊C8—1971no—
267#103—1971no—
268#103—1971no—
269#102—1971no—
270#102—1971no—
271#102—1971no—
272#102—1971no—
273C22⋊C8—1971no—
274C22⋊C8—1971no—
275#97—1971no—
276C24—3941no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—7881no—
282#41—3942no—
283C2×Q8—1971no—
284C2×Q8C22×D19711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—7881no—
292C2×D4—3942no—
293C2×D4—3941no—
294C2×D4—3941no—
295C2×D4—3941no—
296C2×D4—1971no—
297C2×D4—1971no—
298C2×D4C22⋊D19711yescharacteristic
299C2×D4C22×D19711yescharacteristic
300C2×D4C22×D19711yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—3941no—
304C22×C4—3941no—
305C22×C4C22×D19711yescharacteristic
306C22×C4C22×D19711yescharacteristic
307Q16—3942no—
308SD16—7881no—
309SD16—3942no—
310SD16—3942no—
311SD16—1974no—
312D8—7881no—
313D8—3942no—
314D8—3942no—
315D8—3942no—
316M16—3942no—
317C2×C8—3942no—
318C2×C8—1971no—
319C2×C8—1971no—
320C2×C8—1971no—
321C2×C8—1971no—
322C4⋊C4—21no—
323C4⋊C4—3941no—
324C4⋊C4—1971no—
325C4⋊C4C22⋊D19711yescharacteristic
326C4⋊C4C22×D19711yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—3941no—
330C22⋊C4—3941no—
331C22⋊C4—3941no—
332C22⋊C4—3941no—
333C22⋊C4—3941no—
334C22⋊C4C22⋊D19711yescharacteristic
335C22⋊C4C22⋊D19711yescharacteristic
336C42—21no—
337C42C22×D19711yescharacteristic
338C23—21no—
339C23—21no—
340C23—7882no—
341C23—7882no—
342C23—15761no—
343C23—3941no—
344C23—3941no—
345C23C2×C22⋊D19711yescharacteristic
346C23C2×C22⋊D19711yescharacteristic
347Q8—21no—
348Q8—3941no—
349Q8—3941no—
350Q8D4×D19712yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—7882no—
363D4—7882no—
364D4—15761no—
365D4—3941no—
366D4—3941no—
367D4—3941no—
368D4—1972no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—7882no—
378C2×C4—15761no—
379C2×C4—3941no—
380C2×C4—3941no—
381C2×C4C2×C22⋊D19711yescharacteristic
382C2×C4C2×C22⋊D19711yescharacteristic
383C2×C4C2×C22⋊D19711yescharacteristic
384C2×C4C2×C22⋊D19711yescharacteristic
385C2×C4C23×D19711yescharacteristic
386C8—3941no—
387C8—3941no—
388C8—3941no—
389C8—1972no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—7881no—
403C22—7881no—
404C22—7881no—
405C22—7881no—
406C22—7881no—
407C22—15764no—
408C22—3942no—
409C22C22×C22⋊D19711yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—41no—
416C4—7881no—
417C4—7881no—
418C4C2×D4×D19711yescharacteristic
419C4#2026337111yescharacteristic
420C2—41no—
421C2—41no—
422C2—81no—
423C2—81no—
424C2—7882no—
425C2—15761no—
426C2D4×C22⋊D19711yescharacteristic
427C2#2716187711yescharacteristic
428C2#2716188211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#550×C197C2
socleC22×C197C22×D4
hypercenter#550D197
hypocenterC197#3268
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#20263351C2semidirect
2. C2×#20263346C2semidirect
3. #27161725C2semidirect
4. #27161466C2semidirect
5. #27161323C2semidirect
6. #27161313C2semidirect
7. #27161301C2semidirect
8. #27161271C2semidirect
9. #27161241C2semidirect
10. #27161224C2semidirect
11. #27161222C2semidirect
12. #27161212C2semidirect
13. #27161177C2semidirect
14. #27161151C2semidirect
15. #550×C197C2semidirect
20. C2×#15691348C22semidirect
21. C2×#15691346C22semidirect
22. C2×#15691346C22semidirect
30. C22×#13096802C22semidirect
39. #20263259C22semidirect
40. C4⋊#13096802C22semidirect
41. #20263250C22semidirect
42. #20263249C22semidirect
43. #20263247C22semidirect
44. #20263247C22semidirect
48. C22⋊#13096802C22semidirect
49. #20263237C22semidirect
51. C2×#41×C197C22semidirect
53. #127×C197C22semidirect
57. #121×C197C22semidirect
58. C4×Q8×C197C22semidirect
78. #15691345D4semidirect
214. C197#3268semidirect
218. #550D197semidirect
233. C2×#41C2×D197semidirect
249. #127C2×D197semidirect
258. #121C2×D197semidirect
260. C4×Q8C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633512
C2×#202633462
#271617252
#271614662
#271613232
#271613132
#271613012
#271612712
#271612412
#271612242
#271612222
#271612122
#271611772
#271611512
#550×C1972
#3268197