Finite Group Databasebeta

← #38754238

Subgroups — #38754238

Order 25216.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 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#27161471C211yesmaximal, characteristic, complemented
5#27161335C211yesmaximal, characteristic, complemented
6#27161310C211yesmaximal, characteristic, complemented
7#27161294C211yesmaximal, characteristic, complemented
8#27161281C211yesmaximal, characteristic, complemented
9#27161241C211yesmaximal, characteristic, complemented
10#27161239C211yesmaximal, characteristic, complemented
11#27161229C211yesmaximal, characteristic, complemented
12#27161224C211yesmaximal, characteristic, complemented
13#27161189C211yesmaximal, characteristic, complemented
14#27161151C211yesmaximal, characteristic, complemented
15#547×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×D788C2211yescharacteristic
18#20263351—24no—
19#20263346—24no—
20C2×#15691348C2211yescharacteristic, complemented
21C2×#15691348C2211yescharacteristic, complemented
22C2×#15691346C2211yescharacteristic, complemented
23C2×#15691345C2211yescharacteristic
24C2×#15691345C2211yescharacteristic
25C2×#15691345C2211yescharacteristic
26C2×#15691345C2211yescharacteristic
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#20263257C2211yescharacteristic, complemented
40C4⋊#13096802C2211yescharacteristic, complemented
41#20263250C2211yescharacteristic, complemented
42#20263250C2211yescharacteristic, complemented
43#20263249C2211yescharacteristic, complemented
44#20263247C2211yescharacteristic, complemented
45#20263246C2211yescharacteristic
46#20263246C2211yescharacteristic
47C22⋊#13096802C2211yescharacteristic
48C22⋊#13096802C2211yescharacteristic, complemented
49#20263237C2211yescharacteristic, complemented
50C2×#41×C197C2211yescharacteristic
51C2×#41×C197C2211yescharacteristic, complemented
52C4⋊Q8×C197C2211yescharacteristic, complemented
53#123×C197—21no—
54#123×C197—21no—
55C22⋊Q8×C197C2211yescharacteristic
56C22⋊Q8×C197C2211yescharacteristic, complemented
57#121×C197—21no—
58C4×D4×C197C2211yescharacteristic
59C4×D4×C197C2211yescharacteristic, complemented
60C2×C4⋊C4×C197—21no—
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#15691348—41no—
76#15691346—22no—
77#15691345—22no—
78#15691345—22no—
79#15691345—22no—
80#15691345—41no—
81#15691343—22no—
82#15691343D414yescomplemented
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×Q8×C197C2311yescharacteristic
102C2×D4×C197—21no—
103C2×D4×C197—21no—
104C2×D4×C197D411yescharacteristic
105C2×D4×C197C2311yescharacteristic
106C22×C4×C197—21no—
107C22×C4×C197—21no—
108C22×C4×C197—21no—
109C22×C4×C197C2311yescharacteristic
110C22×C4×C197C2311yescharacteristic
111C4⋊C4×C197—21no—
112C4⋊C4×C197—21no—
113C4⋊C4×C197—21no—
114C4⋊C4×C197—21no—
115C4⋊C4×C197D411yescharacteristic
116C4⋊C4×C197C2311yescharacteristic
117C22⋊C4×C197—21no—
118C22⋊C4×C197—21no—
119C22⋊C4×C197—21no—
120C22⋊C4×C197D411yescharacteristic
121C22⋊C4×C197D411yescharacteristic
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×C197C2×D411yescharacteristic
147C23×C197C2×D411yescharacteristic
148Q8×C197—21no—
149Q8×C197—21no—
150Q8×C197—21no—
151Q8×C197—21no—
152D4×C197—21no—
153D4×C197—41no—
154D4×C197—42no—
155D4×C197—42no—
156D4×C197C2×D412yes—
157C2×C4×C197—21no—
158C2×C4×C197—21no—
159C2×C4×C197—21no—
160C2×C4×C197—21no—
161C2×C4×C197—21no—
162C2×C4×C197—21no—
163C2×C4×C197—21no—
164C2×C4×C197—42no—
165C2×C4×C197—42no—
166C2×C4×C197—42no—
167C2×C4×C197C2×D411yescharacteristic
168C2×C4×C197C2×D411yescharacteristic
169C2×C4×C197C2×D411yescharacteristic
170C2×C4×C197C2×D411yescharacteristic
171C2×C4×C197C2411yescharacteristic
172C22×C197—21no—
173C22×C197—21no—
174C22×C197—22no—
175C22×C197—21no—
176C22×C197—21no—
177C22×C197—21no—
178C22×C197—21no—
179C22×C197—41no—
180C22×C197—41no—
181C22×C197C22×D411yescharacteristic
182C2×D197—22no—
183C2×D197—41no—
184C2×D197—41no—
185C2×D197—41no—
186C2×D197—41no—
187C2×D197—41no—
188C2×D197—84no—
189C4×C197—21no—
190C4×C197—21no—
191C4×C197—21no—
192C4×C197—21no—
193C4×C197—41no—
194C4×C197—41no—
195C4×C197—41no—
196C4×C197C22×D411yescharacteristic
197C4×C197H(5,𝔽2)11yescharacteristic
198Dic197—41no—
199Dic197—41no—
200C2×C197—41no—
201C2×C197—41no—
202C2×C197—42no—
203C2×C197D4211yescharacteristic
204C2×C197#57511yescharacteristic
205C2×C197#57711yescharacteristic
206D197—42no—
207D197—81no—
208C197#326711yescharacteristic, complemented, Sylow
209#3267—1971nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1971no—
211C2×#135—1971no—
212#547D19711yescharacteristic, complemented
213#546—1971no—
214#497—1971no—
215#470—1971no—
216#465—1971no—
217#463—1971no—
218#461—1971no—
219#452—1971no—
220#452—1971no—
221#450—1971no—
222#449—1971no—
223#442—1971no—
224#435—1971no—
225C2×#41—1971no—
226C2×#41C2×D19711yescharacteristic
227C2×#41C2×D19711yescharacteristic, complemented
228C22×D4—1971no—
229AGL(1,ℤ/8)—3944no—
230#135—3944no—
231C2×Q16—1971no—
232C2×Q16—1971no—
233C2×SD16—1971no—
234C2×SD16—1971no—
235C2×SD16—1971no—
236C2×SD16—1971no—
237C2×SD16—1971no—
238C2×D8—1971no—
239C2×D8—1971no—
240C2×M16—1971no—
241C22×C8—1971no—
242C4⋊Q8C2×D19711yescharacteristic, complemented
243#124—1971no—
244#123—21no—
245#123—21no—
246#123—3941no—
247C22⋊Q8—1971no—
248C22⋊Q8C2×D19711yescharacteristic
249C22⋊Q8C2×D19711yescharacteristic, complemented
250#121—21no—
251#121—3941no—
252#121—1971no—
253#120—3941no—
254C4×D4—1971no—
255C4×D4C2×D19711yescharacteristic
256C4×D4C2×D19711yescharacteristic, complemented
257C2×C4⋊C4—21no—
258C2×C22⋊C4—3941no—
259#106—1971no—
260C4⋊C8—1971no—
261#103—1971no—
262#103—1971no—
263#103—1971no—
264#102—1971no—
265#102—1971no—
266#102—1971no—
267C22⋊C8—1971no—
268C22⋊C8—1971no—
269#97—1971no—
270C24—3941no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—7881no—
276#41—3942no—
277C2×Q8—1971no—
278C2×Q8C22×D19711yescharacteristic
279C2×Q8C22×D19711yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—7881no—
283C2×D4—3942no—
284C2×D4—3941no—
285C2×D4—3941no—
286C2×D4—3941no—
287C2×D4—1971no—
288C2×D4—1971no—
289C2×D4C22⋊D19711yescharacteristic
290C2×D4C22×D19711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—3941no—
295C22×C4—3941no—
296C22×C4C22×D19711yescharacteristic
297C22×C4C22×D19711yescharacteristic
298Q16—7881no—
299Q16—3942no—
300SD16—7881no—
301SD16—3942no—
302SD16—3942no—
303SD16—3942no—
304SD16—3942no—
305D8—3942no—
306D8—1974no—
307M16—3942no—
308C2×C8—3942no—
309C2×C8—1971no—
310C2×C8—1971no—
311C2×C8—1971no—
312C2×C8—1971no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—3941no—
318C4⋊C4—1971no—
319C4⋊C4C22⋊D19711yescharacteristic
320C4⋊C4C22×D19711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—3941no—
325C22⋊C4—3941no—
326C22⋊C4—3941no—
327C22⋊C4—3941no—
328C22⋊C4—3941no—
329C22⋊C4C22⋊D19711yescharacteristic
330C22⋊C4C22⋊D19711yescharacteristic
331C42C22×D19711yescharacteristic
332C23—21no—
333C23—7882no—
334C23—7882no—
335C23—15761no—
336C23—3941no—
337C23—3941no—
338C23C2×C22⋊D19711yescharacteristic
339C23C2×C22⋊D19711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—3941no—
345Q8—3941no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—7882no—
351D4—7882no—
352D4—15761no—
353D4—3941no—
354D4—3941no—
355D4—3941no—
356D4—1972no—
357D4D4×D19712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—7882no—
369C2×C4—15761no—
370C2×C4—3941no—
371C2×C4—3941no—
372C2×C4C2×C22⋊D19711yescharacteristic
373C2×C4C2×C22⋊D19711yescharacteristic
374C2×C4C2×C22⋊D19711yescharacteristic
375C2×C4C2×C22⋊D19711yescharacteristic
376C2×C4C23×D19711yescharacteristic
377C8—3941no—
378C8—3941no—
379C8—3941no—
380C8—1972no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—7881no—
391C22—7881no—
392C22—7881no—
393C22—7881no—
394C22—7881no—
395C22—15764no—
396C22—3942no—
397C22C22×C22⋊D19711yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—7881no—
406C4—7881no—
407C4C2×D4×D19711yescharacteristic
408C4#2026337111yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—7882no—
413C2—15761no—
414C2D4×C22⋊D19711yescharacteristic
415C2#2716187711yescharacteristic
416C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#547×C197C2
socleC22×C197C22×D4
hypercenter#547D197
hypocenterC197#3267
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#20263351C2semidirect
2. C2×#20263346C2semidirect
3. #27161725C2semidirect
4. #27161471C2semidirect
5. #27161335C2semidirect
6. #27161310C2semidirect
7. #27161294C2semidirect
8. #27161281C2semidirect
9. #27161241C2semidirect
10. #27161239C2semidirect
11. #27161229C2semidirect
12. #27161224C2semidirect
13. #27161189C2semidirect
14. #27161151C2semidirect
15. #547×C197C2semidirect
20. C2×#15691348C22semidirect
21. C2×#15691348C22semidirect
22. C2×#15691346C22semidirect
30. C22×#13096802C22semidirect
39. #20263257C22semidirect
40. C4⋊#13096802C22semidirect
41. #20263250C22semidirect
42. #20263250C22semidirect
43. #20263249C22semidirect
44. #20263247C22semidirect
48. C22⋊#13096802C22semidirect
49. #20263237C22semidirect
51. C2×#41×C197C22semidirect
52. C4⋊Q8×C197C22semidirect
56. C22⋊Q8×C197C22semidirect
59. C4×D4×C197C22semidirect
82. #15691343D4semidirect
208. C197#3267semidirect
212. #547D197semidirect
227. C2×#41C2×D197semidirect
242. C4⋊Q8C2×D197semidirect
249. C22⋊Q8C2×D197semidirect
256. C4×D4C2×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
#271614712
#271613352
#271613102
#271612942
#271612812
#271612412
#271612392
#271612292
#271612242
#271611892
#271611512
#547×C1972
#3267197