Finite Group Databasebeta

← #38754393

Subgroups — #38754393

Order 25216.

526 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263346C211yesmaximal, characteristic, complemented
2C22×#15691343C211yesmaximal, characteristic, complemented
3#27161725C211yesmaximal, characteristic, complemented
4#27161461C211yesmaximal, characteristic, complemented
5#27161318C211yesmaximal, characteristic, complemented
6#27161313C211yesmaximal, characteristic, complemented
7#27161294C211yesmaximal, characteristic, complemented
8#27161291C211yesmaximal, characteristic, complemented
9#27161230C211yesmaximal, characteristic, complemented
10#27161229C211yesmaximal, characteristic, complemented
11#27161224C211yesmaximal, characteristic, complemented
12#27161222C211yesmaximal, characteristic, complemented
13C4×#15691343C211yesmaximal, characteristic, complemented
14D4×#13096802C211yesmaximal, characteristic, complemented
15#546×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic, complemented
17C22×D788C2211yescharacteristic
18#20263346—24no—
19C2×#15691348C2211yescharacteristic, complemented
20C2×#15691346C2211yescharacteristic, complemented
21C2×#15691346C2211yescharacteristic, complemented
22C2×#15691345C2211yescharacteristic, complemented
23C2×#15691345C2211yescharacteristic, complemented
24C2×#15691343—24no—
25C2×#15691343C2211yescharacteristic
26C2×#15691343C2211yescharacteristic
27C2×#15691343C2211yescharacteristic
28C2×#15691343C2211yescharacteristic
29C22×#13096802C2211yescharacteristic
30C22×#13096802C2211yescharacteristic, complemented
31#20263314C2211yescharacteristic, complemented
32#20263311—21no—
33#20263305C2211yescharacteristic
34#20263300C2211yescharacteristic, complemented
35#20263299—21no—
36#20263294—21no—
37C4×D788C2211yescharacteristic, complemented
38C22⋊C4×D197—21no—
39#20263259C2211yescharacteristic, complemented
40C4⋊#13096802C2211yescharacteristic, complemented
41#20263250C2211yescharacteristic, complemented
42#20263249C2211yescharacteristic, complemented
43#20263247C2211yescharacteristic, complemented
44#20263247C2211yescharacteristic, complemented
45#20263246C2211yescharacteristic, complemented
46#20263246C2211yescharacteristic, complemented
47C22⋊#13096802C2211yescharacteristic
48C22⋊#13096802C2211yescharacteristic, complemented
49C4×#13096802C2211yescharacteristic, complemented
50C2×#41×C197C2211yescharacteristic, complemented
51C22×D4×C197C2211yescharacteristic
52#124×C197C2211yescharacteristic, complemented
53#123×C197—21no—
54C22⋊Q8×C197C2211yescharacteristic
55#121×C197—21no—
56#121×C197C2211yescharacteristic, complemented
57#120×C197—21no—
58C4×D4×C197C2211yescharacteristic
59C4×D4×C197C2211yescharacteristic, complemented
60C2×C22⋊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#15691346—22no—
76#15691346—41no—
77#15691345—22no—
78#15691345—41no—
79#15691343—22no—
80#15691343—22no—
81#15691343—22no—
82#15691343D414yescomplemented
83C2×#13096802—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—
96C24×C197—21no—
97#41×C197—22no—
98#41×C197—41no—
99C2×Q8×C197C2311yescharacteristic
100C2×D4×C197—22no—
101C2×D4×C197—21no—
102C2×D4×C197—21no—
103C2×D4×C197—21no—
104C2×D4×C197—41no—
105C2×D4×C197D411yescharacteristic
106C2×D4×C197C2311yescharacteristic
107C2×D4×C197C2311yescharacteristic
108C22×C4×C197—21no—
109C22×C4×C197—21no—
110C22×C4×C197C2311yescharacteristic
111C22×C4×C197C2311yescharacteristic
112C4⋊C4×C197—21no—
113C4⋊C4×C197D411yescharacteristic
114C4⋊C4×C197C2311yescharacteristic
115C22⋊C4×C197—21no—
116C22⋊C4×C197—21no—
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×C197—21no—
147C23×C197—42no—
148C23×C197—42no—
149C23×C197—81no—
150C23×C197C2×D411yescharacteristic
151C23×C197C2×D411yescharacteristic
152Q8×C197—21no—
153Q8×C197—21no—
154D4×C197—22no—
155D4×C197—21no—
156D4×C197—21no—
157D4×C197—21no—
158D4×C197—42no—
159D4×C197—42no—
160D4×C197—81no—
161D4×C197C2×D412yes—
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—81no—
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—22no—
179C22×C197—21no—
180C22×C197—21no—
181C22×C197—41no—
182C22×C197—41no—
183C22×C197—41no—
184C22×C197—41no—
185C22×C197—41no—
186C22×C197—84no—
187C22×C197#13512yescomplemented
188C22×C197C22×D411yescharacteristic
189C2×D197—22no—
190C2×D197—41no—
191C2×D197—41no—
192C2×D197—41no—
193C2×D197—41no—
194C2×D197—41no—
195C2×D197—84no—
196C4×C197—21no—
197C4×C197—21no—
198C4×C197—21no—
199C4×C197—21no—
200C4×C197—41no—
201C4×C197—41no—
202C4×C197C22×D411yescharacteristic
203C4×C197H(5,𝔽2)11yescharacteristic
204Dic197—41no—
205Dic197—41no—
206C2×C197—22no—
207C2×C197—41no—
208C2×C197—42no—
209C2×C197—81no—
210C2×C197D4211yescharacteristic
211C2×C197C2×#13511yescharacteristic
212C2×C197#57611yescharacteristic
213D197—42no—
214D197—81no—
215C197#328011yescharacteristic, complemented, Sylow
216#3280—1971nomaximal, Sylow
217C2×#135—1971no—
218C22×D8—1971no—
219#546—1971no—
220#546D19711yescharacteristic, complemented
221#495—1971no—
222#467—1971no—
223#466—1971no—
224#463—1971no—
225#463—1971no—
226#450—1971no—
227#450—1971no—
228#449—1971no—
229#449—1971no—
230C4×D8—1971no—
231C8×D4—1971no—
232C2×#41—1971no—
233C2×#41C2×D19711yescharacteristic, complemented
234C22×D4—1971no—
235C22×D4C2×D19711yescharacteristic
236#135—3944no—
237C2×Q16—1971no—
238C2×SD16—1971no—
239C2×SD16—1971no—
240C2×SD16—1971no—
241C2×SD16—1971no—
242C2×D8—3944no—
243C2×D8—1971no—
244C2×D8—1971no—
245C2×D8—1971no—
246C2×D8—1971no—
247C22×C8—1971no—
248C22×C8—1971no—
249#124—1971no—
250#124C2×D19711yescharacteristic, complemented
251#123—21no—
252#123—3941no—
253C22⋊Q8—1971no—
254C22⋊Q8C2×D19711yescharacteristic
255#121—21no—
256#121—3941no—
257#121—1971no—
258#121C2×D19711yescharacteristic, complemented
259#120—21no—
260#120—3941no—
261C4×D4—1971no—
262C4×D4C2×D19711yescharacteristic
263C4×D4C2×D19711yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—3941no—
266#107—1971no—
267C4⋊C8—1971no—
268#103—1971no—
269#103—1971no—
270#102—1971no—
271#102—1971no—
272#102—1971no—
273#102—1971no—
274C22⋊C8—1971no—
275C22⋊C8—1971no—
276C4×C8—1971no—
277C24—21no—
278C24—3941no—
279#41—22no—
280#41—41no—
281#41—7881no—
282#41—3942no—
283C2×Q8—1971no—
284C2×Q8C22×D19711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—7881no—
291C2×D4—3942no—
292C2×D4—3941no—
293C2×D4—3941no—
294C2×D4—3941no—
295C2×D4—1971no—
296C2×D4—1971no—
297C2×D4C22⋊D19711yescharacteristic
298C2×D4C22×D19711yescharacteristic
299C2×D4C22×D19711yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—3941no—
303C22×C4—3941no—
304C22×C4C22×D19711yescharacteristic
305C22×C4C22×D19711yescharacteristic
306Q16—3942no—
307SD16—7881no—
308SD16—7881no—
309SD16—3942no—
310SD16—3942no—
311D8—3942no—
312D8—3942no—
313D8—3942no—
314D8—1974no—
315C2×C8—3942no—
316C2×C8—3942no—
317C2×C8—1971no—
318C2×C8—1971no—
319C2×C8—1971no—
320C2×C8—1971no—
321C4⋊C4—21no—
322C4⋊C4—3941no—
323C4⋊C4—1971no—
324C4⋊C4C22⋊D19711yescharacteristic
325C4⋊C4C22×D19711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—3941no—
332C22⋊C4—3941no—
333C22⋊C4—3941no—
334C22⋊C4—3941no—
335C22⋊C4—3941no—
336C22⋊C4C22⋊D19711yescharacteristic
337C22⋊C4C22⋊D19711yescharacteristic
338C42C22×D19711yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—7882no—
345C23—7882no—
346C23—15761no—
347C23—3941no—
348C23—3941no—
349C23C2×C22⋊D19711yescharacteristic
350C23C2×C22⋊D19711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3941no—
354Q8—3941no—
355D4—22no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—7882no—
363D4—7882no—
364D4—15761no—
365D4—3941no—
366D4—3941no—
367D4—3941no—
368D4—1972no—
369D4D4×D19712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—7882no—
379C2×C4—15761no—
380C2×C4—3941no—
381C2×C4—3941no—
382C2×C4C2×C22⋊D19711yescharacteristic
383C2×C4C2×C22⋊D19711yescharacteristic
384C2×C4C2×C22⋊D19711yescharacteristic
385C2×C4C2×C22⋊D19711yescharacteristic
386C2×C4C23×D19711yescharacteristic
387C8—3941no—
388C8—3941no—
389C8—3941no—
390C8—1972no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—7881no—
404C22—7881no—
405C22—7881no—
406C22—7881no—
407C22—7881no—
408C22—15764no—
409C22—3942no—
410C22#2026334612yescomplemented
411C22C22×C22⋊D19711yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—7881no—
419C4—7881no—
420C4C2×D4×D19711yescharacteristic
421C4#2026337111yescharacteristic
422C2—22no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—7882no—
427C2—15761no—
428C2D4×C22⋊D19711yescharacteristic
429C2C2×#2026334611yescharacteristic
430C2#2716188211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#546×C197C2
socleC22×C197C22×D4
hypercenter#546D197
hypocenterC197#3280
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#20263346C2semidirect
2. C22×#15691343C2semidirect
3. #27161725C2semidirect
4. #27161461C2semidirect
5. #27161318C2semidirect
6. #27161313C2semidirect
7. #27161294C2semidirect
8. #27161291C2semidirect
9. #27161230C2semidirect
10. #27161229C2semidirect
11. #27161224C2semidirect
12. #27161222C2semidirect
13. C4×#15691343C2semidirect
14. D4×#13096802C2semidirect
15. #546×C197C2semidirect
16. C2×#15691358C22semidirect
19. C2×#15691348C22semidirect
20. C2×#15691346C22semidirect
21. C2×#15691346C22semidirect
22. C2×#15691345C22semidirect
23. C2×#15691345C22semidirect
30. C22×#13096802C22semidirect
31. #20263314C22semidirect
34. #20263300C22semidirect
37. C4×D788C22semidirect
39. #20263259C22semidirect
40. C4⋊#13096802C22semidirect
41. #20263250C22semidirect
42. #20263249C22semidirect
43. #20263247C22semidirect
44. #20263247C22semidirect
45. #20263246C22semidirect
46. #20263246C22semidirect
48. C22⋊#13096802C22semidirect
49. C4×#13096802C22semidirect
50. C2×#41×C197C22semidirect
52. #124×C197C22semidirect
56. #121×C197C22semidirect
59. C4×D4×C197C22semidirect
82. #15691343D4semidirect
187. C22×C197#135semidirect
215. C197#3280semidirect
220. #546D197semidirect
233. C2×#41C2×D197semidirect
250. #124C2×D197semidirect
258. #121C2×D197semidirect
263. C4×D4C2×D197semidirect
410. C22#20263346semidirect

Maximal subgroups

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

subgroupindex
C2×#202633462
C22×#156913432
#271617252
#271614612
#271613182
#271613132
#271612942
#271612912
#271612302
#271612292
#271612242
#271612222
C4×#156913432
D4×#130968022
#546×C1972
#3280197