Finite Group Databasebeta

← #38754390

Subgroups — #38754390

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×#20263347C211yesmaximal, characteristic, complemented
2C22×#15691343C211yesmaximal, characteristic, complemented
3#27161723C211yesmaximal, characteristic, complemented
4#27161720C211yesmaximal, characteristic, complemented
5#27161460C211yesmaximal, characteristic, complemented
6#27161318C211yesmaximal, characteristic, complemented
7#27161314C211yesmaximal, characteristic, complemented
8#27161296C211yesmaximal, characteristic, complemented
9#27161293C211yesmaximal, characteristic, complemented
10#27161232C211yesmaximal, characteristic, complemented
11#27161225C211yesmaximal, characteristic, complemented
12#27161223C211yesmaximal, characteristic, complemented
13#27161159C211yesmaximal, characteristic, complemented
14#27161142C211yesmaximal, characteristic, complemented
15#450×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic, complemented
17C2×#15691357C2211yescharacteristic, complemented
18C22×D788C2211yescharacteristic
19#20263347—24no—
20C2×#15691348C2211yescharacteristic, complemented
21C2×#15691347C2211yescharacteristic, complemented
22C2×#15691347C2211yescharacteristic, complemented
23C2×#15691343—24no—
24C2×#15691343C2211yescharacteristic
25C2×#15691343C2211yescharacteristic
26C2×#15691343C2211yescharacteristic
27C2×#15691343C2211yescharacteristic
28C22×#13096802C2211yescharacteristic
29C2×C8×D197C2211yescharacteristic, complemented
30#20263315C2211yescharacteristic, complemented
31#20263313C2211yescharacteristic, complemented
32#20263311—21no—
33#20263307—21no—
34#20263305C2211yescharacteristic
35#20263301—21no—
36#20263301C2211yescharacteristic, complemented
37#20263298—21no—
38#20263297C2211yescharacteristic, complemented
39#20263296—21no—
40#20263295—21no—
41D4×Dic197C2211yescharacteristic, complemented
42#20263288C2211yescharacteristic, complemented
43C4×C22⋊D197C2211yescharacteristic
44C2×C22⋊Dic197—21no—
45C22⋊C4×D197—21no—
46#20263259C2211yescharacteristic, complemented
47#20263255C2211yescharacteristic, complemented
48#20263250C2211yescharacteristic, complemented
49#20263249C2211yescharacteristic, complemented
50#20263248C2211yescharacteristic, complemented
51#20263248C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53C8×Dic197C2211yescharacteristic, complemented
54C22×D4×C197C2211yescharacteristic
55C2×SD16×C197C2211yescharacteristic, complemented
56C2×SD16×C197C2211yescharacteristic, complemented
57C22⋊Q8×C197C2211yescharacteristic
58#102×C197C2211yescharacteristic, complemented
59#102×C197C2211yescharacteristic, complemented
60C22⋊C8×C197C2211yescharacteristic
61C23×D197—21no—
62#15691359—22no—
63#15691359—41no—
64#15691357—22no—
65#15691357—41no—
66C2×Dic394C2311yescharacteristic
67C2×D788—22no—
68C2×D788—21no—
69C2×D788—21no—
70C2×D788—21no—
71C2×D788—41no—
72C2×D788C2311yescharacteristic
73C2×D788C2311yescharacteristic
74C2×C22⋊D197—21no—
75C2×C22⋊D197—21no—
76C2×C22⋊D197D411yescharacteristic
77C22×Dic197—21no—
78C22×Dic197—21no—
79C2×C4×D197—21no—
80C2×C4×D197—21no—
81C2×C4×D197C2311yescharacteristic
82#15691348—22no—
83#15691347—22no—
84#15691347—41no—
85#15691343—22no—
86#15691343—22no—
87#15691343—22no—
88#15691343D414yescomplemented
89C2×#13096802—22no—
90C2×#13096802C2311yescharacteristic
91C2×#13096802C2311yescharacteristic
92C8×D197—22no—
93C4⋊Dic197C2311yescharacteristic
94#15691337—21no—
95#15691337D411yescharacteristic
96C22⋊Dic197—21no—
97C22⋊Dic197—21no—
98C22⋊Dic197—21no—
99C22⋊Dic197—21no—
100C22⋊Dic197—21no—
101C22⋊Dic197D411yescharacteristic
102#15691335—21no—
103#15691335—21no—
104#15691335—21no—
105#15691335—21no—
106#15691335D411yescharacteristic
107C4×Dic197C2311yescharacteristic
108C24×C197—21no—
109C2×Q8×C197C2311yescharacteristic
110C2×D4×C197—22no—
111C2×D4×C197—21no—
112C2×D4×C197—41no—
113C2×D4×C197C2311yescharacteristic
114C2×D4×C197C2311yescharacteristic
115C22×C4×C197C2311yescharacteristic
116SD16×C197—22no—
117SD16×C197—41no—
118C2×C8×C197C2311yescharacteristic
119C2×C8×C197C2311yescharacteristic
120C4⋊C4×C197—21no—
121C4⋊C4×C197C2311yescharacteristic
122C22⋊C4×C197—21no—
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—42no—
126C22×D197—42no—
127C22×D197—81no—
128C22×D197C2×D411yescharacteristic
129Dic394—21no—
130Dic394—21no—
131D788—21no—
132D788—21no—
133D788—21no—
134D788—42no—
135D788—42no—
136D788—81no—
137D788C2×D412yes—
138C22⋊D197—22no—
139C22⋊D197—42no—
140C22⋊D197—81no—
141C2×Dic197—21no—
142C2×Dic197—21no—
143C2×Dic197—42no—
144C2×Dic197—81no—
145C2×Dic197C2×D411yescharacteristic
146C2×Dic197C2×D411yescharacteristic
147C2×Dic197C2×D411yescharacteristic
148C4×D197—21no—
149C4×D197—21no—
150C4×D197—42no—
151C4×D197—81no—
152#13096802—21no—
153#13096802C2×D412yes—
154C23×C197—21no—
155C23×C197—21no—
156C23×C197—42no—
157C23×C197—42no—
158C23×C197—81no—
159C23×C197C2×D411yescharacteristic
160Q8×C197—21no—
161Q8×C197—21no—
162D4×C197—21no—
163D4×C197—21no—
164D4×C197—21no—
165D4×C197—42no—
166D4×C197C2×D412yes—
167C2×C4×C197—21no—
168C2×C4×C197—21no—
169C2×C4×C197—21no—
170C2×C4×C197—21no—
171C2×C4×C197C2×D411yescharacteristic
172C2×C4×C197C2411yescharacteristic
173C8×C197—21no—
174C8×C197—21no—
175C22×C197—21no—
176C22×C197—21no—
177C22×C197—22no—
178C22×C197—41no—
179C22×C197—41no—
180C22×C197—41no—
181C22×C197—41no—
182C22×C197—41no—
183C22×C197—84no—
184C22×C197#13512yescomplemented
185C22×C197C22×D411yescharacteristic
186C2×D197—21no—
187C2×D197—22no—
188C2×D197—21no—
189C2×D197—21no—
190C2×D197—41no—
191C2×D197—41no—
192C2×D197—41no—
193C2×D197—41no—
194C2×D197—41no—
195C2×D197—84no—
196C4×C197—21no—
197C4×C197—41no—
198C4×C197—41no—
199C4×C197C22×D411yescharacteristic
200C4×C197H(5,𝔽2)11yescharacteristic
201Dic197—21no—
202Dic197—21no—
203Dic197—21no—
204Dic197—41no—
205Dic197—41no—
206C2×C197—22no—
207C2×C197—42no—
208C2×C197—81no—
209C2×C197D4211yescharacteristic
210C2×C197C2×#13511yescharacteristic
211C2×C197#57611yescharacteristic
212D197—41no—
213D197—42no—
214D197—81no—
215C197#328011yescharacteristic, complemented, Sylow
216#3280—1971nomaximal, Sylow
217C2×#135—1971no—
218C22×D8—1971no—
219#546—1971no—
220#546—1971no—
221#495—1971no—
222#467—1971no—
223#466—1971no—
224#463—1971no—
225#463—1971no—
226#450—1971no—
227#450D19711yescharacteristic, complemented
228#449—1971no—
229#449—1971no—
230C4×D8—1971no—
231C8×D4—1971no—
232C2×#41—1971no—
233C2×#41—1971no—
234C22×D4—1971no—
235C22×D4C2×D19711yescharacteristic
236#135—3944no—
237C2×Q16—1971no—
238C2×SD16—1971no—
239C2×SD16—1971no—
240C2×SD16C2×D19711yescharacteristic, complemented
241C2×SD16C2×D19711yescharacteristic, complemented
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#124—1971no—
251#123—3941no—
252#123—3941no—
253C22⋊Q8—1971no—
254C22⋊Q8C2×D19711yescharacteristic
255#121—3941no—
256#121—3941no—
257#121—1971no—
258#121—1971no—
259#120—3941no—
260#120—3941no—
261C4×D4—1971no—
262C4×D4—1971no—
263C4×D4—1971no—
264C2×C22⋊C4—3941no—
265C2×C22⋊C4—3941no—
266#107—1971no—
267C4⋊C8—1971no—
268#103—1971no—
269#103—1971no—
270#102—1971no—
271#102—1971no—
272#102C2×D19711yescharacteristic, complemented
273#102C2×D19711yescharacteristic, complemented
274C22⋊C8—1971no—
275C22⋊C8C2×D19711yescharacteristic
276C4×C8—1971no—
277C24—21no—
278C24—3941no—
279#41—7881no—
280#41—7881no—
281#41—3942no—
282#41—3942no—
283C2×Q8—1971no—
284C2×Q8C22×D19711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—41no—
288C2×D4—7881no—
289C2×D4—3941no—
290C2×D4—3942no—
291C2×D4—3941no—
292C2×D4—3941no—
293C2×D4—3941no—
294C2×D4—3941no—
295C2×D4—1971no—
296C2×D4—1971no—
297C2×D4—1971no—
298C2×D4C22×D19711yescharacteristic
299C2×D4C22×D19711yescharacteristic
300C22×C4—3941no—
301C22×C4—3941no—
302C22×C4—3941no—
303C22×C4—3941no—
304C22×C4—1971no—
305C22×C4C22×D19711yescharacteristic
306Q16—3942no—
307SD16—22no—
308SD16—41no—
309SD16—7881no—
310SD16—3942no—
311D8—3942no—
312D8—3942no—
313D8—3942no—
314D8—1974no—
315C2×C8—3942no—
316C2×C8—3942no—
317C2×C8—1971no—
318C2×C8—1971no—
319C2×C8C22×D19711yescharacteristic
320C2×C8C22×D19711yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—3941no—
323C4⋊C4—1971no—
324C4⋊C4—1971no—
325C4⋊C4C22×D19711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—3941no—
328C22⋊C4—3941no—
329C22⋊C4—3941no—
330C22⋊C4—3941no—
331C22⋊C4—3941no—
332C22⋊C4—3941no—
333C22⋊C4—3941no—
334C22⋊C4—3941no—
335C22⋊C4—3941no—
336C22⋊C4—1971no—
337C22⋊C4—1971no—
338C42—1971no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—7882no—
345C23—7882no—
346C23—15761no—
347C23—3941no—
348C23—3941no—
349C23—1971no—
350C23D4×D19711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3941no—
354Q8—3941no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—7882no—
360D4—7882no—
361D4—7882no—
362D4—15761no—
363D4—15761no—
364D4—3942no—
365D4—3941no—
366D4—3941no—
367D4—3941no—
368D4—1972no—
369D4D4×D19712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—7882no—
375C2×C4—7882no—
376C2×C4—15761no—
377C2×C4—15761no—
378C2×C4—3941no—
379C2×C4—3941no—
380C2×C4—3941no—
381C2×C4—3941no—
382C2×C4—1971no—
383C2×C4—1971no—
384C2×C4—1971no—
385C2×C4D4×D19711yescharacteristic
386C2×C4C23×D19711yescharacteristic
387C8—21no—
388C8—21no—
389C8—3941no—
390C8—1972no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—84no—
400C22—7881no—
401C22—7881no—
402C22—7881no—
403C22—7881no—
404C22—7881no—
405C22—15764no—
406C22—3941no—
407C22—3941no—
408C22—3942no—
409C22—3941no—
410C22#2026334712yescomplemented
411C22C2×D4×D19711yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—7881no—
416C4—7881no—
417C4—3941no—
418C4—3941no—
419C4—3941no—
420C4C2×D4×D19711yescharacteristic
421C4#2026337111yescharacteristic
422C2—22no—
423C2—42no—
424C2—81no—
425C2—7882no—
426C2—7881no—
427C2—15761no—
428C2#2716171111yescharacteristic
429C2C2×#2026334711yescharacteristic
430C2#2716188311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#450×C197C2
socleC22×C197C22×D4
hypercenter#450D197
hypocenterC197#3280
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#20263347C2semidirect
2. C22×#15691343C2semidirect
3. #27161723C2semidirect
4. #27161720C2semidirect
5. #27161460C2semidirect
6. #27161318C2semidirect
7. #27161314C2semidirect
8. #27161296C2semidirect
9. #27161293C2semidirect
10. #27161232C2semidirect
11. #27161225C2semidirect
12. #27161223C2semidirect
13. #27161159C2semidirect
14. #27161142C2semidirect
15. #450×C197C2semidirect
16. C2×#15691359C22semidirect
17. C2×#15691357C22semidirect
20. C2×#15691348C22semidirect
21. C2×#15691347C22semidirect
22. C2×#15691347C22semidirect
29. C2×C8×D197C22semidirect
30. #20263315C22semidirect
31. #20263313C22semidirect
36. #20263301C22semidirect
38. #20263297C22semidirect
41. D4×Dic197C22semidirect
42. #20263288C22semidirect
46. #20263259C22semidirect
47. #20263255C22semidirect
48. #20263250C22semidirect
49. #20263249C22semidirect
50. #20263248C22semidirect
51. #20263248C22semidirect
52. #20263238C22semidirect
53. C8×Dic197C22semidirect
55. C2×SD16×C197C22semidirect
56. C2×SD16×C197C22semidirect
58. #102×C197C22semidirect
59. #102×C197C22semidirect
88. #15691343D4semidirect
184. C22×C197#135semidirect
215. C197#3280semidirect
227. #450D197semidirect
240. C2×SD16C2×D197semidirect
241. C2×SD16C2×D197semidirect
272. #102C2×D197semidirect
273. #102C2×D197semidirect
410. C22#20263347semidirect

Maximal subgroups

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

subgroupindex
C2×#202633472
C22×#156913432
#271617232
#271617202
#271614602
#271613182
#271613142
#271612962
#271612932
#271612322
#271612252
#271612232
#271611592
#271611422
#450×C1972
#3280197