Finite Group Databasebeta

← #38754316

Subgroups — #38754316

Order 25216.

520 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×#20263353C211yesmaximal, characteristic, complemented
2C2×#20263351C211yesmaximal, characteristic, complemented
3#27161720C211yesmaximal, characteristic, complemented
4#27161717C211yesmaximal, characteristic, complemented
5#27161464C211yesmaximal, characteristic, complemented
6#27161328C211yesmaximal, characteristic, complemented
7#27161327C211yesmaximal, characteristic, complemented
8#27161297C211yesmaximal, characteristic, complemented
9#27161296C211yesmaximal, characteristic, complemented
10#27161240C211yesmaximal, characteristic, complemented
11#27161215C211yesmaximal, characteristic, complemented
12#27161214C211yesmaximal, characteristic, complemented
13#27161159C211yesmaximal, characteristic, complemented
14#27161154C211yesmaximal, characteristic, complemented
15#452×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C22×D788C2211yescharacteristic, complemented
18C2×D4×D197C2211yescharacteristic, complemented
19#20263353—24no—
20#20263351—24no—
21C2×#15691345C2211yescharacteristic, complemented
22C2×#15691345C2211yescharacteristic, complemented
23C2×#15691345C2211yescharacteristic, complemented
24C2×D1576C2211yescharacteristic
25C2×D1576C2211yescharacteristic
26C2×#15691343C2211yescharacteristic
27C2×#15691343C2211yescharacteristic
28C2×#15691342C2211yescharacteristic, complemented
29C2×#15691341C2211yescharacteristic, complemented
30#20263315C2211yescharacteristic, complemented
31#20263315C2211yescharacteristic, complemented
32#20263311—21no—
33#20263309—21no—
34#20263302C2211yescharacteristic, complemented
35#20263301—21no—
36#20263301C2211yescharacteristic
37#20263300C2211yescharacteristic
38#20263298—21no—
39#20263296—21no—
40#20263294—21no—
41D4×Dic197C2211yescharacteristic, complemented
42#20263288C2211yescharacteristic, complemented
43C4×C22⋊D197C2211yescharacteristic
44C2×#15691335—21no—
45C22⋊C4×D197—21no—
46#20263259C2211yescharacteristic, complemented
47#20263255C2211yescharacteristic, complemented
48#20263250C2211yescharacteristic, complemented
49#20263248C2211yescharacteristic, complemented
50#20263248C2211yescharacteristic, complemented
51#20263246C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53C8×Dic197C2211yescharacteristic, complemented
54C2×#41×C197C2211yescharacteristic
55C2×Q16×C197C2211yescharacteristic, complemented
56C2×SD16×C197C2211yescharacteristic, complemented
57C22⋊Q8×C197C2211yescharacteristic, complemented
58#103×C197C2211yescharacteristic, complemented
59#102×C197C2211yescharacteristic, complemented
60C22⋊C8×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C23×D197—21no—
63#15691359—22no—
64#15691359—41no—
65C2×D788—22no—
66C2×D788—21no—
67C2×D788—21no—
68C2×D788—21no—
69C2×D788—41no—
70C2×D788C2311yescharacteristic
71C2×D788C2311yescharacteristic
72C2×D788C2311yescharacteristic
73C2×C22⋊D197—21no—
74C2×C22⋊D197—21no—
75C2×C22⋊D197D411yescharacteristic
76D4×D197—22no—
77D4×D197—41no—
78C22×Dic197—21no—
79C2×C4×D197—21no—
80C2×C4×D197—21no—
81C2×C4×D197C2311yescharacteristic
82#15691345—22no—
83#15691345—22no—
84#15691345—22no—
85D1576—22no—
86D1576—41no—
87#15691343—22no—
88#15691343D414yescomplemented
89#15691342—22no—
90#15691341—22no—
91C2×#13096802C2311yescharacteristic
92C2×#13096802C2311yescharacteristic
93C4⋊Dic197C2311yescharacteristic
94#15691337—21no—
95#15691337D411yescharacteristic
96C22⋊Dic197—21no—
97C22⋊Dic197D411yescharacteristic
98#15691335—21no—
99#15691335—21no—
100#15691335—21no—
101#15691335—21no—
102#15691335—21no—
103#15691335—21no—
104#15691335—21no—
105#15691335—21no—
106#15691335D411yescharacteristic
107C4×Dic197C2311yescharacteristic
108#41×C197—22no—
109#41×C197—41no—
110C2×Q8×C197C2311yescharacteristic
111C2×Q8×C197C2311yescharacteristic
112C2×D4×C197—21no—
113C2×D4×C197C2311yescharacteristic
114C22×C4×C197—21no—
115C22×C4×C197C2311yescharacteristic
116Q16×C197—41no—
117SD16×C197—22no—
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—21no—
126C22×D197—42no—
127C22×D197—42no—
128C22×D197—42no—
129C22×D197—42no—
130C22×D197—81no—
131C22×D197—81no—
132C22×D197C2×D411yescharacteristic
133D788—21no—
134D788—21no—
135D788—21no—
136D788—21no—
137D788—21no—
138D788—42no—
139D788—42no—
140D788—81no—
141D788C2×D412yes—
142C22⋊D197—41no—
143C22⋊D197—42no—
144C22⋊D197—81no—
145C2×Dic197—21no—
146C2×Dic197—81no—
147C2×Dic197C2×D411yescharacteristic
148C2×Dic197C2×D411yescharacteristic
149C2×Dic197C2×D411yescharacteristic
150C4×D197—21no—
151C4×D197—21no—
152C4×D197—42no—
153C4×D197—81no—
154#13096802—21no—
155#13096802C2×D412yes—
156C23×C197—21no—
157C23×C197C2×D411yescharacteristic
158Q8×C197—21no—
159Q8×C197—21no—
160Q8×C197—21no—
161Q8×C197—21no—
162D4×C197—21no—
163D4×C197—42no—
164D4×C197C2×D412yes—
165C2×C4×C197—21no—
166C2×C4×C197—21no—
167C2×C4×C197—21no—
168C2×C4×C197—21no—
169C2×C4×C197—21no—
170C2×C4×C197—42no—
171C2×C4×C197C2×D411yescharacteristic
172C2×C4×C197C2411yescharacteristic
173C8×C197—21no—
174C8×C197—21no—
175C22×C197—21no—
176C22×C197—22no—
177C22×C197—21no—
178C22×C197—21no—
179C22×C197—41no—
180C22×C197—41no—
181C22×C197C22×D411yescharacteristic
182C2×D197—21no—
183C2×D197—22no—
184C2×D197—21no—
185C2×D197—21no—
186C2×D197—41no—
187C2×D197—41no—
188C2×D197—41no—
189C2×D197—41no—
190C2×D197—41no—
191C2×D197—41no—
192C2×D197—41no—
193C2×D197—41no—
194C2×D197—84no—
195C2×D197—84no—
196C4×C197—21no—
197C4×C197—41no—
198C4×C197—41no—
199C4×C197—41no—
200C4×C197C22×D411yescharacteristic
201C4×C197H(5,𝔽2)11yescharacteristic
202Dic197—21no—
203Dic197—21no—
204Dic197—21no—
205Dic197—41no—
206C2×C197—41no—
207C2×C197—42no—
208C2×C197D4211yescharacteristic
209C2×C197#57511yescharacteristic
210C2×C197#57611yescharacteristic
211D197—41no—
212D197—42no—
213D197—81no—
214D197—81no—
215C197#327311yescharacteristic, complemented, Sylow
216#3273—1971nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1971no—
218C2×AGL(1,ℤ/8)—1971no—
219#546—1971no—
220#546—1971no—
221#495—1971no—
222#468—1971no—
223#468—1971no—
224#463—1971no—
225#463—1971no—
226#452—1971no—
227#452D19711yescharacteristic, complemented
228#447—1971no—
229#447—1971no—
230C4×D8—1971no—
231#436—1971no—
232C2×#41—1971no—
233C2×#41C2×D19711yescharacteristic
234C22×D4—1971no—
235C22×D4—1971no—
236AGL(1,ℤ/8)—3944no—
237AGL(1,ℤ/8)—3944no—
238C2×Q16C2×D19711yescharacteristic, complemented
239C2×SD16—1971no—
240C2×SD16—1971no—
241C2×SD16—1971no—
242C2×SD16C2×D19711yescharacteristic, complemented
243C2×D8—1971no—
244C2×D8—1971no—
245C2×D8—1971no—
246C2×D8—1971no—
247C2×M16—1971no—
248C2×M16—1971no—
249#124—1971no—
250#124—1971no—
251#123—3941no—
252#123—3941no—
253C22⋊Q8—1971no—
254C22⋊Q8C2×D19711yescharacteristic, complemented
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#103C2×D19711yescharacteristic, complemented
270#102—1971no—
271#102—1971no—
272#102—1971no—
273#102C2×D19711yescharacteristic, complemented
274C22⋊C8—1971no—
275C22⋊C8C2×D19711yescharacteristic, complemented
276C4×C8—1971no—
277C24—3941no—
278C24—3941no—
279#41—22no—
280#41—41no—
281#41—7881no—
282#41—3942no—
283C2×Q8C22×D19711yescharacteristic
284C2×Q8C22×D19711yescharacteristic
285C2×D4—21no—
286C2×D4—7881no—
287C2×D4—7881no—
288C2×D4—3941no—
289C2×D4—3942no—
290C2×D4—3941no—
291C2×D4—3942no—
292C2×D4—3941no—
293C2×D4—3941no—
294C2×D4—3941no—
295C2×D4—1971no—
296C2×D4—1971no—
297C2×D4—1971no—
298C2×D4—1971no—
299C2×D4C22×D19711yescharacteristic
300C22×C4—21no—
301C22×C4—3941no—
302C22×C4—3941no—
303C22×C4—3941no—
304C22×C4—1971no—
305C22×C4C22×D19711yescharacteristic
306Q16—41no—
307SD16—22no—
308SD16—3942no—
309SD16—3942no—
310SD16—3942no—
311D8—7881no—
312D8—3942no—
313D8—3942no—
314D8—1974no—
315M16—3942no—
316M16—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—7882no—
341C23—7882no—
342C23—7882no—
343C23—7882no—
344C23—15761no—
345C23—15761no—
346C23—3941no—
347C23—3941no—
348C23—3941no—
349C23—1971no—
350C23D4×D19711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—21no—
354Q8—21no—
355D4—21no—
356D4—42no—
357D4—7882no—
358D4—7882no—
359D4—7882no—
360D4—7881no—
361D4—15761no—
362D4—15761no—
363D4—3941no—
364D4—3941no—
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—42no—
376C2×C4—7882no—
377C2×C4—15761no—
378C2×C4—15761no—
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—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—7881no—
398C22—7881no—
399C22—7881no—
400C22—7881no—
401C22—7881no—
402C22—7881no—
403C22—7881no—
404C22—7881no—
405C22—15764no—
406C22—15764no—
407C22—3941no—
408C22—3941no—
409C22—3942no—
410C22—3941no—
411C22C2×D4×D19711yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—7881no—
417C4—3941no—
418C4—3941no—
419C4—3941no—
420C4C2×D4×D19711yescharacteristic
421C4#2026337111yescharacteristic
422C2—41no—
423C2—42no—
424C2—7882no—
425C2—7881no—
426C2—15761no—
427C2—15761no—
428C2#2716171111yescharacteristic
429C2#2716187811yescharacteristic
430C2#2716188111yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#452×C197C2
socleC22×C197C22×D4
hypercenter#452D197
hypocenterC197#3273
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#20263353C2semidirect
2. C2×#20263351C2semidirect
3. #27161720C2semidirect
4. #27161717C2semidirect
5. #27161464C2semidirect
6. #27161328C2semidirect
7. #27161327C2semidirect
8. #27161297C2semidirect
9. #27161296C2semidirect
10. #27161240C2semidirect
11. #27161215C2semidirect
12. #27161214C2semidirect
13. #27161159C2semidirect
14. #27161154C2semidirect
15. #452×C197C2semidirect
17. C22×D788C22semidirect
18. C2×D4×D197C22semidirect
21. C2×#15691345C22semidirect
22. C2×#15691345C22semidirect
23. C2×#15691345C22semidirect
28. C2×#15691342C22semidirect
29. C2×#15691341C22semidirect
30. #20263315C22semidirect
31. #20263315C22semidirect
34. #20263302C22semidirect
41. D4×Dic197C22semidirect
42. #20263288C22semidirect
46. #20263259C22semidirect
47. #20263255C22semidirect
48. #20263250C22semidirect
49. #20263248C22semidirect
50. #20263248C22semidirect
51. #20263246C22semidirect
52. #20263238C22semidirect
53. C8×Dic197C22semidirect
55. C2×Q16×C197C22semidirect
56. C2×SD16×C197C22semidirect
57. C22⋊Q8×C197C22semidirect
58. #103×C197C22semidirect
59. #102×C197C22semidirect
60. C22⋊C8×C197C22semidirect
88. #15691343D4semidirect
215. C197#3273semidirect
227. #452D197semidirect
238. C2×Q16C2×D197semidirect
242. C2×SD16C2×D197semidirect
254. C22⋊Q8C2×D197semidirect
269. #103C2×D197semidirect
273. #102C2×D197semidirect
275. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633532
C2×#202633512
#271617202
#271617172
#271614642
#271613282
#271613272
#271612972
#271612962
#271612402
#271612152
#271612142
#271611592
#271611542
#452×C1972
#3273197