Finite Group Databasebeta

← #38754223

Subgroups — #38754223

Order 25216.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263357C211yesmaximal, characteristic, complemented
2C22×#15691345C211yesmaximal, characteristic, complemented
3#27161725C211yesmaximal, characteristic
4#27161471C211yesmaximal, characteristic, complemented
5#27161327C211yesmaximal, characteristic, complemented
6#27161318C211yesmaximal, characteristic, complemented
7#27161301C211yesmaximal, characteristic, complemented
8#27161281C211yesmaximal, characteristic, complemented
9#27161239C211yesmaximal, characteristic, complemented
10#27161235C211yesmaximal, characteristic, complemented
11#27161229C211yesmaximal, characteristic, complemented
12#27161218C211yesmaximal, characteristic, complemented
13#27161177C211yesmaximal, characteristic, complemented
14#27161151C211yesmaximal, characteristic, complemented
15D4×Q8×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×D788C2211yescharacteristic
18#20263357—24no—
19C2×#15691348C2211yescharacteristic, complemented
20C2×#15691348C2211yescharacteristic, complemented
21C2×#15691348C2211yescharacteristic, complemented
22C2×#15691345—24no—
23C2×#15691345C2211yescharacteristic
24C2×#15691345C2211yescharacteristic
25C2×#15691345C2211yescharacteristic
26C2×#15691345C2211yescharacteristic
27C2×#15691345C2211yescharacteristic
28C2×#15691345C2211yescharacteristic
29C2×#15691342C2211yescharacteristic, complemented
30C22×#13096802C2211yescharacteristic
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#20263250C2211yescharacteristic, complemented
43#20263250C2211yescharacteristic, complemented
44#20263249C2211yescharacteristic, complemented
45#20263246C2211yescharacteristic
46#20263246C2211yescharacteristic
47C22⋊#13096802C2211yescharacteristic
48C22⋊#13096802C2211yescharacteristic, complemented
49#20263237C2211yescharacteristic, complemented
50C22×Q8×C197C2211yescharacteristic
51C22×Q8×C197C2211yescharacteristic, complemented
52C4⋊Q8×C197—21no—
53C4⋊Q8×C197C2211yescharacteristic, complemented
54C22⋊Q8×C197—21no—
55C22⋊Q8×C197—21no—
56C22⋊Q8×C197C2211yescharacteristic
57C22⋊Q8×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#15691348—22no—
76#15691348—41no—
77#15691345—22no—
78#15691345—22no—
79#15691345—22no—
80#15691345—22no—
81#15691345—41no—
82#15691345D414yescomplemented
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—
96C2×Q8×C197—21no—
97C2×Q8×C197—21no—
98C2×Q8×C197—22no—
99C2×Q8×C197—22no—
100C2×Q8×C197—41no—
101C2×Q8×C197—41no—
102C2×Q8×C197C2311yescharacteristic
103C2×Q8×C197C2311yescharacteristic
104C2×Q8×C197C2311yescharacteristic
105C2×D4×C197D411yescharacteristic
106C22×C4×C197—21no—
107C22×C4×C197—21no—
108C22×C4×C197C2311yescharacteristic
109C22×C4×C197C2311yescharacteristic
110C4⋊C4×C197—21no—
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×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×C197C2×D411yescharacteristic
146C23×C197C2×D411yescharacteristic
147Q8×C197—21no—
148Q8×C197—21no—
149Q8×C197—21no—
150Q8×C197—21no—
151Q8×C197—21no—
152Q8×C197—42no—
153Q8×C197—42no—
154Q8×C197C2×D412yes—
155D4×C197—22no—
156C2×C4×C197—21no—
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×C197C2×D411yescharacteristic
167C2×C4×C197C2×D411yescharacteristic
168C2×C4×C197C2×D411yescharacteristic
169C2×C4×C197C2×D411yescharacteristic
170C2×C4×C197C2411yescharacteristic
171C22×C197—21no—
172C22×C197—21no—
173C22×C197—21no—
174C22×C197—21no—
175C22×C197—21no—
176C22×C197#13712yescomplemented
177C22×C197C22×D411yescharacteristic
178C2×D197—22no—
179C2×D197—41no—
180C2×D197—41no—
181C2×D197—41no—
182C2×D197—41no—
183C2×D197—41no—
184C2×D197—84no—
185C4×C197—21no—
186C4×C197—21no—
187C4×C197—22no—
188C4×C197—21no—
189C4×C197—21no—
190C4×C197—41no—
191C4×C197—41no—
192C4×C197—41no—
193C4×C197C22×D411yescharacteristic
194C4×C197H(5,𝔽2)11yescharacteristic
195Dic197—41no—
196Dic197—41no—
197C2×C197—22no—
198C2×C197—41no—
199C2×C197D4211yescharacteristic
200C2×C197C2×#13711yescharacteristic
201C2×C197#57711yescharacteristic
202D197—42no—
203D197—81no—
204C197#326611yescharacteristic, complemented, Sylow
205#3266—1971nomaximal, Sylow
206C2×#137—1971no—
207C22×SD16—1971no—
208D4×Q8D19711yescharacteristic, complemented
209#546—1971no—
210#497—1971no—
211#468—1971no—
212#467—1971no—
213#464—1971no—
214#461—1971no—
215#452—1971no—
216#451—1971no—
217#450—1971no—
218#448—1971no—
219#440—1971no—
220#435—1971no—
221C2×#41—1971no—
222C22×Q8C2×D19711yescharacteristic
223C22×Q8C2×D19711yescharacteristic, complemented
224C22×D4—1971no—
225#137—3944no—
226C2×Q16—1971no—
227C2×Q16—1971no—
228C2×Q16—1971no—
229C2×SD16—3944no—
230C2×SD16—1971no—
231C2×SD16—1971no—
232C2×SD16—1971no—
233C2×SD16—1971no—
234C2×SD16—1971no—
235C2×SD16—1971no—
236C2×M16—1971no—
237C22×C8—1971no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D19711yescharacteristic, complemented
240#124—1971no—
241#123—3941no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—1971no—
245C22⋊Q8C2×D19711yescharacteristic
246C22⋊Q8C2×D19711yescharacteristic, complemented
247#121—3941no—
248#121—1971no—
249#120—3941no—
250C4×Q8C2×D19711yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—1971no—
253C4×D4C2×D19711yescharacteristic
254C2×C22⋊C4—3941no—
255#107—1971no—
256C4⋊C8—1971no—
257#103—1971no—
258#103—1971no—
259#103—1971no—
260#103—1971no—
261#102—1971no—
262#102—1971no—
263C22⋊C8—1971no—
264C22⋊C8—1971no—
265#97—1971no—
266C24—3941no—
267#41—7881no—
268#41—3942no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—1971no—
276C2×Q8C22×D19711yescharacteristic
277C2×Q8C22×D19711yescharacteristic
278C2×Q8C22×D19711yescharacteristic
279C2×D4—7881no—
280C2×D4—3942no—
281C2×D4—3941no—
282C2×D4—3941no—
283C2×D4—3941no—
284C2×D4—1971no—
285C2×D4—1971no—
286C2×D4C22⋊D19711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—3941no—
290C22×C4—3941no—
291C22×C4C22×D19711yescharacteristic
292C22×C4C22×D19711yescharacteristic
293Q16—7881no—
294Q16—3942no—
295Q16—3942no—
296SD16—7881no—
297SD16—3942no—
298SD16—3942no—
299SD16—3942no—
300SD16—3942no—
301SD16—1974no—
302M16—3942no—
303C2×C8—3942no—
304C2×C8—1971no—
305C2×C8—1971no—
306C2×C8—1971no—
307C2×C8—1971no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—3941no—
314C4⋊C4—1971no—
315C4⋊C4C22⋊D19711yescharacteristic
316C4⋊C4C22×D19711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—3941no—
320C22⋊C4—3941no—
321C22⋊C4—3941no—
322C22⋊C4—3941no—
323C22⋊C4—3941no—
324C22⋊C4C22⋊D19711yescharacteristic
325C22⋊C4C22⋊D19711yescharacteristic
326C42—21no—
327C42C22×D19711yescharacteristic
328C23—7882no—
329C23—7882no—
330C23—15761no—
331C23—3941no—
332C23—3941no—
333C23C2×C22⋊D19711yescharacteristic
334C23C2×C22⋊D19711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—3941no—
343Q8—3941no—
344Q8D4×D19712yes—
345D4—22no—
346D4—7882no—
347D4—7882no—
348D4—15761no—
349D4—3941no—
350D4—3941no—
351D4—3941no—
352D4—1972no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—7882no—
364C2×C4—15761no—
365C2×C4—3941no—
366C2×C4—3941no—
367C2×C4C2×C22⋊D19711yescharacteristic
368C2×C4C2×C22⋊D19711yescharacteristic
369C2×C4C2×C22⋊D19711yescharacteristic
370C2×C4C2×C22⋊D19711yescharacteristic
371C2×C4C23×D19711yescharacteristic
372C8—3941no—
373C8—3941no—
374C8—3941no—
375C8—1972no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—7881no—
382C22—7881no—
383C22—7881no—
384C22—7881no—
385C22—7881no—
386C22—15764no—
387C22—3942no—
388C22#2026335712yescomplemented
389C22C22×C22⋊D19711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—7881no—
399C4—7881no—
400C4C2×D4×D19711yescharacteristic
401C4#2026337111yescharacteristic
402C2—22no—
403C2—41no—
404C2—7882no—
405C2—15761no—
406C2D4×C22⋊D19711yescharacteristic
407C2C2×#2026335711yescharacteristic
408C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingD4×Q8×C197C2
socleC22×C197C22×D4
prime radical#27161725C2
hypercenterD4×Q8D197
hypocenterC197#3266
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#20263357C2semidirect
2. C22×#15691345C2semidirect
4. #27161471C2semidirect
5. #27161327C2semidirect
6. #27161318C2semidirect
7. #27161301C2semidirect
8. #27161281C2semidirect
9. #27161239C2semidirect
10. #27161235C2semidirect
11. #27161229C2semidirect
12. #27161218C2semidirect
13. #27161177C2semidirect
14. #27161151C2semidirect
15. D4×Q8×C197C2semidirect
19. C2×#15691348C22semidirect
20. C2×#15691348C22semidirect
21. C2×#15691348C22semidirect
29. C2×#15691342C22semidirect
39. #20263259C22semidirect
40. C4⋊#13096802C22semidirect
41. #20263250C22semidirect
42. #20263250C22semidirect
43. #20263250C22semidirect
44. #20263249C22semidirect
48. C22⋊#13096802C22semidirect
49. #20263237C22semidirect
51. C22×Q8×C197C22semidirect
53. C4⋊Q8×C197C22semidirect
57. C22⋊Q8×C197C22semidirect
58. C4×Q8×C197C22semidirect
82. #15691345D4semidirect
176. C22×C197#137semidirect
204. C197#3266semidirect
208. D4×Q8D197semidirect
223. C22×Q8C2×D197semidirect
239. C4⋊Q8C2×D197semidirect
246. C22⋊Q8C2×D197semidirect
250. C4×Q8C2×D197semidirect
388. C22#20263357semidirect

Maximal subgroups

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

subgroupindex
C2×#202633572
C22×#156913452
#271617252
#271614712
#271613272
#271613182
#271613012
#271612812
#271612392
#271612352
#271612292
#271612182
#271611772
#271611512
D4×Q8×C1972
#3266197