Finite Group Databasebeta

← #38751401

Subgroups — #38751401

Order 25216.

520 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263332C211yesmaximal, characteristic, complemented
2C22×C8×D197C211yesmaximal, characteristic, complemented
3#27161588C211yesmaximal, characteristic
4#27161153C211yesmaximal, characteristic, complemented
5#27161151C211yesmaximal, characteristic, complemented
6#27161149C211yesmaximal, characteristic, complemented
7#27161147C211yesmaximal, characteristic, complemented
8C8×D788C211yesmaximal, characteristic, complemented
9C8×C22⋊D197C211yesmaximal, characteristic, complemented
10C8×C22⋊D197C211yesmaximal, characteristic, complemented
11#27161104C211yesmaximal, characteristic, complemented
12#27161044C211yesmaximal, characteristic, complemented
13#27161041C211yesmaximal, characteristic, complemented
14#27161033C211yesmaximal, characteristic, complemented
15C8×D4×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×C4×D197C2211yescharacteristic
18#20263332—24no—
19C2×#15691342C2211yescharacteristic, complemented
20C2×#15691341C2211yescharacteristic
21C2×#15691341C2211yescharacteristic
22C2×#15691341C2211yescharacteristic, complemented
23C22×#13096802C2211yescharacteristic
24C2×C8×D197—24no—
25C2×C8×D197C2211yescharacteristic
26C2×C8×D197C2211yescharacteristic
27C2×C8×D197C2211yescharacteristic
28C2×C8×D197C2211yescharacteristic
29C2×C8×D197C2211yescharacteristic, complemented
30#20263311C411yescharacteristic
31#20263308C411yescharacteristic
32#20263304C411yescharacteristic
33#20263303C411yescharacteristic
34#20263299C411yescharacteristic
35#20263298C411yescharacteristic
36#20263296C411yescharacteristic
37#20263294C411yescharacteristic
38C4×D788C2211yescharacteristic
39C4×C22⋊D197C2211yescharacteristic
40C4×C22⋊D197C2211yescharacteristic
41#20263283C2211yescharacteristic
42C4⋊#13096802C2211yescharacteristic, complemented
43#20263255C2211yescharacteristic, complemented
44#20263255C2211yescharacteristic, complemented
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic, complemented
47#20263238C2211yescharacteristic
48#20263238C2211yescharacteristic
49#20263238C2211yescharacteristic, complemented
50#20263238C2211yescharacteristic, complemented
51#20263237C2211yescharacteristic, complemented
52#20263236C2211yescharacteristic, complemented
53C8×Dic197C2211yescharacteristic, complemented
54C22×C8×C197C2211yescharacteristic
55C22×C8×C197C2211yescharacteristic, complemented
56C4×D4×C197C2211yescharacteristic
57C4⋊C8×C197C2211yescharacteristic, complemented
58C22⋊C8×C197C2211yescharacteristic
59C22⋊C8×C197C2211yescharacteristic, complemented
60C4×C8×C197C2211yescharacteristic, complemented
61C23×D197C2×C411yescharacteristic
62#15691358—22no—
63#15691358—22no—
64C2×Dic394C2×C411yescharacteristic
65C2×D788C2×C411yescharacteristic
66C2×D788C2×C411yescharacteristic
67C2×C22⋊D197C2×C411yescharacteristic
68C2×C22⋊D197C2×C411yescharacteristic
69C2×C22⋊D197C2×C411yescharacteristic
70C2×C22⋊D197C2×C411yescharacteristic
71C22×Dic197C2×C411yescharacteristic
72C2×C4×D197—22no—
73C2×C4×D197—22no—
74C2×C4×D197C2×C411yescharacteristic
75C2×C4×D197C2×C411yescharacteristic
76C2×C4×D197C2311yescharacteristic
77C2×C4×D197C2311yescharacteristic
78C2×C4×D197C2311yescharacteristic
79#15691342—22no—
80#15691341—22no—
81#15691341—22no—
82#15691341—22no—
83C2×#13096802—22no—
84C2×#13096802C2311yescharacteristic
85C2×#13096802C2311yescharacteristic
86C2×#13096802C2311yescharacteristic
87C2×#13096802C2311yescharacteristic
88C8×D197—22no—
89C8×D197—22no—
90C8×D197—22no—
91C8×D197—22no—
92C8×D197D414yescomplemented
93C4⋊Dic197C2×C411yescharacteristic
94#15691337C2×C411yescharacteristic
95#15691337C2×C411yescharacteristic
96#15691337C2×C411yescharacteristic
97#15691337C2×C411yescharacteristic
98C22⋊Dic197C2×C411yescharacteristic
99C22⋊Dic197C2×C411yescharacteristic
100#15691335C2×C411yescharacteristic
101#15691335C2×C411yescharacteristic
102#15691335C2×C411yescharacteristic
103#15691335C2×C411yescharacteristic
104#15691335C2×C411yescharacteristic
105#15691335C2×C411yescharacteristic
106C4×Dic197C2311yescharacteristic
107C2×D4×C197C2×C411yescharacteristic
108C22×C4×C197C2311yescharacteristic
109C22×C4×C197C2311yescharacteristic
110C2×C8×C197—22no—
111C2×C8×C197—22no—
112C2×C8×C197C2311yescharacteristic
113C2×C8×C197C2311yescharacteristic
114C2×C8×C197C2311yescharacteristic
115C2×C8×C197C2311yescharacteristic
116C4⋊C4×C197C2×C411yescharacteristic
117C22⋊C4×C197C2×C411yescharacteristic
118C22⋊C4×C197C2×C411yescharacteristic
119C42×C197C2311yescharacteristic
120C22×D197—22no—
121C22×D197—22no—
122C22×D197—22no—
123C22×D197C22×C411yescharacteristic
124C22×D197C22×C411yescharacteristic
125C22×D197C22×C411yescharacteristic
126Dic394—22no—
127D788—22no—
128D788—22no—
129C22⋊D197—22no—
130C22⋊D197—22no—
131C22⋊D197—22no—
132C22⋊D197—22no—
133C2×Dic197—22no—
134C2×Dic197C22×C411yescharacteristic
135C2×Dic197C22×C411yescharacteristic
136C2×Dic197C22×C411yescharacteristic
137C2×Dic197C22×C411yescharacteristic
138C2×Dic197C22×C411yescharacteristic
139C4×D197—22no—
140C4×D197—21no—
141C4×D197—21no—
142C4×D197—21no—
143C4×D197—21no—
144C4×D197—22no—
145C4×D197C2×D412yes—
146C4×D197#4112yes—
147#13096802—21no—
148#13096802—21no—
149#13096802—21no—
150#13096802C2×D412yes—
151C23×C197C22×C411yescharacteristic
152C23×C197C22×C411yescharacteristic
153D4×C197—22no—
154C2×C4×C197—21no—
155C2×C4×C197—21no—
156C2×C4×C197—21no—
157C2×C4×C197—21no—
158C2×C4×C197C22×C411yescharacteristic
159C2×C4×C197C22×C411yescharacteristic
160C2×C4×C197C22×C411yescharacteristic
161C2×C4×C197C22×C411yescharacteristic
162C2×C4×C197C2411yescharacteristic
163C8×C197—21no—
164C8×C197—21no—
165C8×C197—21no—
166C8×C197C2×D412yes—
167C22×C197—21no—
168C22×C197—21no—
169C22×C197—21no—
170C22×C197—21no—
171C22×C197—21no—
172C22×C197#13112yescomplemented
173C22×C197C23×C411yescharacteristic
174C2×D197—28no—
175C2×D197—21no—
176C2×D197—21no—
177C2×D197—21no—
178C2×D197—21no—
179C2×D197—21no—
180C2×D197—21no—
181C2×D197—21no—
182C2×D197C4×D412yes—
183C2×D197#13112yescomplemented
184C4×C197—21no—
185C4×C197—21no—
186C4×C197—21no—
187C4×C197—21no—
188C4×C197C22×D411yescharacteristic
189C4×C197C2×#4111yescharacteristic
190Dic197—21no—
191Dic197—21no—
192Dic197—21no—
193Dic197—21no—
194Dic197C4×D412yes—
195C2×C197—22no—
196C2×C197—41no—
197C2×C197C2×C4×D411yescharacteristic
198C2×C197C2×#13111yescharacteristic
199C2×C197C2×#13111yescharacteristic
200D197—22no—
201D197—22no—
202D197—41no—
203C197#292911yescharacteristic, complemented, Sylow
204#2929—1971nomaximal, Sylow
205C2×#131—1971no—
206C23×C8—1971no—
207#525—1971no—
208#435—1971no—
209#435—1971no—
210#435—1971no—
211#435—1971no—
212C8×D4—1971no—
213C8×D4—1971no—
214C8×D4—1971no—
215C8×D4D19711yescharacteristic, complemented
216#424—1971no—
217#408—1971no—
218#407—1971no—
219#405—1971no—
220C2×#41—1971no—
221C23×C4—1971no—
222#131—3944no—
223C2×M16—1971no—
224C2×M16—1971no—
225C2×M16—1971no—
226C2×M16—1971no—
227C22×C8—3944no—
228C22×C8—1971no—
229C22×C8—1971no—
230C22×C8—1971no—
231C22×C8—1971no—
232C22×C8—1971no—
233C22×C8—1971no—
234C22×C8C2×D19711yescharacteristic
235C22×C8C2×D19711yescharacteristic, complemented
236#123—1971no—
237#123—1971no—
238C22⋊Q8—1971no—
239C22⋊Q8—1971no—
240#121—1971no—
241#121—1971no—
242#120—1971no—
243#120—1971no—
244C4×D4—1971no—
245C4×D4—1971no—
246C4×D4—1971no—
247C4×D4C2×D19711yescharacteristic
248#117—1971no—
249C4⋊C8—1971no—
250C4⋊C8—1971no—
251C4⋊C8—1971no—
252C4⋊C8C2×D19711yescharacteristic, complemented
253C22⋊C8—1971no—
254C22⋊C8—1971no—
255C22⋊C8—1971no—
256C22⋊C8—1971no—
257C22⋊C8—1971no—
258C22⋊C8—1971no—
259C22⋊C8C2×D19711yescharacteristic
260C22⋊C8C2×D19711yescharacteristic, complemented
261#97—1971no—
262#97—1971no—
263C4×C8—1971no—
264C4×C8C2×D19711yescharacteristic, complemented
265C24—1971no—
266#41—3942no—
267#41—3942no—
268C2×Q8—1971no—
269C2×D4—1971no—
270C2×D4—1971no—
271C2×D4—1971no—
272C2×D4—1971no—
273C2×D4—1971no—
274C2×D4—1971no—
275C2×D4C4×D19711yescharacteristic
276C22×C4—3942no—
277C22×C4—3942no—
278C22×C4—1971no—
279C22×C4—1971no—
280C22×C4—1971no—
281C22×C4—1971no—
282C22×C4—1971no—
283C22×C4—1971no—
284C22×C4C22×D19711yescharacteristic
285C22×C4C22×D19711yescharacteristic
286M16—3942no—
287M16—3942no—
288M16—3942no—
289M16—3942no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—3942no—
293C2×C8—3942no—
294C2×C8—3942no—
295C2×C8—3942no—
296C2×C8—3942no—
297C2×C8—1971no—
298C2×C8—1971no—
299C2×C8—1971no—
300C2×C8—1971no—
301C2×C8—1974no—
302C2×C8C22×D19711yescharacteristic
303C2×C8C22×D19711yescharacteristic
304C2×C8C22×D19711yescharacteristic
305C2×C8C22×D19711yescharacteristic
306C4⋊C4—1971no—
307C4⋊C4—1971no—
308C4⋊C4—1971no—
309C4⋊C4—1971no—
310C4⋊C4—1971no—
311C4⋊C4C4×D19711yescharacteristic
312C22⋊C4—1971no—
313C22⋊C4—1971no—
314C22⋊C4—1971no—
315C22⋊C4—1971no—
316C22⋊C4—1971no—
317C22⋊C4—1971no—
318C22⋊C4—1971no—
319C22⋊C4—1971no—
320C22⋊C4C4×D19711yescharacteristic
321C22⋊C4C4×D19711yescharacteristic
322C42—1971no—
323C42C22×D19711yescharacteristic
324C23—3942no—
325C23—3942no—
326C23—3942no—
327C23—1971no—
328C23—1971no—
329C23—1971no—
330C23C2×C4×D19711yescharacteristic
331C23C2×C4×D19711yescharacteristic
332Q8—3942no—
333D4—22no—
334D4—3942no—
335D4—3942no—
336D4—3942no—
337D4—3942no—
338D4—3942no—
339D4—3942no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—3942no—
345C2×C4—3942no—
346C2×C4—3942no—
347C2×C4—3941no—
348C2×C4—3941no—
349C2×C4—3941no—
350C2×C4—3941no—
351C2×C4—1971no—
352C2×C4—1971no—
353C2×C4—1971no—
354C2×C4—1971no—
355C2×C4—1971no—
356C2×C4—1972no—
357C2×C4—1972no—
358C2×C4C2×C4×D19711yescharacteristic
359C2×C4C2×C4×D19711yescharacteristic
360C2×C4C2×C4×D19711yescharacteristic
361C2×C4C2×C4×D19711yescharacteristic
362C2×C4C23×D19711yescharacteristic
363C8—21no—
364C8—21no—
365C8—21no—
366C8—3941no—
367C8—3941no—
368C8—3941no—
369C8—1972no—
370C8D4×D19712yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—3948no—
377C22—3941no—
378C22—3941no—
379C22—3941no—
380C22—3941no—
381C22—3941no—
382C22—3941no—
383C22—3941no—
384C22—1972no—
385C22—1972no—
386C22#2026333212yescomplemented
387C22C22×C4×D19711yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—3941no—
393C4—3941no—
394C4—3941no—
395C4—3941no—
396C4—1972no—
397C4C2×D4×D19711yescharacteristic
398C4#41×D19711yescharacteristic
399C2—22no—
400C2—41no—
401C2—7881no—
402C2—3942no—
403C2—3942no—
404C2C4×D4×D19711yescharacteristic
405C2#131×D19711yescharacteristic
406C2C2×#2026333211yescharacteristic
subgroupG/H
centerC2×C8C22×D197
commutator, socleC22×C197C23×C4
FrattiniC2×C4C23×D197
FittingC8×D4×C197C2
prime radical#27161588C2
cosocleC2×C4×C197C24
hypercenterC8×D4D197
hypocenterC197#2929
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C2×C8 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20263332C2semidirect
2. C22×C8×D197C2semidirect
4. #27161153C2semidirect
5. #27161151C2semidirect
6. #27161149C2semidirect
7. #27161147C2semidirect
8. C8×D788C2semidirect
9. C8×C22⋊D197C2semidirect
10. C8×C22⋊D197C2semidirect
11. #27161104C2semidirect
12. #27161044C2semidirect
13. #27161041C2semidirect
14. #27161033C2semidirect
15. C8×D4×C197C2semidirect
19. C2×#15691342C22semidirect
22. C2×#15691341C22semidirect
29. C2×C8×D197C22semidirect
42. C4⋊#13096802C22semidirect
43. #20263255C22semidirect
44. #20263255C22semidirect
46. C22⋊#13096802C22semidirect
49. #20263238C22semidirect
50. #20263238C22semidirect
51. #20263237C22semidirect
52. #20263236C22semidirect
53. C8×Dic197C22semidirect
55. C22×C8×C197C22semidirect
57. C4⋊C8×C197C22semidirect
59. C22⋊C8×C197C22semidirect
60. C4×C8×C197C22semidirect
92. C8×D197D4semidirect
172. C22×C197#131semidirect
183. C2×D197#131semidirect
203. C197#2929semidirect
215. C8×D4D197semidirect
235. C22×C8C2×D197semidirect
252. C4⋊C8C2×D197semidirect
260. C22⋊C8C2×D197semidirect
264. C4×C8C2×D197semidirect
386. C22#20263332semidirect

Maximal subgroups

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

subgroupindex
C2×#202633322
C22×C8×D1972
#271615882
#271611532
#271611512
#271611492
#271611472
C8×D7882
C8×C22⋊D1972
#271611042
#271610442
#271610412
#271610332
C8×D4×C1972
#2929197