Finite Group Databasebeta

← #38745780

Subgroups — #38745780

Order 25216.

632 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263312C211yesmaximal, characteristic
2C2×#20263291C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D197C211yesmaximal, characteristic
4#27160983C211yesmaximal, characteristic, complemented
5#27160979C211yesmaximal, characteristic
6#27160972C211yesmaximal, characteristic, complemented
7#27160971C211yesmaximal, characteristic, complemented
8#27160969C211yesmaximal, characteristic, complemented
9#27160967C211yesmaximal, characteristic, complemented
10#27160950C211yesmaximal, characteristic, complemented
11C4⋊C4×Dic197C211yesmaximal, characteristic, complemented
12#27160930C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic197C211yesmaximal, characteristic, complemented
14#27160926C211yesmaximal, characteristic, complemented
15#385×C197C211yesmaximal, characteristic, complemented
16C22×Dic394C2211yescharacteristic
17C22×C22⋊D197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19#20263312C418yescomplemented
20#20263291—24no—
21C4×C22⋊D197—24no—
22C2×C4⋊Dic197C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×C22⋊Dic197C2211yescharacteristic
27C2×C22⋊Dic197C2211yescharacteristic
28C2×C22⋊Dic197C2211yescharacteristic
29C2×#15691335C2211yescharacteristic
30C2×#15691335C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×#15691335C2211yescharacteristic
33C2×#15691335C2211yescharacteristic
34C2×C4×Dic197C2211yescharacteristic
35C2×C4×Dic197C2211yescharacteristic
36C2×C4×Dic197C2211yescharacteristic
37C2×C4×Dic197C2211yescharacteristic
38C2×C4×Dic197C2211yescharacteristic
39C2×C4×Dic197C2211yescharacteristic
40C2×C4×Dic197C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45#20263233C2211yescharacteristic
46#20263233C2211yescharacteristic
47C23×C4×C197C2211yescharacteristic
48C2×C4⋊C4×C197C2211yescharacteristic
49C2×C4⋊C4×C197C2211yescharacteristic
50C2×C22⋊C4×C197C2211yescharacteristic
51C2×C22⋊C4×C197C2211yescharacteristic
52#95×C197C2211yescharacteristic
53#95×C197C2211yescharacteristic
54C23×D197C2311yescharacteristic
55C2×Dic394—22no—
56C2×Dic394—22no—
57C2×Dic394C2×C414yescomplemented
58C2×C22⋊D197—22no—
59C2×C22⋊D197—22no—
60C2×C22⋊D197C2×C414yes—
61C22×Dic197C2311yescharacteristic
62C22×Dic197C2311yescharacteristic
63C22×Dic197C2311yescharacteristic
64C22×Dic197C2311yescharacteristic
65C22×Dic197C2311yescharacteristic
66C22×Dic197C2311yescharacteristic
67C22×Dic197C2311yescharacteristic
68C2×C4×D197—22no—
69C2×C4×D197—22no—
70C2×C4×D197—41no—
71C4⋊Dic197—22no—
72#15691337—22no—
73#15691337—22no—
74#15691337D414yes—
75C22⋊Dic197—22no—
76C22⋊Dic197—22no—
77C22⋊Dic197C2×C414yescomplemented
78#15691335—22no—
79#15691335—22no—
80#15691335—41no—
81#15691335C2×C414yescomplemented
82#15691335C2×C414yescomplemented
83C4×Dic197—22no—
84C4×Dic197—22no—
85C4×Dic197—22no—
86C4×Dic197—22no—
87C4×Dic197—41no—
88C4×Dic197—41no—
89C4×Dic197C2×C414yescomplemented
90C24×C197C2311yescharacteristic
91C22×C4×C197—22no—
92C22×C4×C197—22no—
93C22×C4×C197—41no—
94C22×C4×C197C2311yescharacteristic
95C22×C4×C197C2311yescharacteristic
96C22×C4×C197C2311yescharacteristic
97C22×C4×C197C2311yescharacteristic
98C22×C4×C197C2311yescharacteristic
99C22×C4×C197C2311yescharacteristic
100C4⋊C4×C197—22no—
101C4⋊C4×C197—22no—
102C22⋊C4×C197—41no—
103C22⋊C4×C197C2×C414yescomplemented
104C22×D197—21no—
105C22×D197—21no—
106C22×D197—21no—
107C22×D197—21no—
108C22×D197—21no—
109C22×D197—21no—
110C22×D197C22×C412yes—
111Dic394—28no—
112C22⋊D197—28no—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C2×Dic197C22×C412yes—
128C2×Dic197C22×C412yes—
129C2×Dic197C22×C412yes—
130C2×Dic197C2×D412yes—
131C2×Dic197C2×D412yes—
132C2×Dic197#4112yes—
133C2×Dic197#4112yes—
134C4×D197—44no—
135C23×C197—21no—
136C23×C197—21no—
137C23×C197—21no—
138C23×C197—21no—
139C23×C197—21no—
140C23×C197—21no—
141C23×C197C22×C412yes—
142C23×C197C2411yescharacteristic
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—21no—
154C2×C4×C197—21no—
155C2×C4×C197—21no—
156C2×C4×C197—21no—
157C2×C4×C197—44no—
158C2×C4×C197C22×C412yes—
159C2×C4×C197C22×C412yes—
160C2×C4×C197C2×D412yes—
161C2×C4×C197#4112yes—
162C22×C197—22no—
163C22×C197—22no—
164C22×C197—22no—
165C22×C197—41no—
166C22×C197—41no—
167C22×C197—41no—
168C22×C197—41no—
169C22×C197C23×C411yescharacteristic
170C22×C197C22×D411yescharacteristic
171C22×C197C2×#4111yescharacteristic
172C22×C197C2×#4111yescharacteristic
173C22×C197C2×#4111yescharacteristic
174C22×C197H(5,𝔽2)11yescharacteristic
175C22×C1972−1+411yescharacteristic
176C2×D197—22no—
177C2×D197—22no—
178C2×D197—22no—
179C2×D197—41no—
180C2×D197—41no—
181C2×D197—41no—
182C2×D197—41no—
183C4×C197—22no—
184C4×C197—22no—
185C4×C197—22no—
186C4×C197—22no—
187C4×C197—41no—
188C4×C197—41no—
189Dic197—22no—
190Dic197—22no—
191Dic197—22no—
192Dic197—22no—
193Dic197—41no—
194Dic197—41no—
195Dic197C4×D414yes—
196C2×C197—42no—
197C2×C197C2×C4×D411yescharacteristic
198C2×C197C4×#4111yescharacteristic
199C2×C197#52011yescharacteristic
200C2×C197#54611yescharacteristic
201C2×C197#54811yescharacteristic
202C2×C197#55611yescharacteristic
203C2×C197#55911yescharacteristic
204D197—42no—
205C197#236711yescharacteristic, complemented, Sylow
206#2367—1971nomaximal, Sylow
207C2×#124—1971no—
208C2×C4×Q8—1971no—
209C2×C4×D4—1971no—
210#391—1971no—
211#390—1971no—
212#388—1971no—
213#388—1971no—
214#388—1971no—
215#388—1971no—
216#385—1971no—
217#385D19711yescharacteristic, complemented
218C4×C4⋊C4—1971no—
219C4×C4⋊C4—1971no—
220C4×C22⋊C4—1971no—
221C4×C22⋊C4—1971no—
222C22×Q8—1971no—
223C22×D4—1971no—
224C23×C4—1971no—
225C23×C4C2×D19711yescharacteristic
226#124—1978no—
227C4×Q8—3944no—
228C4×D4—3944no—
229C2×C4⋊C4—1971no—
230C2×C4⋊C4—1971no—
231C2×C4⋊C4—1971no—
232C2×C4⋊C4—1971no—
233C2×C4⋊C4C2×D19711yescharacteristic
234C2×C4⋊C4C2×D19711yescharacteristic
235C2×C22⋊C4—1971no—
236C2×C22⋊C4—1971no—
237C2×C22⋊C4—1971no—
238C2×C22⋊C4—1971no—
239C2×C22⋊C4—1971no—
240C2×C22⋊C4—1971no—
241C2×C22⋊C4—1971no—
242C2×C22⋊C4—1971no—
243C2×C22⋊C4C2×D19711yescharacteristic
244C2×C22⋊C4C2×D19711yescharacteristic
245C2×C42—1971no—
246C2×C42—1971no—
247C2×C42—1971no—
248C2×C42—1971no—
249C2×C42—1971no—
250C2×C42—1971no—
251C2×C42—1971no—
252#95—1971no—
253#95—1971no—
254#95—1971no—
255#95—1971no—
256#95—1971no—
257#95—1971no—
258#95C2×D19711yescharacteristic
259#95C2×D19711yescharacteristic
260C24—1971no—
261C24C22×D19711yescharacteristic
262C2×Q8—3942no—
263C2×Q8—3942no—
264C2×Q8—1974no—
265C2×D4—3942no—
266C2×D4—3942no—
267C2×D4—1974no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—7881no—
272C22×C4—3942no—
273C22×C4—3942no—
274C22×C4—1971no—
275C22×C4—1971no—
276C22×C4—1971no—
277C22×C4—1971no—
278C22×C4—1971no—
279C22×C4—1971no—
280C22×C4—1971no—
281C22×C4C22×D19711yescharacteristic
282C22×C4C22×D19711yescharacteristic
283C22×C4C22×D19711yescharacteristic
284C22×C4C22×D19711yescharacteristic
285C22×C4C22×D19711yescharacteristic
286C22×C4C22×D19711yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—3942no—
290C4⋊C4—3942no—
291C4⋊C4—3942no—
292C4⋊C4—1974no—
293C22⋊C4—41no—
294C22⋊C4—7881no—
295C22⋊C4—3942no—
296C22⋊C4—3942no—
297C22⋊C4—3942no—
298C22⋊C4—3942no—
299C22⋊C4—1974no—
300C22⋊C4—1974no—
301C22⋊C4—1974no—
302C22⋊C4C4×D19714yescomplemented
303C42—7881no—
304C42—7881no—
305C42—3942no—
306C42—3942no—
307C42—3942no—
308C42—3942no—
309C42—1974no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—3941no—
317C23—3941no—
318C23—3941no—
319C23—3941no—
320C23—3941no—
321C23—3941no—
322C23—1972no—
323C23C2×C4×D19712yes—
324C23C23×D19711yescharacteristic
325Q8—3948no—
326D4—3948no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—44no—
342C2×C4—7884no—
343C2×C4—3941no—
344C2×C4—3941no—
345C2×C4—3941no—
346C2×C4—3941no—
347C2×C4—3941no—
348C2×C4—3941no—
349C2×C4—3941no—
350C2×C4—3941no—
351C2×C4—3941no—
352C2×C4—3941no—
353C2×C4—3941no—
354C2×C4—3941no—
355C2×C4—3941no—
356C2×C4—3941no—
357C2×C4—1972no—
358C2×C4—1972no—
359C2×C4—1972no—
360C2×C4—1972no—
361C2×C4—1972no—
362C2×C4—1972no—
363C2×C4—1972no—
364C2×C4C2×C4×D19712yes—
365C2×C4C2×C4×D19712yes—
366C2×C4D4×D19712yes—
367C2×C4#1569135712yes—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—7881no—
376C22—7881no—
377C22—7881no—
378C22—7881no—
379C22—3942no—
380C22—3942no—
381C22—3942no—
382C22C22×C4×D19711yescharacteristic
383C22C2×D4×D19711yescharacteristic
384C22#41×D19711yescharacteristic
385C22#41×D19711yescharacteristic
386C22C2×#1569135711yescharacteristic
387C22#2026337111yescharacteristic
388C22#2026337411yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—7881no—
396C4—7881no—
397C4—3942no—
398C4—3942no—
399C4—3942no—
400C4—3942no—
401C4—1974no—
402C2—42no—
403C2—7882no—
404C2C4×D4×D19711yescharacteristic
405C2C4×#1569135711yescharacteristic
406C2#2716156011yescharacteristic
407C2#2716171611yescharacteristic
408C2#2716173911yescharacteristic
409C2#2716180011yescharacteristic
410C2#2716181311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#385×C197C2
socle, cosocleC23×C197C24
prime radicalC22×C22⋊D197C22
hypercenter#385D197
hypocenterC197#2367
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#20263291C2semidirect
4. #27160983C2semidirect
6. #27160972C2semidirect
7. #27160971C2semidirect
8. #27160969C2semidirect
9. #27160967C2semidirect
10. #27160950C2semidirect
11. C4⋊C4×Dic197C2semidirect
12. #27160930C2semidirect
13. C4×C22⋊Dic197C2semidirect
14. #27160926C2semidirect
15. #385×C197C2semidirect
19. #20263312C4semidirect
57. C2×Dic394C2×C4semidirect
77. C22⋊Dic197C2×C4semidirect
81. #15691335C2×C4semidirect
82. #15691335C2×C4semidirect
89. C4×Dic197C2×C4semidirect
103. C22⋊C4×C197C2×C4semidirect
205. C197#2367semidirect
217. #385D197semidirect
302. C22⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633122
C2×#202632912
C2×C4×C22⋊D1972
#271609832
#271609792
#271609722
#271609712
#271609692
#271609672
#271609502
C4⋊C4×Dic1972
#271609302
C4×C22⋊Dic1972
#271609262
#385×C1972
#2367197