Finite Group Databasebeta

← #38745666

Subgroups — #38745666

Order 25216.

480 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263291C211yesmaximal, characteristic
2#27160983C211yesmaximal, characteristic
3#27160983C211yesmaximal, characteristic
4#27160966C211yesmaximal, characteristic
5#27160965C211yesmaximal, characteristic
6#27160960C211yesmaximal, characteristic
7#27160948C211yesmaximal, characteristic
8#27160943C211yesmaximal, characteristic
9#27160943C211yesmaximal, characteristic
10C4×#15691337C211yesmaximal, characteristic
11C4×#15691337C211yesmaximal, characteristic
12#27160930C211yesmaximal, characteristic
13#27160930C211yesmaximal, characteristic
14#27160930C211yesmaximal, characteristic
15#387×C197C211yesmaximal, characteristic
16C22×Dic394C2211yescharacteristic
17#20263291C418yescomplemented
18C2×C4⋊Dic197C2211yescharacteristic
19C2×#15691337C2211yescharacteristic
20C2×#15691337C2211yescharacteristic
21C2×#15691337C2211yescharacteristic
22C2×#15691337C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×#15691337C2211yescharacteristic
27C2×C4×Dic197C2211yescharacteristic
28C2×C4×Dic197C2211yescharacteristic
29C2×C4×Dic197C2211yescharacteristic
30C2×C4×Dic197C2211yescharacteristic
31C2×C4×Dic197C2211yescharacteristic
32C2×C4×Dic197C2211yescharacteristic
33C2×C4×Dic197C2211yescharacteristic
34C2×C4×Dic197C2211yescharacteristic
35#20263233C2211yescharacteristic
36#20263233C2211yescharacteristic
37#20263233C2211yescharacteristic
38#20263233C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45C2×C4⋊C4×C197C2211yescharacteristic
46C2×C4⋊C4×C197C2211yescharacteristic
47C2×C42×C197C2211yescharacteristic
48#95×C197C2211yescharacteristic
49#95×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52C2×Dic394—22no—
53C2×Dic394—22no—
54C2×Dic394C2×C414yes—
55C22×Dic197C2311yescharacteristic
56C22×Dic197C2311yescharacteristic
57C22×Dic197C2311yescharacteristic
58C22×Dic197C2311yescharacteristic
59C22×Dic197C2311yescharacteristic
60C22×Dic197C2311yescharacteristic
61C22×Dic197C2311yescharacteristic
62C22×Dic197C2311yescharacteristic
63C4⋊Dic197—41no—
64#15691337—22no—
65#15691337—22no—
66#15691337—22no—
67#15691337—22no—
68#15691337—22no—
69#15691337C2×C414yes—
70#15691337C2×C414yes—
71#15691337Q814yescomplemented
72C4×Dic197—22no—
73C4×Dic197—22no—
74C4×Dic197—41no—
75C4×Dic197—41no—
76C4×Dic197—41no—
77C4×Dic197C2×C414yes—
78C4×Dic197C2×C414yes—
79C4×Dic197C2×C414yes—
80C22×C4×C197C2311yescharacteristic
81C22×C4×C197C2311yescharacteristic
82C22×C4×C197C2311yescharacteristic
83C22×C4×C197C2311yescharacteristic
84C22×C4×C197C2311yescharacteristic
85C22×C4×C197C2311yescharacteristic
86C22×C4×C197C2311yescharacteristic
87C4⋊C4×C197—41no—
88C4⋊C4×C197C2×C414yes—
89C42×C197—41no—
90Dic394—28no—
91C2×Dic197—21no—
92C2×Dic197—21no—
93C2×Dic197—21no—
94C2×Dic197—21no—
95C2×Dic197—21no—
96C2×Dic197—21no—
97C2×Dic197—21no—
98C2×Dic197—21no—
99C2×Dic197—21no—
100C2×Dic197—21no—
101C2×Dic197—21no—
102C2×Dic197—21no—
103C2×Dic197—21no—
104C2×Dic197—21no—
105C2×Dic197—21no—
106C2×Dic197C22×C412yes—
107C2×Dic197C22×C412yes—
108C2×Dic197C22×C412yes—
109C2×Dic197C22×C412yes—
110C2×Dic197C2×Q812yes—
111C2×Dic197C2×Q812yes—
112C2×Dic197#4112yes—
113C2×Dic197#4112yes—
114C2×Dic197#4112yes—
115C23×C197C2411yescharacteristic
116C2×C4×C197—21no—
117C2×C4×C197—21no—
118C2×C4×C197—21no—
119C2×C4×C197—21no—
120C2×C4×C197—21no—
121C2×C4×C197—21no—
122C2×C4×C197—21no—
123C2×C4×C197—21no—
124C2×C4×C197—21no—
125C2×C4×C197—21no—
126C2×C4×C197—21no—
127C2×C4×C197—21no—
128C2×C4×C197—21no—
129C2×C4×C197—21no—
130C2×C4×C197—21no—
131C2×C4×C197—21no—
132C2×C4×C197—21no—
133C2×C4×C197C22×C412yes—
134C2×C4×C197C22×C412yes—
135C2×C4×C197C22×C412yes—
136C2×C4×C197C2×Q812yes—
137C22×C197C23×C411yescharacteristic
138C22×C197C22×Q811yescharacteristic
139C22×C197C2×#4111yescharacteristic
140C22×C197C2×#4111yescharacteristic
141C22×C197C2×#4111yescharacteristic
142C22×C197H(5,𝔽2)11yescharacteristic
143C22×C1972−1+411yescharacteristic
144C4×C197—22no—
145C4×C197—22no—
146C4×C197—22no—
147C4×C197—22no—
148C4×C197—41no—
149C4×C197—41no—
150C4×C197—41no—
151Dic197—22no—
152Dic197—22no—
153Dic197—22no—
154Dic197—41no—
155Dic197—41no—
156Dic197—41no—
157Dic197#11714yes—
158Dic197C4×Q814yes—
159C2×C197C2×#11711yescharacteristic
160C2×C197C2×C4×Q811yescharacteristic
161C2×C197#52011yescharacteristic
162C2×C197#55411yescharacteristic
163C2×C197#55611yescharacteristic
164C2×C197#55711yescharacteristic
165C2×C197#55911yescharacteristic
166C197#235511yescharacteristic, complemented, Sylow
167#2355—1971nomaximal, Sylow
168C2×C4×Q8—1971no—
169#391—1971no—
170#391—1971no—
171#387—1971no—
172#387—1971no—
173#387—1971no—
174#387D19711yescharacteristic
175#384—1971no—
176#382—1971no—
177#382—1971no—
178C4×C4⋊C4—1971no—
179C4×C4⋊C4—1971no—
180C4×C4⋊C4—1971no—
181C4×C4⋊C4—1971no—
182C4×C4⋊C4—1971no—
183C22×Q8—1971no—
184C4×Q8—1978no—
185C2×C4⋊C4—1971no—
186C2×C4⋊C4—1971no—
187C2×C4⋊C4—1971no—
188C2×C4⋊C4—1971no—
189C2×C4⋊C4—1971no—
190C2×C4⋊C4—1971no—
191C2×C4⋊C4—1971no—
192C2×C4⋊C4—1971no—
193C2×C4⋊C4—1971no—
194C2×C4⋊C4C2×D19711yescharacteristic
195C2×C4⋊C4C2×D19711yescharacteristic
196C2×C42—1971no—
197C2×C42—1971no—
198C2×C42—1971no—
199C2×C42—1971no—
200C2×C42—1971no—
201C2×C42—1971no—
202C2×C42—1971no—
203C2×C42—1971no—
204C2×C42C2×D19711yescharacteristic
205#95—1971no—
206#95—1971no—
207#95—1971no—
208#95—1971no—
209#95—1971no—
210#95—1971no—
211#95—1971no—
212#95—1971no—
213#95—1971no—
214#95—1971no—
215#95C2×D19711yescharacteristic
216#95C2×D19711yescharacteristic
217#95C2×D19711yescharacteristic
218#95C2×D19711yescharacteristic
219C2×Q8—3942no—
220C2×Q8—3942no—
221C2×Q8—1974no—
222C22×C4—1971no—
223C22×C4—1971no—
224C22×C4—1971no—
225C22×C4—1971no—
226C22×C4—1971no—
227C22×C4—1971no—
228C22×C4—1971no—
229C22×C4—1971no—
230C22×C4C22×D19711yescharacteristic
231C22×C4C22×D19711yescharacteristic
232C22×C4C22×D19711yescharacteristic
233C22×C4C22×D19711yescharacteristic
234C22×C4C22×D19711yescharacteristic
235C22×C4C22×D19711yescharacteristic
236C22×C4C22×D19711yescharacteristic
237C4⋊C4—41no—
238C4⋊C4—7881no—
239C4⋊C4—3942no—
240C4⋊C4—3942no—
241C4⋊C4—3942no—
242C4⋊C4—3942no—
243C4⋊C4—3942no—
244C4⋊C4—1974no—
245C4⋊C4—1974no—
246C4⋊C4—1974no—
247C4⋊C4C4×D19714yes—
248C42—41no—
249C42—7881no—
250C42—7881no—
251C42—7881no—
252C42—3942no—
253C42—3942no—
254C42—1974no—
255C42—1974no—
256C42—1974no—
257C23C23×D19711yescharacteristic
258Q8—3948no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—3941no—
277C2×C4—3941no—
278C2×C4—3941no—
279C2×C4—3941no—
280C2×C4—3941no—
281C2×C4—3941no—
282C2×C4—3941no—
283C2×C4—3941no—
284C2×C4—3941no—
285C2×C4—3941no—
286C2×C4—3941no—
287C2×C4—3941no—
288C2×C4—3941no—
289C2×C4—3941no—
290C2×C4—3941no—
291C2×C4—1972no—
292C2×C4—1972no—
293C2×C4—1972no—
294C2×C4—1972no—
295C2×C4—1972no—
296C2×C4—1972no—
297C2×C4—1972no—
298C2×C4—1972no—
299C2×C4—1972no—
300C2×C4C2×C4×D19712yes—
301C2×C4C2×C4×D19712yes—
302C2×C4C2×C4×D19712yes—
303C2×C4Q8×D19712yes—
304C22C22×C4×D19711yescharacteristic
305C22C2×Q8×D19711yescharacteristic
306C22#41×D19711yescharacteristic
307C22#41×D19711yescharacteristic
308C22#41×D19711yescharacteristic
309C22#2026337111yescharacteristic
310C22#2026337311yescharacteristic
311C4—22no—
312C4—22no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—7881no—
319C4—7881no—
320C4—7881no—
321C4—3942no—
322C4—3942no—
323C4—3942no—
324C4—1974no—
325C4—1974no—
326C2#117×D19711yescharacteristic
327C2C4×Q8×D19711yescharacteristic
328C2#2716155811yescharacteristic
329C2#2716178311yescharacteristic
330C2#2716179611yescharacteristic
331C2#2716180611yescharacteristic
332C2#2716181311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#387×C197C2
socle, prime radical, cosocleC23×C197C24
hypercenter#387D197
hypocenterC197#2355
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
17. #20263291C4semidirect
71. #15691337Q8semidirect
166. C197#2355semidirect

Maximal subgroups

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

subgroupindex
C2×#202632912
#271609832
#271609662
#271609652
#271609602
#271609482
#271609432
C4×#156913372
#271609302
#387×C1972
#2355197