Finite Group Databasebeta

← #38745455

Subgroups — #38745455

Order 25216.

460 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263324C211yesmaximal, characteristic
2#27160986C211yesmaximal, characteristic
3#27160983C211yesmaximal, characteristic
4#27160983C211yesmaximal, characteristic
5#27160976C211yesmaximal, characteristic
6#27160974C211yesmaximal, characteristic
7#27160966C211yesmaximal, characteristic
8#27160963C211yesmaximal, characteristic
9#27160961C211yesmaximal, characteristic
10#27160960C211yesmaximal, characteristic
11#27160943C211yesmaximal, characteristic
12C4×#15691337C211yesmaximal, characteristic
13#27160930C211yesmaximal, characteristic
14#27160924C211yesmaximal, characteristic
15#391×C197C211yesmaximal, characteristic
16C22×Dic394C2211yescharacteristic
17#20263324C418yescomplemented
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
33#20263233C2211yescharacteristic
34#20263233C2211yescharacteristic
35#20263233C2211yescharacteristic
36#20263233C2211yescharacteristic
37#20263233C2211yescharacteristic
38#20263233C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45C22×Q8×C197C2211yescharacteristic
46C2×C4⋊C4×C197C2211yescharacteristic
47C2×C42×C197C2211yescharacteristic
48#95×C197C2211yescharacteristic
49#95×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52C2×Dic394—41no—
53C2×Dic394—41no—
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—41no—
67#15691337—41no—
68#15691337C2×C414yes—
69#15691337C2×C414yes—
70#15691337C2×C414yes—
71#15691337C2×C414yes—
72C4×Dic197—22no—
73C4×Dic197—41no—
74C4×Dic197—41no—
75C4×Dic197—41no—
76C4×Dic197C2×C414yes—
77C4×Dic197Q814yescomplemented
78C2×Q8×C197—41no—
79C2×Q8×C197—41no—
80C2×Q8×C197C2×C414yes—
81C22×C4×C197C2311yescharacteristic
82C22×C4×C197C2311yescharacteristic
83C22×C4×C197C2311yescharacteristic
84C22×C4×C197C2311yescharacteristic
85C22×C4×C197C2311yescharacteristic
86C22×C4×C197C2311yescharacteristic
87C22×C4×C197C2311yescharacteristic
88C4⋊C4×C197—41no—
89C42×C197—41no—
90Dic394—44no—
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×Dic197—21no—
107C2×Dic197—21no—
108C2×Dic197C22×C412yes—
109C2×Dic197C22×C412yes—
110C2×Dic197C22×C412yes—
111C2×Dic197C22×C412yes—
112C2×Dic197C2×Q812yes—
113C2×Dic197C2×Q812yes—
114C2×Dic197#4112yes—
115C23×C197C2411yescharacteristic
116Q8×C197—44no—
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×C197—21no—
134C2×C4×C197C22×C412yes—
135C2×C4×C197C22×C412yes—
136C2×C4×C197C22×C412yes—
137C2×C4×C197C2×Q812yes—
138C22×C197C23×C411yescharacteristic
139C22×C197C22×Q811yescharacteristic
140C22×C197C2×#4111yescharacteristic
141C22×C197H(5,𝔽2)11yescharacteristic
142C22×C197H(5,𝔽2)11yescharacteristic
143C22×C1972−1+411yescharacteristic
144C22×C1972−1+411yescharacteristic
145C4×C197—22no—
146C4×C197—22no—
147C4×C197—22no—
148C4×C197—22no—
149C4×C197—41no—
150C4×C197—41no—
151C4×C197—41no—
152Dic197—22no—
153Dic197—22no—
154Dic197—22no—
155Dic197—22no—
156Dic197—41no—
157Dic197—41no—
158Dic197—41no—
159Dic197C4×Q814yes—
160C2×C197C2×C4×Q811yescharacteristic
161C2×C197#51811yescharacteristic
162C2×C197#51911yescharacteristic
163C2×C197#54211yescharacteristic
164C2×C197#54211yescharacteristic
165C2×C197#54411yescharacteristic
166C2×C197#54411yescharacteristic
167C197#233111yescharacteristic, complemented, Sylow
168#2331—1971nomaximal, Sylow
169C2×C4⋊Q8—1971no—
170#391—1971no—
171#391—1971no—
172#391—1971no—
173#391D19711yescharacteristic
174#389—1971no—
175#389—1971no—
176#387—1971no—
177#387—1971no—
178#387—1971no—
179#387—1971no—
180#382—1971no—
181C4×C4⋊C4—1971no—
182C4×C4⋊C4—1971no—
183#376—1971no—
184C22×Q8—1971no—
185C22×Q8C2×D19711yescharacteristic
186C4⋊Q8—1978no—
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⋊C4—1971no—
195C2×C4⋊C4—1971no—
196C2×C4⋊C4C2×D19711yescharacteristic
197C2×C42—1971no—
198C2×C42—1971no—
199C2×C42—1971no—
200C2×C42—1971no—
201C2×C42—1971no—
202C2×C42—1971no—
203C2×C42C2×D19711yescharacteristic
204#95—1971no—
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#95—1971no—
216#95C2×D19711yescharacteristic
217#95C2×D19711yescharacteristic
218#95C2×D19711yescharacteristic
219#95C2×D19711yescharacteristic
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—7881no—
223C2×Q8—7881no—
224C2×Q8—1974no—
225C2×Q8C4×D19714yes—
226C22×C4—1971no—
227C22×C4—1971no—
228C22×C4—1971no—
229C22×C4—1971no—
230C22×C4—1971no—
231C22×C4—1971no—
232C22×C4—1971no—
233C22×C4—1971no—
234C22×C4C22×D19711yescharacteristic
235C22×C4C22×D19711yescharacteristic
236C22×C4C22×D19711yescharacteristic
237C22×C4C22×D19711yescharacteristic
238C22×C4C22×D19711yescharacteristic
239C22×C4C22×D19711yescharacteristic
240C22×C4C22×D19711yescharacteristic
241C4⋊C4—41no—
242C4⋊C4—7881no—
243C4⋊C4—7881no—
244C4⋊C4—7881no—
245C4⋊C4—3942no—
246C4⋊C4—3942no—
247C4⋊C4—1974no—
248C4⋊C4—1974no—
249C4⋊C4—1974no—
250C4⋊C4—1974no—
251C42—41no—
252C42—7881no—
253C42—7881no—
254C42—7881no—
255C42—3942no—
256C42—1974no—
257C42—1974no—
258C23C23×D19711yescharacteristic
259Q8—44no—
260Q8—7884no—
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—21no—
277C2×C4—21no—
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—3941no—
292C2×C4—3941no—
293C2×C4—3941no—
294C2×C4—3941no—
295C2×C4—1972no—
296C2×C4—1972no—
297C2×C4—1972no—
298C2×C4—1972no—
299C2×C4—1972no—
300C2×C4—1972no—
301C2×C4—1972no—
302C2×C4C2×C4×D19712yes—
303C2×C4C2×C4×D19712yes—
304C2×C4C2×C4×D19712yes—
305C2×C4Q8×D19712yes—
306C22C22×C4×D19711yescharacteristic
307C22C2×Q8×D19711yescharacteristic
308C22#41×D19711yescharacteristic
309C22#2026337111yescharacteristic
310C22#2026337111yescharacteristic
311C22#2026337311yescharacteristic
312C22#2026337411yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—7881no—
321C4—7881no—
322C4—7881no—
323C4—3942no—
324C4—3942no—
325C4—3942no—
326C4—3942no—
327C4—1974no—
328C2C4×Q8×D19711yescharacteristic
329C2#2716155211yescharacteristic
330C2#2716155511yescharacteristic
331C2#2716169311yescharacteristic
332C2#2716169911yescharacteristic
333C2#2716170611yescharacteristic
334C2#2716170811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#391×C197C2
socle, prime radical, cosocleC23×C197C24
hypercenter#391D197
hypocenterC197#2331
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
17. #20263324C4semidirect
77. C4×Dic197Q8semidirect
167. C197#2331semidirect

Maximal subgroups

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

subgroupindex
C2×#202633242
#271609862
#271609832
#271609762
#271609742
#271609662
#271609632
#271609612
#271609602
#271609432
C4×#156913372
#271609302
#271609242
#391×C1972
#2331197