Finite Group Databasebeta

← #38745586

Subgroups — #38745586

Order 25216.

548 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D197C211yesmaximal, characteristic
2#27160976C211yesmaximal, characteristic, complemented
3#27160974C211yesmaximal, characteristic, complemented
4#27160965C211yesmaximal, characteristic, complemented
5#27160964C211yesmaximal, characteristic, complemented
6#27160962C211yesmaximal, characteristic, complemented
7#27160938C211yesmaximal, characteristic
8#27160938C211yesmaximal, characteristic
9#27160936C211yesmaximal, characteristic
10C4×#15691337C211yesmaximal, characteristic, complemented
11#27160930C211yesmaximal, characteristic, complemented
12C4×#15691335C211yesmaximal, characteristic
13#27160926C211yesmaximal, characteristic
14#27160926C211yesmaximal, characteristic
15#387×C197C211yesmaximal, characteristic, complemented
16C22×C4×D197C2211yescharacteristic
17C22×C4×D197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19C2×C4⋊Dic197C2211yescharacteristic
20C2×C4⋊Dic197C2211yescharacteristic
21C2×#15691337C2211yescharacteristic
22C2×#15691337C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×#15691337C2211yescharacteristic
27C4⋊C4×D197C418yescomplemented
28C2×#15691335C2211yescharacteristic
29C2×#15691335C2211yescharacteristic
30C2×#15691335C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×C4×Dic197C2211yescharacteristic
33C2×C4×Dic197C2211yescharacteristic
34C2×C4×Dic197C2211yescharacteristic
35C2×C4×Dic197C2211yescharacteristic
36C2×C4×Dic197C2211yescharacteristic
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
52C23×D197C2311yescharacteristic
53C22×Dic197C2311yescharacteristic
54C22×Dic197C2311yescharacteristic
55C22×Dic197C2311yescharacteristic
56C22×Dic197C2311yescharacteristic
57C22×Dic197C2311yescharacteristic
58C22×Dic197C2311yescharacteristic
59C22×Dic197C2311yescharacteristic
60C2×C4×D197—22no—
61C2×C4×D197—22no—
62C2×C4×D197—41no—
63C2×C4×D197—41no—
64C2×C4×D197—41no—
65C2×C4×D197—41no—
66C2×C4×D197C2×C414yes—
67C2×C4×D197C2×C414yes—
68C2×C4×D197C2×C414yes—
69C4⋊Dic197—41no—
70C4⋊Dic197C2×C414yes—
71#15691337—22no—
72#15691337—22no—
73#15691337—41no—
74#15691337—41no—
75#15691337C2×C414yes—
76#15691337C2×C414yes—
77#15691335—22no—
78#15691335—22no—
79#15691335—22no—
80#15691335Q814yes—
81C4×Dic197—22no—
82C4×Dic197—22no—
83C4×Dic197—41no—
84C4×Dic197—41no—
85C4×Dic197—41no—
86C22×C4×C197C2311yescharacteristic
87C22×C4×C197C2311yescharacteristic
88C22×C4×C197C2311yescharacteristic
89C22×C4×C197C2311yescharacteristic
90C22×C4×C197C2311yescharacteristic
91C22×C4×C197C2311yescharacteristic
92C22×C4×C197C2311yescharacteristic
93C4⋊C4×C197—41no—
94C4⋊C4×C197C2×C414yes—
95C42×C197—41no—
96C22×D197—21no—
97C22×D197—21no—
98C22×D197—21no—
99C22×D197—21no—
100C22×D197C22×C412yes—
101C22×D197C2×Q812yes—
102C22×D197#4112yes—
103C2×Dic197—21no—
104C2×Dic197—21no—
105C2×Dic197—21no—
106C2×Dic197—21no—
107C2×Dic197—21no—
108C2×Dic197—21no—
109C2×Dic197—21no—
110C2×Dic197—21no—
111C2×Dic197—21no—
112C2×Dic197—21no—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197C22×C412yes—
119C2×Dic197C22×C412yes—
120C2×Dic197C22×C412yes—
121C2×Dic197C2×Q812yes—
122C2×Dic197#4112yes—
123C2×Dic197#4112yes—
124C4×D197—28no—
125C4×D197—44no—
126C4×D197—44no—
127C23×C197C2411yescharacteristic
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×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197C22×C412yes—
146C2×C4×C197C22×C412yes—
147C2×C4×C197C22×C412yes—
148C2×C4×C197C2×Q812yes—
149C22×C197C23×C411yescharacteristic
150C22×C197C22×Q811yescharacteristic
151C22×C197C2×#4111yescharacteristic
152C22×C197C2×#4111yescharacteristic
153C22×C197C2×#4111yescharacteristic
154C22×C197H(5,𝔽2)11yescharacteristic
155C22×C197H(5,𝔽2)11yescharacteristic
156C2×D197—22no—
157C2×D197—22no—
158C2×D197—22no—
159C2×D197—22no—
160C2×D197—22no—
161C2×D197—22no—
162C2×D197C4×Q814yes—
163C4×C197—22no—
164C4×C197—22no—
165C4×C197—22no—
166C4×C197—22no—
167C4×C197—41no—
168C4×C197—41no—
169C4×C197—41no—
170Dic197—22no—
171Dic197—22no—
172Dic197—22no—
173Dic197—41no—
174Dic197—41no—
175Dic197—41no—
176Dic197#11714yes—
177C2×C197C2×#11711yescharacteristic
178C2×C197C2×C4×Q811yescharacteristic
179C2×C197#51811yescharacteristic
180C2×C197#55311yescharacteristic
181C2×C197#55311yescharacteristic
182C2×C197#55411yescharacteristic
183C2×C197#55411yescharacteristic
184D197—24no—
185C197#234511yescharacteristic, complemented, Sylow
186#2345—1971nomaximal, Sylow
187C22×C4⋊C4—1971no—
188#389—1971no—
189#389—1971no—
190#387—1971no—
191#387—1971no—
192#387—1971no—
193#387D19711yescharacteristic, complemented
194#380—1971no—
195#380—1971no—
196#380—1971no—
197C4×C4⋊C4—1971no—
198C4×C4⋊C4—1971no—
199C4×C22⋊C4—1971no—
200C4×C22⋊C4—1971no—
201C4×C22⋊C4—1971no—
202C23×C4—1971no—
203C23×C4—1971no—
204C23×C4—1971no—
205C2×C4⋊C4—1971no—
206C2×C4⋊C4—1971no—
207C2×C4⋊C4—1971no—
208C2×C4⋊C4—1971no—
209C2×C4⋊C4—1971no—
210C2×C4⋊C4—1971no—
211C2×C4⋊C4—1971no—
212C2×C4⋊C4—1971no—
213C2×C4⋊C4—1978no—
214C2×C4⋊C4C2×D19711yescharacteristic
215C2×C4⋊C4C2×D19711yescharacteristic
216C2×C22⋊C4—1971no—
217C2×C22⋊C4—1971no—
218C2×C22⋊C4—1971no—
219C2×C22⋊C4—1971no—
220C2×C42—1971no—
221C2×C42—1971no—
222C2×C42—1971no—
223C2×C42—1971no—
224C2×C42—1971no—
225C2×C42C2×D19711yescharacteristic
226#95—1971no—
227#95—1971no—
228#95—1971no—
229#95—1971no—
230#95—1971no—
231#95—1971no—
232#95—1971no—
233#95—1971no—
234#95C2×D19711yescharacteristic
235#95C2×D19711yescharacteristic
236#95C2×D19711yescharacteristic
237#95C2×D19711yescharacteristic
238C24—1971no—
239C22×C4—7881no—
240C22×C4—7881no—
241C22×C4—7881no—
242C22×C4—7881no—
243C22×C4—3942no—
244C22×C4—3942no—
245C22×C4—1974no—
246C22×C4—1974no—
247C22×C4—1971no—
248C22×C4—1971no—
249C22×C4—1971no—
250C22×C4—1971no—
251C22×C4—1971no—
252C22×C4—1971no—
253C22×C4—1974no—
254C22×C4—1971no—
255C22×C4C22×D19711yescharacteristic
256C22×C4C22×D19711yescharacteristic
257C22×C4C22×D19711yescharacteristic
258C22×C4C22×D19711yescharacteristic
259C22×C4C22×D19711yescharacteristic
260C22×C4C22×D19711yescharacteristic
261C22×C4C22×D19711yescharacteristic
262C4⋊C4—41no—
263C4⋊C4—7881no—
264C4⋊C4—7881no—
265C4⋊C4—7881no—
266C4⋊C4—3942no—
267C4⋊C4—3942no—
268C4⋊C4—1974no—
269C4⋊C4—1974no—
270C4⋊C4—1974no—
271C4⋊C4C4×D19714yes—
272C22⋊C4—3942no—
273C22⋊C4—3942no—
274C22⋊C4—3942no—
275C22⋊C4—1974no—
276C42—41no—
277C42—7881no—
278C42—7881no—
279C42—7881no—
280C42—3942no—
281C42—3942no—
282C23—3941no—
283C23—3941no—
284C23—3941no—
285C23—3941no—
286C23—1972no—
287C23—1972no—
288C23—1972no—
289C23C23×D19711yescharacteristic
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—7884no—
308C2×C4—7884no—
309C2×C4—3948no—
310C2×C4—3941no—
311C2×C4—3941no—
312C2×C4—3941no—
313C2×C4—3941no—
314C2×C4—3941no—
315C2×C4—3941no—
316C2×C4—3941no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4—3941no—
320C2×C4—3941no—
321C2×C4—3941no—
322C2×C4—3941no—
323C2×C4—3941no—
324C2×C4—3941no—
325C2×C4—1972no—
326C2×C4—1972no—
327C2×C4—1972no—
328C2×C4—1972no—
329C2×C4—1972no—
330C2×C4—1972no—
331C2×C4C2×C4×D19712yes—
332C2×C4C2×C4×D19712yes—
333C2×C4C2×C4×D19712yes—
334C2×C4Q8×D19712yes—
335C22—3942no—
336C22—3942no—
337C22—3942no—
338C22—3942no—
339C22—3942no—
340C22—3942no—
341C22—1974no—
342C22C22×C4×D19711yescharacteristic
343C22C2×Q8×D19711yescharacteristic
344C22#41×D19711yescharacteristic
345C22#41×D19711yescharacteristic
346C22#41×D19711yescharacteristic
347C22#2026337111yescharacteristic
348C22#2026337211yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—7881no—
357C4—7881no—
358C4—7881no—
359C4—3942no—
360C4—3942no—
361C4—3942no—
362C4—1974no—
363C2—3944no—
364C2#117×D19711yescharacteristic
365C2C4×Q8×D19711yescharacteristic
366C2#2716155111yescharacteristic
367C2#2716177411yescharacteristic
368C2#2716178011yescharacteristic
369C2#2716178311yescharacteristic
370C2#2716178511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#387×C197C2
socle, cosocleC23×C197C24
prime radicalC23×D197C23
hypercenter#387D197
hypocenterC197#2345
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #27160976C2semidirect
3. #27160974C2semidirect
4. #27160965C2semidirect
5. #27160964C2semidirect
6. #27160962C2semidirect
10. C4×#15691337C2semidirect
11. #27160930C2semidirect
15. #387×C197C2semidirect
27. C4⋊C4×D197C4semidirect
185. C197#2345semidirect
193. #387D197semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1972
#271609762
#271609742
#271609652
#271609642
#271609622
#271609382
#271609362
C4×#156913372
#271609302
C4×#156913352
#271609262
#387×C1972
#2345197