Finite Group Databasebeta

← #38751047

Subgroups — #38751047

Order 25216.

564 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263304C211yesmaximal, characteristic
2C2×C42×D197C211yesmaximal, characteristic
3#27160983C211yesmaximal, characteristic, complemented
4#27160983C211yesmaximal, characteristic, complemented
5#27160976C211yesmaximal, characteristic, complemented
6#27160974C211yesmaximal, characteristic, complemented
7#27160964C211yesmaximal, characteristic, complemented
8#27160963C211yesmaximal, characteristic, complemented
9#27160960C211yesmaximal, characteristic, complemented
10#27160953C211yesmaximal, characteristic
11#27160950C211yesmaximal, characteristic
12#27160941C211yesmaximal, characteristic
13#27160941C211yesmaximal, characteristic
14#27160938C211yesmaximal, characteristic
15#387×C197C211yesmaximal, characteristic, complemented
16C22×Dic394C2211yescharacteristic
17C22×C4×D197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19C22×C4×D197C2211yescharacteristic
20#20263304—24no—
21C2×C4⋊Dic197C2211yescharacteristic
22C2×#15691337C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×#15691337C2211yescharacteristic
27C2×#15691337C2211yescharacteristic
28C2×#15691335C2211yescharacteristic
29C2×#15691335C2211yescharacteristic
30C2×#15691335C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×C4×Dic197C2211yescharacteristic
33C2×C4×Dic197C2211yescharacteristic
34C2×C4×Dic197C2211yescharacteristic
35C42×D197—24no—
36#20263233C2211yescharacteristic
37#20263233C2211yescharacteristic
38#20263233C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45#20263233C2211yescharacteristic
46C2×C4⋊C4×C197C2211yescharacteristic
47C2×C4⋊C4×C197C2211yescharacteristic
48C2×C42×C197C2211yescharacteristic
49#95×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52#95×C197C2211yescharacteristic
53C23×D197C2311yescharacteristic
54C2×Dic394—22no—
55C2×Dic394—22no—
56C2×Dic394—22no—
57C22×Dic197C2311yescharacteristic
58C22×Dic197C2311yescharacteristic
59C22×Dic197C2311yescharacteristic
60C22×Dic197C2311yescharacteristic
61C22×Dic197C2311yescharacteristic
62C22×Dic197C2311yescharacteristic
63C22×Dic197C2311yescharacteristic
64C2×C4×D197—22no—
65C2×C4×D197—22no—
66C2×C4×D197—22no—
67C2×C4×D197—22no—
68C2×C4×D197—22no—
69C2×C4×D197—22no—
70C2×C4×D197—22no—
71C2×C4×D197D414yes—
72C2×C4×D197Q814yescomplemented
73C4⋊Dic197—22no—
74#15691337—22no—
75#15691337—22no—
76#15691337—22no—
77#15691337—22no—
78#15691337—22no—
79#15691337—22no—
80#15691335—22no—
81#15691335—22no—
82#15691335—22no—
83#15691335—22no—
84C4×Dic197—22no—
85C4×Dic197—22no—
86C4×Dic197—22no—
87C22×C4×C197C2311yescharacteristic
88C22×C4×C197C2311yescharacteristic
89C22×C4×C197C2311yescharacteristic
90C22×C4×C197C2311yescharacteristic
91C22×C4×C197C2311yescharacteristic
92C22×C4×C197C2311yescharacteristic
93C22×C4×C197C2311yescharacteristic
94C4⋊C4×C197—22no—
95C4⋊C4×C197—22no—
96C42×C197—22no—
97C22×D197—21no—
98C22×D197—21no—
99C22×D197—21no—
100C22×D197—21no—
101C22×D197C2×D412yes—
102C22×D197C2×Q812yes—
103C22×D197#4112yes—
104Dic394—44no—
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×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197C2×D412yes—
122C2×Dic197C2×Q812yes—
123C2×Dic197#4112yes—
124C2×Dic197#4112yes—
125C2×Dic197#4112yes—
126C4×D197—28no—
127C4×D197—28no—
128C4×D197—28no—
129C23×C197C2411yescharacteristic
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×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197C2×D412yes—
149C2×C4×C197C2×Q812yes—
150C2×C4×C197#4112yes—
151C22×C197C22×D411yescharacteristic
152C22×C197C22×Q811yescharacteristic
153C22×C197C2×#4111yescharacteristic
154C22×C197C2×#4111yescharacteristic
155C22×C197C2×#4111yescharacteristic
156C22×C197C2×#4111yescharacteristic
157C22×C197C2×#4111yescharacteristic
158C2×D197—22no—
159C2×D197—22no—
160C2×D197—22no—
161C2×D197—22no—
162C2×D197—22no—
163C2×D197—22no—
164C2×D197C22⋊Q814yes—
165C4×C197—22no—
166C4×C197—22no—
167C4×C197—22no—
168C4×C197—41no—
169C4×C197—41no—
170C4×C197—41no—
171C4×C197—41no—
172Dic197—22no—
173Dic197—22no—
174Dic197—41no—
175Dic197—41no—
176Dic197—41no—
177Dic197—41no—
178Dic197#12314yes—
179C2×C197C2×C22⋊Q811yescharacteristic
180C2×C197C2×#12311yescharacteristic
181C2×C197#52511yescharacteristic
182C2×C197#52911yescharacteristic
183C2×C197#52911yescharacteristic
184C2×C197#53311yescharacteristic
185C2×C197#53311yescharacteristic
186D197—24no—
187C197#285511yescharacteristic, complemented, Sylow
188#2855—1971nomaximal, Sylow
189C2×C22⋊Q8—1971no—
190C22×C42—1971no—
191#391—1971no—
192#391—1971no—
193#389—1971no—
194#389—1971no—
195#387—1971no—
196#387—1971no—
197#387—1971no—
198#387D19711yescharacteristic, complemented
199#385—1971no—
200#385—1971no—
201#381—1971no—
202#381—1971no—
203#380—1971no—
204C22×Q8—1971no—
205C23×C4—1971no—
206C23×C4—1971no—
207C23×C4—1971no—
208C22⋊Q8—3944no—
209C2×C4⋊C4—1971no—
210C2×C4⋊C4—1971no—
211C2×C4⋊C4—1971no—
212C2×C4⋊C4—1971no—
213C2×C4⋊C4—1971no—
214C2×C4⋊C4—1971no—
215C2×C4⋊C4—1971no—
216C2×C4⋊C4C2×D19711yescharacteristic
217C2×C4⋊C4C2×D19711yescharacteristic
218C2×C22⋊C4—1971no—
219C2×C22⋊C4—1971no—
220C2×C22⋊C4—1971no—
221C2×C22⋊C4—1971no—
222C2×C42—3944no—
223C2×C42—1971no—
224C2×C42—1971no—
225C2×C42—1971no—
226C2×C42C2×D19711yescharacteristic
227#95—1971no—
228#95—1971no—
229#95—1971no—
230#95—1971no—
231#95—1971no—
232#95—1971no—
233#95—1971no—
234#95—1971no—
235#95—1971no—
236#95—1971no—
237#95C2×D19711yescharacteristic
238#95C2×D19711yescharacteristic
239#95C2×D19711yescharacteristic
240#95C2×D19711yescharacteristic
241C24—1971no—
242C2×Q8—3942no—
243C2×Q8—3942no—
244C2×Q8—3942no—
245C22×C4—3942no—
246C22×C4—3942no—
247C22×C4—3942no—
248C22×C4—3942no—
249C22×C4—3942no—
250C22×C4—3942no—
251C22×C4—3942no—
252C22×C4—1971no—
253C22×C4—1971no—
254C22×C4—1971no—
255C22×C4—1971no—
256C22×C4—1971no—
257C22×C4—1971no—
258C22×C4—1974no—
259C22×C4—1974no—
260C22×C4—1971no—
261C22×C4C22×D19711yescharacteristic
262C22×C4C22×D19711yescharacteristic
263C22×C4C22×D19711yescharacteristic
264C22×C4C22×D19711yescharacteristic
265C22×C4C22×D19711yescharacteristic
266C22×C4C22×D19711yescharacteristic
267C22×C4C22×D19711yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—3942no—
271C4⋊C4—3942no—
272C4⋊C4—3942no—
273C4⋊C4—3942no—
274C4⋊C4—3942no—
275C4⋊C4—3942no—
276C4⋊C4—3942no—
277C22⋊C4—3942no—
278C22⋊C4—3942no—
279C22⋊C4—3942no—
280C22⋊C4—3942no—
281C42—22no—
282C42—3942no—
283C42—3942no—
284C42—3942no—
285C23—3941no—
286C23—3941no—
287C23—3941no—
288C23—3941no—
289C23—1972no—
290C23—1972no—
291C23—1972no—
292C23C23×D19711yescharacteristic
293Q8—7884no—
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—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—3948no—
313C2×C4—3948no—
314C2×C4—3948no—
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—3941no—
326C2×C4—3941no—
327C2×C4—3941no—
328C2×C4—3941no—
329C2×C4—3941no—
330C2×C4—3941no—
331C2×C4—1972no—
332C2×C4—1972no—
333C2×C4—1972no—
334C2×C4—1972no—
335C2×C4—1972no—
336C2×C4D4×D19712yes—
337C2×C4Q8×D19712yes—
338C2×C4#1569135812yes—
339C22—3942no—
340C22—3942no—
341C22—3942no—
342C22—3942no—
343C22—3942no—
344C22—3942no—
345C22—1974no—
346C22C2×D4×D19711yescharacteristic
347C22C2×Q8×D19711yescharacteristic
348C22#41×D19711yescharacteristic
349C22#41×D19711yescharacteristic
350C22#41×D19711yescharacteristic
351C22#41×D19711yescharacteristic
352C22C2×#1569135811yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—7881no—
361C4—7881no—
362C4—7881no—
363C4—7881no—
364C4—3942no—
365C4—3942no—
366C4—1974no—
367C2—3944no—
368C2C22⋊Q8×D19711yescharacteristic
369C2#123×D19711yescharacteristic
370C2#2716158811yescharacteristic
371C2#2716160711yescharacteristic
372C2#2716161211yescharacteristic
373C2#2716163311yescharacteristic
374C2#2716163511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#387×C197C2
prime radicalC23×D197C23
hypercenter#387D197
hypocenterC197#2855
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. #27160983C2semidirect
4. #27160983C2semidirect
5. #27160976C2semidirect
6. #27160974C2semidirect
7. #27160964C2semidirect
8. #27160963C2semidirect
9. #27160960C2semidirect
15. #387×C197C2semidirect
72. C2×C4×D197Q8semidirect
187. C197#2855semidirect
198. #387D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633042
C2×C42×D1972
#271609832
#271609762
#271609742
#271609642
#271609632
#271609602
#271609532
#271609502
#271609412
#271609382
#387×C1972
#2855197