Finite Group Databasebeta

← #27144465

Subgroups — #27144465

Order 12416.

528 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690790C211yesmaximal, characteristic
2#20260639C211yesmaximal, characteristic, complemented
3#20260639C211yesmaximal, characteristic, complemented
4#20260633C211yesmaximal, characteristic, complemented
5#20260631C211yesmaximal, characteristic, complemented
6#20260621C211yesmaximal, characteristic, complemented
7#20260620C211yesmaximal, characteristic, complemented
8#20260618C211yesmaximal, characteristic, complemented
9#20260609C211yesmaximal, characteristic
10#20260606C211yesmaximal, characteristic
11#20260594C211yesmaximal, characteristic
12C4×#13095488C211yesmaximal, characteristic
13#20260582C211yesmaximal, characteristic
14#95×D97C211yesmaximal, characteristic
15#387×C97C211yesmaximal, characteristic, complemented
16C22×Dic194C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18C22×C4×D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20#15690790C418yescomplemented
21C2×C4⋊Dic97C2211yescharacteristic
22C2×C4⋊Dic97C2211yescharacteristic
23C2×C4⋊Dic97C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×#13095488C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×C4×Dic97C2211yescharacteristic
33C2×C4×Dic97C2211yescharacteristic
34C2×C4×Dic97C2211yescharacteristic
35#15690717C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45C2×C4⋊C4×C97C2211yescharacteristic
46C2×C4⋊C4×C97C2211yescharacteristic
47C2×C42×C97C2211yescharacteristic
48#95×C97C2211yescharacteristic
49#95×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52C23×D97C2311yescharacteristic
53C2×Dic194—41no—
54C2×Dic194—41no—
55C2×Dic194C2×C414yescomplemented
56C22×Dic97C2311yescharacteristic
57C22×Dic97C2311yescharacteristic
58C22×Dic97C2311yescharacteristic
59C22×Dic97C2311yescharacteristic
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C2×C4×D97—22no—
64C2×C4×D97—41no—
65C2×C4×D97—41no—
66C2×C4×D97—41no—
67C2×C4×D97—41no—
68C2×C4×D97—41no—
69C2×C4×D97—41no—
70C2×C4×D97C2×C414yes—
71C2×C4×D97Q814yescomplemented
72C4⋊Dic97—41no—
73C4⋊Dic97C2×C414yescomplemented
74C4⋊Dic97C2×C414yescomplemented
75#13095490—41no—
76#13095490—41no—
77#13095490—41no—
78#13095490—41no—
79#13095488—22no—
80#13095488—22no—
81#13095488C2×C414yes—
82#13095488C2×C414yes—
83C4×Dic97—41no—
84C4×Dic97—41no—
85C4×Dic97—41no—
86C22×C4×C97C2311yescharacteristic
87C22×C4×C97C2311yescharacteristic
88C22×C4×C97C2311yescharacteristic
89C22×C4×C97C2311yescharacteristic
90C22×C4×C97C2311yescharacteristic
91C22×C4×C97C2311yescharacteristic
92C22×C4×C97C2311yescharacteristic
93C4⋊C4×C97—41no—
94C4⋊C4×C97C2×C414yescomplemented
95C42×C97—41no—
96C23×C97C2411yescharacteristic
97C22×D97—21no—
98C22×D97—21no—
99C22×D97—21no—
100C22×D97—21no—
101C22×D97C22×C412yes—
102C22×D97C2×Q812yes—
103C22×D97#4112yes—
104C2×C4×C97—21no—
105C2×C4×C97—21no—
106C2×C4×C97—21no—
107C2×C4×C97—21no—
108C2×C4×C97—21no—
109C2×C4×C97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97C22×C412yes—
122C2×C4×C97C22×C412yes—
123C2×C4×C97C22×C412yes—
124C2×C4×C97C2×Q812yes—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97—21no—
140C2×Dic97—21no—
141C2×Dic97—21no—
142C2×Dic97C22×C412yes—
143C2×Dic97C22×C412yes—
144C2×Dic97C22×C412yes—
145C2×Dic97C2×Q812yes—
146C4×D97—44no—
147C4×D97—44no—
148C4×D97—44no—
149Dic194—44no—
150C22×C97C23×C411yescharacteristic
151C22×C97C22×Q811yescharacteristic
152C22×C97C2×#4111yescharacteristic
153C22×C97H(5,𝔽2)11yescharacteristic
154C22×C97H(5,𝔽2)11yescharacteristic
155C22×C97H(5,𝔽2)11yescharacteristic
156C22×C972−1+411yescharacteristic
157C2×D97—22no—
158C2×D97—22no—
159C2×D97—22no—
160C2×D97—22no—
161C2×D97—22no—
162C2×D97—22no—
163C2×D97C4×Q814yes—
164C4×C97—22no—
165C4×C97—22no—
166C4×C97—22no—
167C4×C97—22no—
168C4×C97—41no—
169C4×C97—41no—
170C4×C97—41no—
171Dic97—22no—
172Dic97—22no—
173Dic97—22no—
174Dic97—22no—
175Dic97—41no—
176Dic97—41no—
177Dic97—41no—
178C2×C97C2×C4×Q811yescharacteristic
179C2×C97#51811yescharacteristic
180C2×C97#52011yescharacteristic
181C2×C97#53811yescharacteristic
182C2×C97#53911yescharacteristic
183C2×C97#54311yescharacteristic
184C2×C97#54411yescharacteristic
185D97—24no—
186#2329—971nomaximal, Sylow
187C97#232911yescharacteristic, complemented, Sylow
188C2×C22⋊Q8—971no—
189#391—971no—
190#391—971no—
191#389—971no—
192#389—971no—
193#387—971no—
194#387—971no—
195#387—971no—
196#387D9711yescharacteristic, complemented
197#385—971no—
198#385—971no—
199#380—971no—
200C4×C22⋊C4—971no—
201C4×C22⋊C4—971no—
202C2×#95—971no—
203C22×Q8—971no—
204C23×C4—971no—
205C23×C4—971no—
206C23×C4—971no—
207C22⋊Q8—978no—
208C2×C4⋊C4—971no—
209C2×C4⋊C4—971no—
210C2×C4⋊C4—971no—
211C2×C4⋊C4—971no—
212C2×C4⋊C4—971no—
213C2×C4⋊C4—971no—
214C2×C4⋊C4—971no—
215C2×C4⋊C4C2×D9711yescharacteristic
216C2×C4⋊C4C2×D9711yescharacteristic
217C2×C22⋊C4—971no—
218C2×C22⋊C4—971no—
219C2×C22⋊C4—971no—
220C2×C22⋊C4—971no—
221C2×C42—971no—
222C2×C42—971no—
223C2×C42—971no—
224C2×C42C2×D9711yescharacteristic
225#95—971no—
226#95—971no—
227#95—971no—
228#95—971no—
229#95—971no—
230#95—971no—
231#95—971no—
232#95—971no—
233#95—971no—
234#95—971no—
235#95C2×D9711yescharacteristic
236#95C2×D9711yescharacteristic
237#95C2×D9711yescharacteristic
238#95C2×D9711yescharacteristic
239C24—971no—
240C2×Q8—974no—
241C2×Q8—3881no—
242C2×Q8—3881no—
243C22×C4—971no—
244C22×C4—971no—
245C22×C4—971no—
246C22×C4—971no—
247C22×C4—974no—
248C22×C4—974no—
249C22×C4—971no—
250C22×C4—971no—
251C22×C4—971no—
252C22×C4—3881no—
253C22×C4—3881no—
254C22×C4—3881no—
255C22×C4—3881no—
256C22×C4—3881no—
257C22×C4—3881no—
258C22×C4—1942no—
259C22×C4C22×D9711yescharacteristic
260C22×C4C22×D9711yescharacteristic
261C22×C4C22×D9711yescharacteristic
262C22×C4C22×D9711yescharacteristic
263C22×C4C22×D9711yescharacteristic
264C22×C4C22×D9711yescharacteristic
265C22×C4C22×D9711yescharacteristic
266C4⋊C4—41no—
267C4⋊C4—974no—
268C4⋊C4—974no—
269C4⋊C4—3881no—
270C4⋊C4—3881no—
271C4⋊C4—3881no—
272C4⋊C4—3881no—
273C4⋊C4—3881no—
274C4⋊C4C4×D9714yescomplemented
275C22⋊C4—974no—
276C22⋊C4—974no—
277C22⋊C4—1942no—
278C22⋊C4—1942no—
279C42—41no—
280C42—3881no—
281C42—3881no—
282C42—3881no—
283C23—972no—
284C23—972no—
285C23—972no—
286C23—1941no—
287C23—1941no—
288C23—1941no—
289C23—1941no—
290C23C23×D9711yescharacteristic
291Q8—3884no—
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—21no—
308C2×C4—21no—
309C2×C4—972no—
310C2×C4—972no—
311C2×C4—972no—
312C2×C4—972no—
313C2×C4—3884no—
314C2×C4—3884no—
315C2×C4—3884no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4—1941no—
321C2×C4—1941no—
322C2×C4—1941no—
323C2×C4—1941no—
324C2×C4—1941no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4—1941no—
328C2×C4—1941no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4—1941no—
332C2×C4—1941no—
333C2×C4C2×C4×D9712yes—
334C2×C4C2×C4×D9712yes—
335C2×C4C2×C4×D9712yes—
336C2×C4Q8×D9712yes—
337C22—974no—
338C22—1942no—
339C22—1942no—
340C22—1942no—
341C22—1942no—
342C22—1942no—
343C22—1942no—
344C22C22×C4×D9711yescharacteristic
345C22C2×Q8×D9711yescharacteristic
346C22#41×D9711yescharacteristic
347C22#1569085511yescharacteristic
348C22#1569085611yescharacteristic
349C22#1569085611yescharacteristic
350C22#1569085711yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—3881no—
359C4—3881no—
360C4—3881no—
361C4—1942no—
362C4—1942no—
363C4—1942no—
364C4—1942no—
365C2—1944no—
366C2C4×Q8×D9711yescharacteristic
367C2#2026120911yescharacteristic
368C2#2026121411yescharacteristic
369C2#2026132111yescharacteristic
370C2#2026132611yescharacteristic
371C2#2026136011yescharacteristic
372C2#2026136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#387×C97C2
socle, cosocleC23×C97C24
prime radicalC23×D97C23
hypercenter#387D97
hypocenterC97#2329
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #20260639C2semidirect
3. #20260639C2semidirect
4. #20260633C2semidirect
5. #20260631C2semidirect
6. #20260621C2semidirect
7. #20260620C2semidirect
8. #20260618C2semidirect
15. #387×C97C2semidirect
20. #15690790C4semidirect
55. C2×Dic194C2×C4semidirect
71. C2×C4×D97Q8semidirect
73. C4⋊Dic97C2×C4semidirect
74. C4⋊Dic97C2×C4semidirect
94. C4⋊C4×C97C2×C4semidirect
187. C97#2329semidirect
196. #387D97semidirect
274. C4⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907902
#202606392
#202606332
#202606312
#202606212
#202606202
#202606182
#202606092
#202606062
#202605942
C4×#130954882
#202605822
#95×D972
#387×C972
#232997