Finite Group Databasebeta

← #27155837

Subgroups — #27155837

Order 12416.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261468C213yesmaximal, complemented
2#20261464C211yesmaximal, characteristic, complemented
3#20261450C213yesmaximal, complemented
4#20261433C213yesmaximal, complemented
5#20261420C213yesmaximal, complemented
6#20261404C211yesmaximal, characteristic, complemented
7#20261397C211yesmaximal, characteristic, complemented
8#20261392C211yesmaximal, characteristic, complemented
9#20261356C213yesmaximal, complemented
10#20261336C213yesmaximal, complemented
11#20261267C213yesmaximal, complemented
12#20261263C213yesmaximal, complemented
13#20261211C211yesmaximal, characteristic, complemented
14C4×#13095512C211yesmaximal, characteristic, complemented
15#520×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×Q8×D97C2211yescharacteristic
18#15690810C2213yes—
19#15690806C2213yes—
20#15690805C2216yes—
21#15690805C2211yescharacteristic
22#15690803C2213yes—
23#15690801C2213yes—
24#15690801C2213yes—
25#15690800C2213yes—
26#15690798C2213yes—
27#15690797C2213yes—
28#15690796C2213yes—
29#15690796C2216yes—
30#15690795C2213yes—
31#15690793C2216yes—
32#15690792C2213yes—
33#15690792C2216yes—
34#15690790C2213yes—
35#15690789C2213yes—
36#15690788C2213yes—
37#15690785C2213yes—
38#15690784C2213yes—
39#15690784C2216yes—
40#15690783C2213yes—
41#15690781C2213yes—
42#15690781C2213yes—
43Q8×Dic97C2211yescharacteristic
44Q8×Dic97C2211yescharacteristic
45C4×Dic194C2213yes—
46C4×Dic194C2213yes—
47#15690775C2213yes—
48C4×D388C2213yes—
49C4×D388C2213yes—
50#15690772C2213yes—
51C4×C22⋊D97C2213yes—
52C4×C22⋊D97C2213yes—
53C4×C22⋊D97C2216yes—
54#15690769C2216yes—
55#15690768C2213yes—
56#15690768C2213yes—
57#15690767C2213yes—
58C42×D97C2213yes—
59C2×#41×C97C2211yescharacteristic
60C4×Q8×C97C2211yescharacteristic
61C4×Q8×C97C2211yescharacteristic
62C4×D4×C97C2213yes—
63C4×D4×C97C2213yes—
64#117×C97C2213yes—
65C2×C4⋊C4×C97C2213yes—
66#13095512—24no—
67C2×Dic194C2313yes—
68C2×Dic194C2313yes—
69Q8×D97—24no—
70C2×D388C2313yes—
71C2×D388C2313yes—
72C2×C22⋊D97C2313yes—
73C2×C22⋊D97C2316yes—
74C2×C22⋊D97C2313yes—
75C2×C4×D97C2313yes—
76C2×C4×D97C2313yes—
77C2×C4×D97C2313yes—
78C2×C4×D97C2313yes—
79C4⋊Dic97C2313yes—
80C4⋊Dic97C2313yes—
81C4⋊Dic97C2313yes—
82C4⋊Dic97C2316yes—
83C4⋊Dic97C2311yescharacteristic
84#13095490C2313yes—
85#13095490C2313yes—
86#13095490C2311yescharacteristic
87#13095490C2313yes—
88#13095490C2313yes—
89#13095490C2313yes—
90C22⋊Dic97C2313yes—
91C22⋊Dic97C2316yes—
92C22⋊Dic97C2313yes—
93#13095488C2311yescharacteristic
94#13095488C2313yes—
95#13095488C2316yes—
96#13095488C2311yescharacteristic
97#13095488C2313yes—
98#13095488C2316yes—
99#13095488C2311yescharacteristic
100#13095488C2313yes—
101C4×Dic97C2311yescharacteristic
102C4×Dic97C2313yes—
103C4×Dic97C2313yes—
104C4×Dic97C2313yes—
105C4×Dic97C2316yes—
106#41×C97—24no—
107C2×Q8×C97C2311yescharacteristic
108C2×D4×C97C2313yes—
109C22×C4×C97C2313yes—
110C22×C4×C97C2316yes—
111C4⋊C4×C97C2313yes—
112C4⋊C4×C97C2313yes—
113C4⋊C4×C97C2311yescharacteristic
114C4⋊C4×C97C2313yes—
115C22⋊C4×C97C2313yes—
116C22⋊C4×C97C2313yes—
117C42×C97C2313yes—
118C42×C97C2313yes—
119C23×C97C2413yes—
120C22×D97C2411yescharacteristic
121C22×D97C2413yes—
122Q8×C97#4114yes—
123D4×C97—26no—
124C2×C4×C97—26no—
125C2×C4×C97—46no—
126C2×C4×C97C2411yescharacteristic
127C2×C4×C97C2411yescharacteristic
128C2×C4×C97C2413yes—
129C2×C4×C97C2413yes—
130C2×C4×C97C2411yescharacteristic
131C2×C4×C97C2413yes—
132C22⋊D97—43no—
133C22⋊D97—46no—
134C22⋊D97—43no—
135C2×Dic97C2411yescharacteristic
136C2×Dic97C2411yescharacteristic
137C2×Dic97C2413yes—
138C2×Dic97C2411yescharacteristic
139C2×Dic97C2413yes—
140C2×Dic97C2413yes—
141D388—26no—
142D388—43no—
143C4×D97—26no—
144C4×D97—26no—
145C4×D97—43no—
146C4×D97—43no—
147Dic194—26no—
148Dic194—43no—
149C22×C97—23no—
150C22×C97—23no—
151C22×C97—23no—
152C22×C97C2511yescharacteristic
153C2×D97—21no—
154C2×D97—23no—
155C2×D97—23no—
156C2×D97—23no—
157C2×D97—21no—
158C2×D97—21no—
159C4×C97—21no—
160C4×C97—21no—
161C4×C97—21no—
162C4×C97—23no—
163C4×C97—23no—
164C4×C97C2×#4116yes—
165Dic97—23no—
166Dic97—21no—
167Dic97—23no—
168Dic97—23no—
169Dic97—21no—
170Dic97—21no—
171C2×C97—43no—
172C2×C97C22×#4111yescharacteristic
173C2×C97#58511yescharacteristic
174C2×C97#58511yescharacteristic
175D97—41no—
176D97—43no—
177#3524—971nomaximal, Sylow
178C97#352411yescharacteristic, complemented, Sylow
179#559—973no—
180#557—971no—
181#555—973no—
182#553—973no—
183#552—973no—
184#550—971no—
185#548—971no—
186#548—971no—
187#542—973no—
188#540—973no—
189#529—973no—
190#529—973no—
191#520D9711yescharacteristic, complemented
192#519—971no—
193C4×#41—971no—
194C2×#41—971no—
195C2×#41C2×D9711yescharacteristic
196C22×Q8—971no—
197C4⋊Q8—973no—
198#127—973no—
199#126—971no—
200#126—976no—
201#126—973no—
202#125—973no—
203#125—973no—
204#125—973no—
205#124—973no—
206#124—973no—
207#124—976no—
208#124—973no—
209#123—976no—
210#123—973no—
211#123—976no—
212#123—973no—
213C22⋊Q8—973no—
214C22⋊Q8—973no—
215C22⋊Q8—973no—
216#121—973no—
217#121—973no—
218#121—976no—
219#121—973no—
220#121—973no—
221#121—973no—
222C4×Q8—971no—
223C4×Q8—971no—
224C4×Q8—973no—
225C4×Q8—973no—
226C4×Q8—973no—
227C4×Q8C2×D9711yescharacteristic
228C4×Q8C2×D9711yescharacteristic
229C4×D4—973no—
230C4×D4—973no—
231C4×D4—976no—
232C4×D4—973no—
233C4×D4—973no—
234C4×D4—973no—
235C4×D4C2×D9713yes—
236C4×D4C2×D9713yes—
237#117—976no—
238#117—973no—
239#117—973no—
240#117—973no—
241#117C2×D9713yes—
242C2×C4⋊C4C2×D9713yes—
243C2×C42—973no—
244#41—24no—
245#41—1944no—
246C2×Q8—973no—
247C2×Q8—973no—
248C2×Q8—1944no—
249C2×Q8C22×D9711yescharacteristic
250C2×D4—973no—
251C2×D4—976no—
252C2×D4—973no—
253C2×D4—973no—
254C2×D4—973no—
255C2×D4C22×D9713yes—
256C22×C4—973no—
257C22×C4—973no—
258C22×C4—973no—
259C22×C4—973no—
260C22×C4C22×D9713yes—
261C22×C4C22×D9716yes—
262C4⋊C4—973no—
263C4⋊C4—973no—
264C4⋊C4—971no—
265C4⋊C4—973no—
266C4⋊C4—973no—
267C4⋊C4—973no—
268C4⋊C4—973no—
269C4⋊C4—973no—
270C4⋊C4—973no—
271C4⋊C4—976no—
272C4⋊C4—971no—
273C4⋊C4C22×D9713yes—
274C4⋊C4C22×D9713yes—
275C4⋊C4C22×D9713yes—
276C4⋊C4C22×D9711yescharacteristic
277C22⋊C4—973no—
278C22⋊C4—976no—
279C22⋊C4—973no—
280C22⋊C4—971no—
281C22⋊C4—973no—
282C22⋊C4—976no—
283C22⋊C4—971no—
284C22⋊C4—971no—
285C22⋊C4—973no—
286C22⋊C4—976no—
287C22⋊C4—973no—
288C22⋊C4C22×D9713yes—
289C22⋊C4C22×D9713yes—
290C42—971no—
291C42—973no—
292C42—973no—
293C42—976no—
294C42—973no—
295C42C22×D9713yes—
296C42C22×D9713yes—
297C23—971no—
298C23—973no—
299C23C23×D9713yes—
300Q8—3883no—
301Q8—1946no—
302Q8#1309551114yes—
303D4—26no—
304D4—3883no—
305D4—3883no—
306D4—3883no—
307D4—3886no—
308D4—1946no—
309C2×C4—26no—
310C2×C4—46no—
311C2×C4—971no—
312C2×C4—971no—
313C2×C4—973no—
314C2×C4—971no—
315C2×C4—973no—
316C2×C4—973no—
317C2×C4—3883no—
318C2×C4—3883no—
319C2×C4—1946no—
320C2×C4—1946no—
321C2×C4C23×D9713yes—
322C2×C4C23×D9711yescharacteristic
323C2×C4C23×D9711yescharacteristic
324C2×C4C23×D9713yes—
325C2×C4C23×D9711yescharacteristic
326C2×C4C23×D9713yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—1941no—
331C22—1943no—
332C22—1943no—
333C22—1941no—
334C22—1943no—
335C22—1941no—
336C22C24×D9711yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—21no—
342C4—1941no—
343C4—1943no—
344C4—1941no—
345C4—1941no—
346C4—1943no—
347C4—1943no—
348C4C2×#1309551116yes—
349C2—43no—
350C2—3881no—
351C2—3883no—
352C2C22×#1309551111yescharacteristic
353C2#2026157011yescharacteristic
354C2#2026157211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#520×C97C2
hypercenter#520D97
hypocenterC97#3524
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #20261468C2semidirect
2. #20261464C2semidirect
3. #20261450C2semidirect
4. #20261433C2semidirect
5. #20261420C2semidirect
6. #20261404C2semidirect
7. #20261397C2semidirect
8. #20261392C2semidirect
9. #20261356C2semidirect
10. #20261336C2semidirect
11. #20261267C2semidirect
12. #20261263C2semidirect
13. #20261211C2semidirect
14. C4×#13095512C2semidirect
15. #520×C97C2semidirect
178. C97#3524semidirect
191. #520D97semidirect

Maximal subgroups

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

subgroupindex
#202614682
#202614642
#202614502
#202614332
#202614202
#202614042
#202613972
#202613922
#202613562
#202613362
#202612672
#202612632
#202612112
C4×#130955122
#520×C972
#352497