Finite Group Databasebeta

← #27155815

Subgroups — #27155815

Order 12416.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261468C213yesmaximal, complemented
2Q8×Dic194C211yesmaximal, characteristic, complemented
3#20261440C213yesmaximal, complemented
4#20261404C211yesmaximal, characteristic, complemented
5#20261397C211yesmaximal, characteristic, complemented
6#20261392C211yesmaximal, characteristic, complemented
7#20261390C213yesmaximal, complemented
8#20261386C213yesmaximal, complemented
9#20261356C213yesmaximal, complemented
10#20261350C213yesmaximal, complemented
11#20261282C213yesmaximal, complemented
12#20261263C213yesmaximal, complemented
13#20261217C211yesmaximal, characteristic, complemented
14C4×Q8×D97C211yesmaximal, characteristic, complemented
15#520×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095511C2216yes—
18C2×Q8×D97C2211yescharacteristic
19#15690810C2213yes—
20#15690809C2213yes—
21#15690809C2213yes—
22#15690806C2213yes—
23#15690805C2213yes—
24#15690803C2213yes—
25#15690800C2213yes—
26#15690798C2213yes—
27#15690797C2213yes—
28#15690796C2213yes—
29#15690796C2216yes—
30#15690795C2213yes—
31#15690793C2216yes—
32#15690792C2213yes—
33#15690792C2216yes—
34#15690790C2213yes—
35#15690789C2216yes—
36#15690789C2213yes—
37#15690788C2216yes—
38#15690788C2211yescharacteristic
39#15690785C2213yes—
40#15690784C2213yes—
41#15690783C2213yes—
42#15690781C2213yes—
43#15690781C2213yes—
44Q8×Dic97C2211yescharacteristic
45Q8×Dic97C2211yescharacteristic
46C4×Dic194C2213yes—
47C4×Dic194C2213yes—
48#15690775C2213yes—
49C4×D388C2213yes—
50C4×D388C2213yes—
51#15690772C2213yes—
52C4×C22⋊D97C2213yes—
53C4×C22⋊D97C2213yes—
54C4×C22⋊D97C2216yes—
55#15690767C2213yes—
56C4⋊C4×D97C2213yes—
57C4⋊C4×D97C2213yes—
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—
67#13095511—412no—
68C2×Dic194C2313yes—
69C2×Dic194C2311yescharacteristic
70C2×Dic194C2316yes—
71Q8×D97—24no—
72C2×D388C2313yes—
73C2×D388C2313yes—
74C2×C22⋊D97C2313yes—
75C2×C22⋊D97C2313yes—
76C2×C22⋊D97C2316yes—
77C2×C4×D97C2311yescharacteristic
78C2×C4×D97C2313yes—
79C2×C4×D97C2313yes—
80C2×C4×D97C2313yes—
81C2×C4×D97C2316yes—
82C4⋊Dic97C2313yes—
83C4⋊Dic97C2313yes—
84C4⋊Dic97C2313yes—
85C4⋊Dic97C2313yes—
86#13095490C2313yes—
87#13095490C2316yes—
88#13095490C2311yescharacteristic
89#13095490C2311yescharacteristic
90#13095490C2313yes—
91#13095490C2313yes—
92#13095490C2313yes—
93C22⋊Dic97C2313yes—
94C22⋊Dic97C2316yes—
95C22⋊Dic97C2313yes—
96#13095488C2313yes—
97#13095488C2313yes—
98#13095488C2311yescharacteristic
99#13095488C2313yes—
100#13095488C2316yes—
101#13095488C2311yescharacteristic
102#13095488C2313yes—
103C4×Dic97C2313yes—
104C4×Dic97C2313yes—
105C4×Dic97C2313yes—
106C4×Dic97C2313yes—
107#41×C97—24no—
108C2×Q8×C97C2311yescharacteristic
109C2×D4×C97C2313yes—
110C22×C4×C97C2313yes—
111C22×C4×C97C2316yes—
112C4⋊C4×C97C2313yes—
113C4⋊C4×C97C2313yes—
114C4⋊C4×C97C2313yes—
115C4⋊C4×C97C2311yescharacteristic
116C22⋊C4×C97C2313yes—
117C22⋊C4×C97C2313yes—
118C42×C97C2313yes—
119C42×C97C2313yes—
120C23×C97C2413yes—
121C22×D97C2413yes—
122C22×D97C2411yescharacteristic
123Q8×C97#4114yescomplemented
124D4×C97—26no—
125C2×C4×C97—26no—
126C2×C4×C97—46no—
127C2×C4×C97C2411yescharacteristic
128C2×C4×C97C2411yescharacteristic
129C2×C4×C97C2413yes—
130C2×C4×C97C2413yes—
131C2×C4×C97C2411yescharacteristic
132C2×C4×C97C2413yes—
133C22⋊D97—26no—
134C22⋊D97—26no—
135C22⋊D97—46no—
136C2×Dic97C2411yescharacteristic
137C2×Dic97C2413yes—
138C2×Dic97C2413yes—
139C2×Dic97C2413yes—
140C2×Dic97C2411yescharacteristic
141C2×Dic97C2411yescharacteristic
142D388—26no—
143D388—26no—
144C4×D97—26no—
145C4×D97—26no—
146C4×D97—26no—
147C4×D97—22no—
148C4×D97—46no—
149Dic194—26no—
150Dic194—22no—
151Dic194—46no—
152C22×C97—23no—
153C22×C97—23no—
154C22×C97—23no—
155C22×C97C2511yescharacteristic
156C2×D97—21no—
157C2×D97—21no—
158C2×D97—23no—
159C2×D97—23no—
160C2×D97—23no—
161C2×D97H(5,𝔽2)12yes—
162C4×C97—21no—
163C4×C97—23no—
164C4×C97—23no—
165C4×C97—21no—
166C4×C97—21no—
167C4×C97C2×#4116yes—
168Dic97—23no—
169Dic97—21no—
170Dic97—23no—
171Dic97—23no—
172Dic97—21no—
173Dic972−1+412yes—
174C2×C97—43no—
175C2×C97C22×#4111yescharacteristic
176C2×C97C2×H(5,𝔽2)11yescharacteristic
177C2×C97C2×2−1+411yescharacteristic
178D97—22no—
179D97—43no—
180#3522—971nomaximal, Sylow
181C97#352211yescharacteristic, complemented, Sylow
182#559—973no—
183Q82—971no—
184#554—973no—
185#550—971no—
186#548—971no—
187#548—971no—
188#547—973no—
189#547—973no—
190#542—973no—
191#542—973no—
192#532—973no—
193#529—973no—
194#520—971no—
195#520D9711yescharacteristic, complemented
196C2×C4×Q8—971no—
197C2×#41—971no—
198C2×#41—976no—
199C2×#41C2×D9711yescharacteristic
200C22×Q8—971no—
201C4⋊Q8—973no—
202C4⋊Q8—973no—
203C4⋊Q8—973no—
204#127—973no—
205#126—973no—
206#126—973no—
207#125—973no—
208#124—973no—
209#124—973no—
210#124—976no—
211#124—973no—
212#123—976no—
213#123—973no—
214#123—973no—
215#123—976no—
216C22⋊Q8—976no—
217C22⋊Q8—973no—
218C22⋊Q8—971no—
219C22⋊Q8—976no—
220C22⋊Q8—973no—
221#121—973no—
222#121—973no—
223#121—973no—
224#121—973no—
225#121—973no—
226C4×Q8—971no—
227C4×Q8—971no—
228C4×Q8—973no—
229C4×Q8—973no—
230C4×Q8—973no—
231C4×Q8C2×D9711yescharacteristic
232C4×Q8C2×D9711yescharacteristic
233C4×D4—973no—
234C4×D4—973no—
235C4×D4—976no—
236C4×D4—973no—
237C4×D4—973no—
238C4×D4—973no—
239C4×D4C2×D9713yes—
240C4×D4C2×D9713yes—
241#117—973no—
242#117C2×D9713yes—
243C2×C4⋊C4—973no—
244C2×C4⋊C4—973no—
245C2×C4⋊C4C2×D9713yes—
246C2×C42—973no—
247#41—24no—
248#41—38812no—
249#41—1944no—
250C2×Q8—976no—
251C2×Q8—971no—
252C2×Q8—973no—
253C2×Q8—1944no—
254C2×Q8C22×D9711yescharacteristic
255C2×D4—976no—
256C2×D4—973no—
257C2×D4—973no—
258C2×D4—973no—
259C2×D4—973no—
260C2×D4C22×D9713yes—
261C22×C4—976no—
262C22×C4—971no—
263C22×C4—973no—
264C22×C4—973no—
265C22×C4—973no—
266C22×C4C22×D9713yes—
267C22×C4C22×D9716yes—
268C4⋊C4—973no—
269C4⋊C4—976no—
270C4⋊C4—971no—
271C4⋊C4—971no—
272C4⋊C4—973no—
273C4⋊C4—973no—
274C4⋊C4—973no—
275C4⋊C4—973no—
276C4⋊C4—973no—
277C4⋊C4—973no—
278C4⋊C4—973no—
279C4⋊C4C22×D9713yes—
280C4⋊C4C22×D9713yes—
281C4⋊C4C22×D9713yes—
282C4⋊C4C22×D9711yescharacteristic
283C22⋊C4—973no—
284C22⋊C4—976no—
285C22⋊C4—973no—
286C22⋊C4—973no—
287C22⋊C4—973no—
288C22⋊C4—971no—
289C22⋊C4—971no—
290C22⋊C4—973no—
291C22⋊C4—976no—
292C22⋊C4—973no—
293C22⋊C4C22×D9713yes—
294C22⋊C4C22×D9713yes—
295C42—973no—
296C42—973no—
297C42—973no—
298C42—973no—
299C42C22×D9713yes—
300C42C22×D9713yes—
301C23—971no—
302C23—973no—
303C23C23×D9713yes—
304Q8—3886no—
305Q8—1942no—
306Q8—1946no—
307Q8#1309551114yescomplemented
308D4—26no—
309D4—3886no—
310D4—1946no—
311D4—1946no—
312D4—1946no—
313D4—1946no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—971no—
317C2×C4—971no—
318C2×C4—973no—
319C2×C4—973no—
320C2×C4—973no—
321C2×C4—971no—
322C2×C4—3886no—
323C2×C4—1942no—
324C2×C4—1946no—
325C2×C4—1946no—
326C2×C4—1946no—
327C2×C4C23×D9711yescharacteristic
328C2×C4C23×D9713yes—
329C2×C4C23×D9711yescharacteristic
330C2×C4C23×D9713yes—
331C2×C4C23×D9711yescharacteristic
332C2×C4C23×D9713yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—972no—
337C22—1943no—
338C22—1943no—
339C22—1943no—
340C22—1941no—
341C22—1941no—
342C22C24×D9711yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—972no—
349C4—1941no—
350C4—1943no—
351C4—1943no—
352C4—1943no—
353C4—1941no—
354C4C2×#1309551116yes—
355C2—43no—
356C2—3883no—
357C2—1942no—
358C2C22×#1309551111yescharacteristic
359C2H(5,𝔽2)×D9711yescharacteristic
360C22−1+4×D9711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#520×C97C2
hypercenter#520D97
hypocenterC97#3522
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #20261468C2semidirect
2. Q8×Dic194C2semidirect
3. #20261440C2semidirect
4. #20261404C2semidirect
5. #20261397C2semidirect
6. #20261392C2semidirect
7. #20261390C2semidirect
8. #20261386C2semidirect
9. #20261356C2semidirect
10. #20261350C2semidirect
11. #20261282C2semidirect
12. #20261263C2semidirect
13. #20261217C2semidirect
14. C4×Q8×D97C2semidirect
15. #520×C97C2semidirect
123. Q8×C97#41semidirect
181. C97#3522semidirect
195. #520D97semidirect
307. Q8#13095511semidirect

Maximal subgroups

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

subgroupindex
#202614682
Q8×Dic1942
#202614402
#202614042
#202613972
#202613922
#202613902
#202613862
#202613562
#202613502
#202612822
#202612632
#202612172
C4×Q8×D972
#520×C972
#352297