Finite Group Databasebeta

← #27155848

Subgroups — #27155848

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#20261470C213yesmaximal, complemented
2#20261451C213yesmaximal, complemented
3#20261438C213yesmaximal, complemented
4#20261423C213yesmaximal, complemented
5#20261407C211yesmaximal, characteristic, complemented
6#20261397C211yesmaximal, characteristic, complemented
7#20261397C211yesmaximal, characteristic, complemented
8#20261359C213yesmaximal, complemented
9#20261341C213yesmaximal, complemented
10#20261272C213yesmaximal, complemented
11#20261270C213yesmaximal, complemented
12#20261217C211yesmaximal, characteristic, complemented
13#20261211C211yesmaximal, characteristic, complemented
14C4×#13095512C211yesmaximal, characteristic, complemented
15#557×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095512C2211yescharacteristic
18C2×Q8×D97C2211yescharacteristic
19#15690807C2213yes—
20#15690805C2216yes—
21#15690805C2213yes—
22#15690805C2211yescharacteristic
23#15690801C2213yes—
24#15690799C2213yes—
25#15690799C2216yes—
26#15690798C2213yes—
27#15690798C2213yes—
28#15690795C2216yes—
29#15690795C2213yes—
30#15690795C2213yes—
31#15690795C2216yes—
32#15690790C2213yes—
33#15690790C2213yes—
34#15690786C2213yes—
35#15690785C2213yes—
36#15690785C2213yes—
37#15690785C2213yes—
38#15690785C2213yes—
39#15690785C2216yes—
40#15690785C2213yes—
41Q8×Dic97C2211yescharacteristic
42C4×Dic194C2213yes—
43#15690775C2213yes—
44C4×D388C2213yes—
45C4×D388C2213yes—
46C4×D388C2213yes—
47C4×D388C2213yes—
48C4×D388C2213yes—
49#15690772C2216yes—
50#15690772C2213yes—
51#15690772C2213yes—
52#15690768C2213yes—
53#15690768C2216yes—
54#15690768C2213yes—
55#15690767C2213yes—
56#15690767C2213yes—
57C4⋊C4×D97C2213yes—
58C42×D97C2213yes—
59C4⋊Q8×C97C2213yes—
60#125×C97C2213yes—
61#125×C97C2213yes—
62C4×Q8×C97C2211yescharacteristic
63C4×Q8×C97C2211yescharacteristic
64C4×Q8×C97C2211yescharacteristic
65C4×Q8×C97C2213yes—
66#13095512—24no—
67#13095512—24no—
68C2×Dic194C2313yes—
69Q8×D97—24no—
70C2×D388C2313yes—
71C2×D388C2313yes—
72C2×D388C2316yes—
73C2×D388C2313yes—
74C2×D388C2313yes—
75C2×D388C2313yes—
76C2×C4×D97C2313yes—
77C2×C4×D97C2313yes—
78C2×C4×D97C2313yes—
79C2×C4×D97C2313yes—
80C2×C4×D97C2313yes—
81C2×C4×D97C2316yes—
82C4⋊Dic97C2313yes—
83C4⋊Dic97C2313yes—
84C4⋊Dic97C2311yescharacteristic
85C4⋊Dic97C2313yes—
86#13095490C2313yes—
87#13095490C2311yescharacteristic
88#13095490C2313yes—
89#13095490C2313yes—
90#13095488C2311yescharacteristic
91#13095488C2313yes—
92#13095488C2313yes—
93#13095488C2313yes—
94#13095488C2313yes—
95#13095488C2316yes—
96#13095488C2316yes—
97#13095488C2313yes—
98#13095488C2316yes—
99#13095488C2311yescharacteristic
100#13095488C2311yescharacteristic
101#13095488C2313yes—
102#13095488C2313yes—
103C4×Dic97C2311yescharacteristic
104C4×Dic97C2313yes—
105C4×Dic97C2313yes—
106C4×Dic97C2316yes—
107C2×Q8×C97C2313yes—
108C2×Q8×C97C2311yescharacteristic
109C4⋊C4×C97C2313yes—
110C4⋊C4×C97C2313yes—
111C4⋊C4×C97C2313yes—
112C4⋊C4×C97C2316yes—
113C4⋊C4×C97C2313yes—
114C4⋊C4×C97C2313yes—
115C4⋊C4×C97C2311yescharacteristic
116C42×C97C2313yes—
117C42×C97C2313yes—
118C42×C97C2313yes—
119C22×D97C2411yescharacteristic
120C22×D97C2413yes—
121C22×D97C2413yes—
122Q8×C97—43no—
123Q8×C97#4114yes—
124C2×C4×C97C2411yescharacteristic
125C2×C4×C97C2411yescharacteristic
126C2×C4×C97C2411yescharacteristic
127C2×C4×C97C2413yes—
128C2×C4×C97C2413yes—
129C2×C4×C97C2413yes—
130C2×C4×C97C2413yes—
131C2×Dic97C2411yescharacteristic
132C2×Dic97C2413yes—
133C2×Dic97C2411yescharacteristic
134C2×Dic97C2413yes—
135C2×Dic97C2411yescharacteristic
136D388—26no—
137D388—26no—
138D388—43no—
139D388—43no—
140D388—43no—
141D388—46no—
142C4×D97—26no—
143C4×D97—26no—
144C4×D97—26no—
145C4×D97—43no—
146C4×D97—46no—
147C4×D97—43no—
148Dic194—26no—
149C22×C97C2511yescharacteristic
150C2×D97—21no—
151C2×D97—23no—
152C2×D97—23no—
153C2×D97—23no—
154C2×D97—23no—
155C2×D97—23no—
156C2×D97—21no—
157C2×D97—23no—
158C2×D97—21no—
159C4×C97—21no—
160C4×C97—23no—
161C4×C97—23no—
162C4×C97—21no—
163C4×C97—23no—
164C4×C97—21no—
165C4×C97C2×#4116yes—
166Dic97—23no—
167Dic97—21no—
168Dic97—21no—
169Dic97—21no—
170Dic97—23no—
171C2×C97C22×#4111yescharacteristic
172C2×C97#58511yescharacteristic
173C2×C97#58511yescharacteristic
174D97—41no—
175D97—43no—
176D97—43no—
177#3524—971nomaximal, Sylow
178C97#352411yescharacteristic, complemented, Sylow
179#559—973no—
180#557D9711yescharacteristic, complemented
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#520—971no—
192#519—971no—
193C4×#41—971no—
194C2×#41—971no—
195C2×#41—971no—
196C22×Q8—971no—
197C4⋊Q8C2×D9713yes—
198#127—973no—
199#126—971no—
200#126—973no—
201#126—976no—
202#125—973no—
203#125C2×D9713yes—
204#125C2×D9713yes—
205#124—973no—
206#124—976no—
207#124—973no—
208#124—973no—
209#123—976no—
210#123—973no—
211#123—973no—
212#123—976no—
213C22⋊Q8—973no—
214C22⋊Q8—973no—
215C22⋊Q8—973no—
216#121—973no—
217#121—973no—
218#121—973no—
219#121—973no—
220#121—976no—
221#121—973no—
222C4×Q8—971no—
223C4×Q8—973no—
224C4×Q8—973no—
225C4×Q8C2×D9711yescharacteristic
226C4×Q8C2×D9711yescharacteristic
227C4×Q8C2×D9711yescharacteristic
228C4×Q8C2×D9713yes—
229C4×D4—973no—
230C4×D4—973no—
231C4×D4—976no—
232C4×D4—973no—
233C4×D4—973no—
234C4×D4—973no—
235C4×D4—973no—
236C4×D4—973no—
237#117—973no—
238#117—973no—
239#117—976no—
240#117—973no—
241#117—973no—
242C2×C4⋊C4—973no—
243C2×C42—973no—
244#41—1944no—
245#41—1944no—
246C2×Q8—973no—
247C2×Q8—1944no—
248C2×Q8C22×D9713yes—
249C2×Q8C22×D9711yescharacteristic
250C2×D4—973no—
251C2×D4—973no—
252C2×D4—976no—
253C2×D4—973no—
254C2×D4—973no—
255C2×D4—973no—
256C22×C4—973no—
257C22×C4—973no—
258C22×C4—976no—
259C22×C4—973no—
260C22×C4—973no—
261C22×C4—973no—
262C4⋊C4—973no—
263C4⋊C4—971no—
264C4⋊C4—973no—
265C4⋊C4—973no—
266C4⋊C4—973no—
267C4⋊C4—973no—
268C4⋊C4—971no—
269C4⋊C4—973no—
270C4⋊C4C22×D9713yes—
271C4⋊C4C22×D9713yes—
272C4⋊C4C22×D9713yes—
273C4⋊C4C22×D9716yes—
274C4⋊C4C22×D9713yes—
275C4⋊C4C22×D9711yescharacteristic
276C4⋊C4C22×D9713yes—
277C22⋊C4—971no—
278C22⋊C4—973no—
279C22⋊C4—973no—
280C22⋊C4—973no—
281C22⋊C4—976no—
282C22⋊C4—973no—
283C22⋊C4—976no—
284C22⋊C4—971no—
285C22⋊C4—973no—
286C22⋊C4—971no—
287C22⋊C4—973no—
288C22⋊C4—976no—
289C22⋊C4—973no—
290C42—971no—
291C42—973no—
292C42—973no—
293C42—976no—
294C42C22×D9713yes—
295C42C22×D9713yes—
296C42C22×D9713yes—
297C23—971no—
298C23—973no—
299C23—973no—
300Q8—43no—
301Q8—1946no—
302Q8#1309551214yes—
303D4—3883no—
304D4—3886no—
305D4—3883no—
306D4—3883no—
307D4—1946no—
308D4—1946no—
309C2×C4—971no—
310C2×C4—971no—
311C2×C4—973no—
312C2×C4—973no—
313C2×C4—971no—
314C2×C4—3883no—
315C2×C4—3883no—
316C2×C4—3886no—
317C2×C4—1946no—
318C2×C4—1946no—
319C2×C4—1946no—
320C2×C4C23×D9713yes—
321C2×C4C23×D9711yescharacteristic
322C2×C4C23×D9713yes—
323C2×C4C23×D9711yescharacteristic
324C2×C4C23×D9711yescharacteristic
325C2×C4C23×D9713yes—
326C2×C4C23×D9713yes—
327C22—1941no—
328C22—1943no—
329C22—1943no—
330C22—1943no—
331C22—1941no—
332C22—1943no—
333C22—1943no—
334C22—1943no—
335C22—1941no—
336C22C24×D9711yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—1941no—
344C4—1941no—
345C4—1943no—
346C4—1941no—
347C4—1943no—
348C4C2×#1309551216yes—
349C2—3881no—
350C2—3883no—
351C2—3883no—
352C2C22×#1309551211yescharacteristic
353C2#2026157111yescharacteristic
354C2#2026157211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#557×C97C2
hypercenter#557D97
hypocenterC97#3524
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #20261470C2semidirect
2. #20261451C2semidirect
3. #20261438C2semidirect
4. #20261423C2semidirect
5. #20261407C2semidirect
6. #20261397C2semidirect
7. #20261397C2semidirect
8. #20261359C2semidirect
9. #20261341C2semidirect
10. #20261272C2semidirect
11. #20261270C2semidirect
12. #20261217C2semidirect
13. #20261211C2semidirect
14. C4×#13095512C2semidirect
15. #557×C97C2semidirect
178. C97#3524semidirect
180. #557D97semidirect

Maximal subgroups

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

subgroupindex
#202614702
#202614512
#202614382
#202614232
#202614072
#202613972
#202613592
#202613412
#202612722
#202612702
#202612172
#202612112
C4×#130955122
#557×C972
#352497