Finite Group Databasebeta

← #27153476

Subgroups — #27153476

Order 12416.

482 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690829C211yesmaximal, characteristic, complemented
2C2×Q16×D97C211yesmaximal, characteristic, complemented
3#20261393C212yesmaximal, complemented
4#20260973C211yesmaximal, characteristic, complemented
5#20260968C211yesmaximal, characteristic, complemented
6#20260954C212yesmaximal, complemented
7#20260894C212yesmaximal, complemented
8#20260872C212yesmaximal, complemented
9C4×Dic388C211yesmaximal, characteristic, complemented
10C8×D388C211yesmaximal, characteristic, complemented
11#495×C97C211yesmaximal, characteristic, complemented
12C2×#13095510C2212yes—
13C2×Q8×D97C2212yes—
14#15690829—24no—
15C2×Dic388C2211yescharacteristic
16C2×#13095501C2212yes—
17Q16×D97—24no—
18C2×#13095499C2212yes—
19C2×C8×D97C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic
21#15690796—22no—
22#15690787—22no—
23#15690786C2212yes—
24#15690783—22no—
25#15690781C2212yes—
26C4×Dic194C2212yes—
27C4×D388C2211yescharacteristic
28#15690770—22no—
29#15690744C2211yescharacteristic
30C4⋊#11715895C2211yescharacteristic
31#15690735C2212yes—
32#15690734C2212yes—
33#15690731C2212yes—
34#15690722C2211yescharacteristic
35#15690722C2211yescharacteristic
36C2×Q16×C97C2211yescharacteristic
37C2×SD16×C97C2212yes—
38C2×D8×C97C2211yescharacteristic
39#124×C97C2212yes—
40C4×C8×C97C2211yescharacteristic
41#13095510—24no—
42#13095510—42no—
43C2×Dic194—22no—
44C2×Dic194C2312yes—
45Q8×D97—24no—
46Q8×D97—42no—
47C2×D388D411yescharacteristic
48C2×C22⋊D97—22no—
49C2×C22⋊D97—22no—
50C22×Dic97—22no—
51C2×C4×D97—22no—
52C2×C4×D97C2311yescharacteristic
53C2×C4×D97C2311yescharacteristic
54Dic388D414yescomplemented
55#13095501—24no—
56#13095499—24no—
57C2×#11715895C2311yescharacteristic
58C2×#11715895C2311yescharacteristic
59C8×D97—22no—
60C8×D97—22no—
61C4⋊Dic97D411yescharacteristic
62C4⋊Dic97C2312yes—
63#13095490—22no—
64#13095490—22no—
65C22⋊Dic97—22no—
66#13095488—22no—
67#13095488D411yescharacteristic
68#13095488D411yescharacteristic
69C4×Dic97—22no—
70C2×Q8×C97C2312yes—
71C2×D4×C97C2312yes—
72Q16×C97—22no—
73SD16×C97—42no—
74D8×C97—22no—
75C2×C8×C97C2311yescharacteristic
76C2×C8×C97C2311yescharacteristic
77C22⋊C4×C97—22no—
78C22⋊C4×C97—22no—
79C42×C97C2311yescharacteristic
80C23×C97—22no—
81C22×D97C2×D411yescharacteristic
82C22×D97C2×D411yescharacteristic
83Q8×C97—22no—
84Q8×C97—22no—
85D4×C97—22no—
86D4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97C2411yescharacteristic
91C22⋊D97—44no—
92C22⋊D97—44no—
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97—44no—
96C2×Dic97C2×D411yescharacteristic
97C2×Dic97C2×D411yescharacteristic
98D388—22no—
99C4×D97—21no—
100C4×D97—21no—
101C4×D97—21no—
102C4×D97—21no—
103C4×D97—44no—
104Dic194—22no—
105Dic194—44no—
106Dic194C2×D414yes—
107C8×C97—21no—
108C8×C97C2×D412yes—
109#11715895—21no—
110#11715895—21no—
111C22×C97—42no—
112C22×C97—42no—
113C22×C97—42no—
114C22×C97C22×D411yescharacteristic
115C2×D97—21no—
116C2×D97—21no—
117C2×D97—21no—
118C2×D97—21no—
119C2×D97—21no—
120C2×D97#13512yescomplemented
121C4×C97—21no—
122C4×C97—21no—
123C4×C97—42no—
124C4×C97C22×D411yescharacteristic
125C4×C97H(5,𝔽2)11yescharacteristic
126Dic97—24no—
127Dic97—21no—
128Dic97—21no—
129Dic97—42no—
130C2×C97—82no—
131C2×C97D4211yescharacteristic
132C2×C97C2×#13511yescharacteristic
133C2×C97#57811yescharacteristic
134D97—22no—
135D97—41no—
136#3285—971nomaximal, Sylow
137C97#328511yescharacteristic, complemented, Sylow
138C2×#135—971no—
139C22×Q16—971no—
140#548—972no—
141#495D9711yescharacteristic, complemented
142#467—971no—
143#466—971no—
144#464—972no—
145#452—972no—
146#448—972no—
147C4×Q16—971no—
148C8×D4—971no—
149C2×#41—972no—
150C22×Q8—972no—
151#135—1944no—
152C2×Q16—972no—
153C2×Q16—971no—
154C2×Q16—1944no—
155C2×Q16C2×D9711yescharacteristic
156C2×SD16—972no—
157C2×SD16C2×D9712yes—
158C2×D8C2×D9711yescharacteristic
159C22×C8—971no—
160C22×C8—971no—
161#124—1942no—
162#124C2×D9712yes—
163C22⋊Q8—972no—
164C22⋊Q8—1942no—
165#121—972no—
166#121—1942no—
167C4×Q8—972no—
168C4×D4—971no—
169C4×D4—1942no—
170#107—971no—
171C4⋊C8—971no—
172#103—972no—
173#103—972no—
174#102—972no—
175C22⋊C8—971no—
176C22⋊C8—971no—
177C4×C8C2×D9711yescharacteristic
178#41—3882no—
179#41—1944no—
180C2×Q8—972no—
181C2×Q8—3882no—
182C2×Q8—1942no—
183C2×Q8—1944no—
184C2×Q8C22×D9712yes—
185C2×D4—971no—
186C2×D4—1942no—
187C2×D4—1942no—
188C2×D4C22×D9712yes—
189C22×C4—971no—
190C22×C4—971no—
191C22×C4—1942no—
192C22×C4—1942no—
193Q16—22no—
194Q16—974no—
195Q16—1944no—
196SD16—42no—
197SD16—1944no—
198D8—22no—
199C2×C8—971no—
200C2×C8—971no—
201C2×C8—1942no—
202C2×C8—1942no—
203C2×C8C22×D9711yescharacteristic
204C2×C8C22×D9711yescharacteristic
205C4⋊C4—971no—
206C4⋊C4—972no—
207C4⋊C4—1942no—
208C4⋊C4—1942no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—971no—
212C22⋊C4—971no—
213C22⋊C4—1942no—
214C22⋊C4—1942no—
215C42—1942no—
216C42C22×D9711yescharacteristic
217C23—22no—
218C23—971no—
219C23—971no—
220Q8—22no—
221Q8—22no—
222Q8—974no—
223Q8—3884no—
224Q8—1942no—
225D4—22no—
226D4—22no—
227D4—3884no—
228D4—3884no—
229D4—1942no—
230C2×C4—22no—
231C2×C4—971no—
232C2×C4—971no—
233C2×C4—3884no—
234C2×C4—3884no—
235C2×C4—1942no—
236C2×C4—1942no—
237C2×C4—1941no—
238C2×C4—1941no—
239C2×C4—1941no—
240C2×C4—1941no—
241C2×C4D4×D9711yescharacteristic
242C2×C4D4×D9711yescharacteristic
243C2×C4C23×D9711yescharacteristic
244C8—21no—
245C8—1941no—
246C8—1941no—
247C8D4×D9712yes—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—972no—
252C22—1941no—
253C22—1941no—
254C22—1941no—
255C22—1941no—
256C22—1941no—
257C22C2×D4×D9711yescharacteristic
258C4—21no—
259C4—21no—
260C4—42no—
261C4—3882no—
262C4—1941no—
263C4—1941no—
264C4—1944no—
265C4C2×D4×D9711yescharacteristic
266C4#1569085511yescharacteristic
267C2—82no—
268C2—3881no—
269C2—1942no—
270C2#2026137011yescharacteristic
271C2#135×D9711yescharacteristic
272C2#2026154911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#495×C97C2
socleC22×C97C22×D4
hypercenter#495D97
hypocenterC97#3285
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#15690829C2semidirect
2. C2×Q16×D97C2semidirect
3. #20261393C2semidirect
4. #20260973C2semidirect
5. #20260968C2semidirect
6. #20260954C2semidirect
7. #20260894C2semidirect
8. #20260872C2semidirect
9. C4×Dic388C2semidirect
10. C8×D388C2semidirect
11. #495×C97C2semidirect
54. Dic388D4semidirect
120. C2×D97#135semidirect
137. C97#3285semidirect
141. #495D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908292
C2×Q16×D972
#202613932
#202609732
#202609682
#202609542
#202608942
#202608722
C4×Dic3882
C8×D3882
#495×C972
#328597