Finite Group Databasebeta

← #27152108

Subgroups — #27152108

Order 12416.

484 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#20261288C211yesmaximal, characteristic, complemented
2C2×#15690806C211yesmaximal, characteristic
3#20261098C214yesmaximal, complemented
4#20261080C214yesmaximal, complemented
5#20260741C211yesmaximal, characteristic, complemented
6#20260740C211yesmaximal, characteristic, complemented
7#20260739C211yesmaximal, characteristic, complemented
8M16×Dic97C211yesmaximal, characteristic, complemented
9#418×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11C22×C22⋊D97—22no—
12C2×#13095495C2211yescharacteristic
13#15690809C2212yescomplemented
14#15690806—22no—
15#15690806C2212yes—
16#15690806C2212yes—
17#15690801C2212yescomplemented
18#15690787—22no—
19C4×Dic194—22no—
20#15690769C2211yescharacteristic
21C2×C4×Dic97C2211yescharacteristic
22#15690732C2214yescomplemented
23#15690731C2214yescomplemented
24#15690730C2214yescomplemented
25#15690720C2212yescomplemented
26C8×Dic97C2212yescomplemented
27C22×D4×C97C2211yescharacteristic
28C2×M16×C97C2211yescharacteristic
29#117×C97C2211yescharacteristic
30#102×C97C2214yescomplemented
31C23×D97—21no—
32C2×Dic194—22no—
33C2×D388—22no—
34C2×D388—22no—
35C2×D388—21no—
36C2×D388C2312yes—
37C2×C22⋊D97—24no—
38C2×C22⋊D97—24no—
39C2×C22⋊D97—48no—
40C22×Dic97D412yes—
41#13095495—22no—
42C2×#11715895C2312yes—
43C4⋊Dic97—22no—
44C4⋊Dic97C2312yes—
45#13095490—22no—
46#13095490—22no—
47C22⋊Dic97—21no—
48C4×Dic97—21no—
49C4×Dic97D412yes—
50C4×Dic97C2311yescharacteristic
51C4×Dic97C2311yescharacteristic
52C24×C97—21no—
53C2×D4×C97—21no—
54C2×D4×C97—22no—
55C2×D4×C97—22no—
56C2×D4×C97C2312yes—
57C22×C4×C97C2311yescharacteristic
58M16×C97—22no—
59C2×C8×C97C2312yes—
60C4⋊C4×C97C2312yes—
61C22⋊C4×C97—21no—
62C42×C97—21no—
63C23×C97—22no—
64C23×C97—44no—
65C23×C97—42no—
66C23×C97C2×D411yescharacteristic
67C22×D97—22no—
68C22×D97—44no—
69C22×D97—42no—
70D4×C97—22no—
71D4×C97—22no—
72D4×C97—42no—
73C2×C4×C97—22no—
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97#4112yes—
76C2×C4×C97#4112yes—
77C2×C4×C97C2411yescharacteristic
78C22⋊D97—48no—
79C22⋊D97—48no—
80C2×Dic97—24no—
81C2×Dic97—22no—
82C2×Dic97C2×D412yes—
83C2×Dic97C2×D412yes—
84D388—22no—
85D388—22no—
86D388—42no—
87Dic194—44no—
88C8×C97—22no—
89#11715895—22no—
90C22×C97—22no—
91C22×C97—42no—
92C22×C97—44no—
93C22×C97—84no—
94C22×C97#12412yes—
95C22×C97C22×D411yescharacteristic
96C2×D97—42no—
97C2×D97—44no—
98C2×D97—84no—
99C4×C97—42no—
100C4×C97#12412yes—
101C4×C97C2×#4111yescharacteristic
102C4×C97C2×#4111yescharacteristic
103Dic97—22no—
104Dic97—42no—
105Dic97AGL(1,ℤ/8)14yes—
106C2×C97—22no—
107C2×C97—82no—
108C2×C97C2×#12411yescharacteristic
109C2×C97C2×AGL(1,ℤ/8)12yes—
110D97—82no—
111#3138—971nomaximal, Sylow
112C97#313811yescharacteristic, complemented, Sylow
113#533—971no—
114C2×#127—971no—
115#490—974no—
116#486—974no—
117#418—971no—
118#418—971no—
119#418—971no—
120#418D9711yescharacteristic, complemented
121C4×M16—971no—
122C22×D4—971no—
123C22×D4—1942no—
124C22×D4C2×D9711yescharacteristic
125C2×M16—971no—
126C2×M16C2×D9711yescharacteristic
127C4⋊Q8—972no—
128#127—972no—
129#127—972no—
130#127—1942no—
131#125—972no—
132C22⋊Q8—1942no—
133C4×Q8—1942no—
134#117—971no—
135#117C2×D9711yescharacteristic
136C2×C42—971no—
137#102—974no—
138#102—974no—
139#102—974no—
140#102C2×D9714yescomplemented
141#97—972no—
142C4×C8—972no—
143C24—21no—
144C24—1941no—
145C2×Q8—1942no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—972no—
150C2×D4—3888no—
151C2×D4—1944no—
152C2×D4—1944no—
153C2×D4—1941no—
154C2×D4—1942no—
155C2×D4—1942no—
156C2×D4C22×D9712yes—
157C22×C4—972no—
158C22×C4C22×D9711yescharacteristic
159M16—22no—
160M16—1942no—
161C2×C8—972no—
162C2×C8C22×D9712yes—
163C4⋊C4—972no—
164C4⋊C4—1942no—
165C4⋊C4—1942no—
166C4⋊C4—1942no—
167C4⋊C4C22×D9712yes—
168C22⋊C4—21no—
169C22⋊C4—1941no—
170C42—21no—
171C42—972no—
172C42—971no—
173C42—971no—
174C42—1941no—
175C23—22no—
176C23—44no—
177C23—42no—
178C23—3884no—
179C23—3882no—
180C23—1942no—
181C23D4×D9711yescharacteristic
182Q8—3884no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—3888no—
187D4—3888no—
188D4—3882no—
189D4—1942no—
190D4—1942no—
191C2×C4—22no—
192C2×C4—972no—
193C2×C4—972no—
194C2×C4—1944no—
195C2×C4—1942no—
196C2×C4D4×D9711yescharacteristic
197C2×C4#1309551012yes—
198C2×C4#1309551112yes—
199C2×C4C23×D9711yescharacteristic
200C8—22no—
201C8—1942no—
202C22—22no—
203C22—42no—
204C22—44no—
205C22—84no—
206C22—7764no—
207C22—3882no—
208C22—3884no—
209C22#1569079612yes—
210C22C2×D4×D9711yescharacteristic
211C4—42no—
212C4—974no—
213C4—3882no—
214C4—1942no—
215C4#1569079612yes—
216C4C2×#1309551011yescharacteristic
217C4C2×#1309551111yescharacteristic
218C2—22no—
219C2—82no—
220C2—7762no—
221C2C2×#1569079611yescharacteristic
222C2AGL(1,ℤ/8)×D9712yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#418×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690806C2
hypercenter#418D97
hypocenterC97#3138
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #20261288C2semidirect
3. #20261098C2semidirect
4. #20261080C2semidirect
5. #20260741C2semidirect
6. #20260740C2semidirect
7. #20260739C2semidirect
8. M16×Dic97C2semidirect
9. #418×C97C2semidirect
13. #15690809C22semidirect
17. #15690801C22semidirect
22. #15690732C22semidirect
23. #15690731C22semidirect
24. #15690730C22semidirect
25. #15690720C22semidirect
26. C8×Dic97C22semidirect
30. #102×C97C22semidirect
112. C97#3138semidirect
120. #418D97semidirect
140. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612882
C2×#156908062
#202610982
#202610802
#202607412
#202607402
#202607392
M16×Dic972
#418×C972
#313897