Finite Group Databasebeta

← #27152124

Subgroups — #27152124

Order 12416.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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×#15690799C211yesmaximal, characteristic
3#20261096C214yesmaximal, complemented
4#20261091C214yesmaximal, complemented
5#20260744C211yesmaximal, characteristic, complemented
6#20260741C211yesmaximal, characteristic
7#20260739C211yesmaximal, characteristic
8M16×Dic97C211yesmaximal, characteristic, complemented
9#419×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11C2×#13095495C2211yescharacteristic
12#15690809C2212yescomplemented
13#15690801C2212yescomplemented
14#15690799—22no—
15#15690799C2212yes—
16#15690799C2212yes—
17#15690787—22no—
18C4×Dic194—22no—
19#15690769C2211yescharacteristic
20C2×#13095488—22no—
21C2×C4×Dic97C2211yescharacteristic
22#15690734C2214yescomplemented
23#15690732C2214yes—
24#15690730C2214yes—
25#15690720C2212yescomplemented
26C8×Dic97C2212yescomplemented
27C22×Q8×C97C2211yescharacteristic
28C2×M16×C97C2211yescharacteristic
29#117×C97C2211yescharacteristic
30#103×C97C2214yescomplemented
31C23×D97—21no—
32C2×Dic194—22no—
33C2×D388—22no—
34C2×D388—22no—
35C2×D388—21no—
36C2×D388C2312yes—
37C22×Dic97D412yes—
38#13095495—22no—
39C2×#11715895C2312yes—
40C4⋊Dic97—22no—
41C4⋊Dic97C2312yes—
42#13095490—22no—
43#13095490—22no—
44C22⋊Dic97—21no—
45#13095488—24no—
46#13095488—24no—
47C4×Dic97—21no—
48C4×Dic97D412yes—
49C4×Dic97C2311yescharacteristic
50C4×Dic97C2311yescharacteristic
51C2×Q8×C97—21no—
52C2×Q8×C97—22no—
53C2×Q8×C97—22no—
54C2×Q8×C97C2312yes—
55C22×C4×C97—21no—
56C22×C4×C97C2311yescharacteristic
57M16×C97—22no—
58C2×C8×C97C2312yes—
59C4⋊C4×C97C2312yes—
60C22⋊C4×C97—21no—
61C42×C97—21no—
62C23×C97C2×D411yescharacteristic
63C22×D97—22no—
64C22×D97—44no—
65C22×D97—42no—
66Q8×C97—22no—
67Q8×C97—22no—
68Q8×C97—42no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97—42no—
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97#4112yes—
74C2×C4×C97#4112yes—
75C2×C4×C97C2411yescharacteristic
76C2×Dic97—22no—
77C2×Dic97—42no—
78C2×Dic97C2×D412yes—
79C2×Dic97C2×D412yes—
80D388—22no—
81D388—22no—
82D388—42no—
83Dic194—44no—
84C8×C97—22no—
85#11715895—22no—
86C22×C97—22no—
87C22×C97#12412yes—
88C22×C97C22×D411yescharacteristic
89C2×D97—42no—
90C2×D97—44no—
91C2×D97—84no—
92C4×C97—42no—
93C4×C97—42no—
94C4×C97#12412yes—
95C4×C97C2×#4111yescharacteristic
96C4×C97C2×#4111yescharacteristic
97Dic97—22no—
98Dic97—22no—
99Dic97—42no—
100C2×C97—22no—
101C2×C97C2×#12411yescharacteristic
102C2×C97#57512yes—
103D97—82no—
104#3140—971nomaximal, Sylow
105C97#314011yescharacteristic, complemented, Sylow
106#533—971no—
107C2×#124—971no—
108#489—974no—
109#488—974no—
110#419—971no—
111#419D9711yescharacteristic, complemented
112#418—971no—
113#418—971no—
114C4×M16—971no—
115C22×Q8C2×D9711yescharacteristic
116C22×D4—971no—
117C2×M16—971no—
118C2×M16C2×D9711yescharacteristic
119C4⋊Q8—972no—
120#125—972no—
121#124—972no—
122#124—972no—
123#124—1942no—
124C22⋊Q8—1942no—
125C4×Q8—1942no—
126#117—971no—
127#117C2×D9711yescharacteristic
128C2×C22⋊C4—1942no—
129C2×C42—971no—
130#103—974no—
131#103C2×D9714yescomplemented
132#102—974no—
133#102—974no—
134#97—972no—
135C4×C8—972no—
136C24—1941no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—1942no—
141C2×Q8C22×D9712yes—
142C2×D4—972no—
143C2×D4—1941no—
144C2×D4—1942no—
145C2×D4—1942no—
146C22×C4—21no—
147C22×C4—972no—
148C22×C4C22×D9711yescharacteristic
149M16—22no—
150M16—1942no—
151C2×C8—972no—
152C2×C8C22×D9712yes—
153C4⋊C4—972no—
154C4⋊C4—1942no—
155C4⋊C4—1942no—
156C4⋊C4—1942no—
157C4⋊C4C22×D9712yes—
158C22⋊C4—21no—
159C22⋊C4—1944no—
160C22⋊C4—1944no—
161C22⋊C4—1941no—
162C42—21no—
163C42—972no—
164C42—971no—
165C42—971no—
166C42—1941no—
167C23—3884no—
168C23—3882no—
169C23—1942no—
170C23D4×D9711yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—42no—
174Q8—3884no—
175D4—3882no—
176D4—1942no—
177D4—1942no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—972no—
182C2×C4—972no—
183C2×C4—3882no—
184C2×C4—1942no—
185C2×C4D4×D9711yescharacteristic
186C2×C4#1309551012yes—
187C2×C4#1309551112yes—
188C2×C4C23×D9711yescharacteristic
189C8—22no—
190C8—1942no—
191C22—22no—
192C22—7764no—
193C22—3882no—
194C22—3884no—
195C22#1569079612yes—
196C22C2×D4×D9711yescharacteristic
197C4—42no—
198C4—42no—
199C4—3882no—
200C4—1942no—
201C4—1942no—
202C4#1569079612yes—
203C4C2×#1309551011yescharacteristic
204C4C2×#1309551111yescharacteristic
205C2—22no—
206C2—7762no—
207C2C2×#1569079611yescharacteristic
208C2#2026153412yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#419×C97C2
socleC22×C97C22×D4
prime radicalC22×D388C22
hypercenter#419D97
hypocenterC97#3140
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #20261288C2semidirect
3. #20261096C2semidirect
4. #20261091C2semidirect
5. #20260744C2semidirect
8. M16×Dic97C2semidirect
9. #419×C97C2semidirect
12. #15690809C22semidirect
13. #15690801C22semidirect
22. #15690734C22semidirect
25. #15690720C22semidirect
26. C8×Dic97C22semidirect
30. #103×C97C22semidirect
105. C97#3140semidirect
111. #419D97semidirect
131. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612882
C2×#156907992
#202610962
#202610912
#202607442
#202607412
#202607392
M16×Dic972
#419×C972
#314097