Finite Group Databasebeta

← #27157063

Subgroups — #27157063

Order 12416.

312 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 74 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#20261752C211yesmaximal, characteristic, complemented
2#20261738C211yesmaximal, characteristic, complemented
3#20261583C214yesmaximal, complemented
4C2×#15690772C211yesmaximal, characteristic
5C2×#13095520—21no—
6#15690877—21no—
7#15690877—21no—
8C4⋊#11715897C2212yescomplemented
9C22⋊#11715897—21no—
10C4×#11715897C2212yescomplemented
11C22×D388C411yescharacteristic
12C22×C4×D97—21no—
13#15690772—21no—
14#15690772—21no—
15#15690772C412yes—
16#15690772C2212yes—
17C2×#13095488—21no—
18C2×C4×Dic97C2211yescharacteristic
19C2×C4⋊C4×C97C411yescharacteristic
20#13095520—42no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23C23×D97—21no—
24C2×D388—21no—
25C2×D388—24no—
26C2×D388C2×C412yes—
27C22×Dic97D411yescharacteristic
28C22×Dic97D411yescharacteristic
29C2×C4×D97—22no—
30C2×C4×D97—22no—
31C2×C4×D97—42no—
32C2×C4×D97—42no—
33#13095488—22no—
34#13095488—22no—
35C4×Dic97C2×C411yescharacteristic
36C4×Dic97D411yescharacteristic
37C4×Dic97D411yescharacteristic
38C4×Dic97C2311yescharacteristic
39C22×C4×C97—21no—
40C22×C4×C97C2×C411yescharacteristic
41C4⋊C4×C97—21no—
42C4⋊C4×C97C2×C412yes—
43C23×C97C22⋊C411yescharacteristic
44C22×D97—24no—
45C22×D97—22no—
46C22×D97—42no—
47C22×D97—42no—
48C2×C4×C97—22no—
49C2×C4×C97—22no—
50C2×C4×C97—42no—
51C2×C4×C97C22⋊C411yescharacteristic
52C2×C4×C97C22×C411yescharacteristic
53C2×Dic97—41no—
54C2×Dic97C22⋊C411yescharacteristic
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97D812yes—
57C2×Dic97SD1612yes—
58C2×Dic97C2×D411yescharacteristic
59C2×Dic97C2×D411yescharacteristic
60D388—24no—
61D388—22no—
62D388M1614yescomplemented
63C4×D97—44no—
64C4×D97—42no—
65C4×D97—42no—
66#11715897—42no—
67#11715897—42no—
68C22×C97—21no—
69C22×C97—21no—
70C22×C97#10212yes—
71C22×C97C2×C22⋊C411yescharacteristic
72C2×D97—24no—
73C2×D97—48no—
74C2×D97—42no—
75C2×D97—42no—
76C4×C97—21no—
77C4×C97—42no—
78C4×C97C2×M1612yes—
79Dic97—21no—
80Dic97—21no—
81Dic97#10212yes—
82Dic97C2×D811yescharacteristic
83Dic97C2×SD1611yescharacteristic
84C2×C97—22no—
85C2×C97#40711yescharacteristic
86C2×C97C2×#10211yescharacteristic
87C2×C97#42111yescharacteristic
88D97—44no—
89#1486—971nomaximal, Sylow
90C97#148611yescharacteristic, complemented, Sylow
91C2×C4×D4—971no—
92#431—971no—
93#423—971no—
94D4⋊C8—974no—
95C22×D4—971no—
96C23×C4—1941no—
97C2×M16—1941no—
98C4×D4—972no—
99C4×D4—972no—
100C4×D4—1941no—
101C4×D4—1941no—
102C2×C4⋊C4#8760411yescharacteristic
103C2×C22⋊C4—1941no—
104C2×C42—971no—
105C4⋊C8—972no—
106C4⋊C8—1941no—
107C4⋊C8—1941no—
108C22⋊C8—1941no—
109C4×C8—972no—
110C24—1941no—
111C2×D4—972no—
112C2×D4—1944no—
113C2×D4—1941no—
114C22×C4—21no—
115C22×C4—971no—
116C22×C4—971no—
117C22×C4—3882no—
118C22×C4—3882no—
119C22×C4—1942no—
120C22×C4—1942no—
121C22×C4C2×#8760411yescharacteristic
122M16—3882no—
123C2×C8—1942no—
124C2×C8—1942no—
125C4⋊C4—21no—
126C4⋊C4C2×#8760412yes—
127C22⋊C4—1942no—
128C22⋊C4—1942no—
129C42—971no—
130C42—971no—
131C42—971no—
132C42—971no—
133C23—3882no—
134C23—3882no—
135C23—1944no—
136C23—1942no—
137C23C22⋊#8760411yescharacteristic
138D4—974no—
139D4—1944no—
140D4—1942no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—972no—
145C2×C4—972no—
146C2×C4—971no—
147C2×C4—971no—
148C2×C4—971no—
149C2×C4—971no—
150C2×C4—3882no—
151C2×C4—3884no—
152C2×C4—3882no—
153C2×C4—3881no—
154C2×C4C22⋊#8760411yescharacteristic
155C2×C4C22×#8760411yescharacteristic
156C8—3882no—
157C8—3882no—
158C22—21no—
159C22—21no—
160C22—3888no—
161C22—3882no—
162C22—3882no—
163C22—1944no—
164C22#1569087212yes—
165C22C2×C22⋊#8760411yescharacteristic
166C4—21no—
167C4—42no—
168C4—971no—
169C4—972no—
170C4—971no—
171C4—1941no—
172C4—1941no—
173C4D97⋊M1612yes—
174C2—22no—
175C2—3884no—
176C2#2026170711yescharacteristic
177C2C2×#1569087211yescharacteristic
178C2#2026173411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D388C4
cosocleC4×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1486
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20261752C2semidirect
2. #20261738C2semidirect
3. #20261583C2semidirect
8. C4⋊#11715897C22semidirect
10. C4×#11715897C22semidirect
62. D388M16semidirect
90. C97#1486semidirect

Maximal subgroups

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

subgroupindex
#202617522
#202617382
#202615832
C2×#156907722
#148697