Finite Group Databasebeta

← #27157632

Subgroups — #27157632

Order 12416.

504 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 70 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#20261676C212yesmaximal
2#20261613C214yesmaximal, complemented
3C2×C22⋊C4×D97C211yesmaximal, characteristic
4C2×C22⋊#87604—24no—
5C2×C22⋊#87604C2214yes—
6C24×D97C2211yescharacteristic
7C22×C4×D97C412yescomplemented
8C2×C22⋊Dic97C411yescharacteristic, complemented
9C2×#13095488C2212yes—
10C22⋊C4×D97—24no—
11C2×C22⋊C4×C97C411yescharacteristic, complemented
12C22×#87604—24no—
13C22⋊#87604—28no—
14C22⋊#87604—48no—
15C23×D97—24no—
16C23×D97—44no—
17C23×D97C2×C411yescharacteristic
18C23×D97D414yes—
19C23×D97C2311yescharacteristic
20C22×Dic97C2×C412yes—
21C2×C4×D97—22no—
22C2×C4×D97—22no—
23C2×C4×D97—44no—
24C22⋊Dic97—22no—
25#13095488—24no—
26C24×C97C2×C411yescharacteristic
27C22×C4×C97C2×C412yes—
28C22⋊C4×C97—22no—
29C23×C97—22no—
30C23×C97—42no—
31C23×C97C22⋊C412yes—
32C23×C97C22×C411yescharacteristic
33C22×D97—22no—
34C22×D97—24no—
35C22×D97—28no—
36C22×D97—28no—
37C22×D97—44no—
38C22×D97—42no—
39C22×D97—416no—
40C22×D97C22⋊C412yes—
41C22×D97C2×D412yes—
42C22×D97#4114yes—
43C2×#87604—48no—
44C2×#87604—44no—
45C2×C4×C97—22no—
46C2×C4×C97—42no—
47C2×Dic97—22no—
48C2×Dic97—42no—
49C4×D97—44no—
50C4×D97—44no—
51C22×C97—22no—
52C22×C97—22no—
53C22×C97—24no—
54C22×C97—44no—
55C22×C97C2×C22⋊C411yescharacteristic
56C22×C97#11712yes—
57C2×D97—28no—
58C2×D97—22no—
59C2×D97—22no—
60C2×D97—24no—
61C2×D97—44no—
62C2×D97—416no—
63C2×D97#9914yes—
64C2×D97#12314yes—
65#87604—84no—
66C4×C97—42no—
67Dic97—42no—
68C2×C97—22no—
69C2×C97—42no—
70C2×C97#38111yescharacteristic
71C2×C97C2×#9912yes—
72D97—24no—
73D97—24no—
74#1819—971nomaximal, Sylow
75C97#181911yescharacteristic, complemented, Sylow
76C22×C22⋊C4—971no—
77#379—972no—
78#342—974no—
79C25—971no—
80C23×C4—972no—
81C2×C22⋊C4—971no—
82C2×C22⋊C4—972no—
83C2×C22⋊C4—974no—
84C2×C22⋊C4—1944no—
85C2×C22⋊C4—1944no—
86C2×C22⋊C4#8760411yescharacteristic, complemented
87C24—971no—
88C24—974no—
89C24—971no—
90C24—3884no—
91C24—1944no—
92C24C2×#8760411yescharacteristic
93C22×C4—972no—
94C22×C4—3884no—
95C22×C4—1942no—
96C22×C4—1942no—
97C22×C4—1944no—
98C22×C4C2×#8760412yes—
99C22⋊C4—22no—
100C22⋊C4—3888no—
101C22⋊C4—1944no—
102C22⋊C4—1948no—
103C22⋊C4—1942no—
104C23—22no—
105C23—42no—
106C23—974no—
107C23—972no—
108C23—972no—
109C23—3882no—
110C23—3884no—
111C23—38816no—
112C23—1948no—
113C23—1948no—
114C23—1944no—
115C23—1942no—
116C23C22⋊#8760412yes—
117C23C22×#8760411yescharacteristic
118C2×C4—22no—
119C2×C4—42no—
120C2×C4—3884no—
121C2×C4—3884no—
122C2×C4—3884no—
123C2×C4—3888no—
124C2×C4—3882no—
125C2×C4—1942no—
126C22—22no—
127C22—22no—
128C22—24no—
129C22—44no—
130C22—974no—
131C22—974no—
132C22—3884no—
133C22—38816no—
134C22—1948no—
135C22—1942no—
136C22—1944no—
137C22—1942no—
138C22C2×C22⋊#8760411yescharacteristic
139C22#1569088912yes—
140C4—42no—
141C4—7764no—
142C4—3882no—
143C2—22no—
144C2—42no—
145C2—1944no—
146C2—1944no—
147C2#2026167811yescharacteristic
148C2D97⋊#9912yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC24×D97C22
cosocleC23×D97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97#1819
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20261613C2semidirect
7. C22×C4×D97C4semidirect
8. C2×C22⋊Dic97C4semidirect
11. C2×C22⋊C4×C97C4semidirect
75. C97#1819semidirect
86. C2×C22⋊C4#87604semidirect

Maximal subgroups

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

subgroupindex
#202616762
#202616132
C2×C22⋊C4×D972
#181997