Finite Group Databasebeta

← #27157239

Subgroups — #27157239

Order 12416.

246 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 62 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#20261752C212yesmaximal, complemented
2#20261595C212yesmaximal, complemented
3#20261593C212yesmaximal, complemented
4#20261281C211yesmaximal, characteristic
5#15690877C2214yes—
6C22⋊#11715897—22no—
7C4×#11715897—22no—
8C2×#13095511C411yescharacteristic
9#15690808C2211yescharacteristic
10#15690806C2211yescharacteristic
11#15690799C411yescharacteristic
12#15690797C411yescharacteristic
13#15690782—22no—
14C4×C22⋊D97—22no—
15C2×C4×Dic97C2211yescharacteristic
16C2×#41×C97C411yescharacteristic
17C2×#11715897—24no—
18#13095511—42no—
19C2×Dic194C2×C411yescharacteristic
20C2×D388C2×C411yescharacteristic
21C2×C22⋊D97—21no—
22C2×C22⋊D97—22no—
23C22×Dic97D412yes—
24C2×C4×D97—21no—
25#13095490—22no—
26C22⋊Dic97—22no—
27#13095488—22no—
28C4×Dic97C2×C411yescharacteristic
29C4×Dic97D412yes—
30C4×Dic97C2311yescharacteristic
31#41×C97—42no—
32C2×Q8×C97C2×C411yescharacteristic
33C2×D4×C97—21no—
34C2×D4×C97C2×C411yescharacteristic
35C22×C4×C97—21no—
36C22×C4×C97C2×C411yescharacteristic
37C23×C97—21no—
38C23×C97C22⋊C411yescharacteristic
39C22×D97—21no—
40Q8×C97—41no—
41D4×C97—41no—
42D4×C97—42no—
43C2×C4×C97—22no—
44C2×C4×C97—21no—
45C2×C4×C97—42no—
46C2×C4×C97C22⋊C411yescharacteristic
47C2×C4×C97C22×C411yescharacteristic
48C22⋊D97—42no—
49C22⋊D97—44no—
50C2×Dic97—24no—
51C2×Dic97—21no—
52C2×Dic97C22⋊C412yes—
53C2×Dic97C2×D412yes—
54D388—41no—
55C4×D97—42no—
56Dic194—41no—
57#11715897—44no—
58C22×C97—21no—
59C22×C97—22no—
60C22×C97—41no—
61C22×C97—42no—
62C22×C97C2×C22⋊C411yescharacteristic
63C2×D97—41no—
64C2×D97—42no—
65C4×C97—21no—
66C4×C97—41no—
67C4×C97#10012yes—
68Dic97—22no—
69Dic97—41no—
70Dic97#13514yes—
71C2×C97—41no—
72C2×C97—81no—
73C2×C97C2×#10011yescharacteristic
74C2×C97#41612yes—
75D97—81no—
76#1543—971nomaximal, Sylow
77C97#154311yescharacteristic, complemented, Sylow
78#532—971no—
79#431—972no—
80#333—972no—
81#331—972no—
82C2×#41—971no—
83C2×#41#8760411yescharacteristic
84C4⋊Q8—971no—
85#127—971no—
86#124—971no—
87#124—971no—
88#121—1942no—
89C4×D4—1942no—
90C2×C42—971no—
91C4⋊C8—974no—
92C22⋊C8—1942no—
93C4×C8—1942no—
94#41—42no—
95#41—3882no—
96C2×Q8—971no—
97C2×Q8C2×#8760411yescharacteristic
98C2×D4—21no—
99C2×D4—971no—
100C2×D4—1942no—
101C2×D4—1941no—
102C2×D4C2×#8760411yescharacteristic
103C22×C4—21no—
104C22×C4—972no—
105C22×C4—1941no—
106C22×C4C2×#8760411yescharacteristic
107C2×C8—1944no—
108C4⋊C4—1942no—
109C22⋊C4—1942no—
110C22⋊C4—1942no—
111C42—972no—
112C42—971no—
113C42—971no—
114C23—21no—
115C23—1941no—
116C23C22⋊#8760411yescharacteristic
117Q8—41no—
118Q8—3881no—
119D4—41no—
120D4—42no—
121D4—3881no—
122D4—3884no—
123D4—3882no—
124C2×C4—22no—
125C2×C4—21no—
126C2×C4—42no—
127C2×C4—972no—
128C2×C4—972no—
129C2×C4—3882no—
130C2×C4—1944no—
131C2×C4—1941no—
132C2×C4C22⋊#8760411yescharacteristic
133C2×C4C22×#8760411yescharacteristic
134C8—3884no—
135C22—21no—
136C22—22no—
137C22—41no—
138C22—42no—
139C22—3881no—
140C22—3882no—
141C22C2×C22⋊#8760411yescharacteristic
142C4—21no—
143C4—41no—
144C4—974no—
145C4—3881no—
146C4—1942no—
147C4#1569087012yes—
148C2—41no—
149C2—81no—
150C2—7761no—
151C2C2×#1569087011yescharacteristic
152C2#2026172512yes—
subgroupG/H
centerC2×C4C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×#41×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261281C2
cosocleC4×Dic97C23
hypercenterC2×#41#87604
hypocenterC97#1543
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20261752C2semidirect
2. #20261595C2semidirect
3. #20261593C2semidirect
77. C97#1543semidirect

Maximal subgroups

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

subgroupindex
#202617522
#202615952
#202615932
#202612812
#154397