Finite Group Databasebeta

← #27157240

Subgroups — #27157240

Order 12416.

246 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261753C212yesmaximal, complemented
2#20261595C212yesmaximal, complemented
3#20261593C212yesmaximal, complemented
4#20261284C211yesmaximal, characteristic
5#15690877C2214yes—
6#15690867—22no—
7C4×#11715897—22no—
8C2×#13095512C411yescharacteristic
9C2×#13095510C411yescharacteristic
10#15690808C2211yescharacteristic
11#15690806C2211yescharacteristic
12#15690798C411yescharacteristic
13#15690783—22no—
14#15690770—22no—
15C42×D97C2211yescharacteristic
16#124×C97C411yescharacteristic
17C2×#11715897—24no—
18#13095512—42no—
19#13095510—42no—
20C2×Dic194C2×C411yescharacteristic
21C2×D388—21no—
22C2×D388C2×C411yescharacteristic
23C2×C22⋊D97—21no—
24C2×C22⋊D97—22no—
25C22×Dic97—21no—
26C2×C4×D97—21no—
27C2×C4×D97C2×C411yescharacteristic
28C2×C4×D97D412yes—
29#13095490—22no—
30#13095488—22no—
31C4×Dic97D412yes—
32C4×Dic97C2311yescharacteristic
33C2×Q8×C97C2×C411yescharacteristic
34C2×D4×C97C2×C411yescharacteristic
35C22⋊C4×C97—22no—
36C42×C97C2×C411yescharacteristic
37C23×C97—21no—
38C22×D97—21no—
39C22×D97C22⋊C411yescharacteristic
40Q8×C97—41no—
41D4×C97—41no—
42C2×C4×C97—21no—
43C2×C4×C97C22⋊C412yes—
44C2×C4×C97C22×C411yescharacteristic
45C22⋊D97—42no—
46C22⋊D97—44no—
47C2×Dic97—21no—
48C2×Dic97—42no—
49C2×Dic97C22⋊C411yescharacteristic
50C2×Dic97C2×D412yes—
51D388—41no—
52D388—42no—
53C4×D97—22no—
54C4×D97—24no—
55C4×D97—42no—
56Dic194—41no—
57#11715897—44no—
58C22×C97—41no—
59C22×C97—42no—
60C22×C97C2×C22⋊C411yescharacteristic
61C2×D97—21no—
62C2×D97—22no—
63C2×D97—41no—
64C2×D97—42no—
65C4×C97—21no—
66C4×C97—22no—
67C4×C97—41no—
68Dic97—41no—
69Dic97#10012yes—
70Dic97#13514yes—
71C2×C97—81no—
72C2×C97C2×#10011yescharacteristic
73C2×C97#41612yes—
74D97—41no—
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—971no—
84C4⋊Q8—971no—
85#127—971no—
86#124—971no—
87#124#8760411yescharacteristic
88#121—1942no—
89C4×D4—1942no—
90C2×C42—971no—
91C4⋊C8—974no—
92C22⋊C8—1942no—
93C4×C8—1942no—
94#41—3882no—
95#41—3882no—
96C2×Q8—971no—
97C2×Q8C2×#8760411yescharacteristic
98C2×D4—971no—
99C2×D4—1941no—
100C2×D4—1942no—
101C2×D4—1941no—
102C2×D4C2×#8760411yescharacteristic
103C22×C4—972no—
104C22×C4—971no—
105C22×C4—1941no—
106C22×C4—1941no—
107C2×C8—1944no—
108C4⋊C4—1942no—
109C22⋊C4—22no—
110C22⋊C4—1942no—
111C42—972no—
112C42—971no—
113C42C2×#8760411yescharacteristic
114C23—21no—
115C23—971no—
116C23—1941no—
117Q8—41no—
118Q8—3881no—
119D4—41no—
120D4—3881no—
121D4—3882no—
122D4—3884no—
123D4—3882no—
124C2×C4—21no—
125C2×C4—972no—
126C2×C4—971no—
127C2×C4—3882no—
128C2×C4—3882no—
129C2×C4—1942no—
130C2×C4—1944no—
131C2×C4—1941no—
132C2×C4C22⋊#8760412yes—
133C2×C4C22×#8760411yescharacteristic
134C8—3884no—
135C22—41no—
136C22—42no—
137C22—3881no—
138C22—3882no—
139C22—1941no—
140C22—1942no—
141C22C2×C22⋊#8760411yescharacteristic
142C4—21no—
143C4—22no—
144C4—41no—
145C4—972no—
146C4—974no—
147C4—3881no—
148C2—81no—
149C2—7761no—
150C2—3881no—
151C2D97⋊#10011yescharacteristic
152C2#2026172512yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261284C2
cosocleC4×Dic97C23
hypercenter#124#87604
hypocenterC97#1543
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20261753C2semidirect
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
#202617532
#202615952
#202615932
#202612842
#154397