Finite Group Databasebeta

← C2×#10619750 #13048113

Subgroups — C2×#10619750 #13048113

Order 1296.

246 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 27 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
1C2×#66358C211yesmaximal, characteristic, complemented
2#10619750C212yesmaximal, direct product factor
3C2×(C3⋊S4)⋊C3—33nomaximal
4C2×C3×#611—31nomaximal
5C2×#4185—41nomaximal
6#66358C2211yescharacteristic, complemented
7C23×C3×C9D311yescharacteristic, complemented
8C22×C3×D9—31no—
9C2×C22⋊H(3,𝔽3)D313yescomplemented
10(C3⋊S4)⋊C3—36no—
11C3×#611—32no—
12C3×#611—32no—
13C2×C3×Dic9—31no—
14C2×#732—41no—
15#4185—42no—
16C2×C3⋊S4—33no—
17C2×C3×#67—91no—
18C2×#611—31no—
19C22×C3×C9—31no—
20C22×C3×C9—31no—
21C22×C3×C9C2×D311yescharacteristic, complemented
22C2×(C3⋊D3)⋊C3—123no—
23C2×C3×D9—32no—
24C2×C3×D9—61no—
25C2×C3×D9—61no—
26C22⋊H(3,𝔽3)C2×D313yescomplemented
27C3×Dic9—32no—
28#732—41noSylow
29C23×C32C3⋊D311yescharacteristic
30C22×C3×D3—91no—
31C2×C3×A4—33no—
32C3⋊S4—36no—
33C3×#67—92no—
34C3×#67—92no—
35C2×C3×Dic3—91no—
36C23×C9—31no—
37C22×D9—31no—
38#611—32no—
39#611—32no—
40C2×Dic9—31no—
41C2×H(3,𝔽3)—43no—
42C2×C3×C9—31no—
43C2×C3×C9—31no—
44C2×C3×C9S411yescharacteristic, complemented
45(C3⋊D3)⋊C3—126no—
46C3×D9—62no—
47C2×S4—273no—
48C2×C3×D4—271no—
49C2×#67—91no—
50C22×C32—31no—
51C22×C32—31no—
52C22×C32C2×C3⋊D311yescharacteristic
53C2×C3⋊D3—123no—
54C2×C3×D3—92no—
55C2×C3×D3—181no—
56C2×C3×D3—181no—
57C3×A4—33no—
58C3×Dic3—92no—
59C22×C9—31no—
60C22×C9—31no—
61C22×C9—31no—
62C22×C9—61no—
63C22×C9—61no—
64C2×D9—32no—
65C2×D9—61no—
66C2×D9—61no—
67Dic9—32no—
68H(3,𝔽3)—43no—
69C3×C9C2×S411yescharacteristic, complemented
70C23×C3—31no—
71C23×C3—271no—
72C23×C3#27211yescharacteristic
73C22×D3—91no—
74C2×A4—93no—
75S4—276no—
76C3×D4—272no—
77C3×D4—272no—
78C2×C3×C4—271no—
79#67—92no—
80#67—92no—
81C2×Dic3—91no—
82C2×C32—31no—
83C2×C32—31no—
84C2×C32—123no—
85C2×C32C3⋊S411yescharacteristic
86C3⋊D3—126no—
87C3×D3—182no—
88C2×C9—31no—
89C2×C9—31no—
90C2×C9—31no—
91C2×C9—61no—
92C2×C9—61no—
93D9—62no—
94C2×D4—271noSylow
95C22×C3—31no—
96C22×C3—31no—
97C22×C3—31no—
98C22×C3—31no—
99C22×C3—31no—
100C22×C3—61no—
101C22×C3—61no—
102C22×C3—272no—
103C22×C3—541no—
104C22×C3—541no—
105C22×C3C2×#27211yescharacteristic
106C2×D3—92no—
107C2×D3—181no—
108C2×D3—181no—
109C2×D3—1083no—
110A4—93no—
111C3×C4—272no—
112Dic3—92no—
113C32—123no—
114C32C2×C3⋊S411yescharacteristic
115C9—31no—
116C23—271no—
117C23#418511yescharacteristic, complemented
118D4—272no—
119D4—272no—
120C2×C4—271no—
121C2×C3—31no—
122C2×C3—31no—
123C2×C3—31no—
124C2×C3—31no—
125C2×C3—31no—
126C2×C3—61no—
127C2×C3—61no—
128C2×C3—363no—
129C2×C3—542no—
130C2×C3#625811yescharacteristic
131D3—182no—
132D3—1086no—
133C22—31no—
134C22—31no—
135C22—272no—
136C22—541no—
137C22—541no—
138C22C2×#418511yescharacteristic, complemented
139C4—272no—
140C3—31no—
141C3—363no—
142C3C2×#625811yescharacteristic
143C2—31no—
144C2—31no—
145C2—542no—
146C2#1061975011yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#10619750
commutator, cosocle, hypocenter#66358C22
FrattiniC32C2×C3⋊S4
FittingC23×C3×C9D3
socleC23×C3#272
LCSG ▷ #66358
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×#66358C2semidirect
2. #10619750C2direct (N × H)
6. #66358C22semidirect
7. C23×C3×C9D3semidirect
9. C2×C22⋊H(3,𝔽3)D3semidirect
21. C22×C3×C9C2×D3semidirect
26. C22⋊H(3,𝔽3)C2×D3semidirect
44. C2×C3×C9S4semidirect
69. C3×C9C2×S4semidirect
117. C23#4185semidirect
138. C22C2×#4185semidirect
146. C2#10619750direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#663582
#106197502
C2×(C3⋊S4)⋊C33
C2×C3×#6113
C2×#41854