Finite Group Databasebeta

← Dic3×#226 #10593192

Subgroups — Dic3×#226 #10593192

Order 576.

151 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 151 conjugacy classes of subgroups — too many to draw. Only its 31 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×C3×#226C211yesmaximal, characteristic
2Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
3C3⋊#859C211yesmaximal, characteristic
4C4×#226—31nomaximal
5Dic3×Q16—31nomaximal
6C2×C3×SL(2,𝔽3)C2211yescharacteristic
7C3×#226C412yescomplemented
8Dic32—41nomaximal
9C2×#226D311yescharacteristic
10C2×C3×Q16—31no—
11Q8×Dic3—31no—
12Q8×Dic3D311yescharacteristic
13C4×SL(2,𝔽3)—31no—
14#859—31no—
15#859—61no—
16#835—31no—
17#835—31no—
18#818—31no—
19C8×Dic3—31no—
20C2×C3⋊Dic3—41no—
21C2×C3×Dic3—41no—
22C2×C3×Dic3—41no—
23C3×SL(2,𝔽3)C2×C411yescharacteristic
24C4×Q16—91noSylow
25C2×C3×Q8—31no—
26C2×C3×Q8C2×D311yescharacteristic
27C2×SL(2,𝔽3)—21no—
28C2×SL(2,𝔽3)C2×D311yescharacteristic
29#226Dic312yesdirect product factor
30C3×Q16—32no—
31C3×Q16—32no—
32C2×C3×C8—31no—
33C4⋊Dic3—31no—
34C4⋊Dic3—31no—
35C4⋊Dic3—61no—
36C4×Dic3—31no—
37C4×Dic3—61no—
38C4×Dic3—121no—
39C2×C3⋊C8—31no—
40C22×C32—41no—
41C3⋊Dic3—42no—
42C3×Dic3—42no—
43C3×Dic3—42no—
44C2×Q16—31no—
45C4×Q8—31no—
46C4×Q8—91no—
47#107—91no—
48#103—91no—
49#103—91no—
50C4×C8—91no—
51C3×Q8—31no—
52C3×Q8—32no—
53C3×Q8—61no—
54C3×Q8C4×D311yescharacteristic
55C2×C3×C4—31no—
56C2×C3×C4—61no—
57C2×C3×C4—121no—
58C2×Dic3—31no—
59C2×Dic3—41no—
60C2×Dic3—61no—
61C2×Dic3—121no—
62C2×Dic3—241no—
63C2×Dic3S411yescharacteristic
64SL(2,𝔽3)—21no—
65SL(2,𝔽3)C2×Dic311yescharacteristic
66C3×C8—32no—
67C3⋊C8—61no—
68C2×C32—41no—
69C2×C32—41no—
70C2×C32—41no—
71C2×Q8—31no—
72C2×Q8D3211yescharacteristic
73Q16—32no—
74Q16—32no—
75C2×C8—31no—
76C2×C8—91no—
77C4⋊C4—91no—
78C4⋊C4—91no—
79C4⋊C4—181no—
80C42—91no—
81C42—181no—
82C22×C3—41no—
83C22×C3—81no—
84C22×C3C2×S411yescharacteristic
85C3×C4—31no—
86C3×C4—31no—
87C3×C4—62no—
88C3×C4—122no—
89Dic3—42no—
90Dic3—61no—
91Dic3—122no—
92Dic3—121no—
93Dic3—242no—
94Dic3#22612yesdirect product factor
95C32—41noSylow
96Q8—31no—
97Q8—32no—
98Q8—61no—
99Q8D3×Dic311yescharacteristic
100C2×C4—31no—
101C2×C4—31no—
102C2×C4—61no—
103C2×C4—91no—
104C2×C4—181no—
105C8—32no—
106C8—181no—
107C2×C3—41no—
108C2×C3—41no—
109C2×C3—41no—
110C2×C3—81no—
111C2×C3—81no—
112C2×C3—81no—
113C2×C3C4×S411yescharacteristic
114C2×C3C2×#22611yescharacteristic
115C2×C3#98511yescharacteristic
116C22D3×S411yescharacteristic
117C4—32no—
118C4—31no—
119C4—31no—
120C4—62no—
121C4—181no—
122C4—361no—
123C3—41no—
124C3—81no—
125C3C4×#22611yescharacteristic, complemented
126C2#6419811yescharacteristic
127C2D3×#22611yescharacteristic
128C2Dic3×S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C4
FittingC2×C3×Q8C2×D3
socleC22×C3C2×S4
prime radical, cosocleC2×C3×SL(2,𝔽3)C22
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
7. C3×#226C4semidirect
29. #226Dic3direct (N × H)
94. Dic3#226direct (N × H)
125. C3C4×#226semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2262
Dic3×SL(2,𝔽3)2
C3⋊#8592
C4×#2263
Dic3×Q163
Dic324