Finite Group Databasebeta

← SL(2,𝔽3)×#622 #13165844

Subgroups — SL(2,𝔽3)×#622 #13165844

Order 1728.

172 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 28 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
1SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic
2Q8×#622C311yesmaximal, characteristic, complemented
3C2×C32×SL(2,𝔽3)C411yescharacteristic
4C2×C3×#622—41nomaximal
5Q8×C3⋊Dic3C2×C311yescharacteristic
6C4⋊#622—31no—
7C4×#622—31no—
8Dic3×SL(2,𝔽3)—62no—
9C2×C3×C3⋊Dic3—41no—
10C32×SL(2,𝔽3)C811yescharacteristic, complemented
11C3×#622—42no—
12C8×SL(2,𝔽3)—91nomaximal
13C2×C32×Q8C3×C411yescharacteristic
14C2×C3×SL(2,𝔽3)—22no—
15C2×C3×SL(2,𝔽3)—42no—
16C2×#622—31no—
17C2×#622A411yescharacteristic
18C4⋊(C3⋊Dic3)—31no—
19C4×C3⋊Dic3—31no—
20C22×C33—41no—
21C3×C3⋊Dic3—41no—
22C3×C3⋊Dic3—41no—
23Q8×Dic3—62no—
24C4×SL(2,𝔽3)—91no—
25C32×Q8—31no—
26C32×Q8C3×C811yescharacteristic, complemented
27C2×C32×C4—31no—
28C2×C3⋊Dic3—31no—
29C2×C3⋊Dic3C2×A411yescharacteristic
30C2×C3×Dic3—242no—
31C3×SL(2,𝔽3)—22no—
32C3×SL(2,𝔽3)—42no—
33#622—61no—
34#622SL(2,𝔽3)12yesdirect product factor
35C8×Q8—91noSylow
36C2×C33—41no—
37C2×C33—41no—
38C2×C33—41no—
39C2×C3×Q8—22no—
40C2×SL(2,𝔽3)—42no—
41C2×SL(2,𝔽3)#15711yescharacteristic
42C2×C3×C8—361no—
43C4⋊Dic3—182no—
44C4×Dic3—182no—
45C22×C32—82no—
46C22×C32—162no—
47C22×C32C4×A411yescharacteristic
48C32×C4—31no—
49C32×C4—31no—
50C3⋊Dic3—61no—
51C3⋊Dic3C2×SL(2,𝔽3)11yescharacteristic
52C3⋊Dic3#41⋊C311yescharacteristic
53C3×Dic3—242no—
54C3×Dic3—242no—
55C4×Q8—91no—
56C4⋊C8—271no—
57C4×C8—271no—
58C33—41noSylow
59C3×Q8—22no—
60C3×Q8—62no—
61C2×C3×C4—62no—
62C2×C3×C4—361no—
63C2×Dic3—62no—
64C2×Dic3—182no—
65SL(2,𝔽3)—42no—
66SL(2,𝔽3)#62211yescharacteristic, direct product factor
67C3×C8—362no—
68C2×C32—82no—
69C2×C32—82no—
70C2×C32—82no—
71C2×C32—162no—
72C2×C32—162no—
73C2×C32—162no—
74C2×C32C4×SL(2,𝔽3)11yescharacteristic
75C2×C32C8×A411yescharacteristic
76C2×C32#131⋊C311yescharacteristic
77C2×Q8C3×#15711yescharacteristic
78C2×C8—91no—
79C2×C8—271no—
80C4⋊C4—271no—
81C42—271no—
82C22×C3—22no—
83C22×C3—41no—
84C22×C3—162no—
85C3×C4—62no—
86C3×C4—62no—
87C3×C4—361no—
88C3×C4—361no—
89Dic3—62no—
90Dic3—62no—
91Dic3—362no—
92C32—82no—
93C32—162no—
94C32C8×SL(2,𝔽3)11yescharacteristic, complemented
95Q8—31no—
96Q8C3×#62211yescharacteristic, complemented
97C2×C4—31no—
98C2×C4—91no—
99C2×C4—271no—
100C8—92no—
101C8—541no—
102C2×C3—22no—
103C2×C3—22no—
104C2×C3—22no—
105C2×C3—41no—
106C2×C3—41no—
107C2×C3—41no—
108C2×C3—162no—
109C2×C3—162no—
110C2×C3—162no—
111C22A4×#15711yescharacteristic
112C4—31no—
113C4—31no—
114C4—91no—
115C4—91no—
116C4—541no—
117C3—22no—
118C3—41no—
119C3—162no—
120C2#64289⋊C311yescharacteristic
121C2SL(2,𝔽3)×#15711yescharacteristic
122C2A4×#62211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22A4×#157
commutator, hypocenterC32×Q8C3×C8
FittingC2×C32×Q8C3×C4
socleC22×C32C4×A4
prime radicalC2×C32×SL(2,𝔽3)C4
cosocleQ8×C3⋊Dic3C2×C3
LCSG ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. Q8×#622C3semidirect
10. C32×SL(2,𝔽3)C8semidirect
26. C32×Q8C3×C8semidirect
34. #622SL(2,𝔽3)direct (N × H)
66. SL(2,𝔽3)#622direct (N × H)
94. C32C8×SL(2,𝔽3)semidirect
96. Q8C3×#622semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×C3⋊Dic32
Q8×#6223
C2×C3×#6224
C8×SL(2,𝔽3)9