Finite Group Databasebeta

← C42×#226 #11712575

Subgroups — C42×#226 #11712575

Order 768.

647 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 123 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×C4×#226C213yesmaximal
2C42×SL(2,𝔽3)C211yesmaximal, characteristic
3C4×#859C213yesmaximal
4C42×Q16—31nomaximal, Sylow
5C22×#226C2211yescharacteristic
6C2×C4×SL(2,𝔽3)C2213yes—
7C2×#859C2213yes—
8C4×#226C4112yesdirect product factor
9C42×Dic3—41nomaximal
10C2×C4×Q16—33no—
11C42×Q8—31no—
12C42×Q8D311yescharacteristic
13C4×#107—33no—
14C4×#103—33no—
15C4×#103—33no—
16C42×C8—31no—
17C22×SL(2,𝔽3)C2311yescharacteristic
18C2×#226C2×C416yes—
19C2×C3×C42—41no—
20C2×C4×Dic3—43no—
21C2×C4×Dic3—43no—
22C4×SL(2,𝔽3)C2×C416yes—
23#859C2×C416yes—
24C22×Q16—31no—
25C2×C4×Q8—33no—
26C2×C4×Q8C2×D313yes—
27C4×Q16—312no—
28C4×Q16—312no—
29C2×#107—33no—
30C2×#103—33no—
31C2×#103—33no—
32C2×C4×C8—33no—
33C2×C4×C8—33no—
34C4×C4⋊C4—33no—
35C4×C4⋊C4—33no—
36C4×C4⋊C4—63no—
37C43—31no—
38C43—61no—
39#340—36no—
40C22×C3×C4—43no—
41C22×Dic3—41no—
42C22×Dic3—43no—
43C2×SL(2,𝔽3)C22×C413yes—
44#226C4214yesdirect product factor
45C3×C42—44no—
46C4×Dic3—412no—
47C4×Dic3—412no—
48C22×Q8—31no—
49C22×Q8C22×D311yescharacteristic
50C2×Q16—36no—
51C2×Q16—36no—
52C22×C8—31no—
53C22×C8—33no—
54C4×Q8—36no—
55C4×Q8—312no—
56C4×Q8—66no—
57C4×Q8C4×D316yes—
58C2×C4⋊C4—33no—
59C2×C4⋊C4—33no—
60C2×C4⋊C4—63no—
61C2×C42—33no—
62C2×C42—33no—
63C2×C42—63no—
64C2×C42—63no—
65C2×C42S411yescharacteristic
66#107—312no—
67#103—312no—
68#103—36no—
69#103—36no—
70C4×C8—312no—
71C4×C8—66no—
72#95—61no—
73#95—63no—
74C23×C3—41no—
75C2×C3×C4—46no—
76C2×C3×C4—46no—
77C2×C3×C4—46no—
78C2×Dic3—46no—
79C2×Dic3—46no—
80C2×Dic3—412no—
81SL(2,𝔽3)C2×C4211yescharacteristic
82C2×Q8—33no—
83C2×Q8—36no—
84C2×Q8—63no—
85C2×Q8C2×C4×D313yes—
86C22×C4—31no—
87C22×C4—33no—
88C22×C4—61no—
89C22×C4—63no—
90C22×C4C2×S413yes—
91Q16—312no—
92Q16—34no—
93C2×C8—36no—
94C2×C8—36no—
95C2×C8—66no—
96C4⋊C4—36no—
97C4⋊C4—36no—
98C4⋊C4—63no—
99C4⋊C4—63no—
100C4⋊C4—612no—
101C42—36no—
102C42—36no—
103C42—612no—
104C42—66no—
105C42—126no—
106C42#22614yesdirect product factor
107C22×C3—43no—
108C22×C3—41no—
109C22×C3—43no—
110C3×C4—412no—
111Dic3—44no—
112Dic3—412no—
113C23C22×S411yescharacteristic
114Q8—33no—
115Q8—34no—
116Q8—61no—
117Q8—66no—
118Q8C42×D311yescharacteristic
119C2×C4—33no—
120C2×C4—33no—
121C2×C4—36no—
122C2×C4—66no—
123C2×C4—66no—
124C2×C4—66no—
125C2×C4—126no—
126C2×C4C4×S416yes—
127C2×C4C2×#22616yes—
128C2×C4#98516yes—
129C8—34no—
130C8—66no—
131C2×C3—41no—
132C2×C3—43no—
133C2×C3—43no—
134C22C2×C4×S413yes—
135C22C22×#22611yescharacteristic
136C22C2×#98513yes—
137C4—31no—
138C4—33no—
139C4—64no—
140C4—66no—
141C4—126no—
142C4C4×#226112yesdirect product factor
143C3—41noSylow
144C2C42×S411yescharacteristic
145C2C2×C4×#22613yes—
146C2C4×#98513yes—
subgroupG/H
center, hypercenterC2×C42S4
commutator, hypocenterSL(2,𝔽3)C2×C42
Frattini, socleC23C22×S4
FittingC42×Q8D3
prime radical, cosocleC22×SL(2,𝔽3)C23
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
8. C4×#226C4direct (N × H)
44. #226C42direct (N × H)
106. C42#226direct (N × H)
142. C4C4×#226direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#2262
C42×SL(2,𝔽3)2
C4×#8592
C42×Q163
C42×Dic34