Finite Group Databasebeta

← C42×GL(2,𝔽3) #11712574

Subgroups — C42×GL(2,𝔽3) #11712574

Order 768.

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

normal subgroups only

This group has 827 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×GL(2,𝔽3)C213yesmaximal
2C42×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C4×#859C213yesmaximal, complemented
4C42×SD16—31nomaximal, Sylow
5C22×GL(2,𝔽3)C2211yescharacteristic
6C2×C42×D3—41nomaximal
7C2×C4×SL(2,𝔽3)C2213yes—
8C2×#859C2213yes—
9C4×GL(2,𝔽3)C4112yesdirect product factor
10C2×C4×SD16—33no—
11C42×Q8D311yescharacteristic, complemented
12C42×D4—31no—
13C4×#106—33no—
14C4×#103—33no—
15C4×#102—33no—
16C42×C8—31no—
17C22×C4×D3—43no—
18C22×SL(2,𝔽3)C2311yescharacteristic
19C2×GL(2,𝔽3)C2×C416yes—
20C2×C3×C42—41no—
21C2×C4×Dic3—43no—
22C42×D3—48no—
23C4×SL(2,𝔽3)C2×C416yescomplemented
24#859C2×C416yescomplemented
25C22×SD16—31no—
26C2×C4×Q8C2×D313yes—
27C2×C4×D4—33no—
28C22×C42—61no—
29C4×SD16—312no—
30C4×SD16—312no—
31C2×#106—33no—
32C2×#103—33no—
33C2×#102—33no—
34C2×C4×C8—33no—
35C2×C4×C8—33no—
36C4×C4⋊C4—33no—
37C4×C4⋊C4—33no—
38C4×C22⋊C4—63no—
39C43—31no—
40#340—36no—
41C23×D3—41no—
42C22×C3×C4—43no—
43C22×Dic3—43no—
44C2×C4×D3—412no—
45C2×C4×D3—412no—
46C2×C4×D3—412no—
47C2×SL(2,𝔽3)C22×C413yes—
48GL(2,𝔽3)C4214yesdirect product factor
49C3×C42—44no—
50C4×Dic3—412no—
51C22×Q8C22×D311yescharacteristic
52C22×D4—31no—
53C23×C4—63no—
54C2×SD16—36no—
55C2×SD16—36no—
56C22×C8—31no—
57C22×C8—33no—
58C4×Q8—36no—
59C4×Q8C4×D316yescomplemented
60C4×D4—312no—
61C4×D4—66no—
62C2×C4⋊C4—33no—
63C2×C4⋊C4—33no—
64C2×C22⋊C4—63no—
65C2×C42—33no—
66C2×C42—33no—
67C2×C42—63no—
68C2×C42—124no—
69C2×C42S411yescharacteristic
70#106—312no—
71#103—36no—
72#103—36no—
73#102—312no—
74C4×C8—312no—
75C4×C8—66no—
76#95—61no—
77#95—63no—
78C23×C3—41no—
79C22×D3—46no—
80C22×D3—42no—
81C22×D3—46no—
82C2×C3×C4—46no—
83C2×C3×C4—46no—
84C2×C3×C4—46no—
85C2×Dic3—46no—
86C2×Dic3—46no—
87C2×Dic3—46no—
88C4×D3—448no—
89SL(2,𝔽3)C2×C4211yescharacteristic, complemented
90C24—61no—
91C2×Q8—33no—
92C2×Q8C2×C4×D313yes—
93C2×D4—36no—
94C2×D4—63no—
95C22×C4—31no—
96C22×C4—33no—
97C22×C4—63no—
98C22×C4—612no—
99C22×C4—126no—
100C22×C4—126no—
101C22×C4C2×S413yes—
102SD16—312no—
103SD16—34no—
104C2×C8—36no—
105C2×C8—36no—
106C2×C8—66no—
107C4⋊C4—36no—
108C4⋊C4—36no—
109C4⋊C4—63no—
110C4⋊C4—63no—
111C22⋊C4—612no—
112C42—36no—
113C42—36no—
114C42—66no—
115C42—126no—
116C42GL(2,𝔽3)14yesdirect product factor
117C22×C3—43no—
118C22×C3—41no—
119C22×C3—43no—
120C2×D3—44no—
121C2×D3—412no—
122C2×D3—412no—
123C3×C4—412no—
124Dic3—412no—
125C23—66no—
126C23—121no—
127C23—123no—
128C23C22×S411yescharacteristic
129Q8—33no—
130Q8—61no—
131Q8C42×D311yescharacteristic, complemented
132D4—34no—
133D4—66no—
134C2×C4—33no—
135C2×C4—33no—
136C2×C4—36no—
137C2×C4—66no—
138C2×C4—66no—
139C2×C4—123no—
140C2×C4—123no—
141C2×C4—1224no—
142C2×C4C4×S416yes—
143C2×C4C2×GL(2,𝔽3)16yes—
144C2×C4#98516yes—
145C8—34no—
146C8—66no—
147C2×C3—41no—
148C2×C3—43no—
149C2×C3—43no—
150D3—48no—
151C22—64no—
152C22—126no—
153C22—126no—
154C22C2×C4×S413yes—
155C22C22×GL(2,𝔽3)11yescharacteristic
156C22C2×#98513yes—
157C4—31no—
158C4—33no—
159C4—66no—
160C4—126no—
161C4C4×GL(2,𝔽3)112yesdirect product factor
162C3—41noSylow
163C2—124no—
164C2C42×S411yescharacteristic
165C2C2×C4×GL(2,𝔽3)13yes—
166C2C4×#98513yes—
subgroupG/H
center, hypercenterC2×C42S4
commutator, hypocenterSL(2,𝔽3)C2×C42
Frattini, socleC23C22×S4
FittingC42×Q8D3
prime radicalC22×GL(2,𝔽3)C22
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
2. C42×SL(2,𝔽3)C2semidirect
3. C4×#859C2semidirect
9. C4×GL(2,𝔽3)C4direct (N × H)
11. C42×Q8D3semidirect
23. C4×SL(2,𝔽3)C2×C4semidirect
24. #859C2×C4semidirect
48. GL(2,𝔽3)C42direct (N × H)
59. C4×Q8C4×D3semidirect
89. SL(2,𝔽3)C2×C42semidirect
116. C42GL(2,𝔽3)direct (N × H)
131. Q8C42×D3semidirect
161. C4C4×GL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×GL(2,𝔽3)2
C42×SL(2,𝔽3)2
C4×#8592
C42×SD163
C2×C42×D34