Finite Group Databasebeta

← #73076

Subgroups — #73076

Order 384.

140 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 26 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
1#520⋊C3C211yesmaximal, characteristic
2#5426C211yesmaximal, characteristic
3C2×GU(2,𝔽3)C211yesmaximal, characteristic
4#1859—31nomaximal, Sylow
5C2×#41⋊C3C2211yescharacteristic
6C4×SL(2,𝔽3)—21no—
7GU(2,𝔽3)C412yescomplemented
8#859—21no—
9#808—41nomaximal
10#520D311yescharacteristic
11#428—31no—
12C2×#104—31no—
13#417—31no—
14#384—31no—
15#343—31no—
16#339—31no—
17#41⋊C3C2×C411yescharacteristic
18#41⋊C3Q811yescharacteristic
19C2×SL(2,𝔽3)D411yescharacteristic
20C3×C4⋊C4—41no—
21C4⋊Dic3—41no—
22C2×C3⋊C8—41no—
23C2×#41C2×D311yescharacteristic
24C2×M16—31no—
25C2×M16—31no—
26C4×Q8—21no—
27C4×D4—61no—
28#117—31no—
29C2×C4⋊C4—31no—
30C2×C4⋊C4—31no—
31C2×C4⋊C4—62no—
32C2×C42—31no—
33#107—61no—
34#106—61no—
35#104—32no—
36#104—32no—
37#103—61no—
38#102—61no—
39C2×C3×C4—41no—
40C2×C3×C4—81no—
41C2×Dic3—81no—
42SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
43C3⋊C8—42no—
44#41—31no—
45#41—31no—
46#41Dic611yescharacteristic
47#41C4×D311yescharacteristic
48C2×Q8#6711yescharacteristic
49C2×D4—31no—
50C22×C4—31no—
51C22×C4—32no—
52C22×C4—61no—
53C22×C4—61no—
54M16—32no—
55M16—61no—
56M16—61no—
57M16—61no—
58C2×C8—61no—
59C2×C8—61no—
60C4⋊C4—31no—
61C4⋊C4—61no—
62C4⋊C4—61no—
63C4⋊C4—61no—
64C4⋊C4—124no—
65C4⋊C4S411yescharacteristic
66C22⋊C4—61no—
67C42—32no—
68C42—61no—
69C42—61no—
70C22×C3—41no—
71C3×C4—41no—
72C3×C4—41no—
73C3×C4—161no—
74Dic3—161no—
75C23—31no—
76Q8—31no—
77Q8Dic3⋊C411yescharacteristic
78D4—31no—
79D4—31no—
80D4—61no—
81C2×C4—21no—
82C2×C4—31no—
83C2×C4—31no—
84C2×C4—31no—
85C2×C4—31no—
86C2×C4—31no—
87C2×C4—62no—
88C2×C4—62no—
89C2×C4—62no—
90C2×C4—61no—
91C2×C4—122no—
92C2×C4—122no—
93C2×C4—121no—
94C2×C4C2×S411yescharacteristic
95C8—62no—
96C8—121no—
97C2×C3—41no—
98C2×C3—41no—
99C2×C3—41no—
100C22—31no—
101C22—31no—
102C22—61no—
103C22—61no—
104C22GL(2,ℤ/4)11yescharacteristic
105C4—31no—
106C4—31no—
107C4—41no—
108C4—64no—
109C4—121no—
110C4—241no—
111C4A4⋊Q811yescharacteristic
112C4C4×S411yescharacteristic
113C3—41noSylow
114C2—61no—
115C2—61no—
116C2SL(2,ℤ/4)⋊C411yescharacteristic
117C2#542911yescharacteristic
118C2#543011yescharacteristic
subgroupG/H
center, socleC22GL(2,ℤ/4)
commutator#41⋊C3C2×C4
FrattiniC2×C4C2×S4
Fitting#520D3
prime radical, cosocleC2×#41⋊C3C22
hypercenterC4⋊C4S4
hypocenterSL(2,𝔽3)C4⋊C4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
7. GU(2,𝔽3)C4semidirect
42. SL(2,𝔽3)C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#520⋊C32
#54262
C2×GU(2,𝔽3)2
#18593
#8084