Finite Group Databasebeta

← #73087

Subgroups — #73087

Order 384.

126 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 22 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
1D4×SL(2,𝔽3)C211yesmaximal, characteristic
2#5386C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)⋊C8C211yesmaximal, characteristic, complemented
4#1651—31nomaximal, Sylow
5C22×SL(2,𝔽3)—21no—
6C2×#214—41nomaximal
7C4×SL(2,𝔽3)C2211yescharacteristic
8#859—21no—
9D4×Q8D311yescharacteristic
10#498—31no—
11#475—31no—
12#461—31no—
13#332—31no—
14Q8⋊C8—31no—
15D4⋊C8—31no—
16C2×C3×D4—41no—
17C2×Dic6—41no—
18C2×SL(2,𝔽3)—41no—
19C2×SL(2,𝔽3)D411yescharacteristic
20#214—44no—
21C2×C3⋊C8—41no—
22C22×Q8—21no—
23C2×SD16—61no—
24C4⋊Q8—31no—
25C4⋊Q8—31no—
26C22⋊Q8—61no—
27C4×Q8C2×D311yescharacteristic
28C4×D4—31no—
29#106—61no—
30#106—62no—
31C4⋊C8—31no—
32C4⋊C8—31no—
33#103—61no—
34#103—61no—
35C4×C8—31no—
36C23×C3—81no—
37C3×D4—42no—
38C3×D4—81no—
39C2×C3×C4—41no—
40C2×Dic3—81no—
41Dic6—42no—
42Dic6—81no—
43SL(2,𝔽3)SD1611yescharacteristic, complemented
44C3⋊C8—42no—
45C2×Q8—41no—
46C2×Q8—61no—
47C2×Q8—61no—
48C2×Q8—121no—
49C2×Q8#6711yescharacteristic
50C2×D4S411yescharacteristic
51C22×C4—61no—
52SD16—122no—
53C2×C8—61no—
54C2×C8—61no—
55C2×C8—61no—
56C4⋊C4—31no—
57C4⋊C4—32no—
58C4⋊C4—31no—
59C4⋊C4—61no—
60C4⋊C4—61no—
61C22⋊C4—61no—
62C42—31no—
63C22×C3—41no—
64C22×C3—82no—
65C22×C3—161no—
66C22×C3—161no—
67C3×C4—41no—
68C3×C4—41no—
69Dic3—82no—
70C23—21no—
71Q8—31no—
72Q8—121no—
73Q8—121no—
74Q8—121no—
75Q8—121no—
76Q8#21411yescharacteristic, complemented
77D4—21no—
78D4GL(2,𝔽3)12yes—
79C2×C4—31no—
80C2×C4—31no—
81C2×C4—61no—
82C2×C4—121no—
83C2×C4—121no—
84C2×C4—121no—
85C2×C4C2×S411yescharacteristic
86C8—62no—
87C8—121no—
88C8—121no—
89C2×C3—41no—
90C2×C3—41no—
91C2×C3—41no—
92C2×C3—162no—
93C22—22no—
94C22—41no—
95C22—41no—
96C22GL(2,ℤ/4)11yescharacteristic
97C4—31no—
98C4—31no—
99C4—61no—
100C4—121no—
101C4—241no—
102C4C2×GL(2,𝔽3)11yescharacteristic
103C4#98311yescharacteristic
104C3—41noSylow
105C2—42no—
106C2#541411yescharacteristic
107C2#542111yescharacteristic
108C2#543011yescharacteristic
subgroupG/H
center, socleC22GL(2,ℤ/4)
commutator, cosocleC4×SL(2,𝔽3)C22
FrattiniC2×C4C2×S4
FittingD4×Q8D3
prime radicalD4×SL(2,𝔽3)C2
hypercenterC2×D4S4
hypocenterSL(2,𝔽3)SD16
LCSG ▷ C4×SL(2,𝔽3) ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #5386C2semidirect
3. SL(2,𝔽3)⋊C8C2semidirect
43. SL(2,𝔽3)SD16semidirect
76. Q8#214semidirect

Maximal subgroups

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

subgroupindex
D4×SL(2,𝔽3)2
#53862
SL(2,𝔽3)⋊C82
#16513
C2×#2144