Finite Group Databasebeta

← #73059

Subgroups — #73059

Order 384.

170 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 36 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×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#859C211yesmaximal, characteristic
3C2×#859C211yesmaximal, characteristic
4#1865—31nomaximal, Sylow
5C22×SL(2,𝔽3)C2211yescharacteristic
6C2×Dic3⋊C4—41nomaximal
7C4×SL(2,𝔽3)—21no—
8#859—21no—
9#859C412yescomplemented
10C2×C4×Q8D311yescharacteristic
11C2×#106—31no—
12C2×#103—31no—
13C2×#103—31no—
14#389—31no—
15#340—31no—
16#336—31no—
17C22×C3×C4—41no—
18C22×Dic3—41no—
19C22×Dic3—41no—
20C2×SL(2,𝔽3)C2×C411yescharacteristic
21C2×SL(2,𝔽3)D411yescharacteristic
22C2×SL(2,𝔽3)Q811yescharacteristic
23Dic3⋊C4—44no—
24C22×Q8C2×D311yescharacteristic
25C22×C8—31no—
26C22×C8—31no—
27C4×Q8—21no—
28C4×Q8—61no—
29C2×C4⋊C4—31no—
30C2×C4⋊C4—31no—
31C2×C4⋊C4—31no—
32C2×C4⋊C4—61no—
33C2×C42—31no—
34#106—62no—
35#103—32no—
36#103—32no—
37#103—61no—
38#103—61no—
39#95—61no—
40C23×C3—41no—
41C2×C3×C4—42no—
42C2×C3×C4—81no—
43C2×C3×C4—81no—
44C2×Dic3—42no—
45C2×Dic3—42no—
46C2×Dic3—42no—
47C2×Dic3—42no—
48C2×Dic3—81no—
49C2×Dic3—81no—
50SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
51C2×Q8—31no—
52C2×Q8—31no—
53C2×Q8—31no—
54C2×Q8Dic611yescharacteristic
55C2×Q8C4×D311yescharacteristic
56C2×Q8#6711yescharacteristic
57C22×C4—31no—
58C22×C4—31no—
59C22×C4—61no—
60C22×C4—61no—
61C22×C4S411yescharacteristic
62C2×C8—32no—
63C2×C8—61no—
64C2×C8—61no—
65C2×C8—61no—
66C2×C8—61no—
67C2×C8—61no—
68C4⋊C4—32no—
69C4⋊C4—32no—
70C4⋊C4—61no—
71C4⋊C4—61no—
72C4⋊C4—61no—
73C4⋊C4—61no—
74C4⋊C4—122no—
75C42—61no—
76C42—61no—
77C22×C3—41no—
78C22×C3—41no—
79C22×C3—41no—
80C22×C3—41no—
81C22×C3—41no—
82C22×C3—41no—
83C22×C3—41no—
84C3×C4—82no—
85Dic3—44no—
86Dic3—82no—
87C23C2×S411yescharacteristic
88Q8—31no—
89Q8—31no—
90Q8—31no—
91Q8—61no—
92Q8Dic3⋊C411yescharacteristic, complemented
93C2×C4—21no—
94C2×C4—21no—
95C2×C4—31no—
96C2×C4—31no—
97C2×C4—31no—
98C2×C4—31no—
99C2×C4—31no—
100C2×C4—31no—
101C2×C4—62no—
102C2×C4—61no—
103C2×C4—61no—
104C2×C4—61no—
105C2×C4—121no—
106C2×C4—121no—
107C2×C4—121no—
108C2×C4—121no—
109C2×C4—121no—
110C2×C4GL(2,𝔽3)12yes—
111C8—62no—
112C8—121no—
113C2×C3—41no—
114C2×C3—41no—
115C2×C3—41no—
116C2×C3—41no—
117C2×C3—41no—
118C2×C3—41no—
119C2×C3—41no—
120C22A4⋊Q811yescharacteristic
121C22C4×S411yescharacteristic
122C22C2×GL(2,𝔽3)11yescharacteristic
123C22#98311yescharacteristic
124C22#98411yescharacteristic
125C22#98511yescharacteristic
126C22GL(2,ℤ/4)11yescharacteristic
127C4—22no—
128C4—31no—
129C4—31no—
130C4—31no—
131C4—31no—
132C4—122no—
133C4—121no—
134C4—241no—
135C3—41noSylow
136C2#538611yescharacteristic
137C2#538811yescharacteristic
138C2#538911yescharacteristic
139C2C4×GL(2,𝔽3)11yescharacteristic
140C2SL(2,ℤ/4)⋊C411yescharacteristic
141C2#542111yescharacteristic
142C2#542511yescharacteristic
subgroupG/H
center, Frattini, socleC23C2×S4
commutatorC2×SL(2,𝔽3)C2×C4
FittingC2×C4×Q8D3
prime radical, cosocleC22×SL(2,𝔽3)C22
hypercenterC22×C4S4
hypocenterSL(2,𝔽3)C4⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
9. #859C4semidirect
50. SL(2,𝔽3)C4⋊C4semidirect
92. Q8Dic3⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×SL(2,𝔽3)2
C2×#8592
#18653
C2×Dic3⋊C44