Finite Group Databasebeta

← #85562

Subgroups — #85562

Order 384.

188 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 19 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×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic
2#1787—31nomaximal, Sylow
3C22⋊SL(2,𝔽3)C411yescharacteristic, complemented
4C2×SL(2,ℤ/4)—41nomaximal
5#859—42nomaximal
6C23×Q8D311yescharacteristic
7C2×#103—32no—
8C2×#103—32no—
9C2×C22⋊C8—31no—
10#380—31no—
11C22×A4—41no—
12C2×SL(2,𝔽3)—42no—
13C2×SL(2,𝔽3)—81no—
14SL(2,ℤ/4)—42no—
15C22×Q8—31no—
16C22×Q8—32no—
17C22×Q8—32no—
18C22×Q8—61no—
19C22×Q8—61no—
20C22×Q8Dic311yescharacteristic, complemented
21C23×C4—31no—
22C22×C8—32no—
23C2×C4⋊C4—31no—
24C2×C4⋊C4—31no—
25C2×C22⋊C4—61no—
26#103—62no—
27#103—62no—
28#103—62no—
29#103—62no—
30C22⋊C8—61no—
31C22⋊C8—61no—
32#95—61no—
33C2×A4—41no—
34C2×A4—41no—
35C2×A4—41no—
36C2×Dic3—161no—
37SL(2,𝔽3)—42no—
38SL(2,𝔽3)—81no—
39C24S411yescharacteristic
40C2×Q8—21no—
41C2×Q8—32no—
42C2×Q8—32no—
43C2×Q8—32no—
44C2×Q8—32no—
45C2×Q8—32no—
46C2×Q8—61no—
47C2×Q8—61no—
48C2×Q8—62no—
49C2×Q8—62no—
50C2×Q8—62no—
51C2×Q8—61no—
52C2×Q8—61no—
53C2×Q8—61no—
54C2×Q8—62no—
55C2×Q8S412yes—
56C22×C4—31no—
57C22×C4—31no—
58C22×C4—31no—
59C22×C4—31no—
60C22×C4—31no—
61C22×C4—31no—
62C22×C4—62no—
63C22×C4—62no—
64C22×C4—61no—
65C22×C4—61no—
66C2×C8—62no—
67C2×C8—62no—
68C2×C8—62no—
69C4⋊C4—62no—
70C4⋊C4—62no—
71C22⋊C4—121no—
72C22⋊C4—121no—
73C22×C3—161no—
74A4—41no—
75Dic3—162no—
76C23—31no—
77C23—31no—
78C23—31no—
79C23—31no—
80C23#22611yescharacteristic
81C23GL(2,𝔽3)11yescharacteristic
82C23SL(2,ℤ/4)11yescharacteristic
83Q8—21no—
84Q8—32no—
85Q8—32no—
86Q8—32no—
87Q8—61no—
88Q8—62no—
89Q8—61no—
90Q8—62no—
91Q8—61no—
92Q8SL(2,ℤ/4)12yescomplemented
93C2×C4—32no—
94C2×C4—31no—
95C2×C4—31no—
96C2×C4—32no—
97C2×C4—31no—
98C2×C4—31no—
99C2×C4—62no—
100C2×C4—62no—
101C2×C4—62no—
102C2×C4—62no—
103C2×C4—61no—
104C2×C4—61no—
105C2×C4—121no—
106C2×C4—121no—
107C2×C4—121no—
108C2×C4—121no—
109C2×C4—121no—
110C2×C4—121no—
111C8—122no—
112C2×C3—161no—
113C2×C3—161no—
114C2×C3—161no—
115C22—31no—
116C22—31no—
117C22—31no—
118C22—31no—
119C22—31no—
120C22—31no—
121C22—31no—
122C22—31no—
123C22—31no—
124C22—61no—
125C22#85911yescharacteristic, complemented
126C22#102011yescharacteristic
127C4—32no—
128C4—32no—
129C4—61no—
130C4—61no—
131C4—241no—
132C4—241no—
133C3—161noSylow
134C2—31no—
135C2—31no—
136C2—31no—
137C2—31no—
138C2#593011yescharacteristic
139C2#593111yescharacteristic
140C2#593611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#1020
commutator, hypocenterC22⋊SL(2,𝔽3)C4
FittingC23×Q8D3
socleC24S4
prime radical, cosocleC2×C22⋊SL(2,𝔽3)C2
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
3. C22⋊SL(2,𝔽3)C4semidirect
20. C22×Q8Dic3semidirect
92. Q8SL(2,ℤ/4)semidirect
125. C22#859semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊SL(2,𝔽3)2
#17873
C2×SL(2,ℤ/4)4
#8594