Finite Group Databasebeta

← #87545

Subgroups — #87545

Order 384.

1047 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 1047 conjugacy classes of subgroups — too many to draw. Only its 42 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#5948C211yesmaximal, characteristic, complemented
2C2×C22⋊SL(2,𝔽3)C212yesmaximal, complemented
3C22×2−1+4C311yesmaximal, characteristic, complemented, Sylow
4C22⋊SL(2,𝔽3)C2211yescharacteristic
52−1+4⋊C3—44nomaximal
6D4×A4—41nomaximal
7C2×2−1+4—34no—
8C2×2−1+4—38no—
9C2×2−1+4—34no—
10C22×#41—31no—
11C22×#41—32no—
12C22×#41C2×C311yescharacteristic, complemented
13C23×Q8—31no—
14C23×Q8C2×C312yescomplemented
15C22×A4—42no—
16#41⋊C3—44no—
17C2×SL(2,𝔽3)—48no—
18C4×A4—41no—
192−1+4—312no—
202−1+4—324no—
212−1+4—324no—
222−1+4—34no—
232−1+4—38no—
242−1+4—34no—
252−1+4—38no—
262−1+4A414yescomplemented
27C2×#41—34no—
28C2×#41—324no—
29C2×#41—324no—
30C2×#41—36no—
31C2×#41—36no—
32C2×#41—312no—
33C2×#41—34no—
34C22×Q8—38no—
35C22×Q8—312no—
36C22×Q8—312no—
37C22×Q8—38no—
38C22×Q8—31no—
39C22×Q8—32no—
40C22×Q8C22×C311yescharacteristic
41C22×D4—31no—
42C22×D4—32no—
43C22×D4A411yescharacteristic
44C23×C4—32no—
45C23×C4—32no—
46C23×C4—31no—
47C2×A4—41no—
48C2×A4—82no—
49C3×D4—161no—
50SL(2,𝔽3)—44no—
51C24—31no—
52C24C2×A412yes—
53#41—312no—
54#41—324no—
55#41—324no—
56#41—348no—
57#41—324no—
58#41—324no—
59#41—324no—
60#41—312no—
61#41—34no—
62#41—34no—
63#41—34no—
64#41—38no—
65#41C2×A414yescomplemented
66C2×Q8—34no—
67C2×Q8—324no—
68C2×Q8—36no—
69C2×Q8—36no—
70C2×Q8—324no—
71C2×Q8—324no—
72C2×Q8—324no—
73C2×Q8—312no—
74C2×Q8—38no—
75C2×Q8—34no—
76C2×Q8—38no—
77C2×Q8C2×A418yescomplemented
78C2×D4—312no—
79C2×D4—312no—
80C2×D4—312no—
81C2×D4—31no—
82C2×D4—31no—
83C2×D4—32no—
84C22×C4—324no—
85C22×C4—312no—
86C22×C4—312no—
87C22×C4—36no—
88C22×C4—36no—
89C22×C4—32no—
90C22×C4—31no—
91C22×C4C2×A411yescharacteristic
92C22×C3—162no—
93A4—41no—
94C3×C4—161no—
95C23—22no—
96C23—36no—
97C23—32no—
98C23—32no—
99C23—62no—
100C23—63no—
101C23—61no—
102C23C22×A411yescharacteristic
103Q8—38no—
104Q8—324no—
105Q8—312no—
106Q8—34no—
107Q8—34no—
108Q8C22×A414yes—
109D4—34no—
110D4—324no—
111D4—312no—
112D4—32no—
113D4—38no—
114D4—32no—
115D4—31no—
116D4C22⋊A411yescharacteristic
117C2×C4—312no—
118C2×C4—36no—
119C2×C4—38no—
120C2×C4—324no—
121C2×C4—324no—
122C2×C4—312no—
123C2×C4—38no—
124C2×C4—31no—
125C2×C4—34no—
126C2×C4—31no—
127C2×C3—161no—
128C2×C3—322no—
129C22—31no—
130C22—31no—
131C22—32no—
132C22—34no—
133C22—62no—
134C22—62no—
135C22—62no—
136C22—62no—
137C22—66no—
138C22—66no—
139C222−1+4⋊C311yescharacteristic, complemented
140C22C2×C22⋊A412yes—
141C4—38no—
142C4—34no—
143C4—31no—
144C4C2×C22⋊A411yescharacteristic
145C3—161noSylow
146C2—22no—
147C2—31no—
148C2—31no—
149C2—62no—
150C2—64no—
151C2C22×C22⋊A411yescharacteristic
subgroupG/H
center, FrattiniC2C22×C22⋊A4
commutator, cosocle, hypocenterC22×Q8C22×C3
FittingC22×2−1+4C3
socleC23C22×A4
hypercenterD4C22⋊A4
LCSG ▷ C22×Q8
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #5948C2semidirect
2. C2×C22⋊SL(2,𝔽3)C2semidirect
3. C22×2−1+4C3semidirect
12. C22×#41C2×C3semidirect
14. C23×Q8C2×C3semidirect
26. 2−1+4A4semidirect
65. #41C2×A4semidirect
77. C2×Q8C2×A4semidirect
139. C222−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
#59482
C2×C22⋊SL(2,𝔽3)2
C22×2−1+43
2−1+4⋊C34
D4×A44