Finite Group Databasebeta

← #85473

Subgroups — #85473

Order 384.

257 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 32 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#5926C212yesmaximal, complemented
2#578⋊C3C211yesmaximal, characteristic, complemented
3#5431C212yesmaximal, complemented
4#5407C211yesmaximal, characteristic, complemented
5#5404C211yesmaximal, characteristic, complemented
6#3290—31nomaximal, Sylow
72−1+4⋊C3C2212yescomplemented
8#986C2212yescomplemented
9#985C2211yescharacteristic
10#983—22no—
11C2×GL(2,𝔽3)—22no—
12D3×D8—41nomaximal
13#131⋊C3C2211yescharacteristic, complemented
14GU(2,𝔽3)C2211yescharacteristic, complemented
15#578D311yescharacteristic, complemented
16#577—32no—
17#576—31no—
18#495—31no—
19#473—31no—
20#472—31no—
21#471—32no—
22#455—32no—
23#454—32no—
24#443—31no—
25#443—31no—
26D3×D4—42no—
27#41⋊C3C2311yescharacteristic
28C2×SL(2,𝔽3)—22no—
29GL(2,𝔽3)—22no—
30GL(2,𝔽3)D411yescharacteristic
31#226D411yescharacteristic
32C3×D8—41no—
33#213—42no—
34D24—41no—
35D3×C8—41no—
362−1+4C2×D312yescomplemented
37H(5,𝔽2)—32no—
38#137—32no—
39#137—32no—
40#137—62no—
41AGL(1,ℤ/8)—32no—
42AGL(1,ℤ/8)—32no—
43AGL(1,ℤ/8)—62no—
44AGL(1,ℤ/8)—62no—
45#135—31no—
46#135—31no—
47#135—62no—
48#135—61no—
49C2×Q16—31no—
50C2×SD16—32no—
51C2×SD16—62no—
52C2×D8—31no—
53C2×D8—61no—
54#131—31no—
55#131—32no—
56#131C2×D311yescharacteristic, complemented
57#124—32no—
58#108—31no—
59#108—31no—
60#104—32no—
61#104—31no—
62#104—31no—
63#101—32no—
64#100—32no—
65C4×C8—31no—
66C22×D3—82no—
67C3×D4—42no—
68#67—82no—
69D12—42no—
70C4×D3—41no—
71SL(2,𝔽3)C2×D411yescharacteristic
72C3×C8—41no—
73C3⋊C8—41no—
74#41—31no—
75#41—32no—
76#41—62no—
77#41—62no—
78#41—62no—
79#41C22×D311yescharacteristic
80C2×Q8—22no—
81C2×Q8—32no—
82C2×D4—32no—
83C2×D4—62no—
84C2×D4—62no—
85C2×D4—122no—
86Q16—31no—
87Q16—31no—
88Q16—61no—
89Q16—62no—
90SD16—31no—
91SD16—31no—
92SD16—32no—
93SD16—62no—
94SD16—62no—
95SD16—62no—
96SD16—62no—
97SD16—62no—
98D8—31no—
99D8—61no—
100D8—62no—
101D8—62no—
102D8—61no—
103D8S411yescharacteristic
104M16—31no—
105M16—32no—
106M16—31no—
107M16—61no—
108M16—62no—
109C2×C8—31no—
110C2×C8—31no—
111C2×C8—62no—
112C2×C8—61no—
113C22⋊C4—122no—
114C42—31no—
115C22×C3—82no—
116C2×D3—41no—
117C2×D3—82no—
118C2×D3—164no—
119C3×C4—41no—
120Dic3—41no—
121C23—62no—
122C23—122no—
123Q8—31no—
124Q8—32no—
125Q8—62no—
126Q8D3×D411yescharacteristic
127D4—31no—
128D4—31no—
129D4—62no—
130D4—62no—
131D4—61no—
132D4—62no—
133D4—122no—
134D4—122no—
135D4C2×S412yes—
136C2×C4—31no—
137C2×C4—62no—
138C2×C4—61no—
139C2×C4—62no—
140C2×C4—61no—
141C2×C4—122no—
142C8—31no—
143C8—31no—
144C8—31no—
145C8—62no—
146C8—61no—
147C8C2×S411yescharacteristic
148C2×C3—41no—
149C2×C3—162no—
150D3—42no—
151D3—162no—
152C22—22no—
153C22—31no—
154C22—61no—
155C22—122no—
156C22—242no—
157C22—482no—
158C4—31no—
159C4—62no—
160C4—61no—
161C4—121no—
162C4C22×S411yescharacteristic
163C3—41noSylow
164C2—42no—
165C2—61no—
166C2—121no—
167C2—242no—
168C2D4×S411yescharacteristic
subgroupG/H
center, socleC2D4×S4
commutator, cosocle#41⋊C3C23
FrattiniC4C22×S4
Fitting#578D3
hypercenterD8S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #5926C2semidirect
2. #578⋊C3C2semidirect
3. #5431C2semidirect
4. #5407C2semidirect
5. #5404C2semidirect
7. 2−1+4⋊C3C22semidirect
8. #986C22semidirect
13. #131⋊C3C22semidirect
14. GU(2,𝔽3)C22semidirect
15. #578D3semidirect
36. 2−1+4C2×D3semidirect
56. #131C2×D3semidirect

Maximal subgroups

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

subgroupindex
#59262
#578⋊C32
#54312
#54072
#54042
#32903
D3×D84