Finite Group Databasebeta

← #85366

Subgroups — #85366

Order 384.

285 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 46 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×#986C211yesmaximal, characteristic
2#520⋊C3C211yesmaximal, characteristic, complemented
3#5424C211yesmaximal, characteristic, complemented
4#5394C212yesmaximal, complemented
5C4×GL(2,𝔽3)C212yesmaximal, complemented
6#2946—31nomaximal, Sylow
7C2×#41⋊C3C2211yescharacteristic
8#986C414yescomplemented
9C2×GL(2,𝔽3)C2212yes—
10#893—41nomaximal
11C4×SL(2,𝔽3)C2212yescomplemented
12#859C2211yescharacteristic, complemented
13#859C2211yescharacteristic, complemented
14C2×AGL(1,ℤ/8)—31no—
15#520D311yescharacteristic, complemented
16#518—31no—
17#442—32no—
18#440—32no—
19C4×SD16—32no—
20C4×D8—32no—
21#428—31no—
22#416—31no—
23C2×#102—31no—
24#405—31no—
25C2×D12—41no—
26C2×C4×D3—42no—
27#41⋊C3C2×C412yescomplemented
28C2×SL(2,𝔽3)C2311yescharacteristic
29GL(2,𝔽3)C2×C414yescomplemented
30C3×C4⋊C4—41no—
31(C2×D3)⋊C4—42no—
32C4×Dic3—41no—
33C2×#41C2×D311yescharacteristic
34C22×D4—31no—
35AGL(1,ℤ/8)—34no—
36AGL(1,ℤ/8)—34no—
37C2×SD16—32no—
38C2×D8—32no—
39C2×M16—31no—
40C22×C8—31no—
41C4×Q8C2×D312yes—
42C4×D4—32no—
43C4×D4—32no—
44C4×D4—32no—
45C4×D4—61no—
46C4×D4—61no—
47#117—31no—
48#117—31no—
49C2×C4⋊C4—31no—
50C2×C22⋊C4—62no—
51#107—32no—
52#106—32no—
53#103—31no—
54#103—31no—
55#102—32no—
56#102—32no—
57#102—31no—
58#102—31no—
59#97—32no—
60C4×C8—32no—
61C22×D3—42no—
62C2×C3×C4—41no—
63C2×C3×C4—42no—
64C2×Dic3—41no—
65C2×Dic3—41no—
66D12—44no—
67C4×D3—84no—
68SL(2,𝔽3)C22×C411yescharacteristic
69C24—61no—
70#41—32no—
71#41C4×D312yescomplemented
72C2×Q8C22×D311yescharacteristic
73C2×D4—31no—
74C2×D4—32no—
75C2×D4—34no—
76C2×D4—62no—
77C2×D4—61no—
78C22×C4—31no—
79C22×C4—61no—
80C22×C4—62no—
81C22×C4—62no—
82SD16—34no—
83SD16—34no—
84D8—38no—
85M16—34no—
86C2×C8—32no—
87C2×C8—31no—
88C2×C8—31no—
89C2×C8—62no—
90C4⋊C4—32no—
91C4⋊C4—31no—
92C4⋊C4—31no—
93C4⋊C4—31no—
94C4⋊C4S411yescharacteristic
95C22⋊C4—32no—
96C22⋊C4—62no—
97C22⋊C4—61no—
98C22⋊C4—62no—
99C42—32no—
100C42—61no—
101C22×C3—41no—
102C2×D3—44no—
103C2×D3—82no—
104C2×D3—82no—
105C3×C4—42no—
106C3×C4—82no—
107Dic3—42no—
108Dic3—81no—
109C23—31no—
110C23—64no—
111C23—62no—
112C23—241no—
113C23—241no—
114Q8—31no—
115Q8C2×C4×D311yescharacteristic
116D4—32no—
117D4—34no—
118D4—61no—
119D4—62no—
120D4—64no—
121C2×C4—31no—
122C2×C4—32no—
123C2×C4—32no—
124C2×C4—32no—
125C2×C4—61no—
126C2×C4—61no—
127C2×C4—242no—
128C2×C4—241no—
129C2×C4—242no—
130C2×C4C2×S412yes—
131C2×C4C2×S411yescharacteristic
132C8—34no—
133C8—61no—
134C8—61no—
135C2×C3—41no—
136C2×C3—41no—
137C2×C3—41no—
138D3—84no—
139C22—32no—
140C22—61no—
141C22—61no—
142C22—64no—
143C22—122no—
144C22—122no—
145C22—244no—
146C22C22×S411yescharacteristic
147C4—22no—
148C4—31no—
149C4—31no—
150C4—62no—
151C4—121no—
152C4—121no—
153C4C4×S412yes—
154C3—41noSylow
155C2—62no—
156C2—124no—
157C2C2×C4×S411yescharacteristic
158C2#592511yescharacteristic
159C2#592611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, hypocenterSL(2,𝔽3)C22×C4
Fitting#520D3
prime radicalC2×#986C2
cosocleC2×SL(2,𝔽3)C23
hypercenterC4⋊C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #520⋊C3C2semidirect
3. #5424C2semidirect
4. #5394C2semidirect
5. C4×GL(2,𝔽3)C2semidirect
8. #986C4semidirect
11. C4×SL(2,𝔽3)C22semidirect
12. #859C22semidirect
13. #859C22semidirect
15. #520D3semidirect
27. #41⋊C3C2×C4semidirect
29. GL(2,𝔽3)C2×C4semidirect
71. #41C4×D3semidirect

Maximal subgroups

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

subgroupindex
C2×#9862
#520⋊C32
#54242
#53942
C4×GL(2,𝔽3)2
#29463
#8934