Finite Group Databasebeta

← #85348

Subgroups — #85348

Order 384.

229 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 34 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
1C4×#41⋊C3C211yesmaximal, characteristic, complemented
2#5427C212yesmaximal, complemented
3#5394C212yesmaximal, complemented
4C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5#5386C211yesmaximal, characteristic, complemented
6#3115—31nomaximal, Sylow
7C2×#41⋊C3C2211yescharacteristic
8C2×GL(2,𝔽3)C2212yes—
9#875—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11C4×SL(2,𝔽3)C2211yescharacteristic
12#859C2212yes—
13#532—31no—
14C4×#41D311yescharacteristic, complemented
15#482—32no—
16#475—31no—
17#474—31no—
18#460—31no—
19#459—31no—
20#449—32no—
21#442—32no—
22#440—32no—
23#432—31no—
24C2×D12—41no—
25C2×Dic6—41no—
26#41⋊C3—21no—
27C2×SL(2,𝔽3)C2311yescharacteristic
28GL(2,𝔽3)—22no—
29C3×C42—41no—
30(C2×D3)⋊C4—42no—
31(C2×D3)⋊C4—42no—
32C2×#41—61no—
33C2×#41C2×D311yescharacteristic
34C2×SD16—32no—
35C2×D8—32no—
36C4⋊Q8—31no—
37#127—31no—
38#124—61no—
39#121—32no—
40#121—61no—
41C4×Q8C2×D311yescharacteristic
42C4×Q8C2×D311yescharacteristic
43C4×D4—31no—
44C4×D4—31no—
45C4×D4—32no—
46C4×D4—61no—
47#117—31no—
48C2×C42—31no—
49#107—32no—
50#106—32no—
51C4⋊C8—31no—
52C4⋊C8—31no—
53#103—32no—
54#102—32no—
55#102—32no—
56#102—32no—
57C22⋊C8—32no—
58#97—32no—
59C22×D3—42no—
60C2×C3×C4—41no—
61C2×C3×C4—41no—
62C2×C3×C4—41no—
63C2×Dic3—42no—
64D12—82no—
65Dic6—82no—
66SL(2,𝔽3)#4111yescharacteristic, complemented
67#41—21no—
68#41—61no—
69#41—124no—
70C2×Q8—61no—
71C2×Q8C22×D311yescharacteristic
72C2×D4—31no—
73C2×D4—32no—
74C2×D4—61no—
75C2×D4—62no—
76C2×D4—61no—
77C22×C4—31no—
78C22×C4—31no—
79C22×C4—31no—
80C22×C4—62no—
81SD16—62no—
82SD16—62no—
83D8—64no—
84C2×C8—32no—
85C2×C8—32no—
86C4⋊C4—31no—
87C4⋊C4—31no—
88C4⋊C4—32no—
89C4⋊C4—61no—
90C22⋊C4—61no—
91C22⋊C4—62no—
92C22⋊C4—62no—
93C42—31no—
94C42—31no—
95C42—31no—
96C42S411yescharacteristic
97C22×C3—41no—
98C2×D3—82no—
99C2×D3—82no—
100C2×D3—82no—
101C3×C4—42no—
102C3×C4—81no—
103C3×C4—81no—
104Dic3—82no—
105C23—31no—
106C23—62no—
107Q8—31no—
108Q8—122no—
109Q8#23511yescharacteristic
110D4—32no—
111D4—61no—
112D4—62no—
113D4—62no—
114D4—122no—
115D4—124no—
116D4—122no—
117C2×C4—31no—
118C2×C4—31no—
119C2×C4—31no—
120C2×C4—61no—
121C2×C4—61no—
122C2×C4—62no—
123C2×C4—62no—
124C2×C4—121no—
125C2×C4—124no—
126C2×C4C2×S411yescharacteristic
127C2×C4C2×S411yescharacteristic
128C2×C4C2×S411yescharacteristic
129C8—62no—
130C8—62no—
131C2×C3—41no—
132C2×C3—41no—
133C2×C3—41no—
134D3—162no—
135C22—61no—
136C22—61no—
137C22—61no—
138C22—122no—
139C22—122no—
140C22—122no—
141C22C22×S411yescharacteristic
142C4—21no—
143C4—21no—
144C4—31no—
145C4—31no—
146C4—61no—
147C4—61no—
148C4—122no—
149C4#98612yes—
150C3—41noSylow
151C2—121no—
152C2—242no—
153C2#591211yescharacteristic
154C2C2×#98611yescharacteristic
155C2#592311yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingC4×#41D3
hypercenterC42S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#41⋊C3C2semidirect
2. #5427C2semidirect
3. #5394C2semidirect
4. C4⋊GL(2,𝔽3)C2semidirect
5. #5386C2semidirect
14. C4×#41D3semidirect
66. SL(2,𝔽3)#41semidirect

Maximal subgroups

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

subgroupindex
C4×#41⋊C32
#54272
#53942
C4⋊GL(2,𝔽3)2
#53862
#31153
#8754