Finite Group Databasebeta

← #85408

Subgroups — #85408

Order 384.

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

normal subgroups only

This group has 217 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⋊C3)C211yesmaximal, characteristic, complemented
2#5427C212yesmaximal, complemented
3#5424C212yesmaximal, complemented
4C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5#5391C211yesmaximal, characteristic, complemented
6#3305—31nomaximal, Sylow
7C2×#41⋊C3C2212yes—
8C2×GL(2,𝔽3)C2211yescharacteristic
9#892—41nomaximal
10C4×SL(2,𝔽3)C2211yescharacteristic
11#859C2212yes—
12#859C2211yescharacteristic
13#555—31no—
14#550D311yescharacteristic, complemented
15#486—31no—
16#482—32no—
17#477—31no—
18#466—32no—
19#459—31no—
20#449—32no—
21C4×SD16—31no—
22C8×D4—31no—
23#416—32no—
24C2×C4×D3—41no—
25#41⋊C3—22no—
26C2×SL(2,𝔽3)C2311yescharacteristic
27GL(2,𝔽3)—21no—
28C3×C4⋊C4—41no—
29(C2×D3)⋊C4—42no—
30C4⋊Dic3—41no—
31C4×Dic3—42no—
32C2×#41C2×D312yes—
33C2×SD16—31no—
34C2×D8—32no—
35C22×C8—32no—
36C4⋊Q8—31no—
37#127—31no—
38#124—62no—
39#121—32no—
40#121—32no—
41#121—61no—
42C4×Q8C2×D311yescharacteristic
43C4×D4—31no—
44C4×D4—31no—
45#117—32no—
46#117—61no—
47#107—32no—
48#106—31no—
49C4⋊C8—31no—
50#103—31no—
51#103—32no—
52#102—32no—
53#102—32no—
54#102—31no—
55#102—32no—
56C22⋊C8—32no—
57C4×C8—31no—
58C22×D3—41no—
59C2×C3×C4—41no—
60C2×C3×C4—42no—
61C2×Dic3—41no—
62C2×Dic3—42no—
63C4×D3—44no—
64SL(2,𝔽3)#4111yescharacteristic
65#41—22no—
66#41—62no—
67C2×Q8—61no—
68C2×Q8C22×D311yescharacteristic
69C2×D4—32no—
70C2×D4—31no—
71C2×D4—31no—
72C2×D4—61no—
73C2×D4—62no—
74C22×C4—32no—
75C22×C4—61no—
76SD16—61no—
77SD16—61no—
78D8—122no—
79C2×C8—31no—
80C2×C8—31no—
81C2×C8—32no—
82C2×C8—122no—
83C4⋊C4—31no—
84C4⋊C4—31no—
85C4⋊C4—32no—
86C4⋊C4—61no—
87C4⋊C4S411yescharacteristic
88C22⋊C4—32no—
89C22⋊C4—61no—
90C22⋊C4—62no—
91C22⋊C4—62no—
92C42—31no—
93C42—62no—
94C22×C3—41no—
95C2×D3—42no—
96C2×D3—81no—
97C2×D3—81no—
98C3×C4—42no—
99C3×C4—82no—
100Dic3—42no—
101Dic3—82no—
102C23—32no—
103C23—61no—
104Q8—31no—
105Q8—241no—
106Q8#23711yescharacteristic
107D4—62no—
108D4—62no—
109D4—61no—
110D4—61no—
111D4—62no—
112D4—241no—
113D4—242no—
114C2×C4—31no—
115C2×C4—31no—
116C2×C4—62no—
117C2×C4—62no—
118C2×C4—61no—
119C2×C4—62no—
120C2×C4—122no—
121C2×C4C2×S412yes—
122C2×C4C2×S411yescharacteristic
123C8—61no—
124C8—62no—
125C8—61no—
126C2×C3—41no—
127C2×C3—41no—
128C2×C3—41no—
129D3—82no—
130C22—62no—
131C22—62no—
132C22—62no—
133C22—121no—
134C22—121no—
135C22—121no—
136C22C22×S411yescharacteristic
137C4—22no—
138C4—31no—
139C4—31no—
140C4—61no—
141C4—121no—
142C4—122no—
143C4#98512yes—
144C3—41noSylow
145C2—122no—
146C2—241no—
147C2#591411yescharacteristic
148C2C2×#98511yescharacteristic
149C2#592511yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#550D3
hypercenterC4⋊C4S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4⋊(#41⋊C3)C2semidirect
2. #5427C2semidirect
3. #5424C2semidirect
4. C4×GL(2,𝔽3)C2semidirect
5. #5391C2semidirect
14. #550D3semidirect

Maximal subgroups

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

subgroupindex
C4⋊(#41⋊C3)2
#54272
#54242
C4×GL(2,𝔽3)2
#53912
#33053
#8924