Finite Group Databasebeta

← #85419

Subgroups — #85419

Order 384.

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

normal subgroups only

This group has 199 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
1#557⋊C3C211yesmaximal, characteristic, complemented
2#5395C212yesmaximal
3C4×GL(2,𝔽3)C211yesmaximal, characteristic
4C4⋊#226C211yesmaximal, characteristic, complemented
5C4×#226C212yesmaximal, complemented
6#3364—31nomaximal, Sylow
7C2×GL(2,𝔽3)C2211yescharacteristic
8C2×#226C2212yes—
9#892—41nomaximal
10C4×SL(2,𝔽3)C2212yes—
11C4×SL(2,𝔽3)C2211yescharacteristic
12#859C2211yescharacteristic
13#557D311yescharacteristic, complemented
14#556—31no—
15#488—32no—
16#487—31no—
17#479—32no—
18#474—31no—
19#464—32no—
20#462—31no—
21C8×Q8—31no—
22C4×Q16—32no—
23C4×SD16—31no—
24C2×C4×D3—41no—
25C2×SL(2,𝔽3)C2311yescharacteristic
26GL(2,𝔽3)—21no—
27#226—22no—
28C3×C4⋊C4—41no—
29(C2×D3)⋊C4—42no—
30C4⋊Dic3—41no—
31C4×Dic3—42no—
32C2×Q16—32no—
33C2×SD16—31no—
34C4⋊Q8—31no—
35C4⋊Q8—31no—
36#126—62no—
37#125—32no—
38#124—32no—
39C22⋊Q8—61no—
40C4×Q8—31no—
41C4×Q8—32no—
42C4×Q8C2×D311yescharacteristic
43C4×Q8C2×D312yes—
44C4×D4—31no—
45#117—61no—
46#107—32no—
47#106—31no—
48C4⋊C8—31no—
49C4⋊C8—32no—
50#103—32no—
51#103—32no—
52#103—32no—
53#103—31no—
54#102—31no—
55#102—32no—
56C4×C8—31no—
57C4×C8—32no—
58C22×D3—41no—
59C2×C3×C4—41no—
60C2×C3×C4—42no—
61C2×Dic3—41no—
62C2×Dic3—42no—
63C4×D3—44no—
64SL(2,𝔽3)#4111yescharacteristic
65C2×Q8—31no—
66C2×Q8—32no—
67C2×Q8C22×D311yescharacteristic
68C2×D4—31no—
69C22×C4—61no—
70Q16—62no—
71Q16—62no—
72SD16—61no—
73SD16—61no—
74C2×C8—31no—
75C2×C8—32no—
76C2×C8—31no—
77C4⋊C4—31no—
78C4⋊C4—32no—
79C4⋊C4—32no—
80C4⋊C4—31no—
81C4⋊C4—61no—
82C4⋊C4—62no—
83C4⋊C4—61no—
84C4⋊C4—61no—
85C4⋊C4S411yescharacteristic
86C22⋊C4—61no—
87C22⋊C4—62no—
88C22⋊C4—62no—
89C42—31no—
90C42—32no—
91C42—62no—
92C22×C3—41no—
93C2×D3—42no—
94C2×D3—81no—
95C2×D3—81no—
96C3×C4—42no—
97C3×C4—82no—
98Dic3—42no—
99Dic3—82no—
100C23—61no—
101Q8—31no—
102Q8—62no—
103Q8—62no—
104Q8—62no—
105Q8#23911yescharacteristic
106D4—61no—
107D4—61no—
108C2×C4—31no—
109C2×C4—31no—
110C2×C4—32no—
111C2×C4—61no—
112C2×C4—62no—
113C2×C4—122no—
114C2×C4C2×S412yes—
115C2×C4C2×S411yescharacteristic
116C8—61no—
117C8—61no—
118C8—62no—
119C2×C3—41no—
120C2×C3—41no—
121C2×C3—41no—
122D3—82no—
123C22—121no—
124C22—121no—
125C22—121no—
126C22C22×S411yescharacteristic
127C4—22no—
128C4—31no—
129C4—31no—
130C4—62no—
131C4—61no—
132C4—122no—
133C4—121no—
134C4#98512yes—
135C3—41noSylow
136C2—241no—
137C2#591911yescharacteristic
138C2C2×#98511yescharacteristic
139C2#592711yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
Fitting#557D3
prime radicalC2×GL(2,𝔽3)C22
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. #557⋊C3C2semidirect
4. C4⋊#226C2semidirect
5. C4×#226C2semidirect
13. #557D3semidirect

Maximal subgroups

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

subgroupindex
#557⋊C32
#53952
C4×GL(2,𝔽3)2
C4⋊#2262
C4×#2262
#33643
#8924