Finite Group Databasebeta

← #85392

Subgroups — #85392

Order 384.

287 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 36 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×#983C212yesmaximal, complemented
2D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3#5421C212yesmaximal, complemented
4C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C4×#226C211yesmaximal, characteristic, complemented
6#3278—31nomaximal, Sylow
7C22×SL(2,𝔽3)C2212yes—
8#983—24no—
9C2×GL(2,𝔽3)C2212yes—
10C2×#226C2211yescharacteristic
11#940—41nomaximal
12C4×SL(2,𝔽3)C2211yescharacteristic
13#859C2211yescharacteristic
14C2×#137—32no—
15#550—31no—
16D4×Q8D311yescharacteristic, complemented
17#492—31no—
18#468—32no—
19#462—31no—
20#460—31no—
21#452—32no—
22#448—32no—
23C4×Q16—31no—
24#436—31no—
25C2×C3×D4—41no—
26C2×#67—42no—
27C2×#67—42no—
28C2×D12—41no—
29C2×SL(2,𝔽3)—22no—
30C2×SL(2,𝔽3)C2311yescharacteristic
31GL(2,𝔽3)—22no—
32#226D412yescomplemented
33C4×Dic3—41no—
34C2×#41—32no—
35C22×Q8C2×D312yes—
36#137—64no—
37#137—64no—
38C2×Q16—31no—
39C2×Q16—32no—
40C2×SD16—32no—
41C2×SD16—32no—
42C2×SD16—32no—
43C2×M16—32no—
44C4⋊Q8—31no—
45#127—31no—
46#127—61no—
47C22⋊Q8—32no—
48#121—32no—
49#121—62no—
50C4×Q8—31no—
51C4×Q8C2×D311yescharacteristic
52C4×D4—31no—
53C4×D4—61no—
54#107—31no—
55C4⋊C8—31no—
56#103—31no—
57#103—32no—
58#103—31no—
59#102—32no—
60C22⋊C8—32no—
61C4×C8—31no—
62C23×C3—42no—
63C22×D3—42no—
64C3×D4—82no—
65C2×C3×C4—41no—
66#67—84no—
67#67—84no—
68C2×Dic3—41no—
69C2×Dic3—41no—
70D12—82no—
71SL(2,𝔽3)C2×D411yescharacteristic
72#41—64no—
73#41—122no—
74C2×Q8—22no—
75C2×Q8—32no—
76C2×Q8—31no—
77C2×Q8—62no—
78C2×Q8C22×D311yescharacteristic
79C2×D4—32no—
80C2×D4—61no—
81C2×D4—62no—
82C2×D4—62no—
83C2×D4—61no—
84C2×D4S411yescharacteristic
85C22×C4—32no—
86C22×C4—62no—
87Q16—32no—
88Q16—32no—
89Q16—64no—
90SD16—64no—
91SD16—62no—
92SD16—62no—
93SD16—122no—
94M16—64no—
95C2×C8—31no—
96C2×C8—31no—
97C2×C8—32no—
98C4⋊C4—31no—
99C4⋊C4—31no—
100C4⋊C4—31no—
101C4⋊C4—61no—
102C4⋊C4—61no—
103C22⋊C4—32no—
104C22⋊C4—62no—
105C42—31no—
106C42—61no—
107C22×C3—41no—
108C22×C3—82no—
109C22×C3—82no—
110C22×C3—82no—
111C2×D3—82no—
112C2×D3—82no—
113C2×D3—82no—
114C3×C4—42no—
115Dic3—42no—
116Dic3—42no—
117C23—62no—
118C23C2×S412yes—
119Q8—31no—
120Q8—32no—
121Q8—62no—
122Q8—62no—
123Q8—61no—
124Q8D3×D411yescharacteristic
125D4—22no—
126D4—62no—
127D4—62no—
128D4—124no—
129D4—122no—
130D4—241no—
131D4—242no—
132C2×C4—31no—
133C2×C4—31no—
134C2×C4—32no—
135C2×C4—62no—
136C2×C4—62no—
137C2×C4—61no—
138C2×C4—61no—
139C2×C4—124no—
140C2×C4C2×S411yescharacteristic
141C8—32no—
142C8—62no—
143C8—61no—
144C2×C3—41no—
145C2×C3—41no—
146C2×C3—41no—
147C2×C3—162no—
148D3—162no—
149C22—22no—
150C22—22no—
151C22—22no—
152C22—122no—
153C22—122no—
154C22—122no—
155C22C22×S411yescharacteristic
156C4—31no—
157C4—31no—
158C4—62no—
159C4—62no—
160C4—61no—
161C4—121no—
162C4#98312yescomplemented
163C3—41noSylow
164C2—42no—
165C2—242no—
166C2D4×S411yescharacteristic
167C2C2×#98311yescharacteristic
168C2#592611yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingD4×Q8D3
hypercenterC2×D4S4
hypocenterSL(2,𝔽3)C2×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#983C2semidirect
2. D4×SL(2,𝔽3)C2semidirect
3. #5421C2semidirect
4. C4⋊GL(2,𝔽3)C2semidirect
5. C4×#226C2semidirect
16. D4×Q8D3semidirect
32. #226D4semidirect
162. C4#983semidirect

Maximal subgroups

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

subgroupindex
C2×#9832
D4×SL(2,𝔽3)2
#54212
C4⋊GL(2,𝔽3)2
C4×#2262
#32783
#9404