Finite Group Databasebeta

← #11714077

Subgroups — #11714077

Order 768.

425 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C42×SL(2,𝔽3)C211yesmaximal, characteristic
2C4⋊#859C212yesmaximal
3#73059C214yesmaximal
4#33067—31nomaximal, Sylow
5C2×#870—41nomaximal
6C2×C4×SL(2,𝔽3)C2212yes—
7C2×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×#859C2214yes—
9#2859—31no—
10C42×Q8D311yescharacteristic
11#1948—32no—
12#1919—34no—
13#1865—34no—
14#1831—31no—
15#1803—32no—
16C22×SL(2,𝔽3)C2311yescharacteristic
17C2×C3×C42—41no—
18C2×C4⋊Dic3—42no—
19C2×Dic3⋊C4—44no—
20#870—48no—
21C4×SL(2,𝔽3)—22no—
22C4×SL(2,𝔽3)Q812yes—
23#859—24no—
24C2×#125—61no—
25C2×C4×Q8C2×D312yes—
26C2×C4×Q8C2×D311yescharacteristic
27C2×#106—34no—
28C2×C4⋊C8—32no—
29C2×#103—34no—
30#389—34no—
31#389—62no—
32#384—32no—
33#382—61no—
34C4×C4⋊C4—32no—
35C4×C4⋊C4—31no—
36C43—31no—
37#340—34no—
38#340—34no—
39#336—34no—
40C22×C3×C4—41no—
41C22×C3×C4—42no—
42C22×Dic3—44no—
43C2×SL(2,𝔽3)C2×Q811yescharacteristic
44C2×SL(2,𝔽3)#4112yes—
45C3×C42—44no—
46C4⋊Dic3—48no—
47Dic3⋊C4—416no—
48C22×Q8C22×D311yescharacteristic
49C22×C8—34no—
50#125—124no—
51C4×Q8—22no—
52C4×Q8—32no—
53C4×Q8—62no—
54C4×Q8Dic612yes—
55C2×C4⋊C4—31no—
56C2×C4⋊C4—32no—
57C2×C4⋊C4—34no—
58C2×C4⋊C4—62no—
59C2×C4⋊C4—62no—
60C2×C4⋊C4—64no—
61C2×C42—31no—
62C2×C42—32no—
63C2×C42—31no—
64C2×C42—32no—
65C2×C42S411yescharacteristic
66#106—68no—
67C4⋊C8—64no—
68#103—64no—
69#103—64no—
70#95—61no—
71#95—64no—
72C23×C3—41no—
73C2×C3×C4—42no—
74C2×C3×C4—42no—
75C2×C3×C4—44no—
76C2×C3×C4—42no—
77C2×C3×C4—82no—
78C2×C3×C4—82no—
79C2×Dic3—48no—
80C2×Dic3—84no—
81C2×Dic3—84no—
82SL(2,𝔽3)#12511yescharacteristic, complemented
83C2×Q8—32no—
84C2×Q8—31no—
85C2×Q8C2×Dic611yescharacteristic
86C2×Q8#23512yes—
87C22×C4—31no—
88C22×C4—31no—
89C22×C4—32no—
90C22×C4—64no—
91C22×C4C2×S412yes—
92C22×C4C2×S411yescharacteristic
93C2×C8—64no—
94C2×C8—64no—
95C2×C8—64no—
96C4⋊C4—32no—
97C4⋊C4—34no—
98C4⋊C4—61no—
99C4⋊C4—62no—
100C4⋊C4—64no—
101C4⋊C4—64no—
102C4⋊C4—124no—
103C4⋊C4—128no—
104C4⋊C4—124no—
105C42—32no—
106C42—32no—
107C42—62no—
108C42—62no—
109C42—62no—
110C42—122no—
111C42GL(2,𝔽3)14yes—
112C22×C3—41no—
113C22×C3—42no—
114C22×C3—41no—
115C22×C3—41no—
116C22×C3—42no—
117C3×C4—44no—
118C3×C4—84no—
119Dic3—88no—
120C23C22×S411yescharacteristic
121Q8—31no—
122Q8—32no—
123Q8—61no—
124Q8#87011yescharacteristic, complemented
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—31no—
128C2×C4—31no—
129C2×C4—32no—
130C2×C4—32no—
131C2×C4—32no—
132C2×C4—62no—
133C2×C4—62no—
134C2×C4—62no—
135C2×C4—62no—
136C2×C4—124no—
137C2×C4—124no—
138C2×C4—124no—
139C2×C4A4⋊Q812yes—
140C2×C4C2×GL(2,𝔽3)14yes—
141C2×C4C2×GL(2,𝔽3)12yes—
142C2×C4#98412yes—
143C8—124no—
144C2×C3—41no—
145C2×C3—41no—
146C2×C3—42no—
147C2×C3—42no—
148C2×C3—41no—
149C22C2×A4⋊Q811yescharacteristic
150C22#591212yes—
151C22C22×GL(2,𝔽3)11yescharacteristic
152C22C2×#98411yescharacteristic
153C22#592312yes—
154C4—24no—
155C4—32no—
156C4—31no—
157C4—31no—
158C4—62no—
159C4—122no—
160C4—244no—
161C4#538614yes—
162C3—41noSylow
163C2#8528411yescharacteristic
164C2C2×#538611yescharacteristic
165C2#8532512yes—
166C2#8533812yes—
167C2#8533911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC42×Q8D3
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)#125
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
82. SL(2,𝔽3)#125semidirect
124. Q8#870semidirect

Maximal subgroups

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

subgroupindex
C42×SL(2,𝔽3)2
C4⋊#8592
#730592
#330673
C2×#8704