Finite Group Databasebeta

← #11714109

Subgroups — #11714109

Order 768.

485 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 95 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×#5391C212yesmaximal
2C42×SL(2,𝔽3)C211yesmaximal, characteristic
3C4⋊#859C214yesmaximal
4#33198—31nomaximal, Sylow
5C2×C4×SL(2,𝔽3)C2211yescharacteristic
6C2×C4×SL(2,𝔽3)C2212yes—
7C2×#859C2214yes—
8#5391—24no—
9#4935—41nomaximal
10C2×#477—32no—
11#2868—31no—
12C42×Q8D311yescharacteristic
13#1987—32no—
14#1947—34no—
15#1842—31no—
16#1803—34no—
17C22×SL(2,𝔽3)C2311yescharacteristic
18C2×C3×C42—41no—
19C2×C4⋊Dic3—42no—
20C2×C4⋊Dic3—44no—
21C4×SL(2,𝔽3)D412yes—
22C4×SL(2,𝔽3)Q814yescomplemented
23#859—24no—
24C2×C4⋊Q8—32no—
25C2×C4⋊Q8—62no—
26C2×C4×Q8C2×D312yes—
27C2×C4×Q8C2×D311yescharacteristic
28#477—64no—
29#477—64no—
30C2×#107—34no—
31C2×C4⋊C8—32no—
32C2×C4⋊C8—34no—
33C2×#103—34no—
34#384—34no—
35#384—61no—
36#384—61no—
37C4×C4⋊C4—32no—
38C4×C4⋊C4—31no—
39C43—31no—
40#340—38no—
41C22×C3×C4—42no—
42C22×C3×C4—41no—
43C22×Dic3—44no—
44C2×SL(2,𝔽3)C2×D411yescharacteristic
45C2×SL(2,𝔽3)C2×Q812yes—
46C3×C42—44no—
47C4⋊Dic3—84no—
48C4⋊Dic3—88no—
49C22×Q8—62no—
50C22×Q8C22×D311yescharacteristic
51C22×C8—34no—
52C4⋊Q8—64no—
53C4⋊Q8—64no—
54C4⋊Q8—128no—
55C4×Q8—34no—
56C4×Q8—32no—
57C4×Q8Dic614yes—
58C4×Q8D1212yes—
59C2×C4⋊C4—32no—
60C2×C4⋊C4—31no—
61C2×C4⋊C4—34no—
62C2×C4⋊C4—64no—
63C2×C4⋊C4—62no—
64C2×C4⋊C4—64no—
65C2×C42—31no—
66C2×C42—32no—
67C2×C42—32no—
68C2×C42—31no—
69C2×C42S411yescharacteristic
70#107—68no—
71C4⋊C8—68no—
72C4⋊C8—64no—
73#103—64no—
74#103—64no—
75#95—61no—
76C23×C3—41no—
77C2×C3×C4—44no—
78C2×C3×C4—42no—
79C2×C3×C4—42no—
80C2×C3×C4—44no—
81C2×C3×C4—42no—
82C2×C3×C4—44no—
83C2×Dic3—84no—
84C2×Dic3—88no—
85SL(2,𝔽3)C4⋊Q811yescharacteristic, complemented
86C2×Q8—31no—
87C2×Q8—32no—
88C2×Q8—124no—
89C2×Q8—128no—
90C2×Q8C2×Dic612yes—
91C2×Q8C2×D1211yescharacteristic
92C22×C4—31no—
93C22×C4—32no—
94C22×C4—31no—
95C22×C4—64no—
96C22×C4C2×S412yes—
97C22×C4C2×S411yescharacteristic
98C2×C8—64no—
99C2×C8—68no—
100C4⋊C4—32no—
101C4⋊C4—34no—
102C4⋊C4—62no—
103C4⋊C4—61no—
104C4⋊C4—64no—
105C4⋊C4—64no—
106C4⋊C4—128no—
107C4⋊C4—124no—
108C4⋊C4—128no—
109C42—34no—
110C42—34no—
111C42—32no—
112C42—32no—
113C42—62no—
114C42—64no—
115C42#22614yes—
116C22×C3—42no—
117C22×C3—41no—
118C22×C3—41no—
119C22×C3—41no—
120C22×C3—42no—
121C3×C4—44no—
122C3×C4—48no—
123Dic3—164no—
124C23C22×S411yescharacteristic
125Q8—32no—
126Q8—31no—
127Q8—61no—
128Q8—248no—
129Q8#86911yescharacteristic
130C2×C4—32no—
131C2×C4—32no—
132C2×C4—31no—
133C2×C4—31no—
134C2×C4—34no—
135C2×C4—32no—
136C2×C4—62no—
137C2×C4—62no—
138C2×C4—62no—
139C2×C4—124no—
140C2×C4—128no—
141C2×C4A4⋊Q814yes—
142C2×C4C4⋊S412yes—
143C2×C4C2×#22612yes—
144C2×C4C2×#22614yes—
145C2×C4#98412yes—
146C2×C4#98614yes—
147C8—124no—
148C2×C3—41no—
149C2×C3—41no—
150C2×C3—42no—
151C2×C3—42no—
152C2×C3—41no—
153C22C2×A4⋊Q812yes—
154C22C2×C4⋊S411yescharacteristic
155C22C22×#22611yescharacteristic
156C22C2×#98411yescharacteristic
157C22C2×#98612yes—
158C4—31no—
159C4—31no—
160C4—32no—
161C4—64no—
162C4—62no—
163C4—244no—
164C4C4⋊#22614yes—
165C4#539118yescomplemented
166C3—41noSylow
167C2#8528311yescharacteristic
168C2C2×C4⋊#22611yescharacteristic
169C2C2×#539112yes—
170C2#8533712yes—
171C2#8534611yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC42×Q8D3
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)C4⋊Q8
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
22. C4×SL(2,𝔽3)Q8semidirect
85. SL(2,𝔽3)C4⋊Q8semidirect
165. C4#5391semidirect

Maximal subgroups

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

subgroupindex
C2×#53912
C42×SL(2,𝔽3)2
C4⋊#8592
#331983
#49354