Finite Group Databasebeta

← #11714108

Subgroups — #11714108

Order 768.

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

normal subgroups only

This group has 537 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×#5386C212yesmaximal
2C42×SL(2,𝔽3)C211yesmaximal, characteristic
3C4⋊#859C214yesmaximal
4#33196—31nomaximal, Sylow
5C2×#869—41nomaximal
6C2×C4×SL(2,𝔽3)C2211yescharacteristic
7C2×C4×SL(2,𝔽3)C2212yes—
8C2×#859C2214yes—
9#5386—24no—
10C2×#475—32no—
11#2868—31no—
12C42×Q8D311yescharacteristic
13#1988—32no—
14#1948—34no—
15#1842—31no—
16#1803—34no—
17C22×Dic6—42no—
18C22×SL(2,𝔽3)C2311yescharacteristic
19C2×C3×C42—41no—
20C2×C4⋊Dic3—44no—
21#869—48no—
22C4×SL(2,𝔽3)D412yes—
23C4×SL(2,𝔽3)Q814yescomplemented
24#859—24no—
25C2×C4⋊Q8—32no—
26C2×C4⋊Q8—61no—
27C2×C4⋊Q8—61no—
28C2×C4×Q8C2×D312yes—
29C2×C4×Q8C2×D311yescharacteristic
30#475—64no—
31#475—64no—
32C2×#106—34no—
33C2×C4⋊C8—32no—
34C2×C4⋊C8—34no—
35C2×#103—34no—
36#384—34no—
37#384—62no—
38C4×C4⋊C4—32no—
39C4×C4⋊C4—31no—
40C43—31no—
41#340—38no—
42C22×C3×C4—42no—
43C22×C3×C4—41no—
44C22×Dic3—44no—
45C2×Dic6—48no—
46C2×Dic6—84no—
47C2×Dic6—84no—
48C2×SL(2,𝔽3)C2×D411yescharacteristic
49C2×SL(2,𝔽3)C2×Q812yes—
50C3×C42—44no—
51C4⋊Dic3—416no—
52C22×Q8—62no—
53C22×Q8C22×D311yescharacteristic
54C22×C8—34no—
55C4⋊Q8—64no—
56C4⋊Q8—64no—
57C4⋊Q8—124no—
58C4⋊Q8—124no—
59C4×Q8—34no—
60C4×Q8—32no—
61C4×Q8Dic614yescomplemented
62C4×Q8D1212yes—
63C2×C4⋊C4—32no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—34no—
66C2×C4⋊C4—62no—
67C2×C4⋊C4—64no—
68C2×C4⋊C4—64no—
69C2×C42—31no—
70C2×C42—32no—
71C2×C42—31no—
72C2×C42—32no—
73C2×C42S411yescharacteristic
74#106—68no—
75C4⋊C8—68no—
76C4⋊C8—64no—
77#103—64no—
78#103—64no—
79#95—61no—
80C23×C3—41no—
81C2×C3×C4—44no—
82C2×C3×C4—42no—
83C2×C3×C4—44no—
84C2×C3×C4—42no—
85C2×C3×C4—44no—
86C2×C3×C4—42no—
87C2×Dic3—48no—
88C2×Dic3—84no—
89C2×Dic3—84no—
90Dic6—816no—
91SL(2,𝔽3)C4⋊Q811yescharacteristic, complemented
92C2×Q8—31no—
93C2×Q8—32no—
94C2×Q8—124no—
95C2×Q8—124no—
96C2×Q8—124no—
97C2×Q8C2×Dic612yes—
98C2×Q8C2×D1211yescharacteristic
99C22×C4—31no—
100C22×C4—32no—
101C22×C4—31no—
102C22×C4—64no—
103C22×C4C2×S412yes—
104C22×C4C2×S411yescharacteristic
105C2×C8—64no—
106C2×C8—68no—
107C4⋊C4—32no—
108C4⋊C4—34no—
109C4⋊C4—62no—
110C4⋊C4—61no—
111C4⋊C4—64no—
112C4⋊C4—64no—
113C4⋊C4—124no—
114C4⋊C4—128no—
115C4⋊C4—128no—
116C42—34no—
117C42—34no—
118C42—32no—
119C42—32no—
120C42—64no—
121C42—62no—
122C42GL(2,𝔽3)14yes—
123C22×C3—42no—
124C22×C3—41no—
125C22×C3—41no—
126C22×C3—42no—
127C22×C3—41no—
128C3×C4—48no—
129C3×C4—44no—
130Dic3—88no—
131C23C22×S411yescharacteristic
132Q8—32no—
133Q8—31no—
134Q8—61no—
135Q8—248no—
136Q8#86911yescharacteristic, complemented
137C2×C4—32no—
138C2×C4—32no—
139C2×C4—31no—
140C2×C4—31no—
141C2×C4—34no—
142C2×C4—32no—
143C2×C4—62no—
144C2×C4—62no—
145C2×C4—62no—
146C2×C4—124no—
147C2×C4—124no—
148C2×C4—124no—
149C2×C4A4⋊Q814yes—
150C2×C4C4⋊S412yes—
151C2×C4C2×GL(2,𝔽3)14yes—
152C2×C4C2×GL(2,𝔽3)12yes—
153C2×C4#98414yes—
154C2×C4#98612yes—
155C8—124no—
156C2×C3—41no—
157C2×C3—42no—
158C2×C3—41no—
159C2×C3—41no—
160C2×C3—42no—
161C22C2×A4⋊Q812yes—
162C22C2×C4⋊S411yescharacteristic
163C22C22×GL(2,𝔽3)11yescharacteristic
164C22C2×#98412yes—
165C22C2×#98611yescharacteristic
166C4—31no—
167C4—31no—
168C4—32no—
169C4—62no—
170C4—64no—
171C4—244no—
172C4#538618yescomplemented
173C4C4⋊GL(2,𝔽3)14yes—
174C3—41noSylow
175C2#8528311yescharacteristic
176C2C2×#538612yes—
177C2C2×C4⋊GL(2,𝔽3)11yescharacteristic
178C2#8533611yescharacteristic
179C2#8533712yes—
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
23. C4×SL(2,𝔽3)Q8semidirect
61. C4×Q8Dic6semidirect
91. SL(2,𝔽3)C4⋊Q8semidirect
136. Q8#869semidirect
172. C4#5386semidirect

Maximal subgroups

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

subgroupindex
C2×#53862
C42×SL(2,𝔽3)2
C4⋊#8592
#331963
C2×#8694