Finite Group Databasebeta

← #11714259

Subgroups — #11714259

Order 768.

180 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 15 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#73291C211yesmaximal, characteristic
2#12224—31nomaximal, Sylow
3#5420—41nomaximal
4#5419—42nomaximal
5#2811D311yescharacteristic
6#2028—31no—
7#1605—32no—
8#1354—32no—
9#1319—31no—
10C22×SL(2,𝔽3)—41no—
11C22×SL(2,𝔽3)—42no—
12C22×SL(2,𝔽3)—81no—
13C2×#226—42no—
14#859—42no—
15#859—41no—
16#859—41no—
17#544—31no—
18#451—62no—
19#419—61no—
20#395—32no—
21#391—31no—
22#391—61no—
23#343—31no—
24#339—61no—
25#328—62no—
26#324—32no—
27C2×SL(2,𝔽3)—41no—
28C2×SL(2,𝔽3)—42no—
29C2×SL(2,𝔽3)—41no—
30C2×SL(2,𝔽3)—41no—
31C2×SL(2,𝔽3)—81no—
32C2×SL(2,𝔽3)—81no—
33C2×SL(2,𝔽3)—82no—
34C2×SL(2,𝔽3)—81no—
35#226—82no—
36#217—161no—
37C22×Q8—21no—
38C22×Q8S411yescharacteristic
39C22×Q8S412yes—
40C2×Q16—122no—
41C2×M16—61no—
42C4⋊Q8—121no—
43#125—121no—
44C22⋊Q8—62no—
45#117—61no—
46C2×C4⋊C4—31no—
47C2×C4⋊C4—61no—
48C2×C42—31no—
49#103—121no—
50#103—121no—
51#103—122no—
52C22⋊C8—62no—
53#95—32no—
54#95—62no—
55C23×C3—161no—
56C2×Dic3—161no—
57C2×Dic3—161no—
58SL(2,𝔽3)—42no—
59SL(2,𝔽3)—41no—
60SL(2,𝔽3)—81no—
61C2×Q8—21no—
62C2×Q8—31no—
63C2×Q8—41no—
64C2×Q8—42no—
65C2×Q8—41no—
66C2×Q8—61no—
67C2×Q8—62no—
68C2×Q8—82no—
69C2×Q8—81no—
70C2×Q8—81no—
71C2×Q8#22611yescharacteristic, complemented
72C22×C4—31no—
73C22×C4—32no—
74C22×C4—61no—
75Q16—242no—
76Q16—242no—
77M16—121no—
78M16—121no—
79C2×C8—122no—
80C2×C8—121no—
81C4⋊C4—61no—
82C4⋊C4—61no—
83C4⋊C4—62no—
84C4⋊C4—121no—
85C4⋊C4—122no—
86C4⋊C4—241no—
87C22⋊C4—121no—
88C42—121no—
89C42—121no—
90C22×C3—161no—
91C22×C3—161no—
92C22×C3—161no—
93C22×C3—321no—
94C22×C3—321no—
95Dic3—321no—
96Dic3—321no—
97C23#102011yescharacteristic
98Q8—41no—
99Q8—42no—
100Q8—81no—
101Q8—121no—
102Q8—122no—
103Q8—241no—
104Q8—242no—
105C2×C4—31no—
106C2×C4—31no—
107C2×C4—61no—
108C2×C4—61no—
109C2×C4—61no—
110C2×C4—61no—
111C2×C4—62no—
112C2×C4—121no—
113C2×C4—121no—
114C2×C4—122no—
115C2×C4—121no—
116C2×C4—121no—
117C8—241no—
118C8—242no—
119C2×C3—161no—
120C2×C3—161no—
121C2×C3—161no—
122C2×C3—321no—
123C2×C3—321no—
124C22—21no—
125C22—21no—
126C22#593011yescharacteristic
127C22#593311yescharacteristic
128C22#593411yescharacteristic
129C4—61no—
130C4—61no—
131C4—121no—
132C4—121no—
133C4—122no—
134C4—241no—
135C4—241no—
136C3—161noSylow
137C2—21no—
138C2—21no—
139C2#8554911yescharacteristic
140C2#8555411yescharacteristic
141C2#8555711yescharacteristic
subgroupG/H
center, socleC22#5934
commutator, prime radical, cosocle, hypocenter#73291C2
Frattini, hypercenterC23#1020
Fitting#2811D3
LCSG ▷ #73291
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
71. C2×Q8#226semidirect

Maximal subgroups

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

subgroupindex
#732912
#122243
#54204
#54194