Finite Group Databasebeta

← #10683060

Subgroups — #10683060

Order 768.

150 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 25 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#71254C211yesmaximal, characteristic
2C3×#1410C211yesmaximal, characteristic
3#67788C211yesmaximal, characteristic
4#7535—31nomaximal, Sylow
5C3×#492C2211yescharacteristic
6#5080—21no—
7#5076—21no—
8#4680C411yescharacteristic, complemented
9C3×#356—21no—
10#4545—21no—
11C8×C3⋊C8C411yescharacteristic, complemented
12#1705—31no—
13#1410D311yescharacteristic
14#1409—31no—
15C2×C3×SD16—41no—
16C3×C4⋊Q8D411yescharacteristic
17C3×#127D411yescharacteristic
18C2×#214—42no—
19#869—21no—
20C3×#101—41no—
21C3×C4×C8C2×C411yescharacteristic
22#834—41no—
23#817—41no—
24#808—42no—
25C4×C3⋊C8—21no—
26#498—31no—
27#498—61no—
28#492—61no—
29#492C2×D311yescharacteristic
30#356—21no—
31#353—61no—
32C82—31no—
33C2×C3×Q8—21no—
34C2×C3×D4—21no—
35C2×C3×D4—41no—
36C2×Dic6—41no—
37C3×SD16—82no—
38C3×M16—81no—
39C2×C3×C8—21no—
40C3×C4⋊C4—41no—
41C3×C42C22⋊C411yescharacteristic
42#217—81no—
43#214—84no—
44C4⋊Dic3—41no—
45C4⋊Dic3—41no—
46C2×C3⋊C8—22no—
47C2×SD16—41no—
48C2×SD16—122no—
49C4⋊Q8—61no—
50C4⋊Q8#6711yescharacteristic
51#127D1211yescharacteristic
52#106—122no—
53#106—121no—
54#101—41no—
55#99—121no—
56C4×C8—61no—
57C4×C8C4×D311yescharacteristic
58C23×C3—41no—
59C3×Q8—81no—
60C3×D4—81no—
61C3×D4—82no—
62C2×C3×C4—21no—
63C2×C3×C4—41no—
64C2×C3×C4#9911yescharacteristic
65C2×Dic3—81no—
66C2×Dic3—161no—
67Dic6—82no—
68C3×C8—22no—
69C3×C8—161no—
70C3⋊C8—24no—
71C2×Q8—21no—
72C2×Q8—121no—
73C2×D4—21no—
74C2×D4—41no—
75SD16—82no—
76SD16—244no—
77M16—81no—
78C2×C8—21no—
79C2×C8—62no—
80C4⋊C4—41no—
81C4⋊C4—121no—
82C4⋊C4—121no—
83C22⋊C4—241no—
84C42(C2×D3)⋊C411yescharacteristic
85C22×C3—81no—
86C22×C3—161no—
87C22×C3#35311yescharacteristic
88C3×C4—22no—
89C3×C4—21no—
90C3×C4—81no—
91Dic3—161no—
92Dic3—321no—
93C23—41no—
94Q8—81no—
95Q8—242no—
96D4—81no—
97D4—82no—
98C2×C4—21no—
99C2×C4—41no—
100C2×C4—241no—
101C2×C4—481no—
102C2×C4#80611yescharacteristic
103C8—22no—
104C8—64no—
105C8—161no—
106C2×C3—21no—
107C2×C3—161no—
108C2×C3#140911yescharacteristic
109C22—81no—
110C22—161no—
111C22#447411yescharacteristic
112C4—22no—
113C4—21no—
114C4—81no—
115C4—481no—
116C4—961no—
117C3#753511yescharacteristic, complemented, Sylow
118C2—21no—
119C2—161no—
120C2#6754011yescharacteristic
subgroupG/H
centerC2#67540
commutatorC3×C4×C8C2×C4
Frattini#492C2×D3
FittingC3×#1410C2
socleC2×C3#1409
prime radicalC3×#127D4
cosocleC3×#492C22
hypercenter#1410D3
hypocenterC3#7535
LCSG ▷ C3×C4×C8 ▷ C3×C42 ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1410

Complements

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

NH ≅ G/Ntype
8. #4680C4semidirect
11. C8×C3⋊C8C4semidirect
117. C3#7535semidirect

Maximal subgroups

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

subgroupindex
#712542
C3×#14102
#677882
#75353