Finite Group Databasebeta

← #10683057

Subgroups — #10683057

Order 768.

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

normal subgroups only

This group has 128 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#71299C211yesmaximal, characteristic
2C3×#1411C211yesmaximal, characteristic
3#67795C211yesmaximal, characteristic
4#7534—31nomaximal, Sylow
5C3×#494C2211yescharacteristic
6#5093—21no—
7#5080—21no—
8#4680C411yescharacteristic
9C3×#356—21no—
10#4548—21no—
11#4455C411yescharacteristic
12#1716—31no—
13#1411—31no—
14#1411D311yescharacteristic
15C2×C3×Q16—41no—
16C3×C4⋊Q8D411yescharacteristic
17C3×C4⋊Q8D411yescharacteristic
18#869—21no—
19C3×#101—41no—
20C3×C4×C8C2×C411yescharacteristic
21#836—41no—
22#817—41no—
23#810—42no—
24#808—42no—
25C4×C3⋊C8—21no—
26#498—31no—
27#498—61no—
28#494C2×D311yescharacteristic
29#487—61no—
30#356—21no—
31#356—61no—
32#322—31no—
33C2×C3×Q8—21no—
34C2×C3×Q8—21no—
35C2×Dic6—41no—
36C3×Q16—82no—
37C3×M16—81no—
38C2×C3×C8—21no—
39C3×C4⋊C4—41no—
40C3×C4⋊C4—41no—
41C3×C42C22⋊C411yescharacteristic
42C4⋊Dic3—41no—
43C4⋊Dic3—41no—
44#208—81no—
45C2×C3⋊C8—22no—
46C2×Q16—41no—
47C4⋊Q8—61no—
48C4⋊Q8D1211yescharacteristic
49C4⋊Q8#6711yescharacteristic
50#106—122no—
51#106—121no—
52#103—122no—
53#101—41no—
54#101—121no—
55C4×C8—61no—
56C4×C8C4×D311yescharacteristic
57C3×Q8—81no—
58C3×Q8—81no—
59C2×C3×C4—21no—
60C2×C3×C4—41no—
61C2×C3×C4—41no—
62C2×C3×C4#9911yescharacteristic
63C2×Dic3—81no—
64Dic6—82no—
65C3×C8—22no—
66C3×C8—161no—
67C3⋊C8—42no—
68C3⋊C8—161no—
69C2×Q8—21no—
70C2×Q8—21no—
71C2×Q8—121no—
72Q16—82no—
73M16—81no—
74M16—241no—
75C2×C8—21no—
76C2×C8—62no—
77C4⋊C4—41no—
78C4⋊C4—41no—
79C4⋊C4—121no—
80C4⋊C4—121no—
81C42(C2×D3)⋊C411yescharacteristic
82C22×C3#35311yescharacteristic
83C3×C4—22no—
84C3×C4—21no—
85C3×C4—81no—
86C3×C4—81no—
87Dic3—161no—
88Q8—81no—
89Q8—81no—
90Q8—242no—
91C2×C4—21no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—241no—
95C2×C4#80611yescharacteristic
96C8—22no—
97C8—122no—
98C8—161no—
99C8—481no—
100C2×C3—21no—
101C2×C3#140911yescharacteristic
102C22#447411yescharacteristic
103C4—22no—
104C4—21no—
105C4—81no—
106C4—81no—
107C4—481no—
108C3#753411yescharacteristic, complemented, Sylow
109C2—21no—
110C2#6754011yescharacteristic
subgroupG/H
centerC2#67540
commutatorC3×C4×C8C2×C4
Frattini#494C2×D3
FittingC3×#1411C2
socleC2×C3#1409
prime radicalC22×C3#353
cosocleC3×#494C22
hypercenter#1411D3
hypocenterC3#7534
LCSG ▷ C3×C4×C8 ▷ C3×C42 ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #494 ◁ #1411

Complements

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

NH ≅ G/Ntype
108. C3#7534semidirect

Maximal subgroups

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

subgroupindex
#712992
C3×#14112
#677952
#75343