Finite Group Databasebeta

← #10683033

Subgroups — #10683033

Order 768.

138 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 27 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
1C3×#1662C211yesmaximal, characteristic
2#67786C211yesmaximal, characteristic, complemented
3#67785C211yesmaximal, characteristic, complemented
4#7525—31nomaximal, Sylow
5C3×#562C411yescharacteristic
6C3×#491C2211yescharacteristic
7C3×#489—21no—
8C3×#330C411yescharacteristic
9C3×#329—21no—
10#4548—21no—
11#4466—21no—
12#1662D311yescharacteristic
13#1413—31no—
14#1356—31no—
15C3×C4⋊Q8D411yescharacteristic
16C3×#125C2×C411yescharacteristic
17C3×#124—21no—
18C3×#123—41no—
19C3×C22⋊Q8—41no—
20C3×#121—41no—
21C3×#103—41no—
22C3×#103—41no—
23C3×#102—41no—
24C3×#97—21no—
25C3×#97D411yescharacteristic
26#836—41no—
27#812—41no—
28#562Dic311yescharacteristic
29#491C2×D311yescharacteristic
30#489—21no—
31#356—61no—
32#347—61no—
33#330Dic311yescharacteristic
34#329—21no—
35C2×C3×Q8—21no—
36C2×C3×Q8—21no—
37C2×C3×D4—21no—
38C2×C3×D4—41no—
39C22×C3×C4—41no—
40C2×C3×C8—21no—
41C2×C3×C8—41no—
42C3×C4⋊C4—21no—
43C3×C4⋊C4—41no—
44C3×C4⋊C4—41no—
45C3×C4⋊C4—41no—
46C3×C22⋊C4—41no—
47C3×C22⋊C4—41no—
48C3×C22⋊C4—41no—
49C3×C42C22⋊C411yescharacteristic
50#208—81no—
51C3⋊C16—42no—
52C4⋊Q8#6711yescharacteristic
53#125C2×Dic311yescharacteristic
54#124—21no—
55#123—41no—
56C22⋊Q8—41no—
57#121—41no—
58M32—121no—
59#103—41no—
60#103—41no—
61#102—41no—
62#101—121no—
63#97—21no—
64#97#6711yescharacteristic
65C23×C3—41no—
66C3×Q8—81no—
67C3×Q8—81no—
68C3×D4—81no—
69C3×D4—161no—
70C2×C3×C4—21no—
71C2×C3×C4—41no—
72C2×C3×C4—41no—
73C2×C3×C4—41no—
74C2×C3×C4—161no—
75C2×C3×C4#9911yescharacteristic
76C3×C8—41no—
77C3×C8—81no—
78C3⋊C8—161no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×D4—21no—
82C2×D4—41no—
83C22×C4—41no—
84M16—241no—
85C2×C8—21no—
86C2×C8—41no—
87C4⋊C4—21no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—41no—
94C42#21711yescharacteristic
95C16—122no—
96C22×C3—81no—
97C22×C3—81no—
98C22×C3—81no—
99C22×C3AΣL(1,𝔽16)11yescharacteristic
100C3×C4—21no—
101C3×C4—41no—
102C3×C4—81no—
103C3×C4—81no—
104C3×C4—81no—
105C23—41no—
106Q8—81no—
107Q8—81no—
108D4—81no—
109D4—161no—
110C2×C4—21no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—161no—
115C2×C4#83411yescharacteristic
116C8—41no—
117C8—81no—
118C8—481no—
119C2×C3—21no—
120C2×C3—161no—
121C2×C3#140411yescharacteristic
122C22—81no—
123C22—81no—
124C22—81no—
125C22#453611yescharacteristic
126C4—21no—
127C4—41no—
128C4—81no—
129C4—81no—
130C4—81no—
131C3#752511yescharacteristic, complemented, Sylow
132C2—21no—
133C2—161no—
134C2#6777111yescharacteristic
subgroupG/H
centerC2#67771
commutatorC3×#125C2×C4
Frattini#491C2×D3
FittingC3×#1662C2
socleC2×C3#1404
prime radicalC3×#562C4
cosocleC3×#491C22
hypercenter#1662D3
hypocenterC3#7525
LCSG ▷ C3×#125 ▷ C3×C42 ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #491 ◁ #1662

Complements

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

NH ≅ G/Ntype
2. #67786C2semidirect
3. #67785C2semidirect
131. C3#7525semidirect

Maximal subgroups

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

subgroupindex
C3×#16622
#677862
#677852
#75253