Finite Group Databasebeta

← #11715737

Subgroups — #11715737

Order 768.

457 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 22 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#87517C212yesmaximal
2#85666C211yesmaximal, characteristic
3#32949—31nomaximal, Sylow
4#5950C2211yescharacteristic
5#5932—22no—
6#5398—43nomaximal
7#3573D311yescharacteristic
8#3059—33no—
9#3028—32no—
10#1909—36no—
11#1812—33no—
12#997D411yescharacteristic
13A4⋊Q8—46no—
14#131⋊C3—81no—
15C8×A4—43no—
16#585C2×D311yescharacteristic
17C22×Q16—31no—
18#568—33no—
19#568—61no—
20C2×#131—33no—
21C2×#131—61no—
22#536—36no—
23#470—63no—
24#470—63no—
25#467—63no—
26#467—63no—
27#458—66no—
28#458—62no—
29#427—33no—
30#419—36no—
31#410—36no—
32H(3,ℤ/4)—36no—
33#41⋊C3—81no—
34C4×A4—43no—
35SL(2,ℤ/4)—86no—
36Dic12—161no—
372−1+4—61no—
38H(5,𝔽2)—33no—
39H(5,𝔽2)D1211yescharacteristic
40C2×#41—33no—
41C2×#41—61no—
42C22×Q8—32no—
43C2×Q16—63no—
44C2×Q16—63no—
45#131—21no—
46#131—63no—
47#131—66no—
48#131—63no—
49#131—61no—
50C2×M16—33no—
51C2×M16—33no—
52C2×M16—33no—
53C2×M16—63no—
54C22×C8—31no—
55C22×C8—33no—
56C22×C8S413yes—
57C4⋊Q8—66no—
58#124—66no—
59#123—66no—
60#123—66no—
61C22⋊Q8—66no—
62C22⋊Q8—66no—
63#117—36no—
64#107—63no—
65#106—63no—
66#103—66no—
67#103—66no—
68#99—66no—
69#99—66no—
70C2×A4—43no—
71Dic6—162no—
72SL(2,𝔽3)—81no—
73C3×C8—161no—
74#41—21no—
75#41—61no—
76#41—63no—
77#41—66no—
78#41—63no—
79C2×Q8—33no—
80C2×Q8—66no—
81C2×Q8—66no—
82C2×Q8—61no—
83C2×D4—33no—
84C2×D4—33no—
85C2×D4—33no—
86C2×D4—63no—
87C22×C4—31no—
88C22×C4—33no—
89C22×C4—62no—
90C22×C4C2×S413yes—
91Q16—123no—
92Q16—121no—
93M16—63no—
94M16—66no—
95M16—61no—
96C2×C8—33no—
97C2×C8—33no—
98C2×C8—33no—
99C2×C8—33no—
100C2×C8—63no—
101C2×C8—61no—
102C4⋊C4—66no—
103C4⋊C4—126no—
104C4⋊C4—126no—
105C22⋊C4—66no—
106C22⋊C4—126no—
107C22⋊C4—126no—
108C42—66no—
109A4—43no—
110C3×C4—161no—
111Dic3—322no—
112C23—31no—
113C23—33no—
114C23C4⋊S413yes—
115Q8—21no—
116Q8—63no—
117Q8—126no—
118Q8—122no—
119D4—66no—
120D4—63no—
121D4—61no—
122C2×C4—33no—
123C2×C4—33no—
124C2×C4—33no—
125C2×C4—33no—
126C2×C4—63no—
127C2×C4—61no—
128C2×C4—126no—
129C2×C4—126no—
130C8—33no—
131C8—61no—
132C8#102011yescharacteristic
133C2×C3—161no—
134C22—33no—
135C22—43no—
136C22—61no—
137C22—121no—
138C22—123no—
139C4—33no—
140C4—61no—
141C4—246no—
142C4—242no—
143C4C2×#102011yescharacteristic
144C3—161noSylow
145C2—63no—
146C2—121no—
147C2C4⋊#102011yescharacteristic
subgroupG/H
center, socleC2C4⋊#1020
commutator, prime radical, cosocle#5950C22
FrattiniC4C2×#1020
Fitting#3573D3
hypercenterC8#1020
hypocenter#997D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ C8

Maximal subgroups

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

subgroupindex
#875172
#856662
#329493
#53984