Finite Group Databasebeta

← #11711530

Subgroups — #11711530

Order 768.

203 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C4×C8×A4C211yesmaximal, characteristic
2#73104C212yesmaximal
3#12788—31nomaximal, Sylow
4C2×C8×A4C411yescharacteristic
5C2×C8×A4C2211yescharacteristic
6C42×A4C411yescharacteristic
7A4⋊C16—22no—
8#4463—41nomaximal
9C22×C4×C8D311yescharacteristic
10C2×#364—31no—
11C2×#364—31no—
12#2112—32no—
13#1376—32no—
14C2×C4×A4—21no—
15C2×C4×A4C2×C411yescharacteristic
16C8×A4C812yes—
17C8×A4D411yescharacteristic
18C8×A4Q811yescharacteristic
19C3×C4×C8—41no—
20#812—42no—
21C23×C8Dic311yescharacteristic
22C23×C8C2×D311yescharacteristic
23C22×C42Dic311yescharacteristic
24C2×M32—32no—
25C2×M32—32no—
26C2×C4×C8—31no—
27C2×C4×C8—31no—
28C2×C4×C8—31no—
29C2×C4×C8—31no—
30#364—61no—
31#364—61no—
32#364—61no—
33#364—61no—
34C22⋊C16—62no—
35C22⋊C16—62no—
36C22×A4M1611yescharacteristic
37C4×A4—22no—
38C4×A4C4⋊C411yescharacteristic
39C4×A4C2×C811yescharacteristic
40C2×C3×C8—41no—
41C2×C3×C8—41no—
42C3×C42—41no—
43C3⋊C16—82no—
44C23×C4—21no—
45C23×C4C2×Dic311yescharacteristic
46C22×C8—31no—
47C22×C8—31no—
48C22×C8—32no—
49C22×C8—31no—
50C22×C8—31no—
51C22×C8—31no—
52C22×C8—31no—
53C22×C8C3⋊C812yes—
54C22×C8Dic611yescharacteristic
55C22×C8D1211yescharacteristic
56C2×C42—31no—
57C2×C42—31no—
58C2×C42—61no—
59M32—62no—
60M32—62no—
61M32—62no—
62M32—62no—
63C2×C16—62no—
64C2×C16—62no—
65C4×C8—31no—
66C4×C8—31no—
67C4×C8—31no—
68C4×C8—61no—
69C4×C8S411yescharacteristic
70C2×A4—21no—
71C2×A4C4⋊C811yescharacteristic
72C2×C3×C4—41no—
73C2×C3×C4—81no—
74C3×C8—42no—
75C3×C8—41no—
76C3×C8—41no—
77C24#20811yescharacteristic
78C22×C4—22no—
79C22×C4—31no—
80C22×C4—31no—
81C22×C4—31no—
82C22×C4—31no—
83C22×C4—61no—
84C22×C4—61no—
85C22×C4—62no—
86C22×C4C2×C3⋊C811yescharacteristic
87C22×C4C4⋊Dic311yescharacteristic
88C2×C8—31no—
89C2×C8—31no—
90C2×C8—32no—
91C2×C8—32no—
92C2×C8—31no—
93C2×C8—31no—
94C2×C8—31no—
95C2×C8—31no—
96C2×C8—32no—
97C2×C8—31no—
98C2×C8—31no—
99C2×C8—61no—
100C2×C8—61no—
101C2×C8SL(2,ℤ/4)11yescharacteristic
102C2×C8C2×S411yescharacteristic
103C42—32no—
104C42—31no—
105C42—61no—
106C42SL(2,ℤ/4)11yescharacteristic
107C16—122no—
108C16—122no—
109C22×C3—41no—
110A4#36411yescharacteristic, complemented
111C3×C4—41no—
112C3×C4—41no—
113C3×C4—82no—
114C23—21no—
115C23—31no—
116C23—31no—
117C23—61no—
118C23C4⋊(C3⋊C8)11yescharacteristic
119C2×C4—21no—
120C2×C4—31no—
121C2×C4—31no—
122C2×C4—31no—
123C2×C4—31no—
124C2×C4—31no—
125C2×C4—31no—
126C2×C4—31no—
127C2×C4—62no—
128C2×C4—62no—
129C2×C4—61no—
130C2×C4—62no—
131C2×C4—61no—
132C2×C4—62no—
133C2×C4C2×SL(2,ℤ/4)11yescharacteristic
134C8—32no—
135C8—31no—
136C8—31no—
137C8A4⋊C812yes—
138C8A4⋊Q811yescharacteristic
139C8C4⋊S411yescharacteristic
140C2×C3—41no—
141C2×C3—81no—
142C22—31no—
143C22—31no—
144C22—32no—
145C22—31no—
146C22—61no—
147C22—61no—
148C22—61no—
149C22#446311yescharacteristic, complemented
150C22#540911yescharacteristic
151C4—22no—
152C4—31no—
153C4—31no—
154C4—62no—
155C4C2×A4⋊C811yescharacteristic
156C4C4⋊SL(2,ℤ/4)11yescharacteristic
157C3—41noSylow
158C2—21no—
159C2—31no—
160C2—31no—
161C2—61no—
162C2C4⋊(A4⋊C8)11yescharacteristic
subgroupG/H
centerC8C4⋊S4
commutatorC4×A4C2×C8
FrattiniC2×C8C2×S4
FittingC22×C4×C8D3
socleC23C4⋊(C3⋊C8)
prime radicalC22×A4M16
cosocleC2×C8×A4C22
hypercenterC4×C8S4
hypocenterA4#364
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C8 ◁ C2×C8 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
110. A4#364semidirect
149. C22#4463semidirect

Maximal subgroups

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

subgroupindex
C4×C8×A42
#731042
#127883
#44634