Finite Group Databasebeta

← #11714114

Subgroups — #11714114

Order 768.

469 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 77 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#85283C212yesmaximal
2C4×C8×A4C211yesmaximal, characteristic
3#73110C214yesmaximal
4#33207—31nomaximal, Sylow
5C2×A4⋊Q8—22no—
6C2×C8×A4C2212yes—
7C42×A4C2211yescharacteristic
8C4⋊SL(2,ℤ/4)—22no—
9C4⋊SL(2,ℤ/4)C2214yes—
10#4680—41nomaximal
11C2×#498—31no—
12C22×C4×C8D311yescharacteristic
13#2857—32no—
14#1990—32no—
15#1941—34no—
16#1844—31no—
17#1807—34no—
18C2×C4×A4D412yes—
19C2×C4×A4C2311yescharacteristic
20C2×SL(2,ℤ/4)—24no—
21A4⋊Q8—44no—
22#869—42no—
23C8×A4Q814yescomplemented
24C3×C4×C8—41no—
25#817—44no—
26C23×C8C2×D312yes—
27C2×C4⋊Q8—32no—
28C2×C22⋊Q8—62no—
29C22×C42C2×D311yescharacteristic
30#498—62no—
31#498—61no—
32#498—61no—
33C2×#106—34no—
34C2×#106—34no—
35C2×C4×C8—31no—
36C2×C4×C8—32no—
37C2×C4×C8—31no—
38#391—62no—
39#389—34no—
40#389—62no—
41#384—32no—
42#380—34no—
43#380—62no—
44#340—38no—
45C22×A4C2×D411yescharacteristic
46C2×Dic6—82no—
47C4×A4SD1614yes—
48C4×A4C2×Q812yes—
49SL(2,ℤ/4)—44no—
50C2×C3×C8—42no—
51C3×C42—41no—
52C4⋊Dic3—44no—
53C4⋊Dic3—82no—
54C22×Q8—62no—
55C23×C4D1212yes—
56C23×C4C22×D311yescharacteristic
57C22×C8—32no—
58C22×C8—34no—
59C22×C8—32no—
60C22×C8Dic614yescomplemented
61C4⋊Q8—62no—
62C4⋊Q8—62no—
63C4⋊Q8—62no—
64C4⋊Q8—62no—
65C22⋊Q8—124no—
66C22⋊Q8—124no—
67C2×C4⋊C4—34no—
68C2×C4⋊C4—34no—
69C2×C4⋊C4—62no—
70C2×C4⋊C4—64no—
71C2×C4⋊C4—62no—
72C2×C4⋊C4—62no—
73C2×C22⋊C4—64no—
74C2×C42—31no—
75C2×C42—31no—
76C2×C42—32no—
77#106—68no—
78#106—64no—
79#106—64no—
80C4×C8—32no—
81C4×C8—31no—
82C4×C8—61no—
83C4×C8S411yescharacteristic
84#95—64no—
85#95—64no—
86C2×A4C4⋊Q811yescharacteristic
87C2×A4C2×SD1612yes—
88C2×C3×C4—42no—
89C2×C3×C4—41no—
90C2×Dic3—84no—
91Dic6—164no—
92C3×C8—44no—
93C24C2×D1211yescharacteristic
94C2×Q8—124no—
95C2×Q8—122no—
96C2×Q8—124no—
97C2×Q8—122no—
98C22×C4—32no—
99C22×C4—31no—
100C22×C4—31no—
101C22×C4—32no—
102C22×C4—34no—
103C22×C4—32no—
104C22×C4—64no—
105C22×C4—64no—
106C22×C4#20414yes—
107C22×C4C2×Dic612yes—
108C2×C8—32no—
109C2×C8—34no—
110C2×C8—34no—
111C2×C8—34no—
112C2×C8—62no—
113C2×C8C2×S412yes—
114C4⋊C4—68no—
115C4⋊C4—64no—
116C4⋊C4—64no—
117C4⋊C4—124no—
118C4⋊C4—124no—
119C4⋊C4—128no—
120C4⋊C4—122no—
121C4⋊C4—122no—
122C22⋊C4—124no—
123C22⋊C4—124no—
124C42—32no—
125C42—31no—
126C42—61no—
127C42C2×S411yescharacteristic
128C22×C3—41no—
129A4#49811yescharacteristic, complemented
130C3×C4—42no—
131C3×C4—44no—
132Dic3—164no—
133C23—31no—
134C23—32no—
135C23—31no—
136C23#86911yescharacteristic
137C23C2×#20412yes—
138Q8—244no—
139Q8—244no—
140C2×C4—31no—
141C2×C4—32no—
142C2×C4—32no—
143C2×C4—32no—
144C2×C4—34no—
145C2×C4—34no—
146C2×C4—34no—
147C2×C4—32no—
148C2×C4—61no—
149C2×C4—62no—
150C2×C4—124no—
151C2×C4—124no—
152C2×C4—124no—
153C2×C4—124no—
154C2×C4—124no—
155C2×C4—124no—
156C2×C4C4⋊S412yes—
157C2×C4C22×S411yescharacteristic
158C8—34no—
159C8A4⋊Q814yescomplemented
160C2×C3—41no—
161C2×C3—42no—
162C22—31no—
163C22—32no—
164C22—31no—
165C22—31no—
166C22—32no—
167C22—32no—
168C22—61no—
169C22#468011yescharacteristic, complemented
170C22C2×C4⋊S411yescharacteristic
171C4—32no—
172C4—34no—
173C4—244no—
174C4—244no—
175C4#540114yes—
176C4C2×A4⋊Q812yes—
177C3—41noSylow
178C2—32no—
179C2—31no—
180C2—31no—
181C2#8528311yescharacteristic
182C2C2×#540112yes—
subgroupG/H
centerC22C2×C4⋊S4
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4×C8D3
socleC24C2×D12
prime radicalC22×A4C2×D4
hypercenterC4×C8S4
hypocenterA4#498
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
23. C8×A4Q8semidirect
60. C22×C8Dic6semidirect
129. A4#498semidirect
159. C8A4⋊Q8semidirect
169. C22#4680semidirect

Maximal subgroups

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

subgroupindex
#852832
C4×C8×A42
#731102
#332073
#46804