Finite Group Databasebeta

← #11711269

Subgroups — #11711269

Order 768.

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

normal subgroups only

This group has 265 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#85562C212yesmaximal
2#1446⋊C3C211yesmaximal, characteristic
3#10341—31nomaximal, Sylow
4C2×C22⋊SL(2,𝔽3)C2211yescharacteristic
5C4⋊SL(2,ℤ/4)—41nomaximal
6#2659—31no—
7#1787—32no—
8#1446D311yescharacteristic
9#1294—32no—
10#1280—31no—
11C22⋊SL(2,𝔽3)Q811yescharacteristic, complemented
12C2×C4×A4—41no—
13C2×SL(2,ℤ/4)—42no—
14#859—82no—
15C23×Q8C2×D311yescharacteristic
16C2×C22⋊Q8—61no—
17C2×#103—62no—
18C2×#103—62no—
19C2×C22⋊C8—32no—
20#398—62no—
21#395—62no—
22#380—32no—
23#380—61no—
24C2×#95—31no—
25C2×#95—31no—
26C22×A4—41no—
27C2×SL(2,𝔽3)—82no—
28C4×A4—42no—
29SL(2,ℤ/4)—82no—
30C4⋊Dic3—161no—
31C22×Q8—31no—
32C22×Q8—62no—
33C22×Q8—62no—
34C22×Q8—61no—
35C22×Q8Dic611yescharacteristic
36C23×C4—31no—
37C23×C4—31no—
38C23×C4S411yescharacteristic
39C22×C8—62no—
40C22⋊Q8—122no—
41C22⋊Q8—122no—
42C2×C4⋊C4—32no—
43C2×C4⋊C4—32no—
44C2×C4⋊C4—62no—
45C2×C4⋊C4—61no—
46C2×C4⋊C4—61no—
47C2×C4⋊C4—61no—
48C2×C22⋊C4—62no—
49#103—122no—
50#103—122no—
51#103—122no—
52#103—122no—
53C22⋊C8—62no—
54C22⋊C8—62no—
55#95—62no—
56#95—62no—
57#95—62no—
58#95—62no—
59C2×A4—41no—
60C2×A4—41no—
61C2×A4—41no—
62C2×C3×C4—161no—
63C2×Dic3—162no—
64SL(2,𝔽3)—82no—
65C24C2×S411yescharacteristic
66C2×Q8—22no—
67C2×Q8—62no—
68C2×Q8—62no—
69C2×Q8—62no—
70C2×Q8—62no—
71C2×Q8—62no—
72C2×Q8—122no—
73C2×Q8—122no—
74C2×Q8—122no—
75C2×Q8—241no—
76C2×Q8—241no—
77C22×C4—32no—
78C22×C4—32no—
79C22×C4—31no—
80C22×C4—31no—
81C22×C4—32no—
82C22×C4—61no—
83C22×C4—61no—
84C22×C4—61no—
85C22×C4—61no—
86C22×C4—62no—
87C22×C4—62no—
88C22×C4—121no—
89C22×C4—121no—
90C22×C4—121no—
91C22×C4—121no—
92C22×C4#22612yes—
93C2×C8—122no—
94C2×C8—122no—
95C2×C8—122no—
96C4⋊C4—122no—
97C4⋊C4—122no—
98C4⋊C4—122no—
99C4⋊C4—124no—
100C4⋊C4—241no—
101C4⋊C4—241no—
102C22⋊C4—122no—
103C22⋊C4—122no—
104C22×C3—161no—
105A4—41no—
106C3×C4—162no—
107Dic3—322no—
108C23—31no—
109C23—31no—
110C23—31no—
111C23—31no—
112C23A4⋊Q811yescharacteristic
113C23C2×#22611yescharacteristic
114C23#98611yescharacteristic
115Q8—22no—
116Q8—62no—
117Q8—62no—
118Q8—62no—
119Q8—122no—
120Q8—244no—
121C2×C4—32no—
122C2×C4—32no—
123C2×C4—41no—
124C2×C4—62no—
125C2×C4—62no—
126C2×C4—62no—
127C2×C4—62no—
128C2×C4—121no—
129C2×C4—121no—
130C2×C4—122no—
131C2×C4—121no—
132C2×C4—121no—
133C2×C4—121no—
134C2×C4—122no—
135C2×C4—122no—
136C2×C4—122no—
137C2×C4—122no—
138C2×C4—122no—
139C2×C4—122no—
140C2×C4—122no—
141C2×C4—241no—
142C8—242no—
143C2×C3—161no—
144C2×C3—161no—
145C2×C3—161no—
146C22—31no—
147C22—31no—
148C22—31no—
149C22—31no—
150C22—31no—
151C22—31no—
152C22—31no—
153C22—31no—
154C22—31no—
155C22—61no—
156C22#539111yescharacteristic
157C22#1020⋊C211yescharacteristic
158C4—42no—
159C4—62no—
160C4—62no—
161C4—122no—
162C4—242no—
163C4—242no—
164C3—161noSylow
165C2—31no—
166C2—31no—
167C2—31no—
168C2—31no—
169C2#7298911yescharacteristic
170C2#7299811yescharacteristic
171C2#7300311yescharacteristic
subgroupG/H
center, hypercenterC22#1020⋊C2
commutator, prime radical, cosocleC2×C22⋊SL(2,𝔽3)C22
Frattini, socleC24C2×S4
Fitting#1446D3
hypocenterC22⋊SL(2,𝔽3)Q8
LCSG ▷ C2×C22⋊SL(2,𝔽3) ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
11. C22⋊SL(2,𝔽3)Q8semidirect

Maximal subgroups

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

subgroupindex
#855622
#1446⋊C32
#103413
C4⋊SL(2,ℤ/4)4