Finite Group Databasebeta

← #11711270

Subgroups — #11711270

Order 768.

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

normal subgroups only

This group has 277 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
5C2×A4⋊Q8—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—
14A4⋊Q8—44no—
15#859—82no—
16C23×Q8C2×D311yescharacteristic
17C2×C22⋊Q8—61no—
18C2×#103—62no—
19C2×#103—62no—
20C2×C22⋊C8—32no—
21#398—62no—
22#395—62no—
23#380—32no—
24#380—61no—
25C2×#95—31no—
26C2×#95—31no—
27C22×A4—41no—
28C2×Dic6—161no—
29C2×SL(2,𝔽3)—82no—
30C4×A4—42no—
31SL(2,ℤ/4)—44no—
32C22×Q8—31no—
33C22×Q8—62no—
34C22×Q8—62no—
35C22×Q8—61no—
36C22×Q8Dic611yescharacteristic, complemented
37C23×C4—31no—
38C23×C4—31no—
39C23×C4S411yescharacteristic
40C22×C8—62no—
41C22⋊Q8—122no—
42C22⋊Q8—122no—
43C2×C4⋊C4—32no—
44C2×C4⋊C4—32no—
45C2×C4⋊C4—62no—
46C2×C4⋊C4—61no—
47C2×C4⋊C4—61no—
48C2×C4⋊C4—61no—
49C2×C22⋊C4—62no—
50#103—122no—
51#103—122no—
52#103—122no—
53#103—122no—
54C22⋊C8—62no—
55C22⋊C8—62no—
56#95—62no—
57#95—62no—
58#95—62no—
59#95—62no—
60C2×A4—41no—
61C2×A4—41no—
62C2×A4—41no—
63C2×C3×C4—161no—
64C2×Dic3—162no—
65Dic6—164no—
66SL(2,𝔽3)—82no—
67C24C2×S411yescharacteristic
68C2×Q8—22no—
69C2×Q8—62no—
70C2×Q8—62no—
71C2×Q8—62no—
72C2×Q8—62no—
73C2×Q8—62no—
74C2×Q8—122no—
75C2×Q8—122no—
76C2×Q8—122no—
77C2×Q8—241no—
78C2×Q8—241no—
79C22×C4—32no—
80C22×C4—32no—
81C22×C4—31no—
82C22×C4—31no—
83C22×C4—32no—
84C22×C4—61no—
85C22×C4—61no—
86C22×C4—61no—
87C22×C4—61no—
88C22×C4—62no—
89C22×C4—62no—
90C22×C4—121no—
91C22×C4—121no—
92C22×C4—121no—
93C22×C4—121no—
94C22×C4GL(2,𝔽3)12yes—
95C2×C8—122no—
96C2×C8—122no—
97C2×C8—122no—
98C4⋊C4—122no—
99C4⋊C4—122no—
100C4⋊C4—122no—
101C4⋊C4—124no—
102C4⋊C4—241no—
103C4⋊C4—241no—
104C22⋊C4—122no—
105C22⋊C4—122no—
106C22×C3—161no—
107A4—41no—
108C3×C4—162no—
109Dic3—164no—
110C23—31no—
111C23—31no—
112C23—31no—
113C23—31no—
114C23A4⋊Q811yescharacteristic
115C23C2×GL(2,𝔽3)11yescharacteristic
116C23#98411yescharacteristic
117Q8—22no—
118Q8—62no—
119Q8—62no—
120Q8—62no—
121Q8—122no—
122Q8—244no—
123C2×C4—32no—
124C2×C4—32no—
125C2×C4—41no—
126C2×C4—62no—
127C2×C4—62no—
128C2×C4—62no—
129C2×C4—62no—
130C2×C4—121no—
131C2×C4—121no—
132C2×C4—122no—
133C2×C4—121no—
134C2×C4—121no—
135C2×C4—121no—
136C2×C4—122no—
137C2×C4—122no—
138C2×C4—122no—
139C2×C4—122no—
140C2×C4—122no—
141C2×C4—122no—
142C2×C4—122no—
143C2×C4—241no—
144C8—242no—
145C2×C3—161no—
146C2×C3—161no—
147C2×C3—161no—
148C22—31no—
149C22—31no—
150C22—31no—
151C22—31no—
152C22—31no—
153C22—31no—
154C22—31no—
155C22—31no—
156C22—31no—
157C22—61no—
158C22#538611yescharacteristic
159C22#1020⋊C211yescharacteristic
160C4—42no—
161C4—62no—
162C4—62no—
163C4—122no—
164C4—242no—
165C4—242no—
166C3—161noSylow
167C2—31no—
168C2—31no—
169C2—31no—
170C2—31no—
171C2#7298911yescharacteristic
172C2#7299911yescharacteristic
173C2#7300111yescharacteristic
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
36. C22×Q8Dic6semidirect

Maximal subgroups

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

subgroupindex
#855622
#1446⋊C32
#103413
C2×A4⋊Q84