Finite Group Databasebeta

← #11711377

Subgroups — #11711377

Order 768.

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

normal subgroups only

This group has 187 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#2388⋊C3C211yesmaximal, characteristic
2#68002C212yesmaximal, complemented
3#11828—31nomaximal, Sylow
4#5948C411yescharacteristic
5#5409—41nomaximal
6#395⋊C3C2211yescharacteristic
7#393⋊C3C411yescharacteristic
8#2388D311yescharacteristic
9#1800—31no—
10#1465—31no—
11#1354—32no—
12#1345—32no—
13C22⋊SL(2,𝔽3)C2×C411yescharacteristic
14C2×C4×A4—41no—
15A4⋊C8—42no—
16C22×#41Dic311yescharacteristic
17C2×C4×D4—31no—
18#423—61no—
19C2×#100—61no—
20#407—61no—
21#395C2×D311yescharacteristic
22#394—31no—
23#393Dic311yescharacteristic
24#392—31no—
25C4×C22⋊C4—31no—
26#343—32no—
27#324—32no—
28#323—61no—
29C22×A4—41no—
30#41⋊C3—161no—
31C4×A4—41no—
32C4×A4—41no—
33#208—161no—
34C2×#41—121no—
35C2×#41—121no—
36C22×Q8C2×Dic311yescharacteristic
37C22×D4—31no—
38C22×D4—31no—
39C23×C4—31no—
40C23×C4S411yescharacteristic
41C2×M16—62no—
42C2×M16—121no—
43C4×D4—121no—
44C4×D4—121no—
45C2×C4⋊C4—31no—
46C2×C22⋊C4—31no—
47C2×C22⋊C4—31no—
48C2×C22⋊C4—61no—
49C2×C42—31no—
50C4⋊C8—121no—
51C4⋊C8—121no—
52#100—122no—
53C22⋊C8—62no—
54C22⋊C8—121no—
55#95—31no—
56#95—31no—
57C2×A4—41no—
58C2×A4—81no—
59C2×C3×C4—161no—
60SL(2,𝔽3)—161no—
61C3⋊C8—162no—
62C24—31no—
63C24SL(2,ℤ/4)11yescharacteristic
64#41—41no—
65#41—121no—
66#41—121no—
67#41—122no—
68#41—121no—
69C2×Q8—121no—
70C2×D4—61no—
71C2×D4—61no—
72C2×D4—121no—
73C2×D4—121no—
74C2×D4—121no—
75C2×D4—122no—
76C22×C4—31no—
77C22×C4—31no—
78C22×C4—31no—
79C22×C4—61no—
80C22×C4—61no—
81C22×C4—61no—
82C22×C4—61no—
83C22×C4—121no—
84C22×C4—121no—
85C22×C4SL(2,ℤ/4)11yescharacteristic
86C22×C4C2×S411yescharacteristic
87M16—122no—
88M16—122no—
89M16—241no—
90M16—241no—
91C2×C8—122no—
92C2×C8—122no—
93C4⋊C4—121no—
94C22⋊C4—121no—
95C22⋊C4—121no—
96C22⋊C4—121no—
97C22⋊C4—121no—
98C42—61no—
99C42—61no—
100C22×C3—161no—
101A4—41no—
102C3×C4—161no—
103C3×C4—161no—
104C23—21no—
105C23—61no—
106C23—61no—
107C23—61no—
108C23—61no—
109C23—61no—
110C23—121no—
111C23—121no—
112C23C2×SL(2,ℤ/4)11yescharacteristic
113Q8—41no—
114Q8—121no—
115D4—122no—
116D4—122no—
117D4—121no—
118D4—121no—
119D4—122no—
120C2×C4—31no—
121C2×C4—31no—
122C2×C4—31no—
123C2×C4—31no—
124C2×C4—41no—
125C2×C4—61no—
126C2×C4—61no—
127C2×C4—61no—
128C2×C4—61no—
129C2×C4—61no—
130C2×C4—121no—
131C2×C4—122no—
132C2×C4—121no—
133C2×C4—121no—
134C2×C4—121no—
135C2×C4—121no—
136C8—242no—
137C8—242no—
138C2×C3—161no—
139C2×C3—321no—
140C22—31no—
141C22—31no—
142C22—41no—
143C22—121no—
144C22—121no—
145C22—121no—
146C22—121no—
147C22—121no—
148C22—241no—
149C22—241no—
150C22#542311yescharacteristic
151C4—31no—
152C4—41no—
153C4—61no—
154C4—61no—
155C4—121no—
156C4—121no—
157C4#462511yescharacteristic
158C3—161noSylow
159C2—31no—
160C2—31no—
161C2—81no—
162C2—241no—
163C2C2×#462511yescharacteristic
subgroupG/H
centerC2C2×#4625
commutator, hypocenterC22⋊SL(2,𝔽3)C2×C4
FrattiniC22×C4C2×S4
Fitting#2388D3
socleC23C2×SL(2,ℤ/4)
prime radical#2388⋊C3C2
cosocle#395⋊C3C22
hypercenterC4#4625
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #68002C2semidirect

Maximal subgroups

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

subgroupindex
#2388⋊C32
#680022
#118283
#54094