Finite Group Databasebeta

← #11711375

Subgroups — #11711375

Order 768.

192 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 24 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#2397⋊C3C211yesmaximal, characteristic
2#68003C212yesmaximal, complemented
3#11821—31nomaximal, Sylow
4C4⋊SL(2,ℤ/4)—41nomaximal
5#399⋊C3C2211yescharacteristic
6#392⋊C3C411yescharacteristic, complemented
7#376⋊C3C411yescharacteristic, complemented
8#2397D311yescharacteristic
9#1799—31no—
10#1465—31no—
11#1352—32no—
12#1343—32no—
13C2×C4×A4—41no—
14C2×SL(2,ℤ/4)—42no—
15#95⋊C3C2×C411yescharacteristic
16C2×C4×D4—31no—
17C2×C4×D4—31no—
18C2×#99—61no—
19#407—61no—
20#399—31no—
21#399C2×D311yescharacteristic
22#392—31no—
23#392Dic311yescharacteristic, complemented
24#380—61no—
25#376Dic311yescharacteristic, complemented
26#342—32no—
27#324—61no—
28#323—32no—
29C22×A4—41no—
30C4×A4—41no—
31C4×A4—41no—
32SL(2,ℤ/4)—82no—
33C4⋊Dic3—161no—
34C22×D4—31no—
35C22×D4—31no—
36C23×C4—31no—
37C23×C4S411yescharacteristic
38C2×M16—121no—
39C4×D4—121no—
40C4×D4—121no—
41C4×D4—121no—
42C4×D4—121no—
43C2×C4⋊C4—31no—
44C2×C4⋊C4—31no—
45C2×C4⋊C4—121no—
46C2×C22⋊C4—31no—
47C2×C22⋊C4—31no—
48C2×C22⋊C4—61no—
49C2×C22⋊C4—62no—
50C2×C42—31no—
51#99—122no—
52C22⋊C8—62no—
53C22⋊C8—121no—
54#95—31no—
55#95—121no—
56#95C2×Dic311yescharacteristic
57C2×A4—41no—
58C2×A4—41no—
59C2×A4—41no—
60C2×C3×C4—161no—
61C2×Dic3—162no—
62C24—31no—
63C24C2×S411yescharacteristic
64C2×D4—61no—
65C2×D4—61no—
66C2×D4—121no—
67C2×D4—121no—
68C2×D4—121no—
69C2×D4—122no—
70C22×C4—31no—
71C22×C4—31no—
72C22×C4—31no—
73C22×C4—61no—
74C22×C4—61no—
75C22×C4—61no—
76C22×C4—61no—
77C22×C4—121no—
78C22×C4—121no—
79C22×C4—122no—
80C22×C4SL(2,ℤ/4)11yescharacteristic
81C22×C4SL(2,ℤ/4)11yescharacteristic
82M16—241no—
83M16—241no—
84C2×C8—122no—
85C4⋊C4—121no—
86C4⋊C4—121no—
87C4⋊C4—241no—
88C4⋊C4—241no—
89C22⋊C4—121no—
90C22⋊C4—121no—
91C22⋊C4—122no—
92C22⋊C4—122no—
93C22⋊C4—122no—
94C42—61no—
95C42—61no—
96C22×C3—161no—
97A4—41no—
98C3×C4—161no—
99C3×C4—161no—
100Dic3—322no—
101C23—61no—
102C23—61no—
103C23—61no—
104C23—61no—
105C23—61no—
106C23—121no—
107C23—121no—
108C23#98411yescharacteristic
109C23#98611yescharacteristic
110C23C2×SL(2,ℤ/4)11yescharacteristic
111D4—122no—
112D4—122no—
113D4—122no—
114D4—122no—
115C2×C4—31no—
116C2×C4—31no—
117C2×C4—41no—
118C2×C4—61no—
119C2×C4—61no—
120C2×C4—61no—
121C2×C4—61no—
122C2×C4—61no—
123C2×C4—61no—
124C2×C4—121no—
125C2×C4—121no—
126C2×C4—121no—
127C2×C4—121no—
128C2×C4—121no—
129C2×C4—241no—
130C2×C4—242no—
131C2×C4—242no—
132C2×C4—242no—
133C8—242no—
134C2×C3—161no—
135C2×C3—161no—
136C2×C3—161no—
137C22—31no—
138C22—31no—
139C22—41no—
140C22—61no—
141C22—61no—
142C22—121no—
143C22—121no—
144C22—122no—
145C22—121no—
146C22—121no—
147C22—241no—
148C22#542611yescharacteristic
149C4—31no—
150C4—41no—
151C4—121no—
152C4—121no—
153C4—121no—
154C4—482no—
155C4#462611yescharacteristic
156C3—161noSylow
157C2—31no—
158C2—31no—
159C2—41no—
160C2—41no—
161C2—241no—
162C2C2×#462611yescharacteristic
subgroupG/H
centerC2C2×#4626
commutator, hypocenter#95⋊C3C2×C4
FrattiniC24C2×S4
Fitting#2397D3
socleC23C2×SL(2,ℤ/4)
prime radical#2397⋊C3C2
cosocle#399⋊C3C22
hypercenterC4#4626
LCSG ▷ #95⋊C3
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #68003C2semidirect
6. #392⋊C3C4semidirect
7. #376⋊C3C4semidirect
23. #392Dic3semidirect
25. #376Dic3semidirect

Maximal subgroups

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

subgroupindex
#2397⋊C32
#680032
#118213
C4⋊SL(2,ℤ/4)4