Finite Group Databasebeta

← #11711374

Subgroups — #11711374

Order 768.

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

normal subgroups only

This group has 200 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#2388⋊C3C211yesmaximal, characteristic
2#68001C212yesmaximal, complemented
3#11820—31nomaximal, Sylow
4#5948C411yescharacteristic, complemented
5C4⋊SL(2,ℤ/4)—41nomaximal
6#395⋊C3C411yescharacteristic, complemented
7#393⋊C3C2211yescharacteristic
8#2388D311yescharacteristic
9#1799—31no—
10#1466—31no—
11#1351—32no—
12#1343—32no—
13C22⋊SL(2,𝔽3)C2×C411yescharacteristic
14C2×C4×A4—41no—
15C2×SL(2,ℤ/4)—42no—
16C22×#41Dic311yescharacteristic, complemented
17C2×C4×D4—31no—
18#432—61no—
19C2×#99—61no—
20#395Dic311yescharacteristic, complemented
21#394—31no—
22#393C2×D311yescharacteristic
23#392—31no—
24#380—61no—
25C4×C22⋊C4—31no—
26#342—32no—
27#323—32no—
28#323—61no—
29C22×A4—41no—
30#41⋊C3—161no—
31C4×A4—41no—
32C4×A4—41no—
33SL(2,ℤ/4)—82no—
34C4⋊Dic3—161no—
35C2×#41—121no—
36C2×#41—121no—
37C22×Q8C2×Dic311yescharacteristic
38C22×D4—31no—
39C22×D4—31no—
40C23×C4—31no—
41C23×C4S411yescharacteristic
42C4×D4—121no—
43C4×D4—121no—
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—
51C4⋊C8—121no—
52#99—122no—
53C22⋊C8—62no—
54#97—121no—
55#95—31no—
56#95—31no—
57#95—121no—
58C2×A4—41no—
59C2×A4—41no—
60C2×A4—41no—
61C2×C3×C4—161no—
62C2×Dic3—162no—
63SL(2,𝔽3)—161no—
64C24—31no—
65C24C2×S411yescharacteristic
66#41—41no—
67#41—121no—
68#41—121no—
69#41—122no—
70#41—121no—
71C2×Q8—121no—
72C2×D4—61no—
73C2×D4—61no—
74C2×D4—121no—
75C2×D4—121no—
76C2×D4—121no—
77C2×D4—122no—
78C22×C4—31no—
79C22×C4—31no—
80C22×C4—31no—
81C22×C4—61no—
82C22×C4—61no—
83C22×C4—61no—
84C22×C4—61no—
85C22×C4—121no—
86C22×C4—121no—
87C22×C4—122no—
88C22×C4SL(2,ℤ/4)11yescharacteristic
89C22×C4SL(2,ℤ/4)11yescharacteristic
90C2×C8—122no—
91C4⋊C4—121no—
92C4⋊C4—241no—
93C4⋊C4—241no—
94C22⋊C4—121no—
95C22⋊C4—121no—
96C22⋊C4—122no—
97C22⋊C4—122no—
98C22⋊C4—122no—
99C42—61no—
100C42—61no—
101C22×C3—161no—
102A4—41no—
103C3×C4—161no—
104C3×C4—161no—
105Dic3—322no—
106C23—61no—
107C23—61no—
108C23—61no—
109C23—61no—
110C23—61no—
111C23—121no—
112C23—121no—
113C23#98411yescharacteristic
114C23#98611yescharacteristic
115C23C2×SL(2,ℤ/4)11yescharacteristic
116Q8—41no—
117Q8—121no—
118D4—122no—
119D4—122no—
120D4—121no—
121D4—121no—
122D4—122no—
123C2×C4—31no—
124C2×C4—31no—
125C2×C4—41no—
126C2×C4—61no—
127C2×C4—61no—
128C2×C4—61no—
129C2×C4—61no—
130C2×C4—61no—
131C2×C4—61no—
132C2×C4—121no—
133C2×C4—122no—
134C2×C4—121no—
135C2×C4—121no—
136C2×C4—121no—
137C2×C4—121no—
138C2×C4—242no—
139C2×C4—242no—
140C2×C4—242no—
141C8—242no—
142C2×C3—161no—
143C2×C3—161no—
144C2×C3—161no—
145C22—31no—
146C22—31no—
147C22—41no—
148C22—61no—
149C22—61no—
150C22—121no—
151C22—121no—
152C22—121no—
153C22—121no—
154C22—241no—
155C22—241no—
156C22#542611yescharacteristic
157C4—31no—
158C4—41no—
159C4—121no—
160C4—121no—
161C4—121no—
162C4—482no—
163C4#462511yescharacteristic
164C3—161noSylow
165C2—31no—
166C2—31no—
167C2—41no—
168C2—41no—
169C2—241no—
170C2C2×#462511yescharacteristic
subgroupG/H
centerC2C2×#4625
commutator, hypocenterC22⋊SL(2,𝔽3)C2×C4
FrattiniC24C2×S4
Fitting#2388D3
socleC23C2×SL(2,ℤ/4)
prime radical#2388⋊C3C2
cosocle#393⋊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. #68001C2semidirect
4. #5948C4semidirect
6. #395⋊C3C4semidirect
16. C22×#41Dic3semidirect
20. #395Dic3semidirect

Maximal subgroups

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

subgroupindex
#2388⋊C32
#680012
#118203
C4⋊SL(2,ℤ/4)4