Finite Group Databasebeta

← #13173048

Subgroups — #13173048

Order 1728.

678 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 139 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#11724463C212yesmaximal
2C32×C8×A4C211yesmaximal, characteristic
3#10593632—312nomaximal
4#10592815—31nomaximal
5C32×C4×A4C2211yescharacteristic
6C3⋊(C3⋊SL(2,ℤ/4))—22no—
7#88853—41nomaximal
8#64259—324no—
9C22×C32×C8D311yescharacteristic
10#64133—32no—
11C2×#3746—31no—
12C3×C8×A4D3112yes—
13#63645—31no—
14#63643—32no—
15C2×C32×A4D411yescharacteristic
16#6308—42no—
17C33×C8—41no—
18#5398—99no—
19#5116—94no—
20C22×C32×C4C2×D311yescharacteristic
21C2×#634—32no—
22C3×C4×A4C2×D3112yes—
23C3⋊SL(2,ℤ/4)—624no—
24C2×C32×C8—31no—
25C2×C32×C8—31no—
26#3757—62no—
27C4⋊(C3⋊Dic3)—32no—
28#3750—62no—
29#3746—61no—
30#3746—61no—
31#3746—1212no—
32C32×A4Q1611yescharacteristic, complemented
33C33×C4—41no—
34C3⋊(C3⋊Dic3)—82no—
35A4⋊Q8—918no—
36C22×C3×C8C3⋊D314yes—
37#924—98no—
38C2×Dic12—94no—
39C8×A4C3⋊D319yes—
40#818—94no—
41#816—98no—
42C23×C32D1211yescharacteristic
43C2×C3×A4D12112yes—
44C2×C32×C4—31no—
45C2×C32×C4—31no—
46C2×C3⋊Dic3—62no—
47C2×C3⋊Dic3—62no—
48#634—62no—
49#634—62no—
50#634—1224no—
51C32×C8—31no—
52C32×C8—412no—
53C32×C8S411yescharacteristic
54#467—271noSylow
55C2×C33—41no—
56C22×C3×C4C2×C3⋊D314yes—
57C2×Dic6—98no—
58C4×A4C2×C3⋊D319yes—
59SL(2,ℤ/4)—1818no—
60C2×C3×C8—34no—
61C2×C3×C8—34no—
62#217—188no—
63C4⋊Dic3—98no—
64Dic3⋊C4—188no—
65Dic12—184no—
66Dic12—184no—
67Dic12—369no—
68C22×C32—31no—
69C22×C32—31no—
70C22×C32Dic1211yescharacteristic, complemented
71C3×A4Dic12112yescomplemented
72C32×C4—31no—
73C32×C4—412no—
74C32×C4C2×S411yescharacteristic
75C3⋊Dic3—122no—
76C3⋊Dic3—122no—
77C3⋊Dic3—2424no—
78C2×Q16—271no—
79C22×C8C3⋊(C3⋊D3)11yescharacteristic
80C22⋊Q8—272no—
81#107—271no—
82#103—272no—
83C33—41noSylow
84C23×C3C4⋊(C3⋊D3)14yes—
85C2×A4C4⋊(C3⋊D3)19yes—
86C2×C3×C4—34no—
87C2×C3×C4—34no—
88C2×Dic3—188no—
89C2×Dic3—188no—
90Dic6—188no—
91Dic6—188no—
92Dic6—3618no—
93C3×C8—34no—
94C3×C8—49no—
95C3×C8C3⋊S414yes—
96C2×C32—31no—
97C2×C32—31no—
98C2×C32—412no—
99C2×C32C4⋊S411yescharacteristic
100C2×Q8—272no—
101C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
102Q16—541no—
103Q16—541no—
104C2×C8—31no—
105C2×C8—31no—
106C4⋊C4—272no—
107C4⋊C4—542no—
108C22⋊C4—542no—
109C22×C3—34no—
110C22×C3—34no—
111C22×C3#374614yescomplemented
112A4#374619yescomplemented
113C3×C4—34no—
114C3×C4—49no—
115C3×C4C2×C3⋊S414yes—
116Dic3—368no—
117Dic3—368no—
118Dic3—7218no—
119C32—412no—
120C32#539811yescharacteristic, complemented
121C23C4⋊(C3⋊(C3⋊D3))11yescharacteristic
122Q8—542no—
123Q8—542no—
124C2×C4—31no—
125C2×C4—31no—
126C2×C4—542no—
127C2×C4—542no—
128C8—31no—
129C8C3⋊(C3⋊S4)11yescharacteristic
130C2×C3—34no—
131C2×C3—34no—
132C2×C3—49no—
133C2×C3C4⋊(C3⋊S4)14yes—
134C22—31no—
135C22—31no—
136C22#8885311yescharacteristic, complemented
137C4—31no—
138C4—1082no—
139C4—1082no—
140C4C2×C3⋊(C3⋊S4)11yescharacteristic
141C3—49no—
142C3#1059363214yescomplemented
143C2—31no—
144C2—31no—
145C2C4⋊(C3⋊(C3⋊S4))11yescharacteristic
subgroupG/H
centerC2C4⋊(C3⋊(C3⋊S4))
commutator, cosocleC32×C4×A4C22
FrattiniC4C2×C3⋊(C3⋊S4)
FittingC22×C32×C8D3
socleC23×C32D12
prime radicalC2×C32×A4D4
hypercenterC8C3⋊(C3⋊S4)
hypocenterC32×A4Q16
LCSG ▷ C32×C4×A4 ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
32. C32×A4Q16semidirect
70. C22×C32Dic12semidirect
71. C3×A4Dic12semidirect
111. C22×C3#3746semidirect
112. A4#3746semidirect
120. C32#5398semidirect
136. C22#88853semidirect
142. C3#10593632semidirect

Maximal subgroups

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

subgroupindex
#117244632
C32×C8×A42
#105936323
#105928153
#888534