Finite Group Databasebeta

← #13157203

Subgroups — #13157203

Order 1728.

352 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 61 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#11724511C212yesmaximal
2C2×C3×#3743C211yesmaximal, characteristic
3#10590186—31nomaximal
4C2×C3×C4×C3⋊D3C2211yescharacteristic
5#88993—22no—
6#88989—22no—
7#88988C414yes—
8C2×C33×C8C411yescharacteristic
9C2×C3×C3⋊(C3⋊C8)C411yescharacteristic
10C3×#3743—22no—
11#64283—32no—
12C2×#3743D311yescharacteristic
13C2×C3×ΓL(1,𝔽25)—62no—
14C22×C3×C3⋊D3Q811yescharacteristic
15C2×C33×C4C2×C411yescharacteristic
16C2×C3×C3⋊Dic3D411yescharacteristic
17C3×C4×C3⋊D3C2×C412yes—
18C3×C4×C3⋊D3D412yes—
19C3⋊#622—24no—
20C33×C8—21no—
21C3×C3⋊(C3⋊C8)—21no—
22C3⋊#338—91nomaximal
23C2×C4×C3⋊D3C2×D311yescharacteristic
24C2×C3×C4×D3—62no—
25#3792—62no—
26#3788—62no—
27#3787—34no—
28C2×C32×C8—22no—
29C2×C32×C8—42no—
30C2×C32×C8Dic311yescharacteristic
31C2×C3⋊(C3⋊C8)Dic311yescharacteristic
32#3743—22no—
33C2×C3×C3⋊C8—62no—
34C3×ΓL(1,𝔽25)—68no—
35C22×C33C4⋊C411yescharacteristic
36C2×C3×C3⋊D3—41no—
37C2×C3×C3⋊D3C4⋊C412yes—
38C33×C4C4211yescharacteristic
39C33×C4C22⋊C411yescharacteristic
40C3×C3⋊Dic3C22⋊C412yes—
41C2×C3×M16—91no—
42C2×#208—92no—
43C2×ΓL(1,𝔽25)—62no—
44C22×C3⋊D3Dic611yescharacteristic
45C22×C3×D3—62no—
46C2×C32×C4—22no—
47C2×C32×C4—42no—
48C2×C32×C4C2×Dic311yescharacteristic
49C2×C3⋊Dic3D1211yescharacteristic
50C4×C3⋊D3C4×D312yes—
51C4×C3⋊D3#6712yes—
52C2×C3×Dic3—62no—
53C3×C4×D3—64no—
54C3×C4×D3—64no—
55#622—64no—
56C32×C8—21no—
57C32×C8—24no—
58C32×C8—44no—
59C3⋊(C3⋊C8)—21no—
60C3×C3⋊C8—64no—
61#338—271noSylow
62C2×C33—21no—
63C2×C33#9511yescharacteristic
64C3×C3⋊D3—22no—
65C22×C3×C4—91no—
66C2×C4×D3—62no—
67C3×M16—182no—
68C2×C3×C8—22no—
69C2×C3×C8—42no—
70C2×C3×C8—91no—
71C2×C3×C8#15711yescharacteristic
72#208—182no—
73#208—182no—
74C2×C3⋊C8—62no—
75C2×C3⋊C8—94no—
76ΓL(1,𝔽25)—68no—
77C22×C32—22no—
78C22×C32—42no—
79C22×C32C4⋊Dic311yescharacteristic
80C2×C3⋊D3—41no—
81C2×C3⋊D3Dic3⋊C412yes—
82C2×C3×D3—64no—
83C2×C3×D3—124no—
84C32×C4—22no—
85C32×C4—22no—
86C32×C4—42no—
87C32×C4—42no—
88C32×C4C4×Dic311yescharacteristic
89C32×C4#21711yescharacteristic
90C3⋊Dic3(C2×D3)⋊C412yes—
91C3×Dic3—64no—
92C2×M16—91no—
93C2×M16—272no—
94C33#33811yescharacteristic, complemented, Sylow
95C23×C3—91no—
96C22×D3—62no—
97C2×C3×C4—22no—
98C2×C3×C4—42no—
99C2×C3×C4—91no—
100C2×C3×C4—92no—
101C2×C3×C4—92no—
102C2×C3×C4C2×#15711yescharacteristic
103C2×Dic3—62no—
104C4×D3—64no—
105C4×D3—64no—
106C3×C8—24no—
107C3×C8—21no—
108C3×C8—44no—
109C3×C8—181no—
110C3⋊C8—64no—
111C3⋊C8—184no—
112C2×C32—21no—
113C2×C32—24no—
114C2×C32—22no—
115C2×C32—44no—
116C2×C32—42no—
117C2×C32C3⋊#9511yescharacteristic
118C3⋊D3—22no—
119C3×D3—124no—
120C22×C4—91no—
121M16—182no—
122M16—542no—
123M16—542no—
124C2×C8—91no—
125C2×C8—274no—
126C2×C8C3⋊#15711yescharacteristic
127C22×C3—22no—
128C22×C3—42no—
129C22×C3—92no—
130C22×C3—361no—
131C22×C3C4⋊#15711yescharacteristic
132C2×D3—64no—
133C2×D3—124no—
134C3×C4—22no—
135C3×C4—22no—
136C3×C4—42no—
137C3×C4—42no—
138C3×C4—92no—
139C3×C4C4×#15711yescharacteristic
140C3×C4C22⋊#15711yescharacteristic
141Dic3—64no—
142C32—22no—
143C32—42no—
144C32C3⋊#33811yescharacteristic, complemented
145C23—91no—
146C2×C4—91no—
147C2×C4—92no—
148C2×C4—92no—
149C2×C4C2×C3⋊#15711yescharacteristic
150C8—21no—
151C8—181no—
152C8—544no—
153C2×C3—24no—
154C2×C3—21no—
155C2×C3—22no—
156C2×C3—44no—
157C2×C3—42no—
158C2×C3—182no—
159C2×C3#6378111yescharacteristic
160D3—124no—
161C22—92no—
162C22—361no—
163C22C4⋊(C3⋊#157)11yescharacteristic
164C4—92no—
165C4C4×C3⋊#15711yescharacteristic
166C4#8899411yescharacteristic
167C3—22no—
168C3—42no—
169C3#1059018611yescharacteristic, complemented
170C2—21no—
171C2—182no—
172C2#1172310611yescharacteristic
subgroupG/H
centerC4C4×C3⋊#157
commutatorC33×C4C42
FrattiniC2×C4C2×C3⋊#157
FittingC2×C33×C8C4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3Q8
cosocleC2×C3×C4×C3⋊D3C22
hypercenterC2×C8C3⋊#157
hypocenterC33#338
LCSG ▷ C33×C4 ▷ C33
UCSC1 ◁ C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
94. C33#338semidirect
144. C32C3⋊#338semidirect
169. C3#10590186semidirect

Maximal subgroups

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

subgroupindex
#117245112
C2×C3×#37432
#105901863
C3⋊#3389