Finite Group Databasebeta

← #13136665

Subgroups — #13136665

Order 1728.

181 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 181 conjugacy classes of subgroups — too many to draw. Only its 25 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#11720836C211yesmaximal, characteristic
2#11720771C211yesmaximal, characteristic, complemented
3#11720252C211yesmaximal, characteristic, complemented
4C4×((C3⋊D3)⋊C3)⋊C2C411yescharacteristic
5C8×#272C2211yescharacteristic
6#3743⋊C3—21no—
7#88434C411yescharacteristic
8#88430—21no—
9#88410—21no—
10(C3⋊(C3⋊C16))⋊C3—21no—
11D3×ΓL(1,𝔽25)—61no—
12#63543—61no—
13C2×((C3⋊D3)⋊C3)⋊C2C811yescharacteristic
14C4×#272C2×C411yescharacteristic
15C4×(C3⋊D3)⋊C3—21no—
16#6199C811yescharacteristic
17((C3⋊D3)⋊C3)⋊C4—21no—
18C8×H(3,𝔽3)D411yescharacteristic
19#6180D411yescharacteristic
20(C3⋊(C3⋊C8))⋊C3—21no—
21#4514—91nomaximal
22C4×D32—61no—
23#3743—21no—
24C3×ΓL(1,𝔽25)—61no—
25C3×D3×C8—61no—
26C3×D3×C8—61no—
27#3712—61no—
28#3711—61no—
29D3×C3⋊C8—61no—
30C3⋊(C3⋊C16)—21no—
31C3×C3⋊C16—61no—
32C2×#272C2×C811yescharacteristic
33C2×(C3⋊D3)⋊C3—21no—
34((C3⋊D3)⋊C3)⋊C2—41no—
35C4×H(3,𝔽3)C22⋊C411yescharacteristic
36#1084M1611yescharacteristic
37(C3⋊Dic3)⋊C3—21no—
38D3×M16—181no—
39C2×ΓL(1,𝔽25)—91no—
40C3×M32—91no—
41#812—91no—
42#798—181no—
43C2×D32—61no—
44C4×C3⋊D3—21no—
45C3×C4×D3—61no—
46C3×C4×D3—61no—
47C3×C4×D3—61no—
48#624—61no—
49D3×Dic3—61no—
50D3×Dic3—61no—
51C32×C8—21no—
52C32×C8—21no—
53C3⋊(C3⋊C8)—21no—
54C3×C3⋊C8—61no—
55C3×C3⋊C8—61no—
56C3×C3⋊C8—61no—
57#349—271noSylow
58C2×H(3,𝔽3)C22⋊C811yescharacteristic
59#272—21no—
60(C3⋊D3)⋊C3—41no—
61C2×C4×D3—91no—
62C2×C4×D3—181no—
63C3×M16—181no—
64C2×C3×C8—91no—
65#208—181no—
66C2×C3⋊C8—91no—
67ΓL(1,𝔽25)—61no—
68ΓL(1,𝔽25)—181no—
69ΓL(1,𝔽25)—181no—
70ΓL(1,𝔽25)—181no—
71D3×C8—61no—
72D3×C8—61no—
73D3×C8—181no—
74C3×C16—181no—
75C3×C16—181no—
76C3⋊C16—61no—
77C3⋊C16—181no—
78C2×C3⋊D3—21no—
79C2×C3×D3—61no—
80C2×C3×D3—61no—
81C2×C3×D3—61no—
82D32—122no—
83C32×C4—21no—
84C32×C4—21no—
85C3⋊Dic3—21no—
86C3×Dic3—61no—
87C3×Dic3—61no—
88C3×Dic3—61no—
89C2×M16—271no—
90M32—91no—
91M32—271no—
92H(3,𝔽3)#34911yescharacteristic, complemented, Sylow
93C22×D3—91no—
94C22×D3—181no—
95C2×C3×C4—91no—
96C2×C3×C4—181no—
97C2×Dic3—91no—
98C2×Dic3—181no—
99C4×D3—61no—
100C4×D3—61no—
101C4×D3—61no—
102C4×D3—181no—
103C4×D3—181no—
104C4×D3—181no—
105C4×D3—181no—
106C3×C8—61no—
107C3×C8—61no—
108C3×C8—91no—
109C3×C8—181no—
110C3×C8GO+(4,𝔽2)11yescharacteristic
111C3⋊C8—61no—
112C3⋊C8—61no—
113C3⋊C8—61no—
114C3⋊C8—181no—
115C2×C32—21no—
116C2×C32—21no—
117C3⋊D3—41no—
118C3×D3—62no—
119C3×D3—121no—
120C3×D3—121no—
121C22×C4—271no—
122M16—181no—
123M16—541no—
124C2×C8—91no—
125C2×C8—271no—
126C16—181no—
127C16—181no—
128C22×C3—91no—
129C22×C3—181no—
130C2×D3—61no—
131C2×D3—61no—
132C2×D3—61no—
133C2×D3—181no—
134C2×D3—362no—
135C2×D3—361no—
136C3×C4—61no—
137C3×C4—61no—
138C3×C4—91no—
139C3×C4—181no—
140C3×C4D32⋊C411yescharacteristic
141Dic3—61no—
142Dic3—61no—
143Dic3—61no—
144Dic3—181no—
145C32—21no—
146C32—21no—
147C23—271no—
148C2×C4—91no—
149C2×C4—181no—
150C2×C4—271no—
151C2×C4—541no—
152C8—91no—
153C8—181no—
154C8#1088⋊C211yescharacteristic
155C2×C3—61no—
156C2×C3—61no—
157C2×C3—181no—
158C2×C3—361no—
159C2×C3D32⋊C811yescharacteristic
160D3—62no—
161D3—121no—
162D3—121no—
163D3—361no—
164C22—91no—
165C22—181no—
166C22—1081no—
167C4—91no—
168C4—181no—
169C4(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
170C3—61no—
171C3—61no—
172C3#1059001311yescharacteristic
173C2—181no—
174C2—361no—
175C2(((C3⋊D3)⋊C3)⋊C2)⋊C811yescharacteristic
subgroupG/H
centerC4(((C3⋊D3)⋊C3)⋊C2)⋊C4
commutatorC2×#272C2×C8
FrattiniC3×C8GO+(4,𝔽2)
FittingC8×H(3,𝔽3)D4
socleC2×C3D32⋊C8
prime radicalC2×((C3⋊D3)⋊C3)⋊C2C8
cosocleC8×#272C22
hypercenterC8#1088⋊C2
hypocenterH(3,𝔽3)#349
LCSG ▷ C2×#272 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11720771C2semidirect
3. #11720252C2semidirect
92. H(3,𝔽3)#349semidirect

Maximal subgroups

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

subgroupindex
#117208362
#117207712
#117202522
#45149