Finite Group Databasebeta

← #13147384

Subgroups — #13147384

Order 1728.

757 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C22×C4×#272C211yesmaximal, characteristic
2C2×#88631C212yesmaximal, complemented
3#11720784C214yesmaximal, complemented
4C23×#272C411yescharacteristic
5C22×C4×H(3,𝔽3)C411yescharacteristic
6C22×#1084C2211yescharacteristic
7C2×C4×#272—24no—
8C2×C4×#272C412yes—
9C2×C4×#272C2212yes—
10#88631—24no—
11C2×#6188C2214yescomplemented
12C22×C3×C4×D3—62no—
13C23×H(3,𝔽3)C2×C411yescharacteristic
14C22×#272—22no—
15C22×#272—24no—
16C22×#272C2×C412yes—
17C2×C4×H(3,𝔽3)—22no—
18C2×C4×H(3,𝔽3)C2×C412yes—
19C2×#1084C2×C411yescharacteristic
20C2×#1084D414yes—
21C2×#1084C2311yescharacteristic
22C4×#272—24no—
23C4×#272—24no—
24#6188—24no—
25C3×#407—91nomaximal
26C23×C3×D3—62no—
27C22×C32×C4—22no—
28C22×C3×Dic3—62no—
29C2×C3×C4×D3—64no—
30C2×C3×C4×D3—64no—
31C2×C3×C4×D3—616no—
32C22×H(3,𝔽3)—22no—
33C22×H(3,𝔽3)C22⋊C412yes—
34C22×H(3,𝔽3)C22×C411yescharacteristic
35C2×#272—28no—
36C2×#272—22no—
37C2×#272—22no—
38C2×#272M1614yescomplemented
39C4×H(3,𝔽3)—22no—
40#1084C22⋊C412yes—
41#1084C2×D412yes—
42C23×C3×C4—91no—
43C22×C4×D3—62no—
44C2×C3×M16—92no—
45C3×C22⋊C8—94no—
46C23×C32—22no—
47C22×C3×D3—64no—
48C22×C3×D3—616no—
49C22×C3×D3—68no—
50C2×C32×C4—28no—
51C2×C32×C4—24no—
52C2×C3×Dic3—62no—
53C2×C3×Dic3—68no—
54C2×C3×Dic3—62no—
55C3×C4×D3—616no—
56C3×C4×D3—616no—
57#407—91noSylow
58C2×H(3,𝔽3)—22no—
59C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
60C2×H(3,𝔽3)C2×M1612yes—
61#272—24no—
62C24×C3—91no—
63C23×D3—62no—
64C22×C3×C4—62no—
65C22×C3×C4—91no—
66C22×C3×C4—92no—
67C22×C3×C4—92no—
68C22×C3×C4—184no—
69C22×C3×C4#15711yescharacteristic
70C22×Dic3—62no—
71C2×C4×D3—64no—
72C2×C4×D3—64no—
73C2×C4×D3—616no—
74C3×M16—184no—
75C2×C3×C8—94no—
76C22×C32—28no—
77C22×C32—22no—
78C22×C32—24no—
79C2×C3×D3—68no—
80C2×C3×D3—68no—
81C2×C3×D3—632no—
82C2×C3×D3—68no—
83C32×C4—28no—
84C3×Dic3—64no—
85C3×Dic3—64no—
86C23×C4—91no—
87C2×M16—92no—
88C22⋊C8—94no—
89H(3,𝔽3)#40711yescharacteristic, complemented, Sylow
90C23×C3—62no—
91C23×C3—92no—
92C23×C3—184no—
93C23×C3—182no—
94C23×C3C2×#15711yescharacteristic
95C22×D3—64no—
96C22×D3—68no—
97C22×D3—616no—
98C2×C3×C4—22no—
99C2×C3×C4—68no—
100C2×C3×C4—64no—
101C2×C3×C4—91no—
102C2×C3×C4—91no—
103C2×C3×C4—94no—
104C2×C3×C4—184no—
105C2×C3×C4—184no—
106C2×C3×C4C2×#15712yes—
107C2×Dic3—68no—
108C2×Dic3—62no—
109C2×Dic3—62no—
110C4×D3—616no—
111C4×D3—616no—
112C3×C8—184no—
113C2×C32—28no—
114C2×C32—24no—
115C2×C32—22no—
116C3×D3—616no—
117C24—91no—
118C22×C4—91no—
119C22×C4—92no—
120C22×C4—92no—
121C22×C4—184no—
122C22×C4#108811yescharacteristic
123M16—184no—
124C2×C8—94no—
125C22×C3—22no—
126C22×C3—68no—
127C22×C3—64no—
128C22×C3—62no—
129C22×C3—94no—
130C22×C3—188no—
131C22×C3—182no—
132C22×C3—182no—
133C22×C3C22⋊#15712yes—
134C22×C3C22×#15711yescharacteristic
135C2×D3—68no—
136C2×D3—68no—
137C2×D3—632no—
138C2×D3—68no—
139C3×C4—22no—
140C3×C4—68no—
141C3×C4—92no—
142C3×C4—92no—
143Dic3—64no—
144Dic3—64no—
145C32—22no—
146C23—92no—
147C23—184no—
148C23—182no—
149C23C2×#108811yescharacteristic
150C2×C4—22no—
151C2×C4—94no—
152C2×C4—91no—
153C2×C4—91no—
154C2×C4—184no—
155C2×C4—184no—
156C2×C4C2×#108812yes—
157C8—184no—
158C2×C3—22no—
159C2×C3—68no—
160C2×C3—64no—
161C2×C3—62no—
162C2×C3—184no—
163C2×C3#6428312yes—
164C2×C3C2×C22⋊#15711yescharacteristic
165D3—616no—
166C22—22no—
167C22—94no—
168C22—188no—
169C22—182no—
170C22—182no—
171C22C22⋊#108812yes—
172C22C22×#108811yescharacteristic
173C4—22no—
174C4—92no—
175C4—92no—
176C3—62no—
177C3#1059380211yescharacteristic
178C2—22no—
179C2—184no—
180C2#1172244512yes—
181C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
FittingC22×C4×H(3,𝔽3)C4
prime radicalC23×#272C4
cosocleC2×#1084C23
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)#407
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88631C2semidirect
3. #11720784C2semidirect
11. C2×#6188C22semidirect
38. C2×#272M16semidirect
89. H(3,𝔽3)#407semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#886312
#117207842
C3×#4079