Finite Group Databasebeta

← #13147431

Subgroups — #13147431

Order 1728.

757 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C23×#1084C211yesmaximal, characteristic
2C2×#88631C212yesmaximal, complemented
3C22⋊#6188C214yesmaximal, complemented
4C24×H(3,𝔽3)C411yescharacteristic
5C22×#1084—24no—
6C22×#1084C411yescharacteristic
7C22×#1084C412yes—
8C22×#1084C2212yes—
9C22×#1084C2211yescharacteristic
10#88631—24no—
11C2×#6188C2214yescomplemented
12C23×C3×Dic3—62no—
13C23×H(3,𝔽3)—22no—
14C23×H(3,𝔽3)—24no—
15C23×H(3,𝔽3)C2×C411yescharacteristic
16C23×H(3,𝔽3)C2×C412yes—
17C2×#1084—22no—
18C2×#1084—24no—
19C2×#1084—24no—
20C2×#1084C2×C412yes—
21C2×#1084C2×C411yescharacteristic
22C2×#1084D414yes—
23C2×#1084C2311yescharacteristic
24#6188—24no—
25C3×#407—91nomaximal
26C24×C32—22no—
27C22×C3×Dic3—64no—
28C22×C3×Dic3—62no—
29C22×C3×Dic3—62no—
30C22×C3×Dic3—64no—
31C22×C3×Dic3—616no—
32C22×H(3,𝔽3)—28no—
33C22×H(3,𝔽3)—22no—
34C22×H(3,𝔽3)—22no—
35C22×H(3,𝔽3)—22no—
36C22×H(3,𝔽3)C22⋊C412yes—
37C22×H(3,𝔽3)M1614yescomplemented
38C22×H(3,𝔽3)C22×C411yescharacteristic
39#1084—22no—
40#1084C22⋊C412yes—
41#1084C2×D412yes—
42C23×C3×C4—91no—
43C23×Dic3—62no—
44C2×C3×M16—92no—
45C3×C22⋊C8—94no—
46C23×C32—216no—
47C23×C32—28no—
48C23×C32—24no—
49C23×C32—22no—
50C2×C3×Dic3—62no—
51C2×C3×Dic3—64no—
52C2×C3×Dic3—68no—
53C2×C3×Dic3—62no—
54C2×C3×Dic3—616no—
55C2×C3×Dic3—616no—
56C2×C3×Dic3—68no—
57#407—91noSylow
58C2×H(3,𝔽3)—22no—
59C2×H(3,𝔽3)—24no—
60C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
61C2×H(3,𝔽3)C2×M1612yes—
62C24×C3—62no—
63C24×C3#15711yescharacteristic
64C22×C3×C4—91no—
65C22×C3×C4—92no—
66C22×C3×C4—92no—
67C22×C3×C4—91no—
68C22×C3×C4—184no—
69C22×Dic3—62no—
70C22×Dic3—64no—
71C22×Dic3—64no—
72C22×Dic3—62no—
73C22×Dic3—616no—
74C3×M16—184no—
75C2×C3×C8—94no—
76C22×C32—28no—
77C22×C32—232no—
78C22×C32—28no—
79C22×C32—28no—
80C22×C32—22no—
81C22×C32—28no—
82C22×C32—24no—
83C3×Dic3—64no—
84C3×Dic3—64no—
85C3×Dic3—68no—
86C23×C4—91no—
87C2×M16—92no—
88C22⋊C8—94no—
89H(3,𝔽3)#40711yescharacteristic, complemented, Sylow
90C23×C3—24no—
91C23×C3—22no—
92C23×C3—68no—
93C23×C3—616no—
94C23×C3—64no—
95C23×C3—62no—
96C23×C3C2×#15712yes—
97C23×C3C2×#15711yescharacteristic
98C2×C3×C4—91no—
99C2×C3×C4—91no—
100C2×C3×C4—94no—
101C2×C3×C4—92no—
102C2×C3×C4—184no—
103C2×C3×C4—184no—
104C2×C3×C4—182no—
105C2×Dic3—68no—
106C2×Dic3—62no—
107C2×Dic3—62no—
108C2×Dic3—64no—
109C2×Dic3—616no—
110C2×Dic3—616no—
111C2×Dic3—68no—
112C3×C8—184no—
113C2×C32—216no—
114C2×C32—28no—
115C2×C32—24no—
116C2×C32—22no—
117C24#108811yescharacteristic
118C22×C4—91no—
119C22×C4—92no—
120C22×C4—92no—
121C22×C4—91no—
122C22×C4—184no—
123M16—184no—
124C2×C8—94no—
125C22×C3—28no—
126C22×C3—22no—
127C22×C3—22no—
128C22×C3—22no—
129C22×C3—68no—
130C22×C3—632no—
131C22×C3—68no—
132C22×C3—68no—
133C22×C3—68no—
134C22×C3—64no—
135C22×C3—62no—
136C22×C3#379214yes—
137C22×C3C22⋊#15712yes—
138C22×C3C22×#15711yescharacteristic
139C3×C4—92no—
140C3×C4—92no—
141C3×C4—182no—
142Dic3—64no—
143Dic3—64no—
144Dic3—68no—
145C32—22no—
146C23—24no—
147C23—22no—
148C23C2×#108812yes—
149C23C2×#108811yescharacteristic
150C2×C4—94no—
151C2×C4—91no—
152C2×C4—91no—
153C2×C4—92no—
154C2×C4—184no—
155C2×C4—184no—
156C2×C4—182no—
157C8—184no—
158C2×C3—24no—
159C2×C3—22no—
160C2×C3—616no—
161C2×C3—68no—
162C2×C3—64no—
163C2×C3—62no—
164C2×C3C2×#379212yes—
165C2×C3C2×C22⋊#15711yescharacteristic
166C22—28no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22#8863114yescomplemented
171C22C22⋊#108812yes—
172C22C22×#108811yescharacteristic
173C4—92no—
174C4—92no—
175C4—182no—
176C3—62no—
177C3#1059384511yescharacteristic
178C2—24no—
179C2—22no—
180C2C2×#8863112yes—
181C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C22×#157
commutatorC22×H(3,𝔽3)C22×C4
Fitting, prime radicalC24×H(3,𝔽3)C4
cosocleC2×#1084C23
hypercenterC24×C3#157
hypocenterH(3,𝔽3)#407
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C2×#88631C2semidirect
3. C22⋊#6188C2semidirect
11. C2×#6188C22semidirect
37. C22×H(3,𝔽3)M16semidirect
89. H(3,𝔽3)#407semidirect
170. C22#88631semidirect

Maximal subgroups

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

subgroupindex
C23×#10842
C2×#886312
C22⋊#61882
C3×#4079