Finite Group Databasebeta

← #13136923

Subgroups — #13136923

Order 1728.

301 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 43 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#11722448C211yesmaximal, characteristic
2C2×C4×#1088C211yesmaximal, characteristic
3M16×#272C211yesmaximal, characteristic
4C22×#1088—21no—
5C2×C4×#272C2211yescharacteristic
6C22⋊#1088—21no—
7C4⋊#1088C412yescomplemented
8C4×#1088—21no—
9C4×#1088—21no—
10C4×#1088C412yescomplemented
11M16×H(3,𝔽3)C411yescharacteristic, complemented
12#88538C411yescharacteristic, complemented
13#88527—21no—
14C8×#272—21no—
15C3×D3×M16—62no—
16C22×#272D411yescharacteristic
17C2×#1088—24no—
18C2×#1088—22no—
19C2×#1088—22no—
20C2×C4×H(3,𝔽3)C2×C411yescharacteristic
21C2×#1084Q811yescharacteristic
22C4×#272C2×C411yescharacteristic
23C4×#272C2×C411yescharacteristic
24C4×#272D411yescharacteristic
25C4×#272D411yescharacteristic
26C8×H(3,𝔽3)—21no—
27#6180—21no—
28C3×#339—91nomaximal
29C2×C3×C4×D3—62no—
30C32×M16—22no—
31C3×#208—62no—
32C3×ΓL(1,𝔽25)—64no—
33C3×D3×C8—64no—
34C22×H(3,𝔽3)C4⋊C411yescharacteristic
35C2×#272—22no—
36C2×#272C22⋊C411yescharacteristic
37C2×#272C22⋊C411yescharacteristic
38#1088—24no—
39#1088—42no—
40C4×H(3,𝔽3)C4211yescharacteristic
41C4×H(3,𝔽3)C22⋊C411yescharacteristic
42#1084C4⋊C411yescharacteristic
43#1084C4⋊C411yescharacteristic
44C2×C3×M16—91no—
45C3×#117—91no—
46C2×C3×C42—91no—
47D3×M16—62no—
48C22×C3×D3—62no—
49C2×C32×C4—22no—
50C2×C3×Dic3—62no—
51C3×C4×D3—62no—
52C3×C4×D3—62no—
53C3×C4×D3—62no—
54C3×C4×D3—62no—
55C32×C8—24no—
56C3×C3⋊C8—64no—
57#339—91noSylow
58C2×H(3,𝔽3)—21no—
59C2×H(3,𝔽3)#9511yescharacteristic
60#272—21no—
61#272#10412yes—
62C22×C3×C4—91no—
63C22×C3×C4—181no—
64C2×C4×D3—62no—
65C3×M16—62no—
66C3×M16—91no—
67C3×M16—181no—
68C3×M16#15711yescharacteristic
69C2×C3×C8—181no—
70C3×C4⋊C4—92no—
71C3×C22⋊C4—181no—
72C3×C42—92no—
73C3×C42—181no—
74C3×C42—181no—
75#208—62no—
76ΓL(1,𝔽25)—64no—
77D3×C8—64no—
78C22×C32—22no—
79C2×C3×D3—62no—
80C2×C3×D3—62no—
81C2×C3×D3—124no—
82C32×C4—22no—
83C32×C4—22no—
84C3×Dic3—62no—
85C3×Dic3—62no—
86C2×M16—91no—
87#117—91no—
88C2×C42—91no—
89H(3,𝔽3)#33911yescharacteristic, complemented, Sylow
90C23×C3—91no—
91C22×D3—62no—
92C2×C3×C4—62no—
93C2×C3×C4—91no—
94C2×C3×C4—91no—
95C2×C3×C4—91no—
96C2×C3×C4—91no—
97C2×C3×C4—91no—
98C2×C3×C4—182no—
99C2×C3×C4—184no—
100C2×C3×C4—182no—
101C2×C3×C4C2×#15711yescharacteristic
102C2×Dic3—62no—
103C4×D3—62no—
104C4×D3—62no—
105C4×D3—62no—
106C4×D3—62no—
107C3×C8—21no—
108C3×C8—64no—
109C3×C8—181no—
110C3⋊C8—64no—
111C2×C32—22no—
112C2×C32—42no—
113C3×D3—64no—
114C3×D3—122no—
115C22×C4—91no—
116C22×C4—181no—
117M16—91no—
118M16—181no—
119M16#108811yescharacteristic, complemented
120C2×C8—181no—
121C4⋊C4—92no—
122C22⋊C4—181no—
123C42—92no—
124C42—181no—
125C42—181no—
126C22×C3—62no—
127C22×C3—91no—
128C22×C3—91no—
129C22×C3—182no—
130C22×C3C4⋊#15711yescharacteristic
131C2×D3—62no—
132C2×D3—62no—
133C2×D3—124no—
134C3×C4—62no—
135C3×C4—62no—
136C3×C4—91no—
137C3×C4—91no—
138C3×C4—184no—
139C3×C4—362no—
140C3×C4C4×#15711yescharacteristic
141C3×C4C22⋊#15711yescharacteristic
142Dic3—62no—
143Dic3—62no—
144C32—22no—
145C23—91no—
146C2×C4—91no—
147C2×C4—91no—
148C2×C4—91no—
149C2×C4—91no—
150C2×C4—91no—
151C2×C4—182no—
152C2×C4—184no—
153C2×C4—182no—
154C2×C4C2×#108811yescharacteristic
155C8—21no—
156C8—181no—
157C2×C3—21no—
158C2×C3—62no—
159C2×C3—92no—
160C2×C3—122no—
161C2×C3—181no—
162C2×C3#6378111yescharacteristic
163D3—64no—
164D3—122no—
165C22—91no—
166C22—91no—
167C22—182no—
168C22C4⋊#108811yescharacteristic
169C4—91no—
170C4—91no—
171C4—184no—
172C4—362no—
173C4C4×#108811yescharacteristic
174C4C22⋊#108811yescharacteristic
175C3—62no—
176C3#1059018911yescharacteristic
177C2—21no—
178C2—92no—
179C2—181no—
180C2#1172078711yescharacteristic
subgroupG/H
centerC3×C4C22⋊#157
commutatorC4×H(3,𝔽3)C42
FrattiniC2×C3×C4C2×#157
FittingM16×H(3,𝔽3)C4
socleC2×C3#63781
prime radicalC22×#272D4
cosocleC2×C4×#272C22
hypercenterC3×M16#157
hypocenterH(3,𝔽3)#339
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
7. C4⋊#1088C4semidirect
10. C4×#1088C4semidirect
11. M16×H(3,𝔽3)C4semidirect
12. #88538C4semidirect
89. H(3,𝔽3)#339semidirect
119. M16#1088semidirect

Maximal subgroups

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

subgroupindex
#117224482
C2×C4×#10882
M16×#2722
C3×#3399