Finite Group Databasebeta

← #13136924

Subgroups — #13136924

Order 1728.

271 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 41 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×#88627C211yesmaximal, characteristic
3M16×#272C211yesmaximal, characteristic
4C2×C4×#272C2211yescharacteristic
5C22⋊#1088C412yes—
6C2×#6188C412yescomplemented
7C4⋊#1088—21no—
8C4×#1088—21no—
9#88627—21no—
10#88627—21no—
11M16×H(3,𝔽3)C411yescharacteristic, complemented
12#88538C411yescharacteristic, complemented
13#88527—21no—
14C8×#272—21no—
15C3×D3×M16—62no—
16C22×#272C2×C411yescharacteristic
17C2×#1088—22no—
18C2×C4×H(3,𝔽3)C2×C411yescharacteristic
19C2×#1084C2×C411yescharacteristic
20C4×#272D411yescharacteristic
21C4×#272D411yescharacteristic
22C4×#272D411yescharacteristic
23C4×#272Q811yescharacteristic
24#6188—22no—
25C8×H(3,𝔽3)—21no—
26#6180—21no—
27C3×#344—91nomaximal
28C2×C3×C4×D3—62no—
29C32×M16—22no—
30C3×#208—62no—
31C3×ΓL(1,𝔽25)—64no—
32C3×D3×C8—64no—
33C22×H(3,𝔽3)C4211yescharacteristic
34C2×#272—41no—
35C2×#272C22⋊C411yescharacteristic
36C2×#272C4⋊C411yescharacteristic
37#1088—42no—
38C4×H(3,𝔽3)C22⋊C411yescharacteristic
39C4×H(3,𝔽3)C4⋊C411yescharacteristic
40#1084C22⋊C411yescharacteristic
41#1084C4⋊C411yescharacteristic
42C2×C3×M16—91no—
43C2×C3×M16—91no—
44C3×#117—91no—
45D3×M16—62no—
46C22×C3×D3—62no—
47C2×C32×C4—22no—
48C2×C3×Dic3—62no—
49C3×C4×D3—62no—
50C3×C4×D3—62no—
51C3×C4×D3—62no—
52C3×C4×D3—62no—
53C32×C8—24no—
54C3×C3⋊C8—64no—
55#344—91noSylow
56C2×H(3,𝔽3)—21no—
57C2×H(3,𝔽3)#9511yescharacteristic
58#272—21no—
59#272—21no—
60C22×C3×C4—91no—
61C2×C4×D3—62no—
62C3×M16—62no—
63C3×M16—91no—
64C3×M16—181no—
65C3×M16—181no—
66C3×M16—181no—
67C3×M16#15711yescharacteristic
68C2×C3×C8—92no—
69C2×C3×C8—181no—
70C3×C4⋊C4—181no—
71C3×C22⋊C4—92no—
72C3×C42—181no—
73#208—62no—
74ΓL(1,𝔽25)—64no—
75D3×C8—64no—
76C22×C32—22no—
77C2×C3×D3—62no—
78C2×C3×D3—62no—
79C2×C3×D3—124no—
80C32×C4—22no—
81C32×C4—22no—
82C3×Dic3—62no—
83C3×Dic3—62no—
84C2×M16—91no—
85C2×M16—91no—
86#117—91no—
87H(3,𝔽3)#34411yescharacteristic, complemented, Sylow
88C23×C3—91no—
89C22×D3—62no—
90C2×C3×C4—62no—
91C2×C3×C4—91no—
92C2×C3×C4—91no—
93C2×C3×C4—91no—
94C2×C3×C4—91no—
95C2×C3×C4—91no—
96C2×C3×C4—182no—
97C2×C3×C4C2×#15711yescharacteristic
98C2×Dic3—62no—
99C4×D3—62no—
100C4×D3—62no—
101C4×D3—62no—
102C4×D3—62no—
103C3×C8—21no—
104C3×C8—64no—
105C3×C8—182no—
106C3×C8—181no—
107C3⋊C8—64no—
108C2×C32—22no—
109C2×C32—42no—
110C3×D3—64no—
111C3×D3—122no—
112C22×C4—91no—
113M16—91no—
114M16—181no—
115M16—181no—
116M16—181no—
117M16#108811yescharacteristic, complemented
118C2×C8—92no—
119C2×C8—181no—
120C4⋊C4—181no—
121C22⋊C4—92no—
122C42—181no—
123C22×C3—62no—
124C22×C3—91no—
125C22×C3—91no—
126C22×C3—361no—
127C22×C3C4×#15711yescharacteristic
128C2×D3—62no—
129C2×D3—62no—
130C2×D3—124no—
131C3×C4—62no—
132C3×C4—62no—
133C3×C4—91no—
134C3×C4—91no—
135C3×C4—362no—
136C3×C4C4⋊#15711yescharacteristic
137C3×C4C22⋊#15711yescharacteristic
138Dic3—62no—
139Dic3—62no—
140C32—22no—
141C23—91no—
142C2×C4—91no—
143C2×C4—91no—
144C2×C4—91no—
145C2×C4—91no—
146C2×C4—91no—
147C2×C4—182no—
148C2×C4C2×#108811yescharacteristic
149C8—21no—
150C8—182no—
151C8—181no—
152C2×C3—21no—
153C2×C3—62no—
154C2×C3—122no—
155C2×C3—181no—
156C2×C3—181no—
157C2×C3#6378111yescharacteristic
158D3—64no—
159D3—122no—
160C22—91no—
161C22—91no—
162C22—361no—
163C22C4×#108811yescharacteristic
164C4—91no—
165C4—91no—
166C4—362no—
167C4C4⋊#108811yescharacteristic
168C4C22⋊#108811yescharacteristic
169C3—62no—
170C3#1059019111yescharacteristic
171C2—21no—
172C2—181no—
173C2—181no—
174C2#1172078711yescharacteristic
subgroupG/H
center, socleC2×C3#63781
commutatorC22×H(3,𝔽3)C42
FrattiniC2×C3×C4C2×#157
FittingM16×H(3,𝔽3)C4
prime radicalC22×#272C2×C4
cosocleC2×C4×#272C22
hypercenterC3×M16#157
hypocenterH(3,𝔽3)#344
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
6. C2×#6188C4semidirect
11. M16×H(3,𝔽3)C4semidirect
12. #88538C4semidirect
87. H(3,𝔽3)#344semidirect
117. M16#1088semidirect

Maximal subgroups

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

subgroupindex
#117224482
C2×#886272
M16×#2722
C3×#3449