Finite Group Databasebeta

← #13136925

Subgroups — #13136925

Order 1728.

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

normal subgroups only

This group has 271 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#11722445C211yesmaximal, characteristic
2C2×#88626C211yesmaximal, characteristic
3M16×#272C211yesmaximal, characteristic
4C2×C4×#272C2211yescharacteristic
5#88631—21no—
6C2×#6188—21no—
7#88627C412yes—
8#88626—21no—
9#88626—21no—
10#88626C412yes—
11M16×H(3,𝔽3)C411yescharacteristic
12#88538C411yescharacteristic
13#88527—21no—
14C8×#272—21no—
15C3×D3×M16—62no—
16C22×#272Q811yescharacteristic
17C2×C4×H(3,𝔽3)C2×C411yescharacteristic
18C2×#1084D411yescharacteristic
19C4×#272C2×C411yescharacteristic
20C4×#272C2×C411yescharacteristic
21C4×#272D411yescharacteristic
22C4×#272D411yescharacteristic
23#6188—22no—
24#6188—22no—
25C8×H(3,𝔽3)—21no—
26#6180—21no—
27C3×#341—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)C4⋊C411yescharacteristic
34C2×#272—22no—
35C2×#272C4⋊C411yescharacteristic
36C2×#272C4⋊C411yescharacteristic
37C4×H(3,𝔽3)C4211yescharacteristic
38C4×H(3,𝔽3)C22⋊C411yescharacteristic
39#1084C22⋊C411yescharacteristic
40#1084C22⋊C411yescharacteristic
41C2×C3×M16—91no—
42C2×C3×M16—91no—
43C22×C3×C8—91no—
44D3×M16—62no—
45C22×C3×D3—62no—
46C2×C32×C4—22no—
47C2×C3×Dic3—62no—
48C3×C4×D3—62no—
49C3×C4×D3—62no—
50C3×C4×D3—62no—
51C3×C4×D3—62no—
52C32×C8—24no—
53C3×C3⋊C8—64no—
54#341—91noSylow
55C2×H(3,𝔽3)—21no—
56C2×H(3,𝔽3)#9511yescharacteristic
57#272—21no—
58#272#10812yes—
59C22×C3×C4—91no—
60C2×C4×D3—62no—
61C3×M16—62no—
62C3×M16—92no—
63C3×M16—91no—
64C3×M16—181no—
65C3×M16—181no—
66C3×M16#15711yescharacteristic
67C2×C3×C8—92no—
68C2×C3×C8—181no—
69C2×C3×C8—181no—
70C2×C3×C8—181no—
71C2×C3×C8—181no—
72#208—62no—
73ΓL(1,𝔽25)—64no—
74D3×C8—64no—
75C22×C32—22no—
76C2×C3×D3—62no—
77C2×C3×D3—62no—
78C2×C3×D3—124no—
79C32×C4—22no—
80C32×C4—22no—
81C3×Dic3—62no—
82C3×Dic3—62no—
83C2×M16—91no—
84C2×M16—91no—
85C22×C8—91no—
86H(3,𝔽3)#34111yescharacteristic, complemented, Sylow
87C23×C3—91no—
88C22×D3—62no—
89C2×C3×C4—62no—
90C2×C3×C4—91no—
91C2×C3×C4—91no—
92C2×C3×C4—91no—
93C2×C3×C4—91no—
94C2×C3×C4—91no—
95C2×C3×C4C2×#15711yescharacteristic
96C2×Dic3—62no—
97C4×D3—62no—
98C4×D3—62no—
99C4×D3—62no—
100C4×D3—62no—
101C3×C8—21no—
102C3×C8—64no—
103C3×C8—182no—
104C3×C8—182no—
105C3×C8—181no—
106C3⋊C8—64no—
107C2×C32—22no—
108C2×C32—42no—
109C3×D3—64no—
110C3×D3—122no—
111C22×C4—91no—
112M16—92no—
113M16—91no—
114M16—181no—
115M16—181no—
116M16#108811yescharacteristic
117C2×C8—92no—
118C2×C8—181no—
119C2×C8—181no—
120C2×C8—181no—
121C2×C8—181no—
122C22×C3—62no—
123C22×C3—91no—
124C22×C3—91no—
125C22×C3—182no—
126C22×C3C4⋊#15711yescharacteristic
127C2×D3—62no—
128C2×D3—62no—
129C2×D3—124no—
130C3×C4—62no—
131C3×C4—62no—
132C3×C4—91no—
133C3×C4—91no—
134C3×C4C4×#15711yescharacteristic
135C3×C4C22⋊#15711yescharacteristic
136Dic3—62no—
137Dic3—62no—
138C32—22no—
139C23—91no—
140C2×C4—91no—
141C2×C4—91no—
142C2×C4—91no—
143C2×C4—91no—
144C2×C4—91no—
145C2×C4C2×#108811yescharacteristic
146C8—21no—
147C8—182no—
148C8—182no—
149C8—181no—
150C2×C3—21no—
151C2×C3—62no—
152C2×C3—92no—
153C2×C3—122no—
154C2×C3—181no—
155C2×C3#6378111yescharacteristic
156D3—64no—
157D3—122no—
158C22—91no—
159C22—91no—
160C22—182no—
161C22C4⋊#108811yescharacteristic
162C4—91no—
163C4—91no—
164C4C4×#108811yescharacteristic
165C4C22⋊#108811yescharacteristic
166C3—62no—
167C3#1059019011yescharacteristic
168C2—21no—
169C2—92no—
170C2—181no—
171C2#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×#272Q8
cosocleC2×C4×#272C22
hypercenterC3×M16#157
hypocenterH(3,𝔽3)#341
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
86. H(3,𝔽3)#341semidirect

Maximal subgroups

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

subgroupindex
#117224452
C2×#886262
M16×#2722
C3×#3419