Finite Group Databasebeta

← #13154741

Subgroups — #13154741

Order 1728.

272 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 34 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#11723339C211yesmaximal, characteristic
2#11723096C211yesmaximal, characteristic, complemented
3C3×#63764C211yesmaximal, characteristic, complemented
4#10590194—31nomaximal
5#89014—21no—
6#88989—21no—
7#88853C411yescharacteristic
8C3×C8×C3⋊D3C2211yescharacteristic
9C32×D24C411yescharacteristic
10#88794—21no—
11C3×#3708—21no—
12#64121—31no—
13#63807—62no—
14#63771—31no—
15#63764D311yescharacteristic, complemented
16#6308—21no—
17C3×C4×C3⋊D3D411yescharacteristic
18C32×D12—21no—
19#6290—41no—
20D3×C3⋊Dic3—41no—
21C3⋊#622—41no—
22C33×C8C2×C411yescharacteristic
23C3×C3⋊(C3⋊C8)D411yescharacteristic
24#4510—91nomaximal
25#3830—61no—
26#3795—64no—
27#3788—61no—
28C32×D8—31no—
29#3754—61no—
30#3746—31no—
31#3746—62no—
32#3746—122no—
33C8×C3⋊D3C2×D311yescharacteristic
34C3×D24—22no—
35C3×D3×C8—62no—
36#3716—64no—
37#3708—21no—
38C2×C3×C3⋊D3SD1611yescharacteristic
39C2×C32×D3—41no—
40C33×C4C22⋊C411yescharacteristic
41C3⋊(C3⋊Dic3)—41no—
42C3×C3⋊Dic3D811yescharacteristic
43#912—182no—
44#904—91no—
45C2×C3×C16—91no—
46#819—91no—
47C32×D4—61no—
48#638—121no—
49C2×C3⋊Dic3—121no—
50C4×C3⋊D3D1211yescharacteristic
51#634—61no—
52#634—64no—
53#634—124no—
54C3×D12—24no—
55C3×C4×D3—62no—
56#625—124no—
57D3×Dic3—124no—
58#622—121no—
59C32×C8—22no—
60C32×C8—42no—
61C32×C8C4×D311yescharacteristic
62C3⋊(C3⋊C8)#6711yescharacteristic
63C3×C3⋊C8—62no—
64#359—271noSylow
65C2×C33#10211yescharacteristic
66C3×C3⋊D3—21no—
67C32×D3—81no—
68#237—184no—
69#235—181no—
70C3×D8—62no—
71C2×C3×C8—91no—
72#214—184no—
73#208—181no—
74Dic12—91no—
75Dic12—182no—
76Dic12—362no—
77D24#15711yescharacteristic
78#204—181no—
79D3×C8—62no—
80C3×C16—181no—
81C22×C32—121no—
82C2×C3⋊D3#20411yescharacteristic
83C2×C3×D3—44no—
84C2×C3×D3—62no—
85C32×C4—22no—
86C32×C4—42no—
87C32×C4(C2×D3)⋊C411yescharacteristic
88C3⋊Dic3—121no—
89C3⋊Dic3—124no—
90C3⋊Dic3—244no—
91C3⋊Dic3D2411yescharacteristic
92C3×Dic3—62no—
93#135—271no—
94C2×C16—91no—
95#108—271no—
96C33#35911yescharacteristic, complemented, Sylow
97C3×D4—64no—
98C2×C3×C4—91no—
99#67—361no—
100#67—364no—
101C2×Dic3—364no—
102D12—21no—
103C4×D3—62no—
104C4×D3—361no—
105Dic6—184no—
106Dic6—181no—
107Dic6—364no—
108C3×C8—22no—
109C3×C8—42no—
110C3×C8—91no—
111C3×C8C2×#15711yescharacteristic
112C3⋊C8—62no—
113C3⋊C8—361no—
114C2×C32—22no—
115C2×C32—42no—
116C2×C32—241no—
117C2×C32#82111yescharacteristic
118C3⋊D3—21no—
119C3×D3—84no—
120C3×D3—122no—
121#41—541no—
122Q16—271no—
123SD16—541no—
124D8—31no—
125M16—541no—
126C2×C8—91no—
127C16—181no—
128C22×C3—91no—
129C22×C3—124no—
130C2×D3—41no—
131C2×D3—62no—
132C3×C4—22no—
133C3×C4—42no—
134C3×C4—91no—
135C3×C4C22⋊#15711yescharacteristic
136Dic3—62no—
137Dic3—364no—
138Dic3—361no—
139Dic3—724no—
140C32—22no—
141C32—42no—
142C32#451011yescharacteristic, complemented
143Q8—541no—
144D4—61no—
145D4—1081no—
146C2×C4—91no—
147C2×C4—1081no—
148C8—91no—
149C8—1081no—
150C8D3×#15711yescharacteristic
151C2×C3—22no—
152C2×C3—42no—
153C2×C3—181no—
154C2×C3—244no—
155C2×C3#6378211yescharacteristic
156D3—81no—
157D3—122no—
158C22—91no—
159C22—121no—
160C4—91no—
161C4—1081no—
162C4(C2×D3)⋊#15711yescharacteristic
163C3—22no—
164C3—42no—
165C3#1059019411yescharacteristic, complemented
166C2—181no—
167C2—241no—
168C2#1172270711yescharacteristic
subgroupG/H
centerC2#11722707
commutator, FittingC33×C8C2×C4
Frattini, hypercenterC8D3×#157
socleC2×C33#102
prime radical#11723339C2
cosocleC3×C8×C3⋊D3C22
hypocenterC33#359
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11723096C2semidirect
3. C3×#63764C2semidirect
15. #63764D3semidirect
96. C33#359semidirect
142. C32#4510semidirect
165. C3#10590194semidirect

Maximal subgroups

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

subgroupindex
#117233392
#117230962
C3×#637642
#105901943
#45109