Finite Group Databasebeta

← #13154743

Subgroups — #13154743

Order 1728.

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

normal subgroups only

This group has 276 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#11723090C211yesmaximal, characteristic, complemented
3C3×#63768C211yesmaximal, characteristic, complemented
4#10590192—31nomaximal
5#89014—21no—
6C3×C4⋊#157—21no—
7#88853C411yescharacteristic, complemented
8C3×C8×C3⋊D3C2211yescharacteristic
9C32×D24C411yescharacteristic, complemented
10#88794—21no—
11C3⋊#3708—21no—
12#64121—31no—
13#63807—62no—
14#63768D311yescharacteristic, complemented
15#63765—31no—
16C2×C3×#157—41no—
17#6308—21no—
18C3×C4×C3⋊D3D411yescharacteristic
19C32×D12—21no—
20#6290—41no—
21D3×C3⋊Dic3—41no—
22C33×C8C2×C411yescharacteristic
23C3×C3⋊(C3⋊C8)D411yescharacteristic
24#4488—91nomaximal
25#3830—61no—
26#3795—64no—
27C4⋊#157—21no—
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—61no—
38C2×C3×C3⋊D3D811yescharacteristic
39C2×C32×D3—41no—
40C3×#157—81no—
41C33×C4C22⋊C411yescharacteristic
42C3⋊(C3⋊Dic3)—41no—
43C3×C3⋊Dic3SD1611yescharacteristic
44#912—182no—
45#904—91no—
46C3×#106—91no—
47#812—91no—
48C2×#157—41no—
49C32×D4—61no—
50#638—121no—
51C2×C3⋊Dic3—121no—
52C4×C3⋊D3#6711yescharacteristic
53#634—61no—
54#634—64no—
55#634—124no—
56C3×D12—24no—
57C3×C4×D3—62no—
58#625—124no—
59D3×Dic3—124no—
60C32×C8—22no—
61C32×C8—42no—
62C32×C8C4×D311yescharacteristic
63C3⋊(C3⋊C8)D1211yescharacteristic
64C3×C3⋊C8—62no—
65#360—271noSylow
66C2×C33#10211yescharacteristic
67C3×C3⋊D3—21no—
68C32×D3—81no—
69#237—184no—
70#235—181no—
71C3×D8—62no—
72C2×C3×C8—91no—
73C3×C4⋊C4—181no—
74#214—184no—
75Dic12—91no—
76Dic12—182no—
77Dic12—362no—
78D24#15711yescharacteristic, complemented
79#204—181no—
80D3×C8—62no—
81C3⋊C16—181no—
82C22×C32—121no—
83C2×C3⋊D3#21311yescharacteristic
84C2×C3×D3—44no—
85C2×C3×D3—62no—
86#157—81no—
87C32×C4—22no—
88C32×C4—42no—
89C32×C4(C2×D3)⋊C411yescharacteristic
90C3⋊Dic3—121no—
91C3⋊Dic3—124no—
92C3⋊Dic3—244no—
93C3⋊Dic3#21511yescharacteristic
94C3×Dic3—62no—
95#135—271no—
96M32—271no—
97#106—91no—
98C33#36011yescharacteristic, complemented, Sylow
99C3×D4—64no—
100C2×C3×C4—91no—
101C2×C3×C4—361no—
102#67—361no—
103#67—364no—
104C2×Dic3—364no—
105D12—21no—
106C4×D3—62no—
107C4×D3—361no—
108Dic6—184no—
109Dic6—181no—
110Dic6—364no—
111C3×C8—22no—
112C3×C8—42no—
113C3×C8—91no—
114C3×C8C2×#15711yescharacteristic
115C3⋊C8—62no—
116C2×C32—22no—
117C2×C32—42no—
118C2×C32—241no—
119C2×C32#80911yescharacteristic
120C3⋊D3—21no—
121C3×D3—84no—
122C3×D3—122no—
123#41—541no—
124Q16—271no—
125SD16—541no—
126D8—31no—
127C2×C8—91no—
128C4⋊C4—181no—
129C16—541no—
130C22×C3—91no—
131C22×C3—124no—
132C2×D3—41no—
133C2×D3—62no—
134C3×C4—22no—
135C3×C4—42no—
136C3×C4—91no—
137C3×C4—721no—
138C3×C4C22⋊#15711yescharacteristic
139Dic3—62no—
140Dic3—364no—
141Dic3—361no—
142Dic3—724no—
143C32—22no—
144C32—42no—
145C32#448811yescharacteristic, complemented
146Q8—541no—
147D4—61no—
148D4—1081no—
149C2×C4—91no—
150C2×C4—361no—
151C2×C4—1081no—
152C8—91no—
153C8D3×#15711yescharacteristic
154C2×C3—22no—
155C2×C3—42no—
156C2×C3—181no—
157C2×C3—244no—
158C2×C3#6378211yescharacteristic
159D3—81no—
160D3—122no—
161C22—91no—
162C22—121no—
163C4—91no—
164C4—721no—
165C4—1081no—
166C4(C2×D3)⋊#15711yescharacteristic
167C3—22no—
168C3—42no—
169C3#1059019211yescharacteristic, complemented
170C2—181no—
171C2—241no—
172C2#1172270611yescharacteristic
subgroupG/H
centerC2#11722706
commutator, FittingC33×C8C2×C4
Frattini, hypercenterC8D3×#157
socleC2×C33#102
prime radical#11723339C2
cosocleC3×C8×C3⋊D3C22
hypocenterC33#360
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. #11723090C2semidirect
3. C3×#63768C2semidirect
7. #88853C4semidirect
9. C32×D24C4semidirect
14. #63768D3semidirect
78. D24#157semidirect
98. C33#360semidirect
145. C32#4488semidirect
169. C3#10590192semidirect

Maximal subgroups

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

subgroupindex
#117233392
#117230902
C3×#637682
#105901923
#44889