Finite Group Databasebeta

← #13157233

Subgroups — #13157233

Order 1728.

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

normal subgroups only

This group has 368 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
1C3×#64156C211yesmaximal, characteristic
2#11723102C212yesmaximal, complemented
3#10590216—31nomaximal
4C2×C33×Q8C411yescharacteristic
5C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
6C2×C3⋊#622—22no—
7C3×#3752—22no—
8C3×C4×C3⋊Dic3C2211yescharacteristic
9#64156D311yescharacteristic
10C3×#947—62no—
11#63777—32no—
12C22×C3×C3⋊D3—21no—
13C33×Q8—41no—
14C2×C33×C4—21no—
15C2×C33×C4C2×C411yescharacteristic
16C2×C3×C3⋊Dic3D412yes—
17C3×C4⋊(C3⋊D3)—41no—
18C3⋊#622—24no—
19#4540—91nomaximal
20C2×C32×Q8—22no—
21C2×C32×Q8—42no—
22C2×C32×Q8Dic311yescharacteristic
23C2×C4⋊(C3⋊D3)Dic311yescharacteristic
24C2×C3×D12—62no—
25C2×#622—62no—
26#3752—22no—
27C4×C3⋊Dic3C2×D311yescharacteristic
28C3×(C2×D3)⋊C4—68no—
29C3×C4×Dic3—62no—
30C22×C33C22⋊C411yescharacteristic
31C2×C3×C3⋊D3—41no—
32C2×C3×C3⋊D3—42no—
33C33×C4—21no—
34C33×C4—41no—
35C3×C3⋊Dic3—22no—
36C3×#124—91no—
37#947—62no—
38#803—92no—
39C22×C3⋊D3—21no—
40C22×C3×D3—64no—
41C32×Q8—41no—
42C32×Q8—44no—
43C32×Q8—48no—
44C2×C32×C4—21no—
45C2×C32×C4—24no—
46C2×C32×C4—22no—
47C2×C32×C4—44no—
48C2×C32×C4—42no—
49C2×C32×C4C2×Dic311yescharacteristic
50C2×C3⋊Dic3#6712yes—
51C4⋊(C3⋊D3)—41no—
52C2×C3×Dic3—64no—
53C3×D12—124no—
54#622—64no—
55#329—271noSylow
56C2×C33#10011yescharacteristic
57C2×C33#10412yes—
58C3×C3⋊D3—81no—
59C2×C3×Q8—22no—
60C2×C3×Q8—42no—
61C2×C3×Q8#15711yescharacteristic
62C2×C3×D4—91no—
63C2×D12—62no—
64C3×C22⋊C4—182no—
65C3×C42—91no—
66(C2×D3)⋊C4—68no—
67C4×Dic3—62no—
68C2×C3⋊C8—182no—
69C22×C32—22no—
70C22×C32—42no—
71C22×C32#21711yescharacteristic
72C2×C3⋊D3—41no—
73C2×C3⋊D3—42no—
74C2×C3×D3—124no—
75C2×C3×D3—128no—
76C32×C4—21no—
77C32×C4—24no—
78C32×C4—44no—
79C32×C4—44no—
80C32×C4—48no—
81C32×C4—41no—
82C3⋊Dic3—22no—
83C3×Dic3—124no—
84#124—91no—
85#97—272no—
86C33#32911yescharacteristic, complemented, Sylow
87C23×C3—181no—
88C22×D3—64no—
89C3×Q8—41no—
90C3×Q8—44no—
91C3×Q8—48no—
92C3×D4—361no—
93C2×C3×C4—24no—
94C2×C3×C4—21no—
95C2×C3×C4—22no—
96C2×C3×C4—44no—
97C2×C3×C4—42no—
98C2×C3×C4—92no—
99C2×C3×C4C2×#15711yescharacteristic
100C2×Dic3—64no—
101D12—124no—
102C3⋊C8—184no—
103C2×C32—24no—
104C2×C32—22no—
105C2×C32—42no—
106C2×C32—44no—
107C2×C32#83311yescharacteristic
108C2×C32#83712yes—
109C3⋊D3—81no—
110C3×D3—244no—
111C2×Q8C3⋊#15711yescharacteristic
112C2×D4—91no—
113C2×C8—542no—
114C22⋊C4—182no—
115C42—91no—
116C22×C3—22no—
117C22×C3—42no—
118C22×C3—362no—
119C22×C3—361no—
120C22×C3C22⋊#15711yescharacteristic
121C2×D3—124no—
122C2×D3—128no—
123C3×C4—24no—
124C3×C4—21no—
125C3×C4—44no—
126C3×C4—44no—
127C3×C4—48no—
128C3×C4—41no—
129C3×C4—182no—
130Dic3—124no—
131C32—22no—
132C32—42no—
133C32#454011yescharacteristic, complemented
134C23—181no—
135Q8—41no—
136D4—361no—
137C2×C4—21no—
138C2×C4—92no—
139C2×C4C2×C3⋊#15711yescharacteristic
140C8—544no—
141C2×C3—24no—
142C2×C3—22no—
143C2×C3—44no—
144C2×C3—42no—
145C2×C3—721no—
146C2×C3#6378512yes—
147C2×C3#6378811yescharacteristic
148D3—244no—
149C22—362no—
150C22—361no—
151C22#8899411yescharacteristic
152C4—21no—
153C4—41no—
154C4—182no—
155C3—22no—
156C3—42no—
157C3#1059021611yescharacteristic, complemented
158C2—721no—
159C2#1172311012yes—
160C2#1172311311yescharacteristic
subgroupG/H
centerC22#88994
commutatorC2×C33×C4C2×C4
FrattiniC2×C4C2×C3⋊#157
FittingC2×C33×Q8C4
socleC22×C33C22⋊C4
prime radicalC2×C3×C4⋊(C3⋊D3)C4
cosocleC3×C4×C3⋊Dic3C22
hypercenterC2×Q8C3⋊#157
hypocenterC33#329
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11723102C2semidirect
86. C33#329semidirect
133. C32#4540semidirect
157. C3#10590216semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
#117231022
#105902163
#45409