Finite Group Databasebeta

← #13157231

Subgroups — #13157231

Order 1728.

492 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 49 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×#64149C211yesmaximal, characteristic
2#11723104C212yesmaximal, complemented
3#10590214—31nomaximal
4C2×C33×D4C411yescharacteristic
5C2×C3×#638—22no—
6C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
7C2×C3⋊#622—22no—
8C3×C4×C3⋊Dic3C2211yescharacteristic
9#64149D311yescharacteristic
10C3×#940—62no—
11#63779—32no—
12C23×C33—21no—
13C22×C3×C3⋊D3—21no—
14C33×D4—41no—
15C2×C33×C4C2×C411yescharacteristic
16C3×#638—42no—
17C3×#638—42no—
18C2×C3×C3⋊Dic3D412yes—
19C3×C4⋊(C3⋊D3)—41no—
20C3⋊#622—42no—
21#4544—91nomaximal
22C2×C32×D4—22no—
23C2×C32×D4—42no—
24C2×C32×D4Dic311yescharacteristic
25C2×#638—22no—
26C2×C4⋊(C3⋊D3)Dic311yescharacteristic
27C2×C3×#67—68no—
28C2×C3×D12—62no—
29C2×#622—62no—
30C4×C3⋊Dic3C2×D311yescharacteristic
31C3×C4×Dic3—62no—
32C22×C33—41no—
33C22×C33—42no—
34C22×C33C22⋊C411yescharacteristic
35C2×C3×C3⋊D3—41no—
36C2×C3×C3⋊D3—42no—
37C33×C4—21no—
38C3×C3⋊Dic3D812yes—
39C3×C3⋊Dic3SD1612yes—
40C3×#127—91no—
41#940—62no—
42C4⋊(C3⋊C8)—92no—
43C23×C32—21no—
44C23×C32—24no—
45C23×C32—44no—
46C22×C3⋊D3—21no—
47C22×C3×D3—64no—
48C32×D4—41no—
49C32×D4—44no—
50C32×D4—48no—
51C2×C32×C4—22no—
52C2×C32×C4—42no—
53C2×C32×C4C2×Dic311yescharacteristic
54#638—42no—
55#638—42no—
56C2×C3⋊Dic3#6712yes—
57C4⋊(C3⋊D3)—41no—
58C3×#67—128no—
59C3×#67—128no—
60C2×C3×Dic3—64no—
61C3×D12—124no—
62#622—122no—
63#331—271noSylow
64C2×C33—81no—
65C2×C33#10011yescharacteristic
66C2×C33#10212yes—
67C3×C3⋊D3—81no—
68C2×C3×D4—22no—
69C2×C3×D4—42no—
70C2×C3×D4—91no—
71C2×C3×D4—182no—
72C2×C3×D4#15711yescharacteristic
73C2×#67—68no—
74C2×D12—62no—
75C3×C42—91no—
76C4×Dic3—62no—
77C2×C3⋊C8—182no—
78C22×C32—22no—
79C22×C32—42no—
80C22×C32—44no—
81C22×C32—48no—
82C22×C32—42no—
83C22×C32—41no—
84C22×C32—48no—
85C22×C32—88no—
86C22×C32#21711yescharacteristic
87C2×C3⋊D3—42no—
88C2×C3⋊D3—41no—
89C2×C3×D3—124no—
90C2×C3×D3—128no—
91C32×C4—21no—
92C32×C4—24no—
93C32×C4—44no—
94C3⋊Dic3#21312yes—
95C3⋊Dic3#21412yes—
96C3×Dic3—64no—
97C3×Dic3—64no—
98#127—91no—
99C4⋊C8—272no—
100C33#33111yescharacteristic, complemented, Sylow
101C23×C3—24no—
102C23×C3—21no—
103C23×C3—44no—
104C23×C3—181no—
105C22×D3—64no—
106C3×D4—41no—
107C3×D4—44no—
108C3×D4—48no—
109C3×D4—361no—
110C3×D4—362no—
111C3×D4—362no—
112C2×C3×C4—22no—
113C2×C3×C4—42no—
114C2×C3×C4—92no—
115C2×C3×C4C2×#15711yescharacteristic
116#67—128no—
117#67—128no—
118C2×Dic3—64no—
119D12—124no—
120C3⋊C8—362no—
121C2×C32—24no—
122C2×C32—22no—
123C2×C32—42no—
124C2×C32—44no—
125C2×C32—84no—
126C2×C32—88no—
127C2×C32—81no—
128C2×C32#83212yes—
129C2×C32#83311yescharacteristic
130C3⋊D3—81no—
131C3×D3—244no—
132C2×D4—91no—
133C2×D4—182no—
134C2×D4C3⋊#15711yescharacteristic
135C2×C8—542no—
136C42—91no—
137C22×C3—22no—
138C22×C3—48no—
139C22×C3—44no—
140C22×C3—48no—
141C22×C3—42no—
142C22×C3—41no—
143C22×C3—42no—
144C22×C3—88no—
145C22×C3—362no—
146C22×C3—361no—
147C22×C3C22⋊#15711yescharacteristic
148C2×D3—124no—
149C2×D3—128no—
150C3×C4—24no—
151C3×C4—21no—
152C3×C4—44no—
153C3×C4—92no—
154C3×C4—92no—
155Dic3—64no—
156Dic3—64no—
157C32—22no—
158C32—42no—
159C32#454411yescharacteristic, complemented
160C23—21no—
161C23—181no—
162D4—41no—
163D4—361no—
164D4—362no—
165D4—362no—
166C2×C4—92no—
167C2×C4C2×C3⋊#15711yescharacteristic
168C8—1082no—
169C2×C3—24no—
170C2×C3—22no—
171C2×C3—44no—
172C2×C3—42no—
173C2×C3—84no—
174C2×C3—88no—
175C2×C3—81no—
176C2×C3—721no—
177C2×C3#6378212yes—
178C2×C3#6378811yescharacteristic
179D3—244no—
180C22—42no—
181C22—41no—
182C22—362no—
183C22—361no—
184C22#8899411yescharacteristic
185C4—21no—
186C4—92no—
187C4—92no—
188C3—22no—
189C3—42no—
190C3#1059021411yescharacteristic, complemented
191C2—81no—
192C2—721no—
193C2#1172310712yes—
194C2#1172311311yescharacteristic
subgroupG/H
centerC22#88994
commutatorC2×C33×C4C2×C4
FrattiniC2×C4C2×C3⋊#157
FittingC2×C33×D4C4
socleC22×C33C22⋊C4
prime radicalC3×#64149C2
cosocleC3×C4×C3⋊Dic3C22
hypercenterC2×D4C3⋊#157
hypocenterC33#331
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723104C2semidirect
100. C33#331semidirect
159. C32#4544semidirect
190. C3#10590214semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
#117231042
#105902143
#45449