Finite Group Databasebeta

← #13157228

Subgroups — #13157228

Order 1728.

496 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 33 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#11723099C212yesmaximal, complemented
3#10590211—31nomaximal
4C2×C33×D4C411yescharacteristic, complemented
5C2×C3×#638—22no—
6C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
7#88994—22no—
8C3×C4×C3⋊Dic3C411yescharacteristic, complemented
9#64149D311yescharacteristic
10C3×#940—62no—
11#63774—32no—
12C23×C33—21no—
13C22×C3×C3⋊D3D412yes—
14C2×C3⋊#157—42no—
15C33×D4—41no—
16C2×C33×C4C2×C411yescharacteristic
17C3×#638—44no—
18C2×C3×C3⋊Dic3—21no—
19C3×C4⋊(C3⋊D3)—41no—
20#4545—91nomaximal
21C2×C32×D4—22no—
22C2×C32×D4—42no—
23C2×C32×D4Dic311yescharacteristic, complemented
24C2×#638—22no—
25C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
26C2×C3×#67—68no—
27C2×C3×D12—62no—
28C22⋊#157—62no—
29C4×C3⋊Dic3Dic311yescharacteristic, complemented
30C3×C4×Dic3—62no—
31C22×C33—42no—
32C22×C33—41no—
33C22×C33C22⋊C411yescharacteristic
34C2×C3×C3⋊D3—22no—
35C2×C3×C3⋊D3—42no—
36C3⋊#157—82no—
37C33×C4—21no—
38C3×C3⋊Dic3—22no—
39C3×#127—91no—
40#940—62no—
41#834—92no—
42C23×C32—21no—
43C23×C32—24no—
44C23×C32—44no—
45C22×C3⋊D3#6712yes—
46C22×C3×D3—64no—
47C2×#157—122no—
48C32×D4—41no—
49C32×D4—44no—
50C32×D4—48no—
51C2×C32×C4—22no—
52C2×C32×C4—42no—
53C2×C32×C4C2×Dic311yescharacteristic
54#638—44no—
55C2×C3⋊Dic3—21no—
56C4⋊(C3⋊D3)—41no—
57C3×#67—1216no—
58C2×C3×Dic3—64no—
59C3×D12—124no—
60#353—271noSylow
61C2×C33—21no—
62C2×C33—81no—
63C2×C33#9911yescharacteristic
64C3×C3⋊D3—42no—
65C2×C3×D4—22no—
66C2×C3×D4—42no—
67C2×C3×D4—91no—
68C2×C3×D4—182no—
69C2×C3×D4#15711yescharacteristic, complemented
70C2×#67—68no—
71C2×D12—62no—
72C3×C42—91no—
73#217—182no—
74C4×Dic3—62no—
75C22×C32—22no—
76C22×C32—42no—
77C22×C32—42no—
78C22×C32—44no—
79C22×C32—48no—
80C22×C32—41no—
81C22×C32—48no—
82C22×C32—88no—
83C22×C32#21711yescharacteristic
84C2×C3⋊D3—22no—
85C2×C3⋊D3—42no—
86C2×C3×D3—124no—
87C2×C3×D3—128no—
88#157—242no—
89C32×C4—21no—
90C32×C4—24no—
91C32×C4—44no—
92C3⋊Dic3—22no—
93C3×Dic3—68no—
94#127—91no—
95#99—272no—
96C33#35311yescharacteristic, complemented, Sylow
97C23×C3—24no—
98C23×C3—21no—
99C23×C3—44no—
100C23×C3—92no—
101C22×D3—64no—
102C3×D4—41no—
103C3×D4—44no—
104C3×D4—48no—
105C3×D4—361no—
106C3×D4—364no—
107C2×C3×C4—22no—
108C2×C3×C4—42no—
109C2×C3×C4—181no—
110C2×C3×C4C2×#15711yescharacteristic
111#67—1216no—
112C2×Dic3—64no—
113C2×Dic3—362no—
114D12—124no—
115C2×C32—21no—
116C2×C32—24no—
117C2×C32—22no—
118C2×C32—44no—
119C2×C32—42no—
120C2×C32—84no—
121C2×C32—88no—
122C2×C32—81no—
123C2×C32#83411yescharacteristic
124C3⋊D3—42no—
125C3×D3—244no—
126C2×D4—91no—
127C2×D4—182no—
128C2×D4C3⋊#15711yescharacteristic, complemented
129C22⋊C4—542no—
130C42—91no—
131C22×C3—22no—
132C22×C3—48no—
133C22×C3—44no—
134C22×C3—42no—
135C22×C3—48no—
136C22×C3—41no—
137C22×C3—42no—
138C22×C3—88no—
139C22×C3—182no—
140C22×C3—362no—
141C22×C3C22⋊#15711yescharacteristic
142C2×D3—124no—
143C2×D3—128no—
144C3×C4—24no—
145C3×C4—21no—
146C3×C4—44no—
147C3×C4—182no—
148Dic3—68no—
149Dic3—722no—
150C32—22no—
151C32—42no—
152C32#454511yescharacteristic, complemented
153C23—21no—
154C23—92no—
155D4—41no—
156D4—361no—
157D4—364no—
158C2×C4—181no—
159C2×C4—1082no—
160C2×C4C2×C3⋊#15711yescharacteristic
161C2×C3—24no—
162C2×C3—21no—
163C2×C3—22no—
164C2×C3—44no—
165C2×C3—42no—
166C2×C3—84no—
167C2×C3—88no—
168C2×C3—81no—
169C2×C3—362no—
170C2×C3#6378611yescharacteristic
171D3—244no—
172C22—42no—
173C22—41no—
174C22—182no—
175C22—362no—
176C22#8899411yescharacteristic
177C4—21no—
178C4—182no—
179C4—2162no—
180C3—22no—
181C3—42no—
182C3#1059021111yescharacteristic, complemented
183C2—21no—
184C2—81no—
185C2—362no—
186C2#1172311111yescharacteristic
subgroupG/H
centerC2#11723111
commutatorC2×C33×C4C2×C4
FrattiniC2×C4C2×C3⋊#157
FittingC2×C33×D4C4
socleC2×C33#99
prime radicalC3×#64149C2
cosocleC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×D4C3⋊#157
hypocenterC33#353
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723099C2semidirect
4. C2×C33×D4C4semidirect
8. C3×C4×C3⋊Dic3C4semidirect
23. C2×C32×D4Dic3semidirect
29. C4×C3⋊Dic3Dic3semidirect
69. C2×C3×D4#157semidirect
96. C33#353semidirect
128. C2×D4C3⋊#157semidirect
152. C32#4545semidirect
182. C3#10590211semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
#117230992
#105902113
#45459