Finite Group Databasebeta

← #13158825

Subgroups — #13158825

Order 1728.

820 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 164 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×#64083C211yesmaximal, characteristic
2C32×#870C211yesmaximal, characteristic, complemented
3#11722173C211yesmaximal, characteristic, complemented
4#11722033C214yesmaximal, complemented
5#10592766—31nomaximal
6#10590279—34nomaximal
7C2×C3×C4⋊(C3⋊D3)—22no—
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C33×C42C2211yescharacteristic
10C2×C3⋊(C3⋊(C3⋊C8))C2212yes—
11C32×C4⋊Dic3C2212yes—
12C32×Dic3⋊C4—22no—
13C32×#125—31no—
14#64083D311yescharacteristic
15C3×#874—34no—
16C3×#870D314yescomplemented
17#63636—34no—
18#63630—31no—
19#63630—34no—
20#63630—64no—
21#63562—316no—
22C22×C3×C3⋊D3—22no—
23C2×C33×C4D412yes—
24C2×C33×C4C2311yescharacteristic
25C3×C4⋊(C3⋊D3)—24no—
26C3×C4⋊(C3⋊D3)—44no—
27C2×C32×Dic3—22no—
28C3⋊(C3⋊(C3⋊C8))—22no—
29#5071—91no—
30#5067—94no—
31C2×C4⋊(C3⋊D3)—22no—
32C2×C4⋊(C3⋊D3)C2×D312yes—
33C2×C3×D12—38no—
34C2×C3×D12—68no—
35C32×C4⋊C4—32no—
36C32×C4⋊C4—62no—
37C32×C42—24no—
38C32×C42C2×D314yes—
39C32×C42C2×D311yescharacteristic
40C2×C3⋊(C3⋊C8)—32no—
41C2×C3⋊(C3⋊C8)—38no—
42C2×C3⋊(C3⋊C8)—68no—
43C3×C4⋊Dic3C2×D318yes—
44C3×Dic3⋊C4—28no—
45C22×C33C2×D411yescharacteristic
46C2×C3×C3⋊D3—42no—
47C2×C3×C3⋊D3—44no—
48C33×C4—22no—
49C33×C4#4112yes—
50C32×Dic3—42no—
51C3×#127—91no—
52C3×#125—34no—
53#874—34no—
54#870C3⋊D311yescharacteristic, complemented
55#832—94no—
56#809—916no—
57#803—94no—
58#803—91no—
59#803—184no—
60C22×C3⋊D3—22no—
61C22×C3×D3—68no—
62C2×C32×C4—28no—
63C2×C32×C4—24no—
64C2×C32×C4—62no—
65C2×C32×C4#6718yes—
66C2×C32×C4#6712yes—
67C2×C32×C4C22×D314yes—
68C2×C32×C4C22×D311yescharacteristic
69C4⋊(C3⋊D3)—24no—
70C4⋊(C3⋊D3)—44no—
71C2×C3×Dic3—28no—
72C3×D12—616no—
73C3×D12—1216no—
74C3⋊(C3⋊C8)—616no—
75C3⋊(C3⋊C8)—62no—
76C3⋊(C3⋊C8)—68no—
77#490—271noSylow
78C2×C33#12411yescharacteristic
79C2×C33AGL(1,ℤ/8)12yes—
80C3×C3⋊D3—82no—
81C2×C3×D4—92no—
82C2×C3×D4—182no—
83C2×D12—38no—
84C2×D12—68no—
85C3×C4⋊C4—38no—
86C3×C4⋊C4—68no—
87C3×C42—24no—
88C3×C42D3214yes—
89C3×C42C2×C3⋊D311yescharacteristic
90C4⋊Dic3C2×C3⋊D312yes—
91Dic3⋊C4—22no—
92C2×C3⋊C8—98no—
93C2×C3⋊C8—92no—
94C2×C3⋊C8—188no—
95C22×C32—24no—
96C22×C32C2×#6714yes—
97C22×C32C2×#6711yescharacteristic
98C2×C3⋊D3—44no—
99C2×C3⋊D3—42no—
100C2×C3×D3—128no—
101C2×C3×D3—1216no—
102C32×C4—22no—
103C32×C4—28no—
104C32×C4—28no—
105C32×C4—48no—
106C32×C4—122no—
107C32×C4#23712yes—
108C32×C4#23918yes—
109C3×Dic3—48no—
110#127—91no—
111#125—31no—
112#102—274no—
113#97—271no—
114C33#49011yescharacteristic, complemented, Sylow
115C23×C3—182no—
116C22×D3—68no—
117C3×D4—184no—
118C3×D4—364no—
119C2×C3×C4—28no—
120C2×C3×C4—24no—
121C2×C3×C4—68no—
122C2×C3×C4#62518yes—
123C2×C3×C4#63812yes—
124C2×C3×C4C2×D3214yes—
125C2×C3×C4C22×C3⋊D311yescharacteristic
126C2×Dic3—22no—
127D12—616no—
128D12—1216no—
129C3⋊C8—1816no—
130C3⋊C8—188no—
131C3⋊C8—182no—
132C2×C32—28no—
133C2×C32—24no—
134C2×C32#93212yes—
135C2×C32#93611yescharacteristic
136C2×C32#94714yes—
137C2×C32#94918yes—
138C3⋊D3—82no—
139C3×D3—248no—
140C2×D4—92no—
141C2×D4—182no—
142C2×C8—272no—
143C4⋊C4—32no—
144C4⋊C4—62no—
145C42D3×C3⋊D311yescharacteristic
146C22×C3—24no—
147C22×C3—362no—
148C22×C3—364no—
149C22×C3C2×#62514yes—
150C22×C3C2×#63811yescharacteristic
151C2×D3—1216no—
152C2×D3—128no—
153C3×C4—28no—
154C3×C4—22no—
155C3×C4—28no—
156C3×C4—48no—
157C3×C4—128no—
158C3×C4#379618yes—
159C3×C4#383212yes—
160Dic3—42no—
161C32—24no—
162C32#506714yescomplemented
163C32#507111yescharacteristic, complemented
164C23—182no—
165D4—184no—
166D4—364no—
167C2×C4—62no—
168C2×C4#629012yes—
169C2×C4C2×D3×C3⋊D311yescharacteristic
170C8—542no—
171C2×C3—28no—
172C2×C3—24no—
173C2×C3—722no—
174C2×C3#6382318yes—
175C2×C3#6386314yes—
176C2×C3#6415611yescharacteristic
177C2×C3#6415812yes—
178D3—248no—
179C22—364no—
180C22—362no—
181C22C2×#629011yescharacteristic
182C4—22no—
183C4—122no—
184C4#8901712yes—
185C3—24no—
186C3#1059027914yescomplemented
187C3#1059276611yescharacteristic, complemented
188C2—722no—
189C2#1172336512yes—
190C2#1172343611yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC3×#64083C2
hypercenterC42D3×C3⋊D3
hypocenterC33#490
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C32×#870C2semidirect
3. #11722173C2semidirect
4. #11722033C2semidirect
16. C3×#870D3semidirect
54. #870C3⋊D3semidirect
114. C33#490semidirect
162. C32#5067semidirect
163. C32#5071semidirect
186. C3#10590279semidirect
187. C3#10592766semidirect

Maximal subgroups

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

subgroupindex
C3×#640832
C32×#8702
#117221732
#117220332
#105927663
#105902793