Finite Group Databasebeta

← #13158837

Subgroups — #13158837

Order 1728.

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

normal subgroups only

This group has 884 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
1#11723911C211yesmaximal, characteristic, complemented
2C32×#874C211yesmaximal, characteristic
3C3×#63630C211yesmaximal, characteristic, complemented
4#11722032C214yesmaximal, complemented
5#10592770—31nomaximal
6#10590275—34nomaximal
7C2×C32×D12—22no—
8C2×C32×D12C2212yes—
9C33×C42C2211yescharacteristic
10C4⋊(C3⋊(C3⋊Dic3))C2212yes—
11#88857—22no—
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×#127—31no—
14#64079—31no—
15#64079—34no—
16#64079—64no—
17C3×#874D314yes—
18#63659—34no—
19#63630D311yescharacteristic, complemented
20C3×#803—34no—
21#63563—316no—
22C22×C32×D3—22no—
23C2×C33×C4D412yes—
24C2×C33×C4C2311yescharacteristic
25C2×C3⋊(C3⋊Dic3)—22no—
26C32×D12—24no—
27C32×D12—44no—
28C3×C3⋊(C3⋊C8)—22no—
29#5071—94no—
30#4713—91no—
31C2×C32×D4—32no—
32C2×C32×D4—62no—
33C2×C3×D12—28no—
34C2×C3×D12C2×D318yes—
35C32×C42—24no—
36C32×C42C2×D311yescharacteristic
37C32×C42C2×D314yes—
38C4⋊(C3⋊Dic3)—32no—
39C4⋊(C3⋊Dic3)—38no—
40C4⋊(C3⋊Dic3)—68no—
41#3750—616no—
42#3750—62no—
43#3750—68no—
44C2×C3⋊(C3⋊C8)C2×D312yes—
45C2×C3×C3⋊C8—38no—
46C22×C33C2×D411yescharacteristic
47C2×C32×D3—42no—
48C2×C32×D3—44no—
49C33×C4—22no—
50C33×C4#4112yes—
51C3⋊(C3⋊Dic3)—42no—
52C3×#127—34no—
53#874C3⋊D311yescharacteristic
54#870—94no—
55#870—91no—
56#870—184no—
57C3×#97—91no—
58#832—916no—
59#821—94no—
60#803—34no—
61C23×C32—62no—
62C22×C3×D3—28no—
63C32×D4—64no—
64C32×D4—124no—
65C2×C32×C4—28no—
66C2×C32×C4—24no—
67C2×C32×C4D1212yes—
68C2×C32×C4#6718yes—
69C2×C32×C4C22×D314yes—
70C2×C32×C4C22×D311yescharacteristic
71C2×C3⋊Dic3—616no—
72C2×C3⋊Dic3—62no—
73C2×C3⋊Dic3—68no—
74C3×D12—216no—
75C3×D12—416no—
76C3⋊(C3⋊C8)—22no—
77C3×C3⋊C8—68no—
78#490—271noSylow
79C2×C33#12411yescharacteristic
80C2×C33AGL(1,ℤ/8)12yes—
81C32×D3—82no—
82C2×C3×D4—38no—
83C2×C3×D4—68no—
84C2×D12—22no—
85C2×D12C2×C3⋊D312yes—
86C2×C3×C8—92no—
87C3×C42—24no—
88C3×C42D3214yes—
89C3×C42C2×C3⋊D311yescharacteristic
90C4⋊Dic3—98no—
91C4⋊Dic3—92no—
92C4⋊Dic3—188no—
93Dic3⋊C4—1816no—
94Dic3⋊C4—188no—
95Dic3⋊C4—182no—
96C2×C3⋊C8—38no—
97C22×C32—24no—
98C22×C32—122no—
99C22×C32—124no—
100C22×C32C2×D1211yescharacteristic
101C22×C32C2×#6714yes—
102C2×C3×D3—416no—
103C2×C3×D3—48no—
104C32×C4—22no—
105C32×C4—28no—
106C32×C4—28no—
107C32×C4—48no—
108C32×C4#23512yes—
109C32×C4#23718yes—
110C3⋊Dic3—1216no—
111C3⋊Dic3—122no—
112C3⋊Dic3—128no—
113#127—31no—
114#125—271no—
115#102—274no—
116#97—91no—
117C33#49011yescharacteristic, complemented, Sylow
118C23×C3—68no—
119C22×D3—22no—
120C3×D4—616no—
121C3×D4—1216no—
122C2×C3×C4—28no—
123C2×C3×C4—24no—
124C2×C3×C4#62618yes—
125C2×C3×C4#63812yes—
126C2×C3×C4C2×D3214yes—
127C2×C3×C4C22×C3⋊D311yescharacteristic
128C2×Dic3—1816no—
129C2×Dic3—188no—
130C2×Dic3—182no—
131D12—24no—
132D12—44no—
133C3×C8—182no—
134C3⋊C8—68no—
135C2×C32—28no—
136C2×C32—24no—
137C2×C32—242no—
138C2×C32#87511yescharacteristic
139C2×C32#90812yes—
140C2×C32#93218yes—
141C2×C32#93614yes—
142C3×D3—88no—
143C2×D4—32no—
144C2×D4—62no—
145C2×C8—92no—
146C4⋊C4—272no—
147C4⋊C4—542no—
148C42D3×C3⋊D311yescharacteristic
149C22×C3—24no—
150C22×C3—1216no—
151C22×C3—128no—
152C22×C3C2×#62614yes—
153C22×C3C2×#63811yescharacteristic
154C2×D3—44no—
155C2×D3—42no—
156C3×C4—28no—
157C3×C4—22no—
158C3×C4—28no—
159C3×C4—48no—
160C3×C4#379518yes—
161C3×C4#383012yes—
162Dic3—3616no—
163Dic3—368no—
164Dic3—362no—
165C32—24no—
166C32#471311yescharacteristic, complemented
167C32#507114yescomplemented
168C23—62no—
169D4—64no—
170D4—124no—
171C2×C4—542no—
172C2×C4#629112yes—
173C2×C4C2×D3×C3⋊D311yescharacteristic
174C8—182no—
175C2×C3—28no—
176C2×C3—24no—
177C2×C3—248no—
178C2×C3#6383018yes—
179C2×C3#6386014yes—
180C2×C3#6414112yes—
181C2×C3#6414511yescharacteristic
182D3—82no—
183C22—124no—
184C22—122no—
185C22C2×#629111yescharacteristic
186C4—22no—
187C4—1082no—
188C4#8901412yes—
189C3—24no—
190C3#1059027514yescomplemented
191C3#1059277011yescharacteristic, complemented
192C2—242no—
193C2#1172337812yes—
194C2#1172342911yescharacteristic
subgroupG/H
centerC22C2×#6291
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC32×#874C2
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
1. #11723911C2semidirect
3. C3×#63630C2semidirect
4. #11722032C2semidirect
19. #63630D3semidirect
117. C33#490semidirect
166. C32#4713semidirect
167. C32#5071semidirect
190. C3#10590275semidirect
191. C3#10592770semidirect

Maximal subgroups

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

subgroupindex
#117239112
C32×#8742
C3×#636302
#117220322
#105927703
#105902753