Finite Group Databasebeta

← #13137324

Subgroups — #13137324

Order 1728.

441 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 75 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#11721716C211yesmaximal, characteristic, complemented
2#11721385C211yesmaximal, characteristic
3#63630⋊C3C211yesmaximal, characteristic, complemented
4#11720308C214yesmaximal, complemented
5#10590282—31nomaximal
6#10590279—31nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)—22no—
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C42×H(3,𝔽3)C2211yescharacteristic
10C4⋊#1084C2212yes—
11#88540—22no—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#64083D311yescharacteristic
14C3×#874—31no—
15C3×#870—31no—
16C3×#870—31no—
17C3×#870—61no—
18#63630D311yescharacteristic, complemented
19C3×#803—31no—
20#63564—34no—
21#63562—34no—
22C22×(C3⋊D3)⋊C3—22no—
23C2×C4×H(3,𝔽3)D412yes—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084—22no—
26C4⋊((C3⋊D3)⋊C3)—24no—
27C4⋊((C3⋊D3)⋊C3)—44no—
28(C3⋊(C3⋊C8))⋊C3—22no—
29#5071—91no—
30#5067—91no—
31#4713—91no—
32C2×C4⋊(C3⋊D3)—22no—
33C2×C4⋊(C3⋊D3)C2×D312yes—
34C2×C3×D12—32no—
35C2×C3×D12—62no—
36C32×C42—21no—
37C32×C42C2×D311yescharacteristic
38C32×C42C2×D311yescharacteristic
39C2×C3⋊(C3⋊C8)C2×D312yes—
40C3×C4⋊Dic3—32no—
41C3×C4⋊Dic3—32no—
42C3×C4⋊Dic3—62no—
43C3×Dic3⋊C4—64no—
44C3×Dic3⋊C4—62no—
45C3×Dic3⋊C4—62no—
46C2×C3×C3⋊C8—32no—
47C22×H(3,𝔽3)C2×D411yescharacteristic
48C2×(C3⋊D3)⋊C3—42no—
49C2×(C3⋊D3)⋊C3—44no—
50C4×H(3,𝔽3)—22no—
51C4×H(3,𝔽3)#4112yes—
52#1084—42no—
53C3×#127—91no—
54C3×#125—91no—
55#874—31no—
56#874—91no—
57#870—31no—
58#870—31no—
59#870—61no—
60C3×#97—91no—
61#832—94no—
62#821—94no—
63#809—94no—
64#803—31no—
65#803—91no—
66C22×C3⋊D3—22no—
67C22×C3×D3—62no—
68C2×C32×C4—22no—
69C2×C32×C4—21no—
70C2×C32×C4D1212yes—
71C2×C32×C4#6712yes—
72C2×C32×C4C22×D311yescharacteristic
73C2×C32×C4C22×D311yescharacteristic
74C4⋊(C3⋊D3)—24no—
75C4⋊(C3⋊D3)—44no—
76C2×C3×Dic3—64no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C3×D12—64no—
80C3×D12—124no—
81C3⋊(C3⋊C8)—22no—
82C3×C3⋊C8—62no—
83#490—271noSylow
84C2×H(3,𝔽3)#12411yescharacteristic
85C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
86(C3⋊D3)⋊C3—82no—
87C2×C3×D4—92no—
88C2×C3×D4—182no—
89C2×D12—32no—
90C2×D12—62no—
91C2×D12—92no—
92C2×D12—182no—
93C2×C3×C8—92no—
94C3×C4⋊C4—92no—
95C3×C4⋊C4—182no—
96C3×C42—31no—
97C3×C42—31no—
98C3×C42—61no—
99C3×C42D3211yescharacteristic
100C4⋊Dic3—32no—
101C4⋊Dic3—32no—
102C4⋊Dic3—62no—
103Dic3⋊C4—62no—
104Dic3⋊C4—62no—
105Dic3⋊C4—64no—
106C2×C3⋊C8—32no—
107C2×C3⋊C8—92no—
108C22×C32—21no—
109C22×C32C2×D1211yescharacteristic
110C22×C32C2×#6711yescharacteristic
111C2×C3⋊D3—44no—
112C2×C3⋊D3—42no—
113C2×C3×D3—122no—
114C2×C3×D3—124no—
115C32×C4—22no—
116C32×C4—22no—
117C32×C4—22no—
118C32×C4—42no—
119C32×C4#23512yes—
120C32×C4#23712yes—
121C3×Dic3—124no—
122C3×Dic3—122no—
123C3×Dic3—122no—
124#127—91no—
125#125—91no—
126#102—274no—
127#97—91no—
128H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
129C23×C3—182no—
130C22×D3—62no—
131C22×D3—182no—
132C3×D4—184no—
133C3×D4—364no—
134C2×C3×C4—32no—
135C2×C3×C4—32no—
136C2×C3×C4—31no—
137C2×C3×C4—31no—
138C2×C3×C4—62no—
139C2×C3×C4—61no—
140C2×C3×C4—182no—
141C2×C3×C4#62612yes—
142C2×C3×C4C2×D3211yescharacteristic
143C2×Dic3—62no—
144C2×Dic3—62no—
145C2×Dic3—64no—
146D12—64no—
147D12—124no—
148D12—184no—
149D12—364no—
150C3×C8—182no—
151C3⋊C8—62no—
152C3⋊C8—182no—
153C2×C32—22no—
154C2×C32—21no—
155C2×C32#87511yescharacteristic
156C2×C32#90812yes—
157C2×C32#93212yes—
158C2×C32#93611yescharacteristic
159C3⋊D3—82no—
160C3×D3—242no—
161C2×D4—92no—
162C2×D4—182no—
163C2×C8—92no—
164C4⋊C4—92no—
165C4⋊C4—182no—
166C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
167C22×C3—31no—
168C22×C3—31no—
169C22×C3—61no—
170C22×C3—362no—
171C22×C3—364no—
172C22×C3C2×#62611yescharacteristic
173C2×D3—124no—
174C2×D3—122no—
175C2×D3—362no—
176C2×D3—364no—
177C3×C4—22no—
178C3×C4—32no—
179C3×C4—32no—
180C3×C4—62no—
181C3×C4—62no—
182C3×C4—62no—
183C3×C4—122no—
184C3×C4—362no—
185C3×C4#379512yes—
186Dic3—122no—
187Dic3—122no—
188Dic3—124no—
189C32—21no—
190C32#471311yescharacteristic, complemented
191C32#507111yescharacteristic, complemented
192C23—182no—
193D4—184no—
194D4—364no—
195C2×C4—182no—
196C2×C4#620012yes—
197C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
198C8—182no—
199C2×C3—32no—
200C2×C3—32no—
201C2×C3—31no—
202C2×C3—31no—
203C2×C3—62no—
204C2×C3—61no—
205C2×C3—722no—
206C2×C3#6383012yes—
207C2×C3#6386011yescharacteristic
208D3—242no—
209D3—722no—
210C22—364no—
211C22—362no—
212C22C2×#620011yescharacteristic
213C4—22no—
214C4—362no—
215C4#8864812yes—
216C3—31no—
217C3—31no—
218C3—61no—
219C3#1059027511yescharacteristic
220C2—722no—
221C2#1172107112yes—
222C2#1172111011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radical#11721385C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#490
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721716C2semidirect
3. #63630⋊C3C2semidirect
4. #11720308C2semidirect
18. #63630D3semidirect
128. H(3,𝔽3)#490semidirect
190. C32#4713semidirect
191. C32#5071semidirect

Maximal subgroups

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

subgroupindex
#117217162
#117213852
#63630⋊C32
#117203082
#105902823
#105902793