Finite Group Databasebeta

← #13137363

Subgroups — #13137363

Order 1728.

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

normal subgroups only

This group has 493 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#11721720C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic, complemented
3#63630⋊C3C211yesmaximal, characteristic, complemented
4#11720324C214yesmaximal, complemented
5#10590279—31nomaximal
6#10590275—31nomaximal
7C2×C4⋊#272—22no—
8C2×C4⋊#272C2212yes—
9C42×H(3,𝔽3)C2211yescharacteristic
10C4⋊((C3⋊Dic3)⋊C3)C2212yes—
11#3750⋊C3—22no—
12C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
13#64079D311yescharacteristic, complemented
14C3×#874—31no—
15C3×#874—31no—
16C3×#874—61no—
17C3×#870—31no—
18#63630D311yescharacteristic, complemented
19C3×#803—31no—
20#63563—34no—
21#63562—34no—
22C22×#272—22no—
23C2×C4×H(3,𝔽3)D412yes—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C4⋊#272—24no—
26C4⋊#272—44no—
27C2×(C3⋊Dic3)⋊C3—22no—
28(C3⋊(C3⋊C8))⋊C3—22no—
29#5071—91no—
30#5067—91no—
31#4713—91no—
32C2×C3×D12—32no—
33C2×C3×D12—32no—
34C2×C3×D12—62no—
35C2×C3×D12—64no—
36C2×C3×D12—62no—
37C2×C3×D12—62no—
38C32×C42—21no—
39C32×C42C2×D311yescharacteristic
40C32×C42C2×D311yescharacteristic
41C4⋊(C3⋊Dic3)C2×D312yes—
42#3750—22no—
43C2×C3⋊(C3⋊C8)C2×D312yes—
44C3×C4⋊Dic3—32no—
45C3×Dic3⋊C4—62no—
46C2×C3×C3⋊C8—32no—
47C22×H(3,𝔽3)C2×D411yescharacteristic
48C2×#272—42no—
49C2×#272—44no—
50C4×H(3,𝔽3)—22no—
51C4×H(3,𝔽3)#4112yes—
52(C3⋊Dic3)⋊C3—42no—
53C3×#127—91no—
54C3×#125—91no—
55#874—31no—
56#874—31no—
57#874—61no—
58#870—31no—
59#870—91no—
60C3×#97—91no—
61#832—94no—
62#821—94no—
63#809—94no—
64#803—31no—
65#803—91no—
66C22×C3×D3—64no—
67C22×C3×D3—62no—
68C22×C3×D3—62no—
69C2×C32×C4—22no—
70C2×C32×C4—21no—
71C2×C32×C4D1212yes—
72C2×C32×C4#6712yes—
73C2×C32×C4C22×D311yescharacteristic
74C2×C32×C4C22×D311yescharacteristic
75C2×C3⋊Dic3—22no—
76C2×C3×Dic3—62no—
77C3×D12—64no—
78C3×D12—64no—
79C3×D12—124no—
80C3×D12—128no—
81C3×D12—124no—
82C3×D12—124no—
83C3⋊(C3⋊C8)—22no—
84C3×C3⋊C8—62no—
85#490—271noSylow
86C2×H(3,𝔽3)#12411yescharacteristic
87C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
88#272—82no—
89C2×C3×D4—92no—
90C2×C3×D4—182no—
91C2×D12—32no—
92C2×D12—32no—
93C2×D12—62no—
94C2×D12—62no—
95C2×D12—62no—
96C2×D12—64no—
97C2×C3×C8—92no—
98C3×C4⋊C4—92no—
99C3×C4⋊C4—182no—
100C3×C42—31no—
101C3×C42—31no—
102C3×C42—61no—
103C3×C42D3211yescharacteristic
104C4⋊Dic3—32no—
105C4⋊Dic3—92no—
106Dic3⋊C4—62no—
107Dic3⋊C4—182no—
108C2×C3⋊C8—32no—
109C2×C3⋊C8—92no—
110C22×C32—21no—
111C22×C32C2×D1211yescharacteristic
112C22×C32C2×#6711yescharacteristic
113C2×C3×D3—128no—
114C2×C3×D3—124no—
115C2×C3×D3—122no—
116C2×C3×D3—124no—
117C2×C3×D3—122no—
118C2×C3×D3—124no—
119C32×C4—24no—
120C32×C4—22no—
121C32×C4—22no—
122C32×C4—22no—
123C32×C4#23512yes—
124C32×C4#23912yes—
125C3⋊Dic3—42no—
126C3×Dic3—122no—
127#127—91no—
128#125—91no—
129#102—274no—
130#97—91no—
131H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
132C23×C3—182no—
133C22×D3—62no—
134C22×D3—62no—
135C22×D3—64no—
136C3×D4—184no—
137C3×D4—364no—
138C2×C3×C4—32no—
139C2×C3×C4—32no—
140C2×C3×C4—31no—
141C2×C3×C4—31no—
142C2×C3×C4—62no—
143C2×C3×C4—61no—
144C2×C3×C4—182no—
145C2×C3×C4#62612yes—
146C2×C3×C4C2×D3211yescharacteristic
147C2×Dic3—62no—
148C2×Dic3—182no—
149D12—64no—
150D12—64no—
151D12—124no—
152D12—124no—
153D12—124no—
154D12—128no—
155C3×C8—182no—
156C3⋊C8—62no—
157C3⋊C8—182no—
158C2×C32—22no—
159C2×C32—21no—
160C2×C32#87511yescharacteristic
161C2×C32#90812yes—
162C2×C32#94711yescharacteristic
163C2×C32#94912yes—
164C3×D3—244no—
165C3×D3—242no—
166C3×D3—242no—
167C2×D4—92no—
168C2×D4—182no—
169C2×C8—92no—
170C4⋊C4—92no—
171C4⋊C4—182no—
172C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
173C22×C3—31no—
174C22×C3—31no—
175C22×C3—61no—
176C22×C3—364no—
177C22×C3—362no—
178C22×C3C2×#62611yescharacteristic
179C2×D3—122no—
180C2×D3—122no—
181C2×D3—124no—
182C2×D3—124no—
183C2×D3—128no—
184C2×D3—124no—
185C3×C4—22no—
186C3×C4—32no—
187C3×C4—32no—
188C3×C4—62no—
189C3×C4—62no—
190C3×C4—64no—
191C3×C4—62no—
192C3×C4—362no—
193C3×C4#379912yes—
194Dic3—122no—
195Dic3—362no—
196C32—21no—
197C32#471311yescharacteristic, complemented
198C32#506711yescharacteristic, complemented
199C23—182no—
200D4—184no—
201D4—364no—
202C2×C4—182no—
203C2×C4#620012yes—
204C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
205C8—182no—
206C2×C3—32no—
207C2×C3—32no—
208C2×C3—31no—
209C2×C3—31no—
210C2×C3—62no—
211C2×C3—61no—
212C2×C3—722no—
213C2×C3#6382512yes—
214C2×C3#6386411yescharacteristic
215D3—242no—
216D3—242no—
217D3—244no—
218C22—364no—
219C22—362no—
220C22C2×#620011yescharacteristic
221C4—22no—
222C4—362no—
223C4#8865212yes—
224C3—31no—
225C3—31no—
226C3—61no—
227C3#1059028211yescharacteristic
228C2—722no—
229C2#1172106812yes—
230C2#1172111411yescharacteristic
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#11721720C2
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
2. #64079⋊C3C2semidirect
3. #63630⋊C3C2semidirect
4. #11720324C2semidirect
13. #64079D3semidirect
18. #63630D3semidirect
131. H(3,𝔽3)#490semidirect
197. C32#4713semidirect
198. C32#5067semidirect

Maximal subgroups

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

subgroupindex
#117217202
#64079⋊C32
#63630⋊C32
#117203242
#105902793
#105902753