Finite Group Databasebeta

← #10592412⋊C3 #13141536

Subgroups — #10592412⋊C3 #13141536

Order 1728.

425 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 92 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#11721385C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
4#63648⋊C3C214yesmaximal, complemented
5#10592412C311yesmaximal, characteristic, complemented
6C3×#4713—31nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)—22no—
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×C8×H(3,𝔽3)C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12#3750⋊C3—22no—
13#64083C2×C311yescharacteristic
14#64079C2×C311yescharacteristic, complemented
15C3×#874—31no—
16C3×#870—31no—
17C32×#97—21no—
18C32×#97D311yescharacteristic, complemented
19C32×#97C2×C311yescharacteristic, complemented
20#63648C2×C314yescomplemented
21C3×#821—34no—
22C22×(C3⋊D3)⋊C3—22no—
23C2×C4×H(3,𝔽3)D412yes—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×(C3⋊Dic3)⋊C3—22no—
26C4⋊((C3⋊D3)⋊C3)—24no—
27C4⋊((C3⋊D3)⋊C3)—44no—
28C8×H(3,𝔽3)—22no—
29C3×#490—91no—
30#4713—31no—
31#4713—91no—
32C2×C4⋊(C3⋊D3)—22no—
33C2×C4⋊(C3⋊D3)C22×C312yes—
34C2×C3×D12—32no—
35C2×C3×D12—62no—
36C2×C32×C8—22no—
37C2×C32×C8C2×D312yes—
38C2×C32×C8C22×C312yes—
39C32×C42—21no—
40C32×C42C2×D311yescharacteristic
41C32×C42C22×C311yescharacteristic
42C4⋊(C3⋊Dic3)C22×C312yes—
43#3750—22no—
44C3×C4⋊Dic3—32no—
45C3×Dic3⋊C4—62no—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C2×(C3⋊D3)⋊C3—42no—
48C2×(C3⋊D3)⋊C3—44no—
49C4×H(3,𝔽3)—22no—
50C4×H(3,𝔽3)#4112yes—
51(C3⋊Dic3)⋊C3—42no—
52C3×#127—91no—
53C3×#125—91no—
54#874—31no—
55#874—91no—
56#870—31no—
57#870—91no—
58C3×#102—94no—
59C3×#97—31no—
60C3×#97—31no—
61C3×#97—61no—
62C3×#97C3×D311yescharacteristic
63#821—34no—
64#821—94no—
65C22×C3⋊D3—22no—
66C22×C3×D3—62no—
67C2×C32×C4—22no—
68C2×C32×C4—21no—
69C2×C32×C4D1212yes—
70C2×C32×C4C3×D412yes—
71C2×C32×C4C22×D311yescharacteristic
72C2×C32×C4C23×C311yescharacteristic
73C2×C3⋊Dic3—22no—
74C4⋊(C3⋊D3)—24no—
75C4⋊(C3⋊D3)—44no—
76C2×C3×Dic3—62no—
77C3×D12—64no—
78C3×D12—124no—
79C32×C8—24no—
80C32×C8—22no—
81C32×C8—22no—
82#490—91noSylow
83C2×H(3,𝔽3)#12411yescharacteristic
84C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
85(C3⋊D3)⋊C3—82no—
86C2×C3×D4—92no—
87C2×C3×D4—182no—
88C2×D12—32no—
89C2×D12—62no—
90C2×D12—92no—
91C2×D12—182no—
92C2×C3×C8—32no—
93C2×C3×C8—32no—
94C2×C3×C8—62no—
95C2×C3×C8C2×C3×D312yes—
96C3×C4⋊C4—92no—
97C3×C4⋊C4—182no—
98C3×C42—31no—
99C3×C42—31no—
100C3×C42—61no—
101C3×C42C2×C3×D311yescharacteristic
102C4⋊Dic3—32no—
103C4⋊Dic3—92no—
104Dic3⋊C4—62no—
105Dic3⋊C4—182no—
106C22×C32—21no—
107C22×C32C2×D1211yescharacteristic
108C22×C32C2×C3×D411yescharacteristic
109C2×C3⋊D3—42no—
110C2×C3⋊D3—44no—
111C2×C3×D3—122no—
112C2×C3×D3—124no—
113C32×C4—22no—
114C32×C4—22no—
115C32×C4—22no—
116C32×C4—42no—
117C32×C4#23512yes—
118C32×C4C3×#4112yes—
119C3⋊Dic3—42no—
120C3×Dic3—122no—
121#127—91no—
122#125—91no—
123#102—94no—
124#97(C3⋊D3)⋊C311yescharacteristic, complemented
125H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
126C23×C3—182no—
127C22×D3—62no—
128C22×D3—182no—
129C3×D4—184no—
130C3×D4—364no—
131C2×C3×C4—32no—
132C2×C3×C4—32no—
133C2×C3×C4—31no—
134C2×C3×C4—31no—
135C2×C3×C4—62no—
136C2×C3×C4—61no—
137C2×C3×C4—182no—
138C2×C3×C4C3×D1212yes—
139C2×C3×C4C22×C3×D311yescharacteristic
140C2×Dic3—62no—
141C2×Dic3—182no—
142D12—64no—
143D12—124no—
144D12—184no—
145D12—364no—
146C3×C8—22no—
147C3×C8—62no—
148C3×C8—62no—
149C3×C8—64no—
150C2×C32—22no—
151C2×C32—21no—
152C2×C32#87511yescharacteristic
153C2×C32#90812yes—
154C2×C32C3×#12411yescharacteristic
155C2×C32C3×AGL(1,ℤ/8)12yes—
156C3⋊D3—82no—
157C3×D3—242no—
158C2×D4—92no—
159C2×D4—182no—
160C2×C8C2×(C3⋊D3)⋊C312yes—
161C4⋊C4—92no—
162C4⋊C4—182no—
163C42C2×(C3⋊D3)⋊C311yescharacteristic
164C22×C3—31no—
165C22×C3—31no—
166C22×C3—61no—
167C22×C3—362no—
168C22×C3—364no—
169C22×C3C2×C3×D1211yescharacteristic
170C2×D3—124no—
171C2×D3—122no—
172C2×D3—362no—
173C2×D3—364no—
174C3×C4—22no—
175C3×C4—32no—
176C3×C4—32no—
177C3×C4—62no—
178C3×C4—62no—
179C3×C4—62no—
180C3×C4—122no—
181C3×C4—362no—
182C3×C4C3×#23512yes—
183Dic3—122no—
184Dic3—362no—
185C32—21no—
186C32#471311yescharacteristic, complemented
187C32C3×#49011yescharacteristic, complemented
188C23—182no—
189D4—184no—
190D4—364no—
191C2×C4—182no—
192C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
193C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
194C8—22no—
195C2×C3—32no—
196C2×C3—32no—
197C2×C3—31no—
198C2×C3—31no—
199C2×C3—62no—
200C2×C3—61no—
201C2×C3—722no—
202C2×C3C3×#87511yescharacteristic
203C2×C3C3×#90812yes—
204D3—242no—
205D3—722no—
206C22—364no—
207C22—362no—
208C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
209C4—22no—
210C4—362no—
211C4#3828⋊C312yes—
212C3—31no—
213C3—31no—
214C3—61no—
215C3C3×#471311yescharacteristic
216C2—722no—
217C2#64084⋊C311yescharacteristic
218C2#64117⋊C312yes—
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C3×D12
prime radical#11721385C2
hypercenter#97(C3⋊D3)⋊C3
hypocenterC32C3×#490
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
2. #64079⋊C3C2semidirect
3. H(3,𝔽3)×#97C2semidirect
4. #63648⋊C3C2semidirect
5. #10592412C3semidirect
14. #64079C2×C3semidirect
18. C32×#97D3semidirect
19. C32×#97C2×C3semidirect
20. #63648C2×C3semidirect
124. #97(C3⋊D3)⋊C3semidirect
125. H(3,𝔽3)#490semidirect
186. C32#4713semidirect
187. C32C3×#490semidirect

Maximal subgroups

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

subgroupindex
#117213852
#64079⋊C32
H(3,𝔽3)×#972
#63648⋊C32
#105924123
C3×#47133