Finite Group Databasebeta

← #10588394⋊C3 #13141553

Subgroups — #10588394⋊C3 #13141553

Order 1728.

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

normal subgroups only

This group has 350 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#11721394C211yesmaximal, characteristic
2#63432⋊C3C211yesmaximal, characteristic, complemented
33−1+2×#97C211yesmaximal, characteristic, complemented
4#63380⋊C3C214yesmaximal, complemented
5C3×#4713—31nomaximal
6#10588394C311yesmaximal, characteristic, complemented
7C2×C4⋊AGL(1,ℤ/9)—22no—
8C2×C4⋊AGL(1,ℤ/9)C2212yes—
9C2×C8×3−1+2C2212yes—
10C42×3−1+2C2211yescharacteristic
11C4⋊(3−1+2⋊C4)C2212yes—
12(3−1+2⋊C4)⋊C4—22no—
13C3×#874—31no—
14C3×#870—31no—
15C32×#97D311yescharacteristic
16C3×#821—34no—
17#63436C2×C311yescharacteristic
18#63432C2×C311yescharacteristic, complemented
19C9×#97—21no—
20C9×#97C2×C311yescharacteristic, complemented
21#63380C2×C314yescomplemented
22C22×AGL(1,ℤ/9)—22no—
23C2×C4×3−1+2D412yes—
24C2×C4×3−1+2C2311yescharacteristic
25C2×3−1+2⋊C4—22no—
26C4⋊AGL(1,ℤ/9)—24no—
27C4⋊AGL(1,ℤ/9)—44no—
28C8×3−1+2—22no—
29C3×#490—91no—
30#4713—31no—
31C2×C3×D12—32no—
32C2×C3×D12—62no—
33C2×C32×C8C2×D312yes—
34C32×C42C2×D311yescharacteristic
35C3×C4⋊Dic3—32no—
36C3×Dic3⋊C4—62no—
37C2×D36—22no—
38C2×D36C22×C312yes—
39C2×C8×C9—22no—
40C2×C8×C9C22×C312yes—
41C42×C9—21no—
42C42×C9C22×C311yescharacteristic
43C4⋊Dic9C22×C312yes—
44Dic9⋊C4—22no—
45C22×3−1+2C2×D411yescharacteristic
46C2×AGL(1,ℤ/9)—42no—
47C2×AGL(1,ℤ/9)—44no—
48C4×3−1+2—22no—
49C4×3−1+2#4112yes—
503−1+2⋊C4—42no—
51C3×#127—91no—
52C3×#125—91no—
53#874—31no—
54#870—31no—
55C3×#102—94no—
56C3×#97—31no—
57C3×#97C3×D311yescharacteristic
58#821—34no—
59C22×C3×D3—62no—
60C2×C32×C4D1212yes—
61C2×C32×C4C22×D311yescharacteristic
62C2×C3×Dic3—62no—
63C3×D12—64no—
64C3×D12—124no—
65C22×D9—22no—
66C32×C8—22no—
67C2×C4×C9—22no—
68C2×C4×C9—21no—
69C2×C4×C9C3×D412yes—
70C2×C4×C9C23×C311yescharacteristic
71C2×Dic9—22no—
72D36—24no—
73D36—44no—
74C8×C9—24no—
75C8×C9—22no—
76#490—91noSylow
77C2×3−1+2#12411yescharacteristic
78C2×3−1+2AGL(1,ℤ/8)12yes—
79AGL(1,ℤ/9)—82no—
80C2×C3×D4—92no—
81C2×C3×D4—182no—
82C2×D12—32no—
83C2×D12—62no—
84C2×C3×C8—32no—
85C2×C3×C8C2×C3×D312yes—
86C3×C4⋊C4—92no—
87C3×C4⋊C4—182no—
88C3×C42—31no—
89C3×C42C2×C3×D311yescharacteristic
90C4⋊Dic3—32no—
91Dic3⋊C4—62no—
92C22×C32C2×D1211yescharacteristic
93C2×C3×D3—122no—
94C2×C3×D3—124no—
95C32×C4—22no—
96C32×C4#23512yes—
97C3×Dic3—122no—
98C22×C9—21no—
99C22×C9C2×C3×D411yescharacteristic
100C2×D9—42no—
101C2×D9—44no—
102C4×C9—22no—
103C4×C9—22no—
104C4×C9—42no—
105C4×C9C3×#4112yes—
106Dic9—42no—
107#127—91no—
108#125—91no—
109#102—94no—
110#97AGL(1,ℤ/9)11yescharacteristic, complemented
1113−1+2#49011yescharacteristic, complemented, Sylow
112C23×C3—182no—
113C22×D3—62no—
114C3×D4—184no—
115C3×D4—364no—
116C2×C3×C4—32no—
117C2×C3×C4—31no—
118C2×C3×C4—182no—
119C2×C3×C4C3×D1212yes—
120C2×C3×C4C22×C3×D311yescharacteristic
121C2×Dic3—62no—
122D12—64no—
123D12—124no—
124C3×C8—22no—
125C3×C8—62no—
126C2×C32#87511yescharacteristic
127C2×C32#90812yes—
128C3×D3—242no—
129C2×C9—22no—
130C2×C9—21no—
131C2×C9C3×#12411yescharacteristic
132C2×C9C3×AGL(1,ℤ/8)12yes—
133D9—82no—
134C2×D4—92no—
135C2×D4—182no—
136C2×C8C2×AGL(1,ℤ/9)12yes—
137C4⋊C4—92no—
138C4⋊C4—182no—
139C42C2×AGL(1,ℤ/9)11yescharacteristic
140C22×C3—31no—
141C22×C3—362no—
142C22×C3—364no—
143C22×C3C2×C3×D1211yescharacteristic
144C2×D3—124no—
145C2×D3—122no—
146C3×C4—22no—
147C3×C4—32no—
148C3×C4—62no—
149C3×C4—362no—
150C3×C4C3×#23512yes—
151Dic3—122no—
152C32#471311yescharacteristic
153C9—21no—
154C9C3×#49011yescharacteristic, complemented
155C23—182no—
156D4—184no—
157D4—364no—
158C2×C4—182no—
159C2×C4C4⋊AGL(1,ℤ/9)12yes—
160C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
161C8—22no—
162C2×C3—32no—
163C2×C3—31no—
164C2×C3—722no—
165C2×C3C3×#87511yescharacteristic
166C2×C3C3×#90812yes—
167D3—242no—
168C22—364no—
169C22—362no—
170C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
171C4—22no—
172C4—362no—
173C4#3697⋊C312yes—
174C3—31no—
175C3C3×#471311yescharacteristic
176C2—722no—
177C2#63437⋊C311yescharacteristic
178C2#63470⋊C312yes—
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#97C2
socleC22×C3C2×C3×D12
prime radical#11721394C2
hypercenter#97AGL(1,ℤ/9)
hypocenterC9C3×#490
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
2. #63432⋊C3C2semidirect
3. 3−1+2×#97C2semidirect
4. #63380⋊C3C2semidirect
6. #10588394C3semidirect
18. #63432C2×C3semidirect
20. C9×#97C2×C3semidirect
21. #63380C2×C3semidirect
110. #97AGL(1,ℤ/9)semidirect
111. 3−1+2#490semidirect
154. C9C3×#490semidirect

Maximal subgroups

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

subgroupindex
#117213942
#63432⋊C32
3−1+2×#972
#63380⋊C32
C3×#47133
#105883943