Finite Group Databasebeta

← #10588749⋊C3 #13142963

Subgroups — #10588749⋊C3 #13142963

Order 1728.

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

normal subgroups only

This group has 316 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
13−1+2×#125C211yesmaximal, characteristic, complemented
2#63437⋊C3C211yesmaximal, characteristic
3#63369⋊C3C212yesmaximal
4#63368⋊C3C212yesmaximal, complemented
5C4×3−1+2⋊C8C211yesmaximal, characteristic, complemented
6C3×#5068—31nomaximal
7#10588749C311yesmaximal, characteristic, complemented
8C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
9C2×3−1+2⋊Q8C2211yescharacteristic
10C4⋊C4×3−1+2—22no—
11C4⋊C4×3−1+2C2212yes—
12C42×3−1+2C2211yescharacteristic
13(C2×AGL(1,ℤ/9))⋊C4—22no—
14C2×3−1+2⋊C8C2212yes—
15C32×#125D311yescharacteristic
16C3×#875—31no—
17C3×#810—32no—
18C3×#809—32no—
19C3×C4×C3⋊C8—31no—
20C9×#125—21no—
21C9×#125C2×C311yescharacteristic, complemented
22#63437C2×C311yescharacteristic
23#63369C2×C312yes—
24#63368C2×C312yescomplemented
25C4×C9⋊C8C2×C311yescharacteristic, complemented
26C22×AGL(1,ℤ/9)—21no—
27C2×C4×3−1+2—22no—
28C2×C4×3−1+2D412yes—
29C2×C4×3−1+2C2311yescharacteristic
30C2×3−1+2⋊C4—21no—
31C4⋊AGL(1,ℤ/9)—22no—
323−1+2⋊Q8—22no—
333−1+2⋊C8—22no—
34C3×#488—91no—
35#5068—31no—
36C2×C3×D12—31no—
37C2×C3×Dic6—31no—
38C32×C4⋊C4—22no—
39C32×C4⋊C4C2×D312yes—
40C32×C42C2×D311yescharacteristic
41C3×(C2×D3)⋊C4—62no—
42C2×C3×C3⋊C8—32no—
43C2×D36C22×C311yescharacteristic
44C2×Dic18C22×C311yescharacteristic
45C9×C4⋊C4—24no—
46C9×C4⋊C4—22no—
47C9×C4⋊C4—22no—
48C9×C4⋊C4C22×C312yes—
49C42×C9—21no—
50C42×C9C22×C311yescharacteristic
51(C2×D9)⋊C4—22no—
52C2×C9⋊C8C22×C312yes—
53C22×3−1+2C2×D411yescharacteristic
54C2×AGL(1,ℤ/9)—41no—
55C2×AGL(1,ℤ/9)—42no—
56C4×3−1+2—22no—
57C4×3−1+2—42no—
58C4×3−1+2#4112yes—
593−1+2⋊C4—41no—
60C3×#125—31no—
61C3×#125C3×D311yescharacteristic
62C3×#124—91no—
63#875—31no—
64C3×#103—92no—
65C3×#102—92no—
66C3×C4×C8—91no—
67#810—32no—
68#809—32no—
69C4×C3⋊C8—31no—
70C22×C3×D3—61no—
71C2×C32×C4—22no—
72C2×C32×C4#6712yes—
73C2×C32×C4C22×D311yescharacteristic
74C2×C3×Dic3—61no—
75C3×D12—62no—
76C3×Dic6—62no—
77C22×D9—21no—
78C3×C3⋊C8—62no—
79C2×C4×C9—24no—
80C2×C4×C9—22no—
81C2×C4×C9—22no—
82C2×C4×C9—21no—
83C2×C4×C9C3×D412yes—
84C2×C4×C9C23×C311yescharacteristic
85C2×Dic9—21no—
86D36—22no—
87Dic18—22no—
88C9⋊C8—22no—
89#488—91noSylow
90C2×3−1+2#12411yescharacteristic
91C2×3−1+2#13512yes—
92AGL(1,ℤ/9)—81no—
93C2×C3×Q8—91no—
94C2×C3×D4—91no—
95C2×D12—31no—
96C2×Dic6—31no—
97C2×C3×C8—92no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—32no—
100C3×C4⋊C4—62no—
101C3×C4⋊C4C2×C3×D312yes—
102C3×C22⋊C4—182no—
103C3×C42—31no—
104C3×C42C2×C3×D311yescharacteristic
105(C2×D3)⋊C4—62no—
106C2×C3⋊C8—32no—
107C22×C32C2×#6711yescharacteristic
108C2×C3×D3—121no—
109C2×C3×D3—122no—
110C32×C4—22no—
111C32×C4—42no—
112C32×C4#23912yes—
113C3×Dic3—121no—
114C22×C9—21no—
115C22×C9C2×C3×D411yescharacteristic
116C2×D9—41no—
117C2×D9—42no—
118C4×C9—22no—
119C4×C9—22no—
120C4×C9—42no—
121C4×C9—44no—
122C4×C9—42no—
123C4×C9C3×#4112yes—
124Dic9—41no—
125#125AGL(1,ℤ/9)11yescharacteristic, complemented
126#124—91no—
127#103—92no—
128#102—92no—
129C4×C8—91no—
1303−1+2#48811yescharacteristic, complemented, Sylow
131C23×C3—181no—
132C22×D3—61no—
133C3×Q8—182no—
134C3×D4—182no—
135C2×C3×C4—22no—
136C2×C3×C4—32no—
137C2×C3×C4—31no—
138C2×C3×C4—62no—
139C2×C3×C4—181no—
140C2×C3×C4C3×#6712yes—
141C2×C3×C4C22×C3×D311yescharacteristic
142C2×Dic3—61no—
143D12—62no—
144Dic6—62no—
145C3×C8—182no—
146C3⋊C8—62no—
147C2×C32#94711yescharacteristic
148C2×C32#95012yes—
149C3×D3—241no—
150C2×C9—21no—
151C2×C9—22no—
152C2×C9C3×#12411yescharacteristic
153C2×C9C3×#13512yes—
154D9—81no—
155C2×Q8—91no—
156C2×D4—91no—
157C2×C8—92no—
158C4⋊C4—22no—
159C4⋊C4C2×AGL(1,ℤ/9)12yes—
160C22⋊C4—182no—
161C42C2×AGL(1,ℤ/9)11yescharacteristic
162C22×C3—31no—
163C22×C3—361no—
164C22×C3—362no—
165C22×C3C2×C3×#6711yescharacteristic
166C2×D3—121no—
167C2×D3—122no—
168C3×C4—22no—
169C3×C4—32no—
170C3×C4—42no—
171C3×C4—62no—
172C3×C4—122no—
173C3×C4—361no—
174C3×C4C3×#23912yes—
175Dic3—121no—
176C32#506811yescharacteristic
177C9—21no—
178C9C3×#48811yescharacteristic, complemented
179C23—181no—
180Q8—182no—
181D4—182no—
182C2×C4—22no—
183C2×C4—181no—
184C2×C4#611⋊C312yes—
185C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
186C8—182no—
187C2×C3—31no—
188C2×C3—32no—
189C2×C3—721no—
190C2×C3C3×#94711yescharacteristic
191C2×C3C3×#95012yes—
192D3—241no—
193C22—361no—
194C22—362no—
195C22C2×#611⋊C311yescharacteristic
196C4—22no—
197C4—42no—
198C4—361no—
199C4#8872412yes—
200C3—31no—
201C3C3×#506811yescharacteristic
202C2—721no—
203C2#63509⋊C311yescharacteristic
204C2#63513⋊C312yes—
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#125C2
socleC22×C3C2×C3×#67
prime radicalC2×C4⋊AGL(1,ℤ/9)C22
hypercenter#125AGL(1,ℤ/9)
hypocenterC9C3×#488
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#125C2semidirect
4. #63368⋊C3C2semidirect
5. C4×3−1+2⋊C8C2semidirect
7. #10588749C3semidirect
21. C9×#125C2×C3semidirect
24. #63368C2×C3semidirect
25. C4×C9⋊C8C2×C3semidirect
125. #125AGL(1,ℤ/9)semidirect
130. 3−1+2#488semidirect
178. C9C3×#488semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1252
#63437⋊C32
#63369⋊C32
#63368⋊C32
C4×3−1+2⋊C82
C3×#50683
#105887493