Finite Group Databasebeta

← #10588744⋊C3 #13142959

Subgroups — #10588744⋊C3 #13142959

Order 1728.

286 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 100 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
2#63432⋊C3C211yesmaximal, characteristic
3#63367⋊C3C212yesmaximal
4#63366⋊C3C212yesmaximal
5C4×3−1+2⋊C8C211yesmaximal, characteristic
6C3×#5065—31nomaximal
7#10588744C311yesmaximal, characteristic, complemented
8C4⋊C4×3−1+2—22no—
9C4⋊C4×3−1+2C2212yes—
10C42×3−1+2C2211yescharacteristic
11C4⋊(3−1+2⋊C4)C2212yes—
12(3−1+2⋊C4)⋊C4—22no—
13C2×3−1+2⋊C8C2211yescharacteristic
14C2×3−1+2⋊C8C2211yescharacteristic
15C32×#125D311yescharacteristic
16C3×#870—31no—
17C3×#808—32no—
18C3×#807—32no—
19C3×C4×C3⋊C8—31no—
20C9×#125—21no—
21C9×#125C2×C311yescharacteristic
22#63432C2×C311yescharacteristic
23#63367C2×C312yes—
24#63366C2×C312yes—
25C4×C9⋊C8C2×C311yescharacteristic
26C2×C4×3−1+2—22no—
27C2×C4×3−1+2D412yes—
28C2×C4×3−1+2C2311yescharacteristic
29C2×3−1+2⋊C4—22no—
303−1+2⋊C8Q812yes—
313−1+2⋊C8Q812yes—
32C3×#499—91no—
33#5065—31no—
34C32×C4⋊C4—22no—
35C32×C4⋊C4C2×D312yes—
36C32×C42C2×D311yescharacteristic
37C3×C4⋊Dic3—32no—
38C3×Dic3⋊C4—62no—
39C2×C3×C3⋊C8—31no—
40C2×C3×C3⋊C8—31no—
41C9×C4⋊C4—24no—
42C9×C4⋊C4—22no—
43C9×C4⋊C4—22no—
44C9×C4⋊C4C22×C312yes—
45C42×C9—21no—
46C42×C9C22×C311yescharacteristic
47C4⋊Dic9C22×C312yes—
48Dic9⋊C4—22no—
49C2×C9⋊C8C22×C311yescharacteristic
50C2×C9⋊C8C22×C311yescharacteristic
51C22×3−1+2C2×D411yescharacteristic
52C4×3−1+2—22no—
53C4×3−1+2—42no—
54C4×3−1+2C2×Q811yescharacteristic
55C4×3−1+2C2×Q811yescharacteristic
563−1+2⋊C4—42no—
57C3×#125—31no—
58C3×#125—91no—
59C3×#125C3×D311yescharacteristic
60#870—31no—
61C3×#107—92no—
62C3×#106—92no—
63C3×C4×C8—91no—
64#808—32no—
65#807—32no—
66C4×C3⋊C8—31no—
67C2×C32×C4—22no—
68C2×C32×C4#6712yes—
69C2×C32×C4C22×D311yescharacteristic
70C2×C3×Dic3—62no—
71C3×C3⋊C8—32no—
72C3×C3⋊C8—32no—
73C2×C4×C9—24no—
74C2×C4×C9—22no—
75C2×C4×C9—22no—
76C2×C4×C9—21no—
77C2×C4×C9C3×D412yes—
78C2×C4×C9C23×C311yescharacteristic
79C2×Dic9—22no—
80C9⋊C8C3×Q812yes—
81C9⋊C8C3×Q812yes—
82#499—91noSylow
83C2×3−1+2C4⋊Q811yescharacteristic
84C2×3−1+2#13512yes—
85C2×C3×C8—91no—
86C2×C3×C8—91no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—32no—
89C3×C4⋊C4—62no—
90C3×C4⋊C4—92no—
91C3×C4⋊C4—182no—
92C3×C4⋊C4C2×C3×D312yes—
93C3×C42—31no—
94C3×C42C2×C3×D311yescharacteristic
95C4⋊Dic3—32no—
96Dic3⋊C4—62no—
97C2×C3⋊C8—31no—
98C2×C3⋊C8—31no—
99C22×C32C2×#6711yescharacteristic
100C32×C4—22no—
101C32×C4—42no—
102C32×C4D3×Q811yescharacteristic
103C32×C4D3×Q811yescharacteristic
104C3×Dic3—122no—
105C22×C9—21no—
106C22×C9C2×C3×D411yescharacteristic
107C4×C9—22no—
108C4×C9—21no—
109C4×C9—21no—
110C4×C9—42no—
111C4×C9—44no—
112C4×C9—42no—
113C4×C9C2×C3×Q811yescharacteristic
114C4×C9C2×C3×Q811yescharacteristic
115Dic9—42no—
116#125—91no—
117#125AGL(1,ℤ/9)11yescharacteristic
118#107—92no—
119#106—92no—
120C4×C8—91no—
1213−1+2#49911yescharacteristic, complemented, Sylow
122C2×C3×C4—22no—
123C2×C3×C4—32no—
124C2×C3×C4—31no—
125C2×C3×C4—62no—
126C2×C3×C4—182no—
127C2×C3×C4C3×#6712yes—
128C2×C3×C4C22×C3×D311yescharacteristic
129C2×Dic3—62no—
130C3×C8—92no—
131C3×C8—92no—
132C3⋊C8—32no—
133C3⋊C8—32no—
134C2×C32#94411yescharacteristic
135C2×C32#95012yes—
136C2×C9—21no—
137C2×C9—22no—
138C2×C9C3×C4⋊Q811yescharacteristic
139C2×C9C3×#13512yes—
140C2×C8—91no—
141C2×C8—91no—
142C4⋊C4—22no—
143C4⋊C4—92no—
144C4⋊C4—182no—
145C4⋊C4C2×AGL(1,ℤ/9)12yes—
146C42C2×AGL(1,ℤ/9)11yescharacteristic
147C22×C3—31no—
148C22×C3C2×C3×#6711yescharacteristic
149C3×C4—22no—
150C3×C4—31no—
151C3×C4—31no—
152C3×C4—42no—
153C3×C4—62no—
154C3×C4—122no—
155C3×C4—362no—
156C3×C4C3×D3×Q811yescharacteristic
157C3×C4C3×D3×Q811yescharacteristic
158Dic3—122no—
159C32#506511yescharacteristic
160C9—21no—
161C9C3×#49911yescharacteristic, complemented
162C2×C4—22no—
163C2×C4—182no—
164C2×C4#611⋊C312yes—
165C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
166C8—92no—
167C8—92no—
168C2×C3—31no—
169C2×C3—32no—
170C2×C3C3×#94411yescharacteristic
171C2×C3C3×#95012yes—
172C22C2×#611⋊C311yescharacteristic
173C4—22no—
174C4—42no—
175C4—362no—
176C4Q8×AGL(1,ℤ/9)11yescharacteristic
177C4Q8×AGL(1,ℤ/9)11yescharacteristic
178C3—31no—
179C3C3×#506511yescharacteristic
180C2#63506⋊C311yescharacteristic
181C2#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 radicalC22×C32C2×#67
hypercenter#125AGL(1,ℤ/9)
hypocenterC9C3×#499
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
7. #10588744C3semidirect
121. 3−1+2#499semidirect
161. C9C3×#499semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1252
#63432⋊C32
#63367⋊C32
#63366⋊C32
C4×3−1+2⋊C82
C3×#50653
#105887443