Finite Group Databasebeta

← #10588746⋊C3 #13142960

Subgroups — #10588746⋊C3 #13142960

Order 1728.

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

normal subgroups only

This group has 298 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
2#63431⋊C3C211yesmaximal, characteristic
3#63368⋊C3C214yesmaximal
4#63362⋊C3C211yesmaximal, characteristic
5C3×#5069—31nomaximal
6#10588746C311yesmaximal, characteristic, complemented
7C2×3−1+2⋊Q8C2212yes—
8C4⋊C4×3−1+2—22no—
9C4⋊C4×3−1+2C2212yes—
10C42×3−1+2C2211yescharacteristic
11C4⋊(3−1+2⋊C4)—22no—
12C2×3−1+2⋊C8C2212yes—
13C32×#125D311yescharacteristic
14C3×#869—31no—
15C3×#810—34no—
16C3×#803—31no—
17C9×#125—21no—
18C9×#125C2×C311yescharacteristic
19#63431C2×C311yescharacteristic
20#63368C2×C314yes—
21#63362C2×C311yescharacteristic
22C2×C4×3−1+2—22no—
23C2×C4×3−1+2D412yes—
24C2×C4×3−1+2C2311yescharacteristic
25C2×3−1+2⋊C4—22no—
263−1+2⋊Q8—24no—
273−1+2⋊C8—22no—
28C3×#491—91no—
29#5069—31no—
30C2×C3×Dic6—32no—
31C32×C4⋊C4—22no—
32C32×C4⋊C4C2×D312yes—
33C32×C42C2×D311yescharacteristic
34C3×C4⋊Dic3—62no—
35C2×C3×C3⋊C8—32no—
36C2×Dic18C22×C312yes—
37C9×C4⋊C4—24no—
38C9×C4⋊C4—22no—
39C9×C4⋊C4—22no—
40C9×C4⋊C4C22×C312yes—
41C42×C9—21no—
42C42×C9C22×C311yescharacteristic
43C4⋊Dic9—22no—
44C2×C9⋊C8C22×C312yes—
45C22×3−1+2C2×D411yescharacteristic
46C4×3−1+2—22no—
47C4×3−1+2—42no—
48C4×3−1+2#4112yes—
493−1+2⋊C4—42no—
50C3×C4⋊Q8—91no—
51C3×#125—31no—
52C3×#125C3×D311yescharacteristic
53#869—31no—
54C3×#103—94no—
55C3×#97—91no—
56#810—34no—
57#803—31no—
58C2×C32×C4—22no—
59C2×C32×C4#6712yes—
60C2×C32×C4C22×D311yescharacteristic
61C2×C3×Dic3—62no—
62C3×Dic6—64no—
63C3×C3⋊C8—62no—
64C2×C4×C9—24no—
65C2×C4×C9—22no—
66C2×C4×C9—22no—
67C2×C4×C9—21no—
68C2×C4×C9C3×D412yes—
69C2×C4×C9C23×C311yescharacteristic
70C2×Dic9—22no—
71Dic18—24no—
72C9⋊C8—22no—
73#491—91noSylow
74C2×3−1+2#12411yescharacteristic
75C2×3−1+2#13712yes—
76C2×C3×Q8—92no—
77C2×Dic6—32no—
78C2×C3×C8—92no—
79C3×C4⋊C4—22no—
80C3×C4⋊C4—32no—
81C3×C4⋊C4—62no—
82C3×C4⋊C4—182no—
83C3×C4⋊C4C2×C3×D312yes—
84C3×C42—31no—
85C3×C42C2×C3×D311yescharacteristic
86C4⋊Dic3—62no—
87C2×C3⋊C8—32no—
88C22×C32C2×#6711yescharacteristic
89C32×C4—22no—
90C32×C4—42no—
91C32×C4#23912yes—
92C3×Dic3—122no—
93C22×C9—21no—
94C22×C9C2×C3×D411yescharacteristic
95C4×C9—22no—
96C4×C9—22no—
97C4×C9—42no—
98C4×C9—44no—
99C4×C9—42no—
100C4×C9C3×#4112yes—
101Dic9—42no—
102C4⋊Q8—91no—
103#125AGL(1,ℤ/9)11yescharacteristic
104#103—94no—
105#97—91no—
1063−1+2#49111yescharacteristic, complemented, Sylow
107C3×Q8—184no—
108C2×C3×C4—22no—
109C2×C3×C4—32no—
110C2×C3×C4—31no—
111C2×C3×C4—62no—
112C2×C3×C4—182no—
113C2×C3×C4C3×#6712yes—
114C2×C3×C4C22×C3×D311yescharacteristic
115C2×Dic3—62no—
116Dic6—64no—
117C3×C8—182no—
118C3⋊C8—62no—
119C2×C32#94711yescharacteristic
120C2×C32#95112yes—
121C2×C9—21no—
122C2×C9—22no—
123C2×C9C3×#12411yescharacteristic
124C2×C9C3×#13712yes—
125C2×Q8—92no—
126C2×C8—92no—
127C4⋊C4—22no—
128C4⋊C4—182no—
129C4⋊C4C2×AGL(1,ℤ/9)12yes—
130C42C2×AGL(1,ℤ/9)11yescharacteristic
131C22×C3—31no—
132C22×C3C2×C3×#6711yescharacteristic
133C3×C4—22no—
134C3×C4—32no—
135C3×C4—42no—
136C3×C4—62no—
137C3×C4—122no—
138C3×C4—362no—
139C3×C4C3×#23912yes—
140Dic3—122no—
141C32#506911yescharacteristic
142C9—21no—
143C9C3×#49111yescharacteristic, complemented
144Q8—184no—
145C2×C4—22no—
146C2×C4—182no—
147C2×C4#611⋊C312yes—
148C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
149C8—182no—
150C2×C3—31no—
151C2×C3—32no—
152C2×C3C3×#94711yescharacteristic
153C2×C3C3×#95112yes—
154C22C2×#611⋊C311yescharacteristic
155C4—22no—
156C4—42no—
157C4—362no—
158C4#8872412yes—
159C3—31no—
160C3C3×#506911yescharacteristic
161C2#63509⋊C311yescharacteristic
162C2#63511⋊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×#491
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
6. #10588746C3semidirect
106. 3−1+2#491semidirect
143. C9C3×#491semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1252
#63431⋊C32
#63368⋊C32
#63362⋊C32
C3×#50693
#105887463