Finite Group Databasebeta

← #10588762⋊C3 #13143058

Subgroups — #10588762⋊C3 #13143058

Order 1728.

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

normal subgroups only

This group has 304 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×C4⋊Q8C211yesmaximal, characteristic
2#63432⋊C3C211yesmaximal, characteristic
3#63395⋊C3C214yesmaximal
4#63362⋊C3C211yesmaximal, characteristic
5C3×#5082—31nomaximal
6#10588762C311yesmaximal, characteristic, complemented
7C2×Q8×3−1+2C2212yes—
8C4⋊C4×3−1+2—22no—
9C42×3−1+2C2211yescharacteristic
10C4⋊(3−1+2⋊C4)C2212yes—
11(3−1+2⋊C4)⋊C4—22no—
12C2×3−1+2⋊C8C2212yes—
13C32×C4⋊Q8D311yescharacteristic
14C3×#870—31no—
15C3×#835—34no—
16C3×#803—31no—
17C9×C4⋊Q8—21no—
18C9×C4⋊Q8C2×C311yescharacteristic
19#63432C2×C311yescharacteristic
20#63395C2×C314yes—
21#63362C2×C311yescharacteristic
22Q8×3−1+2—24no—
23C2×C4×3−1+2—22no—
24C2×C4×3−1+2D412yes—
25C2×C4×3−1+2C2311yescharacteristic
26C2×3−1+2⋊C4—22no—
273−1+2⋊C8—22no—
28C3×#491—91no—
29#5082—31no—
30C2×C32×Q8C2×D312yes—
31C32×C4⋊C4—22no—
32C32×C42C2×D311yescharacteristic
33C3×C4⋊Dic3—32no—
34C3×Dic3⋊C4—62no—
35C2×C3×C3⋊C8—32no—
36C2×Q8×C9—22no—
37C2×Q8×C9C22×C312yes—
38C9×C4⋊C4—24no—
39C9×C4⋊C4—22no—
40C42×C9—21no—
41C42×C9C22×C311yescharacteristic
42C4⋊Dic9C22×C312yes—
43Dic9⋊C4—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—31no—
51C3×C4⋊Q8C3×D311yescharacteristic
52C3×#125—91no—
53#870—31no—
54C3×#103—94no—
55C3×#97—91no—
56#835—34no—
57#803—31no—
58C32×Q8—24no—
59C2×C32×C4—22no—
60C2×C32×C4#6712yes—
61C2×C32×C4C22×D311yescharacteristic
62C2×C3×Dic3—62no—
63C3×C3⋊C8—62no—
64Q8×C9—24no—
65Q8×C9—44no—
66C2×C4×C9—24no—
67C2×C4×C9—22no—
68C2×C4×C9—22no—
69C2×C4×C9—21no—
70C2×C4×C9C3×D412yes—
71C2×C4×C9C23×C311yescharacteristic
72C2×Dic9—22no—
73C9⋊C8—22no—
74#491—91noSylow
75C2×3−1+2#12411yescharacteristic
76C2×3−1+2#13712yes—
77C2×C3×Q8—32no—
78C2×C3×Q8C2×C3×D312yes—
79C2×C3×C8—92no—
80C3×C4⋊C4—22no—
81C3×C4⋊C4—62no—
82C3×C4⋊C4—92no—
83C3×C4⋊C4—182no—
84C3×C42—31no—
85C3×C42C2×C3×D311yescharacteristic
86C4⋊Dic3—32no—
87Dic3⋊C4—62no—
88C2×C3⋊C8—32no—
89C22×C32C2×#6711yescharacteristic
90C32×C4—22no—
91C32×C4—42no—
92C32×C4#23712yes—
93C3×Dic3—122no—
94C22×C9—21no—
95C22×C9C2×C3×D411yescharacteristic
96C4×C9—24no—
97C4×C9—22no—
98C4×C9—22no—
99C4×C9—44no—
100C4×C9—42no—
101C4×C9C3×#4112yes—
102Dic9—42no—
103C4⋊Q8AGL(1,ℤ/9)11yescharacteristic
104#125—91no—
105#103—94no—
106#97—91no—
1073−1+2#49111yescharacteristic, complemented, Sylow
108C3×Q8—24no—
109C3×Q8—64no—
110C2×C3×C4—22no—
111C2×C3×C4—32no—
112C2×C3×C4—31no—
113C2×C3×C4—62no—
114C2×C3×C4—182no—
115C2×C3×C4C3×#6712yes—
116C2×C3×C4C22×C3×D311yescharacteristic
117C2×Dic3—62no—
118C3×C8—182no—
119C3⋊C8—62no—
120C2×C32#93611yescharacteristic
121C2×C32#94212yes—
122C2×C9—21no—
123C2×C9—22no—
124C2×C9C3×#12411yescharacteristic
125C2×C9C3×#13712yes—
126C2×Q8C2×AGL(1,ℤ/9)12yes—
127C2×C8—92no—
128C4⋊C4—22no—
129C4⋊C4—92no—
130C4⋊C4—182no—
131C42C2×AGL(1,ℤ/9)11yescharacteristic
132C22×C3—31no—
133C22×C3C2×C3×#6711yescharacteristic
134C3×C4—22no—
135C3×C4—32no—
136C3×C4—42no—
137C3×C4—62no—
138C3×C4—122no—
139C3×C4—362no—
140C3×C4C3×#23712yes—
141Dic3—122no—
142C32#508211yescharacteristic
143C9—21no—
144C9C3×#49111yescharacteristic, complemented
145Q8—24no—
146C2×C4—22no—
147C2×C4—182no—
148C2×C4#611⋊C312yes—
149C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
150C8—182no—
151C2×C3—31no—
152C2×C3—32no—
153C2×C3C3×#93611yescharacteristic
154C2×C3C3×#94212yes—
155C22C2×#611⋊C311yescharacteristic
156C4—22no—
157C4—42no—
158C4—362no—
159C4#3699⋊C312yes—
160C3—31no—
161C3C3×#508211yescharacteristic
162C2#63498⋊C311yescharacteristic
163C2#63505⋊C312yes—
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C4⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC22×C32C2×#67
hypercenterC4⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#491
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
6. #10588762C3semidirect
107. 3−1+2#491semidirect
144. 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×C4⋊Q82
#63432⋊C32
#63395⋊C32
#63362⋊C32
C3×#50823
#105887623