Finite Group Databasebeta

← #10588760⋊C3 #13143055

Subgroups — #10588760⋊C3 #13143055

Order 1728.

328 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 120 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#63431⋊C3C211yesmaximal, characteristic
3#63367⋊C3C214yesmaximal
4C4×3−1+2⋊C8C211yesmaximal, characteristic
5C3×#5080—31nomaximal
6#10588760C311yesmaximal, characteristic, complemented
7C2×Q8×3−1+2—21no—
8C2×3−1+2⋊Q8—21no—
9C4⋊C4×3−1+2—21no—
10C4⋊C4×3−1+2C2212yes—
11C42×3−1+2C2211yescharacteristic
12C4⋊(3−1+2⋊C4)—21no—
13C4⋊(3−1+2⋊C4)C2212yes—
14C2×3−1+2⋊C8C2212yes—
15C32×C4⋊Q8D311yescharacteristic
16C3×#869—31no—
17C3×#808—34no—
18C3×C4×C3⋊C8—31no—
19C9×C4⋊Q8—21no—
20C9×C4⋊Q8C2×C311yescharacteristic
21#63431C2×C311yescharacteristic
22#63367C2×C314yes—
23C4×C9⋊C8C2×C311yescharacteristic
24Q8×3−1+2—42no—
25C2×C4×3−1+2—22no—
26C2×C4×3−1+2D411yescharacteristic
27C2×C4×3−1+2D411yescharacteristic
28C2×C4×3−1+2C2311yescharacteristic
29C2×3−1+2⋊C4—22no—
303−1+2⋊Q8—42no—
313−1+2⋊C8Q814yescomplemented
32C3×#498—91no—
33#5080—31no—
34C2×C32×Q8—21no—
35C2×C3×Dic6—61no—
36C32×C4⋊C4—21no—
37C32×C4⋊C4C2×D312yes—
38C32×C42C2×D311yescharacteristic
39C3×C4⋊Dic3—32no—
40C3×C4⋊Dic3—61no—
41C2×C3×C3⋊C8—32no—
42C2×Q8×C9—22no—
43C2×Q8×C9—21no—
44C2×Dic18—21no—
45C9×C4⋊C4—22no—
46C9×C4⋊C4—22no—
47C9×C4⋊C4—21no—
48C9×C4⋊C4C22×C312yes—
49C42×C9—21no—
50C42×C9C22×C311yescharacteristic
51C4⋊Dic9—21no—
52C4⋊Dic9C22×C312yes—
53C2×C9⋊C8C22×C312yes—
54C22×3−1+2C2×D411yescharacteristic
55C4×3−1+2—42no—
56C4×3−1+2SD1612yes—
57C4×3−1+2SD1612yes—
58C4×3−1+2C2×Q812yes—
593−1+2⋊C4—42no—
60C3×C4⋊Q8—31no—
61C3×C4⋊Q8—91no—
62C3×C4⋊Q8C3×D311yescharacteristic
63#869—31no—
64C3×#106—94no—
65C3×C4×C8—91no—
66#808—34no—
67C4×C3⋊C8—31no—
68C32×Q8—42no—
69C2×C32×C4—22no—
70C2×C32×C4#6711yescharacteristic
71C2×C32×C4#6711yescharacteristic
72C2×C32×C4C22×D311yescharacteristic
73C2×C3×Dic3—62no—
74C3×Dic6—122no—
75C3×C3⋊C8—34no—
76Q8×C9—44no—
77Q8×C9—42no—
78C2×C4×C9—24no—
79C2×C4×C9—22no—
80C2×C4×C9—21no—
81C2×C4×C9—21no—
82C2×C4×C9—21no—
83C2×C4×C9C3×D411yescharacteristic
84C2×C4×C9C3×D411yescharacteristic
85C2×C4×C9C23×C311yescharacteristic
86C2×Dic9—22no—
87Dic18—42no—
88C9⋊C8C3×Q814yescomplemented
89#498—91noSylow
90C2×3−1+2C4⋊Q811yescharacteristic
91C2×3−1+2C2×SD1611yescharacteristic
92C2×3−1+2C2×SD1611yescharacteristic
93C2×C3×Q8—21no—
94C2×C3×Q8—61no—
95C2×C3×Q8—181no—
96C2×Dic6—61no—
97C2×C3×C8—92no—
98C3×C4⋊C4—21no—
99C3×C4⋊C4—32no—
100C3×C4⋊C4—61no—
101C3×C4⋊C4—92no—
102C3×C4⋊C4—181no—
103C3×C4⋊C4C2×C3×D312yes—
104C3×C42—31no—
105C3×C42C2×C3×D311yescharacteristic
106C4⋊Dic3—32no—
107C4⋊Dic3—61no—
108C2×C3⋊C8—32no—
109C22×C32C2×#6711yescharacteristic
110C32×C4—42no—
111C32×C4#21412yes—
112C32×C4#21512yes—
113C32×C4D3×Q812yes—
114C3×Dic3—122no—
115C22×C9—21no—
116C22×C9C2×C3×D411yescharacteristic
117C4×C9—22no—
118C4×C9—22no—
119C4×C9—22no—
120C4×C9—44no—
121C4×C9—42no—
122C4×C9C3×SD1612yes—
123C4×C9C3×SD1612yes—
124C4×C9C2×C3×Q812yes—
125Dic9—42no—
126C4⋊Q8—91no—
127C4⋊Q8AGL(1,ℤ/9)11yescharacteristic
128#106—94no—
129C4×C8—91no—
1303−1+2#49811yescharacteristic, complemented, Sylow
131C3×Q8—42no—
132C3×Q8—122no—
133C3×Q8—362no—
134C2×C3×C4—22no—
135C2×C3×C4—31no—
136C2×C3×C4—31no—
137C2×C3×C4—31no—
138C2×C3×C4—62no—
139C2×C3×C4—182no—
140C2×C3×C4C3×#6711yescharacteristic
141C2×C3×C4C3×#6711yescharacteristic
142C2×C3×C4C22×C3×D311yescharacteristic
143C2×Dic3—62no—
144Dic6—122no—
145C3×C8—94no—
146C3⋊C8—34no—
147C2×C32C2×#21411yescharacteristic
148C2×C32C2×#21511yescharacteristic
149C2×C32#94411yescharacteristic
150C2×C9—21no—
151C2×C9—21no—
152C2×C9—21no—
153C2×C9C3×C4⋊Q811yescharacteristic
154C2×C9C2×C3×SD1611yescharacteristic
155C2×C9C2×C3×SD1611yescharacteristic
156C2×Q8—21no—
157C2×Q8—181no—
158C2×C8—92no—
159C4⋊C4—21no—
160C4⋊C4—92no—
161C4⋊C4—181no—
162C4⋊C4C2×AGL(1,ℤ/9)12yes—
163C42C2×AGL(1,ℤ/9)11yescharacteristic
164C22×C3—31no—
165C22×C3C2×C3×#6711yescharacteristic
166C3×C4—32no—
167C3×C4—32no—
168C3×C4—32no—
169C3×C4—42no—
170C3×C4—122no—
171C3×C4—362no—
172C3×C4C3×#21412yes—
173C3×C4C3×#21512yes—
174C3×C4C3×D3×Q812yes—
175Dic3—122no—
176C32#508011yescharacteristic
177C9—21no—
178C9C3×#49811yescharacteristic, complemented
179Q8—42no—
180Q8—362no—
181C2×C4—22no—
182C2×C4—182no—
183C2×C4#611⋊C311yescharacteristic
184C2×C4#611⋊C311yescharacteristic
185C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
186C8—94no—
187C2×C3—31no—
188C2×C3—31no—
189C2×C3—31no—
190C2×C3C2×C3×#21411yescharacteristic
191C2×C3C2×C3×#21511yescharacteristic
192C2×C3C3×#94411yescharacteristic
193C22C2×#611⋊C311yescharacteristic
194C4—42no—
195C4—362no—
196C4#3672⋊C312yes—
197C4#3675⋊C312yes—
198C4Q8×AGL(1,ℤ/9)12yes—
199C3—31no—
200C3C3×#508011yescharacteristic
201C2C2×#3672⋊C311yescharacteristic
202C2C2×#3675⋊C311yescharacteristic
203C2#63506⋊C311yescharacteristic
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×#498
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. #10588760C3semidirect
31. 3−1+2⋊C8Q8semidirect
88. C9⋊C8C3×Q8semidirect
130. 3−1+2#498semidirect
178. C9C3×#498semidirect

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
#63431⋊C32
#63367⋊C32
C4×3−1+2⋊C82
C3×#50803
#105887603