Finite Group Databasebeta

← #10588558⋊C3 #13142238

Subgroups — #10588558⋊C3 #13142238

Order 1728.

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

normal subgroups only

This group has 276 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
1#63452⋊C3C212yesmaximal
23−1+2×#107C211yesmaximal, characteristic
3#63378⋊C3C211yesmaximal, characteristic
4C8×3−1+2⋊C4C211yesmaximal, characteristic
5#63367⋊C3C212yesmaximal
6C3×#4878—31nomaximal
7#10588558C311yesmaximal, characteristic, complemented
8C2×C8×3−1+2C2211yescharacteristic
9C4⋊C4×3−1+2C2212yes—
10C4⋊(3−1+2⋊C4)—22no—
11C4⋊(3−1+2⋊C4)C2212yes—
12(3−1+2⋊C4)⋊C4—22no—
13C4×3−1+2⋊C4C2211yescharacteristic
14C2×3−1+2⋊C8C2211yescharacteristic
15C3×#890—32no—
16C32×#107D311yescharacteristic
17C3×#818—31no—
18C3×C8×Dic3—31no—
19C3×#808—32no—
20#63452C2×C312yes—
21C9×#107—21no—
22C9×#107C2×C311yescharacteristic
23#63378C2×C311yescharacteristic
24C8×Dic9C2×C311yescharacteristic
25#63367C2×C312yes—
26C2×C4×3−1+2—22no—
27C2×C4×3−1+2C2311yescharacteristic
28C2×3−1+2⋊C4—22no—
29C2×3−1+2⋊C4D411yescharacteristic
30C2×3−1+2⋊C4D411yescharacteristic
31C8×3−1+2Q812yes—
323−1+2⋊C8Q812yes—
33C3×#499—91no—
34#4878—31no—
35C2×C32×C8C2×D311yescharacteristic
36C32×C4⋊C4C2×D312yes—
37C3×C4⋊Dic3—32no—
38C3×C4⋊Dic3—62no—
39C3×Dic3⋊C4—62no—
40C3×C4×Dic3—31no—
41C2×C3×C3⋊C8—31no—
42C2×C8×C9—21no—
43C2×C8×C9C22×C311yescharacteristic
44C9×C4⋊C4—22no—
45C9×C4⋊C4C22×C312yes—
46C4⋊Dic9—22no—
47C4⋊Dic9C22×C312yes—
48Dic9⋊C4—22no—
49C4×Dic9C22×C311yescharacteristic
50C2×C9⋊C8C22×C311yescharacteristic
51C22×3−1+2C2×D411yescharacteristic
52C4×3−1+2—42no—
53C4×3−1+2C2×Q811yescharacteristic
54C4×3−1+2C2×Q811yescharacteristic
553−1+2⋊C4—21no—
563−1+2⋊C4—21no—
573−1+2⋊C4—42no—
58C3×#125—92no—
59#890—32no—
60C3×#107—31no—
61C3×#107—91no—
62C3×#107C3×D311yescharacteristic
63C3×#106—92no—
64C3×C4×C8—91no—
65#818—31no—
66C8×Dic3—31no—
67#808—32no—
68C2×C32×C4—22no—
69C2×C32×C4C22×D311yescharacteristic
70C2×C3×Dic3—31no—
71C2×C3×Dic3—31no—
72C2×C3×Dic3—62no—
73C32×C8Dic612yes—
74C3×C3⋊C8—32no—
75C2×C4×C9—22no—
76C2×C4×C9—21no—
77C2×C4×C9—42no—
78C2×C4×C9C23×C311yescharacteristic
79C2×Dic9—22no—
80C2×Dic9C3×D411yescharacteristic
81C2×Dic9C3×D411yescharacteristic
82C8×C9—22no—
83C8×C9C3×Q812yes—
84C9⋊C8C3×Q812yes—
85#499—91noSylow
86C2×3−1+2C4⋊Q811yescharacteristic
87C2×3−1+2#13511yescharacteristic
88C2×3−1+2#13511yescharacteristic
89C2×C3×C8—31no—
90C2×C3×C8—91no—
91C2×C3×C8C2×C3×D311yescharacteristic
92C3×C4⋊C4—32no—
93C3×C4⋊C4—92no—
94C3×C4⋊C4—182no—
95C3×C4⋊C4—182no—
96C3×C4⋊C4C2×C3×D312yes—
97C3×C42—91no—
98C4⋊Dic3—32no—
99C4⋊Dic3—62no—
100Dic3⋊C4—62no—
101C4×Dic3—31no—
102C2×C3⋊C8—31no—
103C22×C32D3×D411yescharacteristic
104C32×C4—42no—
105C32×C4C2×Dic611yescharacteristic
106C32×C4D3×Q811yescharacteristic
107C3×Dic3—61no—
108C3×Dic3—61no—
109C3×Dic3—122no—
110C22×C9—21no—
111C22×C9C2×C3×D411yescharacteristic
112C4×C9—21no—
113C4×C9—21no—
114C4×C9—42no—
115C4×C9—82no—
116C4×C9C2×C3×Q811yescharacteristic
117C4×C9C2×C3×Q811yescharacteristic
118Dic9—21no—
119Dic9—21no—
120Dic9—42no—
121#125—92no—
122#107—91no—
123#107AGL(1,ℤ/9)11yescharacteristic
124#106—92no—
125C4×C8—91no—
1263−1+2#49911yescharacteristic, complemented, Sylow
127C2×C3×C4—22no—
128C2×C3×C4—31no—
129C2×C3×C4—62no—
130C2×C3×C4—91no—
131C2×C3×C4—91no—
132C2×C3×C4—182no—
133C2×C3×C4C22×C3×D311yescharacteristic
134C2×Dic3—31no—
135C2×Dic3—31no—
136C2×Dic3—62no—
137C3×C8—32no—
138C3×C8—92no—
139C3×C8C3×Dic612yes—
140C3⋊C8—32no—
141C2×C32#88811yescharacteristic
142C2×C32#91211yescharacteristic
143C2×C32#91911yescharacteristic
144C2×C9—21no—
145C2×C9—21no—
146C2×C9—21no—
147C2×C9C3×C4⋊Q811yescharacteristic
148C2×C9C3×#13511yescharacteristic
149C2×C9C3×#13511yescharacteristic
150C2×C8—91no—
151C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
152C4⋊C4—92no—
153C4⋊C4—182no—
154C4⋊C4—182no—
155C4⋊C4C2×AGL(1,ℤ/9)12yes—
156C42—91no—
157C22×C3—31no—
158C22×C3C3×D3×D411yescharacteristic
159C3×C4—31no—
160C3×C4—31no—
161C3×C4—42no—
162C3×C4—122no—
163C3×C4—181no—
164C3×C4—181no—
165C3×C4—362no—
166C3×C4C2×C3×Dic611yescharacteristic
167C3×C4C3×D3×Q811yescharacteristic
168Dic3—61no—
169Dic3—61no—
170Dic3—122no—
171C32#487811yescharacteristic
172C9—21no—
173C9C3×#49911yescharacteristic, complemented
174C2×C4—22no—
175C2×C4—91no—
176C2×C4—91no—
177C2×C4—182no—
178C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
179C8—92no—
180C83−1+2⋊Q812yes—
181C2×C3—31no—
182C2×C3—31no—
183C2×C3—31no—
184C2×C3C3×#88811yescharacteristic
185C2×C3C3×#91211yescharacteristic
186C2×C3C3×#91911yescharacteristic
187C22D4×AGL(1,ℤ/9)11yescharacteristic
188C4—42no—
189C4—181no—
190C4—181no—
191C4—362no—
192C4C2×3−1+2⋊Q811yescharacteristic
193C4Q8×AGL(1,ℤ/9)11yescharacteristic
194C3—31no—
195C3C3×#487811yescharacteristic
196C2#63450⋊C311yescharacteristic
197C2#63474⋊C311yescharacteristic
198C2#63481⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×C32D3×D4
hypercenter#107AGL(1,ℤ/9)
hypocenterC9C3×#499
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
7. #10588558C3semidirect
126. 3−1+2#499semidirect
173. C9C3×#499semidirect

Maximal subgroups

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

subgroupindex
#63452⋊C32
3−1+2×#1072
#63378⋊C32
C8×3−1+2⋊C42
#63367⋊C32
C3×#48783
#105885583