Finite Group Databasebeta

← 3−1+2⋊#564 #13153193

Subgroups — 3−1+2⋊#564 #13153193

Order 1728.

544 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 202 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#63452⋊C3C214yesmaximal
3#63451⋊C3C214yesmaximal
4#63451⋊C3C214yesmaximal
5#63432⋊C3C212yesmaximal
6C3×C3⋊#564—31nomaximal
7C9⋊#564C311yesmaximal, characteristic, complemented
8C4⋊C4×3−1+2C2212yes—
9C4⋊C4×3−1+2C2214yes—
10C42×3−1+2C2211yescharacteristic
11C4⋊(3−1+2⋊C4)C2212yes—
12C4⋊(3−1+2⋊C4)C2214yes—
13C4⋊(3−1+2⋊C4)C2212yes—
14(3−1+2⋊C4)⋊C4C2214yes—
15(3−1+2⋊C4)⋊C4C2214yes—
16(3−1+2⋊C4)⋊C4C2214yes—
17(3−1+2⋊C4)⋊C4C2214yes—
18C4×3−1+2⋊C4C2214yes—
19C32×#125D311yescharacteristic
20C3×#890—34no—
21C3×#889—34no—
22C3×#889—34no—
23C3×#870—32no—
24C9×#125—21no—
25C9×#125C2×C311yescharacteristic
26#63452C2×C314yes—
27#63451C2×C314yes—
28#63451C2×C314yes—
29#63432C2×C312yes—
30C2×C4×3−1+2C2314yes—
31C2×C4×3−1+2C2311yescharacteristic
32C2×C4×3−1+2C2312yes—
33C2×3−1+2⋊C4C2314yes—
34C2×3−1+2⋊C4C2314yes—
35C3×#564—91no—
36C3⋊#564—31no—
37C32×C4⋊C4C2×D312yes—
38C32×C4⋊C4C2×D314yes—
39C32×C42C2×D311yescharacteristic
40C3×C4⋊Dic3—32no—
41C3×C4⋊Dic3—34no—
42C3×C4⋊Dic3—32no—
43C3×Dic3⋊C4—34no—
44C3×Dic3⋊C4—34no—
45C3×Dic3⋊C4—34no—
46C3×Dic3⋊C4—34no—
47C3×C4×Dic3—34no—
48C9×C4⋊C4—22no—
49C9×C4⋊C4—24no—
50C9×C4⋊C4C22×C312yes—
51C9×C4⋊C4C22×C314yes—
52C42×C9—21no—
53C42×C9C22×C311yescharacteristic
54C4⋊Dic9C22×C312yes—
55C4⋊Dic9C22×C312yes—
56C4⋊Dic9C22×C314yes—
57Dic9⋊C4C22×C314yes—
58Dic9⋊C4C22×C314yes—
59Dic9⋊C4C22×C314yes—
60Dic9⋊C4C22×C314yes—
61C4×Dic9C22×C314yes—
62C22×3−1+2C2411yescharacteristic
63C4×3−1+2—21no—
64C4×3−1+2—24no—
65C4×3−1+2—22no—
663−1+2⋊C4—24no—
673−1+2⋊C4—24no—
68C3×#125—31no—
69C3×#125—92no—
70C3×#125—94no—
71C3×#125—94no—
72C3×#125—94no—
73C3×#125C3×D311yescharacteristic
74#890—34no—
75#889—34no—
76#889—34no—
77#870—32no—
78C2×C32×C4C22×D314yes—
79C2×C32×C4C22×D311yescharacteristic
80C2×C32×C4C22×D312yes—
81C2×C3×Dic3—34no—
82C2×C3×Dic3—34no—
83C2×C4×C9—21no—
84C2×C4×C9—22no—
85C2×C4×C9—24no—
86C2×C4×C9C23×C314yes—
87C2×C4×C9C23×C311yescharacteristic
88C2×C4×C9C23×C312yes—
89C2×Dic9C23×C314yes—
90C2×Dic9C23×C314yes—
91#564—91noSylow
92C2×3−1+22−1+411yescharacteristic
93C2×3−1+22−1+412yes—
94C3×C4⋊C4—32no—
95C3×C4⋊C4—34no—
96C3×C4⋊C4—94no—
97C3×C4⋊C4—94no—
98C3×C4⋊C4—94no—
99C3×C4⋊C4—94no—
100C3×C4⋊C4—92no—
101C3×C4⋊C4—92no—
102C3×C4⋊C4—94no—
103C3×C4⋊C4C2×C3×D312yes—
104C3×C4⋊C4C2×C3×D314yes—
105C3×C42—31no—
106C3×C42—94no—
107C3×C42C2×C3×D311yescharacteristic
108C4⋊Dic3—32no—
109C4⋊Dic3—32no—
110C4⋊Dic3—34no—
111Dic3⋊C4—34no—
112Dic3⋊C4—34no—
113Dic3⋊C4—34no—
114Dic3⋊C4—34no—
115C4×Dic3—34no—
116C22×C32C23×D311yescharacteristic
117C32×C4—21no—
118C32×C4—24no—
119C32×C4—22no—
120C3×Dic3—64no—
121C3×Dic3—64no—
122C22×C9—21no—
123C22×C9C24×C311yescharacteristic
124C4×C9—22no—
125C4×C9—21no—
126C4×C9—24no—
127C4×C9—22no—
128C4×C9—42no—
129C4×C9—44no—
130Dic9—24no—
131Dic9—24no—
132#125—94no—
133#125—94no—
134#125—94no—
135#125—92no—
136#125AGL(1,ℤ/9)11yescharacteristic
1373−1+2#56411yescharacteristic, complemented, Sylow
138C2×C3×C4—31no—
139C2×C3×C4—32no—
140C2×C3×C4—34no—
141C2×C3×C4—94no—
142C2×C3×C4—94no—
143C2×C3×C4C22×C3×D314yes—
144C2×C3×C4C22×C3×D311yescharacteristic
145C2×C3×C4C22×C3×D312yes—
146C2×Dic3—34no—
147C2×Dic3—34no—
148C2×C32#100711yescharacteristic
149C2×C32#101012yes—
150C2×C9—21no—
151C2×C9—22no—
152C2×C9C3×2−1+411yescharacteristic
153C2×C9C3×2−1+412yes—
154C4⋊C4—94no—
155C4⋊C4—94no—
156C4⋊C4—94no—
157C4⋊C4—94no—
158C4⋊C4—94no—
159C4⋊C4—92no—
160C4⋊C4—92no—
161C4⋊C4C2×AGL(1,ℤ/9)12yes—
162C4⋊C4C2×AGL(1,ℤ/9)14yes—
163C42—94no—
164C42C2×AGL(1,ℤ/9)11yescharacteristic
165C22×C3—31no—
166C22×C3C23×C3×D311yescharacteristic
167C3×C4—21no—
168C3×C4—24no—
169C3×C4—22no—
170C3×C4—61no—
171C3×C4—62no—
172C3×C4—64no—
173C3×C4—184no—
174C3×C4—184no—
175Dic3—64no—
176Dic3—64no—
177C32C3⋊#56411yescharacteristic
178C9—21no—
179C9C3×#56411yescharacteristic, complemented
180C2×C4—94no—
181C2×C4—94no—
182C2×C4C22×AGL(1,ℤ/9)14yes—
183C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
184C2×C4C22×AGL(1,ℤ/9)12yes—
185C2×C3—31no—
186C2×C3—32no—
187C2×C3C3×#100711yescharacteristic
188C2×C3C3×#101012yes—
189C22C23×AGL(1,ℤ/9)11yescharacteristic
190C4—21no—
191C4—24no—
192C4—22no—
193C4—184no—
194C4—184no—
195C3—31no—
196C3C3×C3⋊#56411yescharacteristic
197C2#1172263011yescharacteristic
198C2#1172264212yes—
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#125C2
prime radicalC22×C32C23×D3
hypercenter#125AGL(1,ℤ/9)
hypocenterC9C3×#564
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
7. C9⋊#564C3semidirect
137. 3−1+2#564semidirect
179. C9C3×#564semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1252
#63452⋊C32
#63451⋊C32
#63432⋊C32
C3×C3⋊#5643
C9⋊#5643