Finite Group Databasebeta

← C23×#611⋊C3 #13164970

Subgroups — C23×#611⋊C3 #13164970

Order 1728.

4460 conjugacy classes of subgroups; only the 1204 normal subgroups are shown.

normal subgroups only

This group has 4460 conjugacy classes of subgroups — too many to draw. Only its 1204 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
1C25×3−1+2C211yesmaximal, characteristic, complemented
2C24×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
3C22×#611⋊C3C2128yesmaximal, direct product factor
4C23×3−1+2⋊C4C211yesmaximal, characteristic, complemented
5C23×C3×#67—31nomaximal
6C23×#611C311yesmaximal, characteristic, complemented
7C24×3−1+2—28no—
8C24×3−1+2C2211yescharacteristic
9C24×3−1+2C22114yescomplemented
10C23×AGL(1,ℤ/9)—28no—
11C23×AGL(1,ℤ/9)C22114yescomplemented
12C2×#611⋊C3C221112yesdirect product factor
13C22×3−1+2⋊C4C22114yescomplemented
14C25×C32D311yescharacteristic
15C24×C3×D3—31no—
16C22×C3×#67—328no—
17C23×C3×Dic3—31no—
18C25×C9—21no—
19C25×C9C2×C311yescharacteristic, complemented
20C24×D9C2×C311yescharacteristic, complemented
21C22×#611C2×C3128yescomplemented
22C23×Dic9C2×C311yescharacteristic, complemented
23C23×3−1+2—256no—
24C23×3−1+2D418yescomplemented
25C23×3−1+2C2317yes—
26C23×3−1+2C23128yescomplemented
27C22×AGL(1,ℤ/9)—256no—
28C22×AGL(1,ℤ/9)C23128yescomplemented
29#611⋊C3C23164yesdirect product factor
30C2×3−1+2⋊C4C23128yescomplemented
31C23×C3×D4—91no—
32C23×#67—31no—
33C24×C32—28no—
34C24×C32C2×D3114yes—
35C24×C32C2×D311yescharacteristic
36C23×C3×D3—314no—
37C23×C3×D3—68no—
38C2×C3×#67—3112no—
39C22×C3×Dic3—314no—
40C24×C9—216no—
41C24×C9—21no—
42C24×C9—214no—
43C24×C9—28no—
44C24×C9C22×C311yescharacteristic
45C24×C9C22×C3114yescomplemented
46C23×D9—28no—
47C23×D9C22×C3114yescomplemented
48C2×#611C22×C31112yescomplemented
49C22×Dic9C22×C3114yescomplemented
50C22×3−1+2—256no—
51C22×3−1+2C2×D4128yescomplemented
52C22×3−1+2C2417yes—
53C22×3−1+2C2418yescomplemented
54C2×AGL(1,ℤ/9)—256no—
55C2×AGL(1,ℤ/9)C2418yescomplemented
563−1+2⋊C4C2418yescomplemented
57C25×C3—31no—
58C25×C3—91no—
59C25×C3C3×D311yescharacteristic
60C24×D3—31no—
61C22×C3×D4—928no—
62C23×C3×C4—91no—
63C22×#67—328no—
64C23×Dic3—31no—
65C23×C32—256no—
66C23×C32#6718yes—
67C23×C32C22×D3128yes—
68C23×C32C22×D317yes—
69C22×C3×D3—328no—
70C22×C3×D3—656no—
71C3×#67—364no—
72C2×C3×Dic3—328no—
73C23×C9—2112no—
74C23×C9—256no—
75C23×C9—28no—
76C23×C9—27no—
77C23×C9—228no—
78C23×C9C3×D418yescomplemented
79C23×C9C23×C317yes—
80C23×C9C23×C3128yescomplemented
81C22×D9—256no—
82C22×D9C23×C3128yescomplemented
83#611C23×C3164yescomplemented
84C2×Dic9C23×C3128yescomplemented
85C23×D4—91noSylow
86C2×3−1+2—28no—
87C2×3−1+2C22×D4114yescomplemented
88C2×3−1+2C2511yescharacteristic
89AGL(1,ℤ/9)—28no—
90C24×C3—28no—
91C24×C3—31no—
92C24×C3—314no—
93C24×C3—68no—
94C24×C3—914no—
95C24×C3—188no—
96C24×C3C2×C3×D311yescharacteristic
97C24×C3C2×C3×D3114yes—
98C23×D3—314no—
99C23×D3—68no—
100C2×C3×D4—9112no—
101C22×C3×C4—914no—
102C2×#67—3112no—
103C22×Dic3—314no—
104C22×C32—256no—
105C22×C32C2×#67128yes—
106C22×C32C23×D318yes—
107C22×C32C23×D317yes—
108C2×C3×D3—38no—
109C2×C3×D3—656no—
110C3×Dic3—38no—
111C22×C9—2112no—
112C22×C9—256no—
113C22×C9—27no—
114C22×C9—28no—
115C22×C9—228no—
116C22×C9C2×C3×D4128yescomplemented
117C22×C9C24×C317yes—
118C22×C9C24×C318yescomplemented
119C2×D9—256no—
120C2×D9C24×C318yescomplemented
121Dic9C24×C318yescomplemented
122C25—91no—
123C25AGL(1,ℤ/9)11yescharacteristic, complemented
124C22×D4—928no—
125C23×C4—91no—
1263−1+2C23×D411yescharacteristic, complemented, Sylow
127C23×C3—256no—
128C23×C3—38no—
129C23×C3—37no—
130C23×C3—328no—
131C23×C3—656no—
132C23×C3—928no—
133C23×C3—1856no—
134C23×C3C3×#6718yes—
135C23×C3C22×C3×D317yes—
136C23×C3C22×C3×D3128yes—
137C22×D3—328no—
138C22×D3—656no—
139C3×D4—964no—
140C2×C3×C4—928no—
141#67—364no—
142C2×Dic3—328no—
143C2×C32—28no—
144C2×C32C22×#67114yes—
145C2×C32C24×D311yescharacteristic
146C3×D3—68no—
147C2×C9—216no—
148C2×C9—28no—
149C2×C9—214no—
150C2×C9—21no—
151C2×C9C22×C3×D4114yescomplemented
152C2×C9C25×C311yescharacteristic
153D9—28no—
154C24—28no—
155C24—914no—
156C24—188no—
157C24C2×AGL(1,ℤ/9)11yescharacteristic
158C24C2×AGL(1,ℤ/9)114yescomplemented
159C2×D4—9112no—
160C22×C4—914no—
161C22×C3—256no—
162C22×C3—37no—
163C22×C3—38no—
164C22×C3—328no—
165C22×C3—656no—
166C22×C3—98no—
167C22×C3—1856no—
168C22×C3C2×C3×#67128yes—
169C22×C3C23×C3×D317yes—
170C22×C3C23×C3×D318yes—
171C2×D3—38no—
172C2×D3—656no—
173C3×C4—98no—
174Dic3—38no—
175C32C23×#6711yescharacteristic
176C9—21no—
177C9C23×C3×D411yescharacteristic, complemented
178C23—256no—
179C23—928no—
180C23—1856no—
181C23#611⋊C318yesdirect product factor
182C23C22×AGL(1,ℤ/9)17yes—
183C23C22×AGL(1,ℤ/9)128yescomplemented
184D4—964no—
185C2×C4—928no—
186C2×C3—28no—
187C2×C3—31no—
188C2×C3—314no—
189C2×C3—68no—
190C2×C3—188no—
191C2×C3C22×C3×#67114yes—
192C2×C3C24×C3×D311yescharacteristic
193D3—68no—
194C22—256no—
195C22—98no—
196C22—1856no—
197C22C2×#611⋊C3128yesdirect product factor
198C22C23×AGL(1,ℤ/9)17yes—
199C22C23×AGL(1,ℤ/9)18yescomplemented
200C4—98no—
201C3—31no—
202C3C23×C3×#6711yescharacteristic
203C2—28no—
204C2—188no—
205C2C22×#611⋊C3114yesdirect product factor
206C2C24×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC24C2×AGL(1,ℤ/9)
commutator, cosocleC2×C9C25×C3
FrattiniC2×C3C24×C3×D3
FittingC25×3−1+2C2
socleC24×C3C2×C3×D3
hypercenterC25AGL(1,ℤ/9)
hypocenterC9C23×C3×D4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C24 ◁ C25

Complements

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

NH ≅ G/Ntype
1. C25×3−1+2C2semidirect
2. C24×AGL(1,ℤ/9)C2semidirect
3. C22×#611⋊C3C2direct (N × H)
4. C23×3−1+2⋊C4C2semidirect
6. C23×#611C3semidirect
9. C24×3−1+2C22semidirect
11. C23×AGL(1,ℤ/9)C22semidirect
12. C2×#611⋊C3C22direct (N × H)
13. C22×3−1+2⋊C4C22semidirect
19. C25×C9C2×C3semidirect
20. C24×D9C2×C3semidirect
21. C22×#611C2×C3semidirect
22. C23×Dic9C2×C3semidirect
24. C23×3−1+2D4semidirect
26. C23×3−1+2C23semidirect
28. C22×AGL(1,ℤ/9)C23semidirect
29. #611⋊C3C23direct (N × H)
30. C2×3−1+2⋊C4C23semidirect
45. C24×C9C22×C3semidirect
47. C23×D9C22×C3semidirect
48. C2×#611C22×C3semidirect
49. C22×Dic9C22×C3semidirect
51. C22×3−1+2C2×D4semidirect
53. C22×3−1+2C24semidirect
55. C2×AGL(1,ℤ/9)C24semidirect
56. 3−1+2⋊C4C24semidirect
78. C23×C9C3×D4semidirect
80. C23×C9C23×C3semidirect
82. C22×D9C23×C3semidirect
83. #611C23×C3semidirect
84. C2×Dic9C23×C3semidirect
87. C2×3−1+2C22×D4semidirect
116. C22×C9C2×C3×D4semidirect
118. C22×C9C24×C3semidirect
120. C2×D9C24×C3semidirect
121. Dic9C24×C3semidirect
123. C25AGL(1,ℤ/9)semidirect
126. 3−1+2C23×D4semidirect
151. C2×C9C22×C3×D4semidirect
158. C24C2×AGL(1,ℤ/9)semidirect
177. C9C23×C3×D4semidirect
181. C23#611⋊C3direct (N × H)
183. C23C22×AGL(1,ℤ/9)semidirect
197. C22C2×#611⋊C3direct (N × H)
199. C22C23×AGL(1,ℤ/9)semidirect
205. C2C22×#611⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C25×3−1+22
C24×AGL(1,ℤ/9)2
C22×#611⋊C32
C23×3−1+2⋊C42
C23×C3×#673
C23×#6113