Finite Group Databasebeta

← C23×#6195 #13164806

Subgroups — C23×#6195 #13164806

Order 1728.

4834 conjugacy classes of subgroups; only the 913 normal subgroups are shown.

normal subgroups only

This group has 4834 conjugacy classes of subgroups — too many to draw. Only its 913 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
1C24×C3⋊D9C211yesmaximal, characteristic, complemented
2C24×D3×C9C211yesmaximal, characteristic, complemented
3C23×C3×Dic9C211yesmaximal, characteristic, complemented
4C22×#6195C2128yesmaximal, direct product factor
5C23×#626—31nomaximal
6C23×#611—31nomaximal
7C24×C3×C9C2211yescharacteristic
8C23×C3⋊D9—28no—
9C23×C3⋊D9C22114yescomplemented
10C23×D3×C9—28no—
11C23×D3×C9C22114yescomplemented
12C22×C3×Dic9C22114yescomplemented
13C2×#6195C221112yesdirect product factor
14C24×C3⋊D3—31no—
15C24×C3×D3D311yescharacteristic
16C23×C3×Dic3—31no—
17C22×#626—328no—
18C25×C9—31no—
19C24×D9—31no—
20C24×D9—61no—
21C22×#611—328no—
22C23×Dic9D311yescharacteristic, complemented
23C23×C3×C9D418yescomplemented
24C23×C3×C9C2317yes—
25C22×C3⋊D9—256no—
26C22×C3⋊D9C23128yescomplemented
27C22×D3×C9—256no—
28C22×D3×C9C23128yescomplemented
29C2×C3×Dic9C23128yescomplemented
30#6195C23164yesdirect product factor
31C23×#67—91no—
32C23×D12—91no—
33C24×C32C2×D311yescharacteristic
34C23×C3⋊D3—314no—
35C23×C3⋊D3—68no—
36C23×C3×D3—28no—
37C23×C3×D3C2×D3114yes—
38C22×C3×Dic3—314no—
39C2×#626—3112no—
40C24×C9—21no—
41C24×C9—314no—
42C24×C9—68no—
43C24×C9C2×D311yescharacteristic
44C23×D9—314no—
45C23×D9—614no—
46C23×D9—616no—
47C23×D9—68no—
48C2×#611—3112no—
49C22×Dic9C2×D3114yescomplemented
50C22×C3×C9C2×D4128yescomplemented
51C22×C3×C9C2417yes—
52C2×C3⋊D9—256no—
53C2×C3⋊D9C2418yescomplemented
54C2×D3×C9—256no—
55C2×D3×C9C2418yescomplemented
56C3×Dic9C2418yescomplemented
57C25×C3—31no—
58C24×D3—91no—
59C24×D3—91no—
60C24×D3—181no—
61C24×D3D911yescharacteristic, complemented
62C23×C3×C4—91no—
63C22×#67—928no—
64C23×Dic3—31no—
65C22×D12—928no—
66C23×C32#6718yes—
67C23×C32C22×D317yes—
68C22×C3⋊D3—328no—
69C22×C3⋊D3—656no—
70C22×C3×D3—256no—
71C22×C3×D3C22×D3128yes—
72C2×C3×Dic3—328no—
73#626—364no—
74C23×C9—28no—
75C23×C9—27no—
76C23×C9—328no—
77C23×C9—656no—
78C23×C9D1218yescomplemented
79C23×C9C22×D317yes—
80C22×D9—328no—
81C22×D9—628no—
82C22×D9—6112no—
83C22×D9—656no—
84#611—364no—
85C2×Dic9C22×D3128yescomplemented
86C23×D4—271noSylow
87C2×C3×C9C22×D4114yescomplemented
88C2×C3×C9C2511yescharacteristic
89C3⋊D9—28no—
90D3×C9—28no—
91C24×C3—21no—
92C24×C3—314no—
93C24×C3—68no—
94C24×C3C2×D911yescharacteristic
95C24×C3D3211yescharacteristic
96C23×D3—28no—
97C23×D3—914no—
98C23×D3—914no—
99C23×D3—1814no—
100C23×D3—1816no—
101C23×D3—188no—
102C23×D3—188no—
103C23×D3C2×D9114yescomplemented
104C22×C3×C4—914no—
105C2×#67—9112no—
106C22×Dic3—314no—
107C2×D12—9112no—
108C22×C32C2×#67128yes—
109C22×C32C23×D317yes—
110C2×C3⋊D3—38no—
111C2×C3⋊D3—656no—
112C2×C3×D3—256no—
113C2×C3×D3C23×D318yes—
114C3×Dic3—38no—
115C22×C9—27no—
116C22×C9—228no—
117C22×C9—38no—
118C22×C9—656no—
119C22×C9C2×D12128yescomplemented
120C22×C9C23×D317yes—
121C2×D9—38no—
122C2×D9—68no—
123C2×D9—6112no—
124C2×D9—656no—
125Dic9C23×D318yescomplemented
126C25—31no—
127C25—271no—
128C22×D4—2728no—
129C23×C4—91no—
130C3×C9C23×D411yescharacteristic, complemented, Sylow
131C23×C3—28no—
132C23×C3—27no—
133C23×C3—328no—
134C23×C3—656no—
135C23×C3#61118yescomplemented
136C23×C3C22×D917yes—
137C23×C3#62618yes—
138C23×C3C2×D3217yes—
139C22×D3—256no—
140C22×D3—928no—
141C22×D3—928no—
142C22×D3—1828no—
143C22×D3—18112no—
144C22×D3—1856no—
145C22×D3—1856no—
146C22×D3C22×D9128yescomplemented
147C2×C3×C4—928no—
148#67—964no—
149C2×Dic3—328no—
150D12—964no—
151C2×C32C22×#67114yes—
152C2×C32C24×D311yescharacteristic
153C3⋊D3—68no—
154C3×D3—28no—
155C2×C9—214no—
156C2×C9—21no—
157C2×C9—68no—
158C2×C9C22×D12114yescomplemented
159C2×C9C24×D311yescharacteristic
160D9—616no—
161D9—68no—
162C24—314no—
163C24—68no—
164C24—2714no—
165C24—548no—
166C24D3×D911yescharacteristic
167C2×D4—27112no—
168C22×C4—914no—
169C22×C3—27no—
170C22×C3—228no—
171C22×C3—38no—
172C22×C3—656no—
173C22×C3C2×#611128yescomplemented
174C22×C3C23×D917yes—
175C22×C3C2×#626128yes—
176C22×C3C22×D3217yes—
177C2×D3—256no—
178C2×D3—98no—
179C2×D3—98no—
180C2×D3—188no—
181C2×D3—18112no—
182C2×D3—1856no—
183C2×D3—1856no—
184C2×D3C23×D918yescomplemented
185C3×C4—98no—
186Dic3—38no—
187C32C23×#6711yescharacteristic
188C9—21no—
189C9C23×D1211yescharacteristic, complemented
190C23—328no—
191C23—656no—
192C23—2728no—
193C23—5456no—
194C23#619518yesdirect product factor
195C23C2×D3×D917yes—
196D4—2764no—
197C2×C4—928no—
198C2×C3—21no—
199C2×C3—214no—
200C2×C3—68no—
201C2×C3C22×#611114yescomplemented
202C2×C3C24×D911yescharacteristic
203C2×C3C22×#626114yes—
204C2×C3C23×D3211yescharacteristic
205D3—28no—
206D3—1816no—
207D3—188no—
208D3—188no—
209C22—38no—
210C22—656no—
211C22—278no—
212C22—5456no—
213C22C2×#6195128yesdirect product factor
214C22C22×D3×D917yes—
215C4—98no—
216C3—21no—
217C3C23×#61111yescharacteristic, complemented
218C3C23×#62611yescharacteristic
219C2—68no—
220C2—548no—
221C2C22×#6195114yesdirect product factor
222C2C23×D3×D911yescharacteristic
subgroupG/H
center, hypercenterC24D3×D9
commutator, cosocleC2×C3×C9C25
FrattiniC2×C3C23×D32
FittingC24×C3×C9C22
socleC24×C32C2×D3
hypocenterC3×C9C23×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×C3⋊D9C2semidirect
2. C24×D3×C9C2semidirect
3. C23×C3×Dic9C2semidirect
4. C22×#6195C2direct (N × H)
9. C23×C3⋊D9C22semidirect
11. C23×D3×C9C22semidirect
12. C22×C3×Dic9C22semidirect
13. C2×#6195C22direct (N × H)
22. C23×Dic9D3semidirect
23. C23×C3×C9D4semidirect
26. C22×C3⋊D9C23semidirect
28. C22×D3×C9C23semidirect
29. C2×C3×Dic9C23semidirect
30. #6195C23direct (N × H)
49. C22×Dic9C2×D3semidirect
50. C22×C3×C9C2×D4semidirect
53. C2×C3⋊D9C24semidirect
55. C2×D3×C9C24semidirect
56. C3×Dic9C24semidirect
61. C24×D3D9semidirect
78. C23×C9D12semidirect
85. C2×Dic9C22×D3semidirect
87. C2×C3×C9C22×D4semidirect
103. C23×D3C2×D9semidirect
119. C22×C9C2×D12semidirect
125. Dic9C23×D3semidirect
130. C3×C9C23×D4semidirect
135. C23×C3#611semidirect
146. C22×D3C22×D9semidirect
158. C2×C9C22×D12semidirect
173. C22×C3C2×#611semidirect
184. C2×D3C23×D9semidirect
189. C9C23×D12semidirect
194. C23#6195direct (N × H)
201. C2×C3C22×#611semidirect
213. C22C2×#6195direct (N × H)
217. C3C23×#611semidirect
221. C2C22×#6195direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3⋊D92
C24×D3×C92
C23×C3×Dic92
C22×#61952
C23×#6263
C23×#6113