Finite Group Databasebeta

← C23×#6194 #13164805

Subgroups — C23×#6194 #13164805

Order 1728.

4322 conjugacy classes of subgroups; only the 913 normal subgroups are shown.

normal subgroups only

This group has 4322 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×D3×C9C211yesmaximal, characteristic, complemented
2C24×C3×D9C211yesmaximal, characteristic, complemented
3C23×C3⋊Dic9C211yesmaximal, characteristic, complemented
4C22×#6194C2128yesmaximal, direct product factor
5C23×#625—31nomaximal
6C23×#611—31nomaximal
7C24×C3×C9C2211yescharacteristic
8C23×D3×C9—28no—
9C23×D3×C9C22114yescomplemented
10C23×C3×D9—28no—
11C23×C3×D9C22114yescomplemented
12C22×C3⋊Dic9C22114yescomplemented
13C2×#6194C221112yesdirect product factor
14C24×C3×D3—31no—
15C24×C3×D3D311yescharacteristic
16C23×C3⋊Dic3—31no—
17C22×#625—328no—
18C25×C9—31no—
19C24×D9D311yescharacteristic, complemented
20C22×#611—328no—
21C23×Dic9—31no—
22C23×Dic9—61no—
23C23×C3×C9D418yescomplemented
24C23×C3×C9C2317yes—
25C22×D3×C9—256no—
26C22×D3×C9C23128yescomplemented
27C22×C3×D9—256no—
28C22×C3×D9C23128yescomplemented
29C2×C3⋊Dic9C23128yescomplemented
30#6194C23164yesdirect product factor
31C23×#67—91no—
32C23×#67—91no—
33C24×C32C2×D311yescharacteristic
34C23×C3×D3—28no—
35C23×C3×D3—314no—
36C23×C3×D3—68no—
37C23×C3×D3C2×D3114yes—
38C22×C3⋊Dic3—314no—
39C2×#625—3112no—
40C24×C9—21no—
41C24×C9—314no—
42C24×C9—68no—
43C24×C9C2×D311yescharacteristic
44C23×D9—28no—
45C23×D9C2×D3114yescomplemented
46C2×#611—3112no—
47C22×Dic9—314no—
48C22×Dic9—614no—
49C22×C3×C9C2×D4128yescomplemented
50C22×C3×C9C2417yes—
51C2×D3×C9—256no—
52C2×D3×C9C2418yescomplemented
53C2×C3×D9—256no—
54C2×C3×D9C2418yescomplemented
55C3⋊Dic9C2418yescomplemented
56C25×C3—31no—
57C25×C3—91no—
58C24×D3—31no—
59C24×D3D911yescharacteristic, complemented
60C22×#67—928no—
61C22×#67—928no—
62C23×Dic3—91no—
63C23×Dic3—91no—
64C23×Dic3—181no—
65C23×C32#6718yes—
66C23×C32C22×D317yes—
67C22×C3×D3—256no—
68C22×C3×D3—328no—
69C22×C3×D3—656no—
70C22×C3×D3C22×D3128yes—
71C2×C3⋊Dic3—328no—
72#625—364no—
73C23×C9—28no—
74C23×C9—27no—
75C23×C9—328no—
76C23×C9—656no—
77C23×C9#6718yescomplemented
78C23×C9C22×D317yes—
79C22×D9—256no—
80C22×D9C22×D3128yescomplemented
81#611—364no—
82C2×Dic9—328no—
83C2×Dic9—628no—
84C23×D4—271noSylow
85C2×C3×C9C22×D4114yescomplemented
86C2×C3×C9C2511yescharacteristic
87D3×C9—28no—
88C3×D9—28no—
89C24×C3—21no—
90C24×C3—314no—
91C24×C3—68no—
92C24×C3—914no—
93C24×C3—188no—
94C24×C3C2×D911yescharacteristic
95C24×C3D3211yescharacteristic
96C23×D3—28no—
97C23×D3—314no—
98C23×D3—68no—
99C23×D3C2×D9114yescomplemented
100C2×#67—9112no—
101C2×#67—9112no—
102C22×Dic3—914no—
103C22×Dic3—914no—
104C22×Dic3—1814no—
105C22×C32C2×#67128yes—
106C22×C32C23×D317yes—
107C2×C3×D3—256no—
108C2×C3×D3—38no—
109C2×C3×D3—656no—
110C2×C3×D3C23×D318yes—
111C3⋊Dic3—38no—
112C22×C9—27no—
113C22×C9—228no—
114C22×C9—38no—
115C22×C9—656no—
116C22×C9C2×#67128yescomplemented
117C22×C9C23×D317yes—
118C2×D9—256no—
119C2×D9C23×D318yescomplemented
120Dic9—38no—
121Dic9—68no—
122C25—31no—
123C25—91no—
124C22×D4—2728no—
125C23×C4—271no—
126C3×C9C23×D411yescharacteristic, complemented, Sylow
127C23×C3—28no—
128C23×C3—27no—
129C23×C3—328no—
130C23×C3—656no—
131C23×C3—928no—
132C23×C3—1856no—
133C23×C3#61118yescomplemented
134C23×C3C22×D917yes—
135C23×C3#62518yes—
136C23×C3C2×D3217yes—
137C22×D3—256no—
138C22×D3—328no—
139C22×D3—656no—
140C22×D3C22×D9128yescomplemented
141#67—964no—
142#67—964no—
143C2×Dic3—928no—
144C2×Dic3—928no—
145C2×Dic3—1828no—
146C2×C32C22×#67114yes—
147C2×C32C24×D311yescharacteristic
148C3×D3—28no—
149C3×D3—68no—
150C2×C9—214no—
151C2×C9—21no—
152C2×C9—68no—
153C2×C9C22×#67114yescomplemented
154C2×C9C24×D311yescharacteristic
155D9—28no—
156C24—314no—
157C24—68no—
158C24—914no—
159C24—188no—
160C24D3×D911yescharacteristic
161C2×D4—27112no—
162C22×C4—2714no—
163C22×C3—27no—
164C22×C3—228no—
165C22×C3—38no—
166C22×C3—656no—
167C22×C3—98no—
168C22×C3—1856no—
169C22×C3C2×#611128yescomplemented
170C22×C3C23×D917yes—
171C22×C3C2×#625128yes—
172C22×C3C22×D3217yes—
173C2×D3—256no—
174C2×D3—38no—
175C2×D3—656no—
176C2×D3C23×D918yescomplemented
177Dic3—98no—
178Dic3—98no—
179Dic3—188no—
180C32C23×#6711yescharacteristic
181C9—21no—
182C9C23×#6711yescharacteristic, complemented
183C23—328no—
184C23—656no—
185C23—928no—
186C23—1856no—
187C23#619418yesdirect product factor
188C23C2×D3×D917yes—
189D4—2764no—
190C2×C4—2728no—
191C2×C3—21no—
192C2×C3—214no—
193C2×C3—68no—
194C2×C3—188no—
195C2×C3C22×#611114yescomplemented
196C2×C3C24×D911yescharacteristic
197C2×C3C22×#625114yes—
198C2×C3C23×D3211yescharacteristic
199D3—28no—
200D3—68no—
201C22—38no—
202C22—656no—
203C22—98no—
204C22—1856no—
205C22C2×#6194128yesdirect product factor
206C22C22×D3×D917yes—
207C4—278no—
208C3—21no—
209C3C23×#61111yescharacteristic, complemented
210C3C23×#62511yescharacteristic
211C2—68no—
212C2—188no—
213C2C22×#6194114yesdirect product factor
214C2C23×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×D3×C9C2semidirect
2. C24×C3×D9C2semidirect
3. C23×C3⋊Dic9C2semidirect
4. C22×#6194C2direct (N × H)
9. C23×D3×C9C22semidirect
11. C23×C3×D9C22semidirect
12. C22×C3⋊Dic9C22semidirect
13. C2×#6194C22direct (N × H)
19. C24×D9D3semidirect
23. C23×C3×C9D4semidirect
26. C22×D3×C9C23semidirect
28. C22×C3×D9C23semidirect
29. C2×C3⋊Dic9C23semidirect
30. #6194C23direct (N × H)
45. C23×D9C2×D3semidirect
49. C22×C3×C9C2×D4semidirect
52. C2×D3×C9C24semidirect
54. C2×C3×D9C24semidirect
55. C3⋊Dic9C24semidirect
59. C24×D3D9semidirect
77. C23×C9#67semidirect
80. C22×D9C22×D3semidirect
85. C2×C3×C9C22×D4semidirect
99. C23×D3C2×D9semidirect
116. C22×C9C2×#67semidirect
119. C2×D9C23×D3semidirect
126. C3×C9C23×D4semidirect
133. C23×C3#611semidirect
140. C22×D3C22×D9semidirect
153. C2×C9C22×#67semidirect
169. C22×C3C2×#611semidirect
176. C2×D3C23×D9semidirect
182. C9C23×#67semidirect
187. C23#6194direct (N × H)
195. C2×C3C22×#611semidirect
205. C22C2×#6194direct (N × H)
209. C3C23×#611semidirect
213. C2C22×#6194direct (N × H)

Maximal subgroups

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

subgroupindex
C24×D3×C92
C24×C3×D92
C23×C3⋊Dic92
C22×#61942
C23×#6253
C23×#6113