Finite Group Databasebeta

← C23×#6294 #13181690

Subgroups — C23×#6294 #13181690

Order 1728.

6802 conjugacy classes of subgroups; only the 1559 normal subgroups are shown.

normal subgroups only

This group has 6802 conjugacy classes of subgroups — too many to draw. Only its 1559 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×C3⋊D3C211yesmaximal, characteristic
2C23×C3×C3⋊Dic3C212yesmaximal, complemented
3C22×#6294C2128yesmaximal, direct product factor
4C23×#624—31nomaximal
5C23×D3×Dic3—32nomaximal
6C24×C33C2211yescharacteristic
7C23×C3×C3⋊D3C4116yescomplemented
8C23×C3×C3⋊D3C22114yes—
9C22×C3×C3⋊Dic3C22128yescomplemented
10C2×#6294C221112yesdirect product factor
11C24×C3⋊D3D311yescharacteristic
12C24×C3×D3—32no—
13C24×C3×D3—61no—
14C23×C3⋊Dic3D312yescomplemented
15C23×C3×Dic3—32no—
16C23×C3×Dic3—32no—
17C23×C3×Dic3—62no—
18C22×#624—328no—
19C22×D3×Dic3—356no—
20C23×C33C2×C418yescomplemented
21C23×C33C2317yes—
22C22×C3×C3⋊D3C2×C41112yescomplemented
23C22×C3×C3⋊D3C23128yes—
24C2×C3×C3⋊Dic3C23156yescomplemented
25#6294C23164yesdirect product factor
26C24×Dic3—91no—
27C23×C4×D3—92no—
28C24×C32—22no—
29C24×C32—21no—
30C24×C32—41no—
31C24×C32C2×D312yes—
32C24×C32C2×D311yescharacteristic
33C23×C3⋊D3Dic3116yescomplemented
34C23×C3⋊D3C2×D3114yes—
35C23×C3×D3—332no—
36C23×C3×D3—328no—
37C23×C3×D3—616no—
38C23×C3×D3—614no—
39C22×C3⋊Dic3C2×D3128yescomplemented
40C22×C3×Dic3—328no—
41C22×C3×Dic3—328no—
42C22×C3×Dic3—628no—
43C2×#624—3112no—
44C2×D3×Dic3—3224no—
45C22×C33C22×C4128yescomplemented
46C22×C33C2417yes—
47C2×C3×C3⋊D3C22×C41112yescomplemented
48C2×C3×C3⋊D3C2418yes—
49C3×C3⋊Dic3C24116yescomplemented
50C25×C3—91no—
51C24×D3—32no—
52C24×D3—61no—
53C23×C3×C4—92no—
54C23×Dic3—32no—
55C23×Dic3—32no—
56C23×Dic3—62no—
57C23×Dic3—928no—
58C22×C4×D3—956no—
59C23×C32—216no—
60C23×C32—28no—
61C23×C32—27no—
62C23×C32—214no—
63C23×C32—48no—
64C23×C32—47no—
65C23×C32C4×D3116yescomplemented
66C23×C32C2×Dic318yescomplemented
67C23×C32C22×D3114yes—
68C23×C32C22×D317yes—
69C22×C3⋊D3C2×Dic31112yescomplemented
70C22×C3⋊D3C22×D3128yes—
71C22×C3×D3—3224no—
72C22×C3×D3—356no—
73C22×C3×D3—6112no—
74C22×C3×D3—628no—
75C2×C3⋊Dic3C22×D3156yescomplemented
76C2×C3×Dic3—356no—
77C2×C3×Dic3—356no—
78C2×C3×Dic3—656no—
79#624—364no—
80D3×Dic3—3128no—
81C24×C4—271noSylow
82C2×C33C23×C4114yescomplemented
83C2×C33C2511yescharacteristic
84C3×C3⋊D3C23×C4116yescomplemented
85C24×C3—22no—
86C24×C3—21no—
87C24×C3—41no—
88C24×C3—916no—
89C24×C3—914no—
90C24×C3D3212yes—
91C24×C3D3211yescharacteristic
92C23×D3—332no—
93C23×D3—328no—
94C23×D3—616no—
95C23×D3—614no—
96C22×C3×C4—928no—
97C22×Dic3—328no—
98C22×Dic3—328no—
99C22×Dic3—628no—
100C22×Dic3—9112no—
101C2×C4×D3—9224no—
102C22×C32—214no—
103C22×C32—27no—
104C22×C32—256no—
105C22×C32—228no—
106C22×C32—428no—
107C22×C32—47no—
108C22×C32C2×C4×D3156yescomplemented
109C22×C32C22×Dic3128yescomplemented
110C22×C32C23×D3114yes—
111C22×C32C23×D317yes—
112C2×C3⋊D3C22×Dic31112yescomplemented
113C2×C3⋊D3C23×D318yes—
114C2×C3×D3—3224no—
115C2×C3×D3—316no—
116C2×C3×D3—6112no—
117C2×C3×D3—68no—
118C3⋊Dic3C23×D3116yescomplemented
119C3×Dic3—316no—
120C3×Dic3—316no—
121C3×Dic3—616no—
122C25—91no—
123C23×C4—92no—
124C23×C4—2728no—
125C33C24×C411yescharacteristic, complemented, Sylow
126C23×C3—216no—
127C23×C3—28no—
128C23×C3—214no—
129C23×C3—27no—
130C23×C3—48no—
131C23×C3—47no—
132C23×C3—928no—
133C23×C3—9112no—
134C23×C3D3×Dic3116yescomplemented
135C23×C3#62418yescomplemented
136C23×C3C2×D32114yes—
137C23×C3C2×D3217yes—
138C22×D3—356no—
139C22×D3—3224no—
140C22×D3—6112no—
141C22×D3—628no—
142C2×C3×C4—956no—
143C2×Dic3—356no—
144C2×Dic3—356no—
145C2×Dic3—656no—
146C2×Dic3—964no—
147C4×D3—9128no—
148C2×C32—228no—
149C2×C32—214no—
150C2×C32—22no—
151C2×C32—21no—
152C2×C32—414no—
153C2×C32—41no—
154C2×C32C22×C4×D3128yescomplemented
155C2×C32C23×Dic3114yescomplemented
156C2×C32C24×D312yes—
157C2×C32C24×D311yescharacteristic
158C3⋊D3C23×Dic3116yescomplemented
159C3×D3—332no—
160C3×D3—616no—
161C24—916no—
162C24—914no—
163C24#111311yescharacteristic
164C22×C4—928no—
165C22×C4—27112no—
166C22×C3—214no—
167C22×C3—27no—
168C22×C3—228no—
169C22×C3—256no—
170C22×C3—47no—
171C22×C3—428no—
172C22×C3—9112no—
173C22×C3—98no—
174C22×C3C2×D3×Dic3156yescomplemented
175C22×C3C2×#624128yescomplemented
176C22×C3C22×D32114yes—
177C22×C3C22×D3217yes—
178C2×D3—3224no—
179C2×D3—316no—
180C2×D3—6112no—
181C2×D3—68no—
182C3×C4—916no—
183Dic3—316no—
184Dic3—316no—
185Dic3—616no—
186C32—22no—
187C32—21no—
188C32—41no—
189C32C23×C4×D312yescomplemented
190C32C24×Dic311yescharacteristic, complemented
191C23—928no—
192C23—9112no—
193C23#629418yesdirect product factor
194C23C2×#111317yes—
195C2×C4—956no—
196C2×C4—2764no—
197C2×C3—214no—
198C2×C3—228no—
199C2×C3—22no—
200C2×C3—21no—
201C2×C3—414no—
202C2×C3—41no—
203C2×C3—916no—
204C2×C3C22×D3×Dic3128yescomplemented
205C2×C3C22×#624114yescomplemented
206C2×C3C23×D3212yes—
207C2×C3C23×D3211yescharacteristic
208D3—332no—
209D3—616no—
210C22—9112no—
211C22—98no—
212C22C2×#6294128yesdirect product factor
213C22C22×#111317yes—
214C4—916no—
215C3—21no—
216C3—22no—
217C3—41no—
218C3C23×D3×Dic312yescomplemented
219C3C23×#62411yescharacteristic, complemented
220C2—916no—
221C2C22×#6294114yesdirect product factor
222C2C23×#111311yescharacteristic
subgroupG/H
center, hypercenterC24#1113
commutator, hypocenterC33C24×C4
FrattiniC2C23×#1113
Fitting, socleC24×C33C22
prime radicalC24×C3×C3⋊D3C2
cosocleC2×C33C25
LCSG ▷ C33
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×C3×C3⋊Dic3C2semidirect
3. C22×#6294C2direct (N × H)
7. C23×C3×C3⋊D3C4semidirect
9. C22×C3×C3⋊Dic3C22semidirect
10. C2×#6294C22direct (N × H)
14. C23×C3⋊Dic3D3semidirect
20. C23×C33C2×C4semidirect
22. C22×C3×C3⋊D3C2×C4semidirect
24. C2×C3×C3⋊Dic3C23semidirect
25. #6294C23direct (N × H)
33. C23×C3⋊D3Dic3semidirect
39. C22×C3⋊Dic3C2×D3semidirect
45. C22×C33C22×C4semidirect
47. C2×C3×C3⋊D3C22×C4semidirect
49. C3×C3⋊Dic3C24semidirect
65. C23×C32C4×D3semidirect
66. C23×C32C2×Dic3semidirect
69. C22×C3⋊D3C2×Dic3semidirect
75. C2×C3⋊Dic3C22×D3semidirect
82. C2×C33C23×C4semidirect
84. C3×C3⋊D3C23×C4semidirect
108. C22×C32C2×C4×D3semidirect
109. C22×C32C22×Dic3semidirect
112. C2×C3⋊D3C22×Dic3semidirect
118. C3⋊Dic3C23×D3semidirect
125. C33C24×C4semidirect
134. C23×C3D3×Dic3semidirect
135. C23×C3#624semidirect
154. C2×C32C22×C4×D3semidirect
155. C2×C32C23×Dic3semidirect
158. C3⋊D3C23×Dic3semidirect
174. C22×C3C2×D3×Dic3semidirect
175. C22×C3C2×#624semidirect
189. C32C23×C4×D3semidirect
190. C32C24×Dic3semidirect
193. C23#6294direct (N × H)
204. C2×C3C22×D3×Dic3semidirect
205. C2×C3C22×#624semidirect
212. C22C2×#6294direct (N × H)
218. C3C23×D3×Dic3semidirect
219. C3C23×#624semidirect
221. C2C22×#6294direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×C3⋊D32
C23×C3×C3⋊Dic32
C22×#62942
C23×#6243
C23×D3×Dic33