Finite Group Databasebeta

← C23×#3798 #11919204

Subgroups — C23×#3798 #11919204

Order 1152.

13721 conjugacy classes of subgroups; only the 3844 normal subgroups are shown.

normal subgroups only

This group has 13721 conjugacy classes of subgroups — too many to draw. Only its 3844 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
1C23×C4×C3⋊D3C211yesmaximal, characteristic, complemented
2C23×C3×C4×D3C212yesmaximal, complemented
3C23×#627C211yesmaximal, characteristic, complemented
4C23×#626C212yesmaximal, complemented
5C23×#625C211yesmaximal, characteristic, complemented
6C22×#3798C2156yesmaximal, direct product factor
7C23×#235—32nomaximal
8C24×C3⋊D3C2211yescharacteristic
9C24×C3×D3C2212yes—
10C23×C32×C4C2211yescharacteristic
11C23×C3⋊Dic3C2211yescharacteristic
12C22×C4×C3⋊D3C22128yescomplemented
13C23×C3×Dic3C2212yes—
14C22×C3×C4×D3C22156yescomplemented
15C22×#627C22128yescomplemented
16C22×#626C22156yescomplemented
17C22×#625C22128yescomplemented
18C2×#3798C221448yesdirect product factor
19C24×C3×C4—32no—
20C23×#67—32no—
21C23×#67—32no—
22C22×#235—3112no—
23C23×D12—32no—
24C23×C4×D3—32no—
25C23×C4×D3—61no—
26C23×C4×D3D312yescomplemented
27C23×Dic6—32no—
28C24×C32C2311yescharacteristic
29C23×C3⋊D3—28no—
30C23×C3⋊D3C23114yes—
31C23×C3×D3—216no—
32C23×C3×D3C23128yes—
33C22×C32×C4C23114yes—
34C22×C3⋊Dic3C23114yes—
35C2×C4×C3⋊D3C231112yescomplemented
36C22×C3×Dic3C23128yes—
37C2×C3×C4×D3C231224yescomplemented
38C2×#627C231112yescomplemented
39C2×#626C231224yescomplemented
40C2×#625C231112yescomplemented
41#3798C231512yesdirect product factor
42C23×#41—91noSylow
43C25×C3—32no—
44C24×D3—32no—
45C24×D3—61no—
46C24×D3C2×D312yes—
47C23×C3×C4—21no—
48C23×C3×C4—32no—
49C23×C3×C4—356no—
50C23×C3×C4C2×D312yes—
51C22×#67—356no—
52C22×#67—356no—
53C23×Dic3—32no—
54C23×Dic3—61no—
55C23×Dic3C2×D312yes—
56C2×#235—3896no—
57C22×D12—356no—
58C22×C4×D3—356no—
59C22×C4×D3—628no—
60C22×C4×D3C2×D3156yescomplemented
61C22×Dic6—356no—
62C23×C32#4118yescomplemented
63C23×C32C2417yes—
64C22×C3⋊D3—256no—
65C22×C3⋊D3C24128yes—
66C22×C3×D3—2112no—
67C22×C3×D3C24156yes—
68C2×C32×C4C24128yes—
69C2×C3⋊Dic3C24128yes—
70C4×C3⋊D3C24164yescomplemented
71C2×C3×Dic3C24156yes—
72C3×C4×D3C241128yescomplemented
73#627C24164yescomplemented
74#626C241128yescomplemented
75#625C24164yescomplemented
76C22×#41—956no—
77C23×Q8—91no—
78C23×D4—92no—
79C23×D4—91no—
80C24×C4—32no—
81C24×C4—91no—
82C24×C3—21no—
83C24×C3—328no—
84C24×C3—616no—
85C24×C3C22×D312yes—
86C23×D3—216no—
87C23×D3—328no—
88C23×D3—614no—
89C23×D3—616no—
90C23×D3—616no—
91C23×D3C22×D3128yes—
92C22×C3×C4—214no—
93C22×C3×C4—328no—
94C22×C3×C4—3224no—
95C22×C3×C4C22×D3128yes—
96C2×#67—3224no—
97C2×#67—3224no—
98C22×Dic3—328no—
99C22×Dic3—614no—
100C22×Dic3C22×D3128yes—
101#235—31024no—
102C2×D12—3224no—
103C2×C4×D3—3224no—
104C2×C4×D3—6112no—
105C2×C4×D3C22×D31224yescomplemented
106C2×Dic6—3224no—
107C22×C32C2×#41128yescomplemented
108C22×C32C2517yes—
109C2×C3⋊D3—256no—
110C2×C3⋊D3C2518yes—
111C2×C3×D3—2112no—
112C2×C3×D3C25116yes—
113C32×C4C2518yes—
114C3⋊Dic3C2518yes—
115C3×Dic3C25116yes—
116C25—32no—
117C25—91no—
118C2×#41—9448no—
119C22×Q8—928no—
120C22×D4—956no—
121C22×D4—928no—
122C23×C4—32no—
123C23×C4—356no—
124C23×C4—91no—
125C23×C4—928no—
126C23×C4D3211yescharacteristic
127C23×C3—27no—
128C23×C3—28no—
129C23×C3—356no—
130C23×C3—6112no—
131C23×C3#235116yescomplemented
132C23×C3C23×D3114yes—
133C22×D3—2112no—
134C22×D3—356no—
135C22×D3—628no—
136C22×D3—6112no—
137C22×D3—6112no—
138C22×D3C23×D3156yes—
139C2×C3×C4—228no—
140C2×C3×C4—356no—
141C2×C3×C4—3128no—
142C2×C3×C4C23×D3156yes—
143#67—3128no—
144#67—3128no—
145C2×Dic3—356no—
146C2×Dic3—628no—
147C2×Dic3C23×D3156yes—
148D12—3128no—
149C4×D3—3128no—
150C4×D3—664no—
151C4×D3C23×D31128yescomplemented
152Dic6—3128no—
153C2×C32C22×#41114yescomplemented
154C2×C32C2611yescharacteristic
155C3⋊D3—28no—
156C3×D3—216no—
157C24—328no—
158C24—616no—
159C24—914no—
160C24—188no—
161C24C2×D3211yescharacteristic
162#41—9512no—
163C2×Q8—9112no—
164C2×D4—9224no—
165C2×D4—9112no—
166C22×C4—328no—
167C22×C4—3224no—
168C22×C4—914no—
169C22×C4—9112no—
170C22×C4C2×D32114yes—
171C22×C3—228no—
172C22×C3—27no—
173C22×C3—316no—
174C22×C3—6112no—
175C22×C3C2×#235156yescomplemented
176C22×C3C24×D3114yes—
177C2×D3—2112no—
178C2×D3—316no—
179C2×D3—68no—
180C2×D3—6112no—
181C2×D3—6112no—
182C2×D3C24×D3116yes—
183C3×C4—28no—
184C3×C4—316no—
185C3×C4C24×D3116yes—
186Dic3—316no—
187Dic3—68no—
188Dic3C24×D3116yes—
189C32C23×#4111yescharacteristic, complemented, Sylow
190C23—356no—
191C23—6112no—
192C23—928no—
193C23—1856no—
194C23#379818yesdirect product factor
195C23C22×D3217yes—
196Q8—964no—
197D4—9128no—
198D4—964no—
199C2×C4—356no—
200C2×C4—3128no—
201C2×C4—928no—
202C2×C4—964no—
203C2×C4C22×D32128yes—
204C2×C3—214no—
205C2×C3—21no—
206C2×C3—616no—
207C2×C3C22×#235128yescomplemented
208C2×C3C25×D312yes—
209D3—216no—
210D3—616no—
211D3—616no—
212C22—316no—
213C22—6112no—
214C22—98no—
215C22—1856no—
216C22C2×#3798128yesdirect product factor
217C22C23×D3217yes—
218C4—316no—
219C4—98no—
220C4C23×D3218yes—
221C3—21no—
222C3C23×#23512yescomplemented
223C2—616no—
224C2—188no—
225C2C22×#3798114yesdirect product factor
226C2C24×D3211yescharacteristic
subgroupG/H
center, hypercenterC23×C4D32
commutator, cosocleC2×C32C26
FrattiniC2C24×D32
FittingC23×C32×C4C22
socleC24×C32C23
hypocenterC32C23×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C23×C4×C3⋊D3C2semidirect
2. C23×C3×C4×D3C2semidirect
3. C23×#627C2semidirect
4. C23×#626C2semidirect
5. C23×#625C2semidirect
6. C22×#3798C2direct (N × H)
12. C22×C4×C3⋊D3C22semidirect
14. C22×C3×C4×D3C22semidirect
15. C22×#627C22semidirect
16. C22×#626C22semidirect
17. C22×#625C22semidirect
18. C2×#3798C22direct (N × H)
26. C23×C4×D3D3semidirect
35. C2×C4×C3⋊D3C23semidirect
37. C2×C3×C4×D3C23semidirect
38. C2×#627C23semidirect
39. C2×#626C23semidirect
40. C2×#625C23semidirect
41. #3798C23direct (N × H)
60. C22×C4×D3C2×D3semidirect
62. C23×C32#41semidirect
70. C4×C3⋊D3C24semidirect
72. C3×C4×D3C24semidirect
73. #627C24semidirect
74. #626C24semidirect
75. #625C24semidirect
105. C2×C4×D3C22×D3semidirect
107. C22×C32C2×#41semidirect
131. C23×C3#235semidirect
151. C4×D3C23×D3semidirect
153. C2×C32C22×#41semidirect
175. C22×C3C2×#235semidirect
189. C32C23×#41semidirect
194. C23#3798direct (N × H)
207. C2×C3C22×#235semidirect
216. C22C2×#3798direct (N × H)
222. C3C23×#235semidirect
225. C2C22×#3798direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×C3⋊D32
C23×C3×C4×D32
C23×#6272
C23×#6262
C23×#6252
C22×#37982
C23×#2353