Finite Group Databasebeta

← C2×#83118 #11001744

Subgroups — C2×#83118 #11001744

Order 768.

3666 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 3666 conjugacy classes of subgroups — too many to draw. Only its 667 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
1C22×C3×#126C211yesmaximal, characteristic, complemented
2C22×#874C211yesmaximal, characteristic, complemented
3#83118C2116yesmaximal, direct product factor
4C2×#4991C216yesmaximal, complemented
5C2×#4960C216yesmaximal, complemented
6C2×#4937C211yesmaximal, characteristic, complemented
7C2×#2818—31nomaximal, Sylow
8C23×#67C2214yes—
9C23×D12C2216yes—
10C2×C3×#126—28no—
11C2×C3×#126C2218yescomplemented
12C22×C3×C4⋊C4C2213yes—
13C22×C3×C22⋊C4C2213yes—
14C22×C3×C42C2211yescharacteristic
15C22×(C2×D3)⋊C4C22112yes—
16C2×#874—28no—
17C2×#874C2218yescomplemented
18C2×C3⋊#95C2216yes—
19#4991C22148yescomplemented
20#4960C22148yescomplemented
21#4937C2218yescomplemented
22C22×#127—31no—
23C22×#126D311yescharacteristic, complemented
24#2818—316no—
25C2×#394—36no—
26C2×#392—36no—
27C2×#383—31no—
28C25×C3C2311yescharacteristic
29C24×D3C2314yes—
30C23×C3×C4C2313yes—
31C23×C3×C4C2313yes—
32C22×#67—424no—
33C22×#67C23116yes—
34C23×Dic3C2314yes—
35C22×D12—224no—
36C22×D12—424no—
37C22×D12C23124yes—
38C3×#126—232no—
39C2×C3×C4⋊C4—212no—
40C2×C3×C4⋊C4C23112yes—
41C2×C3×C22⋊C4—212no—
42C2×C3×C22⋊C4C23112yes—
43C2×C3×C42D418yescomplemented
44C2×C3×C42C2314yes—
45C2×(C2×D3)⋊C4—424no—
46C2×(C2×D3)⋊C4C23148yes—
47#874—232no—
48C3⋊#95C23124yes—
49C23×D4—34no—
50C23×D4—36no—
51C2×#127—38no—
52C2×#127—68no—
53C2×#126—28no—
54C2×#126C2×D318yescomplemented
55C22×C4⋊C4C2×D313yes—
56C22×C22⋊C4—312no—
57C22×C22⋊C4C2×D313yes—
58C22×C42C2×D311yescharacteristic
59#394—348no—
60#392—348no—
61#383—38no—
62C2×#95—36no—
63C24×C3—22no—
64C24×C3—212no—
65C24×C3C2412yes—
66C24×C3C2411yescharacteristic
67C23×D3—28no—
68C23×D3—224no—
69C23×D3—224no—
70C23×D3C2418yes—
71C22×C3×C4—212no—
72C22×C3×C4—26no—
73C22×C3×C4—212no—
74C22×C3×C4C2×D4112yes—
75C22×C3×C4C2416yes—
76C22×C3×C4C2416yes—
77C2×#67—448no—
78C2×#67—464no—
79C22×Dic3—224no—
80C22×Dic3C2418yes—
81C2×D12—248no—
82C2×D12—448no—
83C2×D12—496no—
84C3×C4⋊C4—224no—
85C3×C22⋊C4—224no—
86C3×C42C2×D4116yescomplemented
87(C2×D3)⋊C4—448no—
88C25—34no—
89C25C22×D311yescharacteristic
90C22×D4—316no—
91C22×D4—324no—
92C22×D4—624no—
93C22×D4—1224no—
94C22×D4—1224no—
95C23×C4—34no—
96C23×C4C22×D313yes—
97C23×C4C22×D313yes—
98#127—632no—
99#126—232no—
100C2×C4⋊C4—212no—
101C2×C4⋊C4C22×D3112yes—
102C2×C22⋊C4—212no—
103C2×C22⋊C4—348no—
104C2×C22⋊C4—1224no—
105C2×C22⋊C4C22×D3112yes—
106C2×C42#6718yescomplemented
107C2×C42C22×D314yes—
108#95—324no—
109C23×C3—22no—
110C23×C3—212no—
111C23×C3—412no—
112C23×C3—416no—
113C23×C3C22×D412yes—
114C23×C3H(5,𝔽2)112yes—
115C23×C3C2511yescharacteristic
116C22×D3—28no—
117C22×D3—224no—
118C22×D3—224no—
119C22×D3—448no—
120C22×D3—416no—
121C22×D3—448no—
122C2×C3×C4—224no—
123C2×C3×C4—212no—
124C2×C3×C4—26no—
125C2×C3×C4—212no—
126C2×C3×C4—412no—
127C2×C3×C4C22×D4112yes—
128#67—464no—
129C2×Dic3—224no—
130C2×Dic3—416no—
131D12—496no—
132C24—212no—
133C24—22no—
134C24—38no—
135C24—624no—
136C24—624no—
137C24—68no—
138C24C23×D312yes—
139C24C23×D311yescharacteristic
140C2×D4—648no—
141C2×D4—1248no—
142C2×D4—1296no—
143C2×D4—1248no—
144C2×D4—1264no—
145C22×C4—212no—
146C22×C4—212no—
147C22×C4—26no—
148C22×C4—38no—
149C22×C4—624no—
150C22×C4C2×#67112yes—
151C22×C4C23×D316yes—
152C22×C4C23×D316yes—
153C4⋊C4—224no—
154C22⋊C4—224no—
155C22⋊C4—1248no—
156C42C2×#67116yescomplemented
157C22×C3—46no—
158C22×C3—48no—
159C22×C3—88no—
160C22×C3#535112yes—
161C22×C3#560116yes—
162C22×C3C23×D411yescharacteristic
163C22×C3C2×H(5,𝔽2)16yes—
164C2×D3—424no—
165C2×D3—48no—
166C2×D3—424no—
167C2×D3—832no—
168C3×C4—212no—
169C3×C4—46no—
170Dic3—48no—
171C23—212no—
172C23—22no—
173C23—416no—
174C23—412no—
175C23—624no—
176C23—624no—
177C23—68no—
178C23—1216no—
179C23—1248no—
180C23—1248no—
181C23#1009112yes—
182C23C22×#6712yes—
183C23C24×D311yescharacteristic
184D4—1296no—
185D4—1264no—
186C2×C4—212no—
187C2×C4—224no—
188C2×C4—212no—
189C2×C4—26no—
190C2×C4—412no—
191C2×C4—624no—
192C2×C4—1216no—
193C2×C4C22×#67112yes—
194C2×C3—82no—
195C2×C3#281818yescomplemented
196C2×C3C2×#53513yes—
197C2×C3C2×#56014yes—
198D3—88no—
199C22—48no—
200C22—46no—
201C22—88no—
202C22—128no—
203C22—1224no—
204C22—1224no—
205C22—2432no—
206C22#5711116yes—
207C22#5830112yes—
208C22C2×#100916yes—
209C22C23×#6711yescharacteristic
210C4—212no—
211C4—46no—
212C4—128no—
213C3C2×#281811yescharacteristic, complemented, Sylow
214C2—82no—
215C2—248no—
216C2#8311818yesdirect product factor
217C2C2×#571114yes—
218C2C2×#583013yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC22×C3×#126C2
socleC24×C3C24
hypercenterC22×#126D3
hypocenterC3C2×#2818
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
1. C22×C3×#126C2semidirect
2. C22×#874C2semidirect
3. #83118C2direct (N × H)
4. C2×#4991C2semidirect
5. C2×#4960C2semidirect
6. C2×#4937C2semidirect
11. C2×C3×#126C22semidirect
17. C2×#874C22semidirect
19. #4991C22semidirect
20. #4960C22semidirect
21. #4937C22semidirect
23. C22×#126D3semidirect
43. C2×C3×C42D4semidirect
54. C2×#126C2×D3semidirect
86. C3×C42C2×D4semidirect
106. C2×C42#67semidirect
156. C42C2×#67semidirect
195. C2×C3#2818semidirect
213. C3C2×#2818semidirect
216. C2#83118direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1262
C22×#8742
#831182
C2×#49912
C2×#49602
C2×#49372
C2×#28183