Finite Group Databasebeta

← C2×#74316 #11001742

Subgroups — C2×#74316 #11001742

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×#874C211yesmaximal, characteristic, complemented
2C22×#876C211yesmaximal, characteristic, complemented
3C2×C3×#383C211yesmaximal, characteristic, complemented
4#74316C2116yesmaximal, direct product factor
5C2×#4671C216yesmaximal, complemented
6C2×#4672C216yesmaximal, complemented
7C2×#2818—31nomaximal, Sylow
8C23×D12C2216yes—
9C23×D12C2214yes—
10C22×C3×C42C2211yescharacteristic
11C22×(C2×D3)⋊C4C2213yes—
12C22×(C2×D3)⋊C4C22112yes—
13C22×Dic3⋊C4C2213yes—
14C2×#876—28no—
15C2×#876C2218yescomplemented
16C2×#874—28no—
17C2×#874C2218yescomplemented
18C3×#383C2218yescomplemented
19C2×C3×#95C2216yes—
20#4672C22148yescomplemented
21#4671C22148yescomplemented
22C22×#127—31no—
23C22×#126—31no—
24#2818—316no—
25C2×#394—36no—
26C2×#392—36no—
27C2×#383D311yescharacteristic, complemented
28C24×D3C2311yescharacteristic
29C24×D3C2314yes—
30C23×C3×C4C2314yes—
31C23×C3×C4C2313yes—
32C23×Dic3C2313yes—
33C22×D12—224no—
34C22×D12—424no—
35C22×D12—424no—
36C22×D12C23124yes—
37C22×D12C23116yes—
38C2×C3×C42D418yescomplemented
39C2×C3×C42C2314yes—
40C2×(C2×D3)⋊C4—212no—
41C2×(C2×D3)⋊C4—424no—
42C2×(C2×D3)⋊C4C23112yes—
43C2×(C2×D3)⋊C4C23148yes—
44C2×Dic3⋊C4—212no—
45C2×Dic3⋊C4C23112yes—
46#876—232no—
47#874—232no—
48C3×#95C23124yes—
49C23×D4—36no—
50C23×D4—34no—
51C2×#127—38no—
52C2×#127—68no—
53C2×#126—38no—
54C2×#126—68no—
55C22×C4⋊C4—33no—
56C22×C22⋊C4—33no—
57C22×C22⋊C4—312no—
58C22×C42C2×D311yescharacteristic
59#394—348no—
60#392—348no—
61#383C2×D318yescomplemented
62C2×#95C2×D316yes—
63C24×C3C2411yescharacteristic
64C23×D3—22no—
65C23×D3—28no—
66C23×D3—224no—
67C23×D3—224no—
68C23×D3—212no—
69C23×D3C2412yes—
70C23×D3C2418yes—
71C22×C3×C4—212no—
72C22×C3×C4—224no—
73C22×C3×C4C2×D4112yes—
74C22×C3×C4C2418yes—
75C22×C3×C4C2416yes—
76C22×Dic3—26no—
77C22×Dic3—212no—
78C22×Dic3C2416yes—
79C2×D12—248no—
80C2×D12—448no—
81C2×D12—448no—
82C2×D12—496no—
83C2×D12—464no—
84C3×C42C2×D4116yescomplemented
85(C2×D3)⋊C4—224no—
86(C2×D3)⋊C4—448no—
87Dic3⋊C4—224no—
88C25—31no—
89C25—34no—
90C22×D4—324no—
91C22×D4—316no—
92C22×D4—624no—
93C22×D4—1224no—
94C22×D4—1224no—
95C23×C4—33no—
96C23×C4C22×D314yes—
97C23×C4C22×D313yes—
98#127—632no—
99#126—632no—
100C2×C4⋊C4—312no—
101C2×C4⋊C4—612no—
102C2×C22⋊C4—312no—
103C2×C22⋊C4—348no—
104C2×C22⋊C4—612no—
105C2×C22⋊C4—1224no—
106C2×C42D1218yescomplemented
107C2×C42C22×D314yes—
108#95C22×D3124yes—
109C23×C3C22×D412yes—
110C23×C3H(5,𝔽2)112yes—
111C23×C3C2511yescharacteristic
112C22×D3—22no—
113C22×D3—28no—
114C22×D3—224no—
115C22×D3—224no—
116C22×D3—212no—
117C22×D3—448no—
118C22×D3—412no—
119C22×D3—416no—
120C22×D3—416no—
121C22×D3—448no—
122C2×C3×C4—224no—
123C2×C3×C4—212no—
124C2×C3×C4—224no—
125C2×C3×C4—416no—
126C2×C3×C4C22×D4112yes—
127C2×Dic3—26no—
128C2×Dic3—212no—
129C2×Dic3—412no—
130D12—496no—
131D12—464no—
132C24—32no—
133C24—38no—
134C24—62no—
135C24—68no—
136C24—624no—
137C24—612no—
138C24—624no—
139C24C23×D311yescharacteristic
140C2×D4—648no—
141C2×D4—1248no—
142C2×D4—1248no—
143C2×D4—1296no—
144C2×D4—1264no—
145C22×C4—212no—
146C22×C4—224no—
147C22×C4—36no—
148C22×C4—66no—
149C22×C4—612no—
150C22×C4C2×D12112yes—
151C22×C4C23×D318yes—
152C22×C4C23×D316yes—
153C4⋊C4—624no—
154C22⋊C4—624no—
155C22⋊C4—1248no—
156C42C2×D12116yescomplemented
157C22×C3#535112yes—
158C22×C3#560116yes—
159C22×C3C23×D411yescharacteristic
160C22×C3C2×H(5,𝔽2)16yes—
161C2×D3—424no—
162C2×D3—46no—
163C2×D3—48no—
164C2×D3—48no—
165C2×D3—424no—
166C2×D3—832no—
167C2×D3—88no—
168C3×C4—212no—
169C3×C4—48no—
170Dic3—46no—
171C23—62no—
172C23—68no—
173C23—624no—
174C23—612no—
175C23—624no—
176C23—1216no—
177C23—1216no—
178C23—1248no—
179C23—1248no—
180C23—1212no—
181C23C22×D1212yes—
182C23#1009112yes—
183C23C24×D311yescharacteristic
184D4—1296no—
185D4—1264no—
186C2×C4—212no—
187C2×C4—224no—
188C2×C4—224no—
189C2×C4—416no—
190C2×C4—66no—
191C2×C4—612no—
192C2×C4—1212no—
193C2×C4C22×D12112yes—
194C2×C3#281818yescomplemented
195C2×C3C2×#53513yes—
196C2×C3C2×#56014yes—
197D3—88no—
198D3—82no—
199C22—128no—
200C22—128no—
201C22—1224no—
202C22—1224no—
203C22—126no—
204C22—248no—
205C22—2432no—
206C22#5525112yes—
207C22#5711116yes—
208C22C23×D1211yescharacteristic
209C22C2×#100916yes—
210C4—212no—
211C4—48no—
212C4—126no—
213C3C2×#281811yescharacteristic, complemented, Sylow
214C2—242no—
215C2—248no—
216C2#7431618yesdirect product factor
217C2C2×#552513yes—
218C2C2×#571114yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#383C2
socleC24×C3C24
hypercenterC2×#383D3
hypocenterC3C2×#2818
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#383

Complements

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

NH ≅ G/Ntype
1. C22×#874C2semidirect
2. C22×#876C2semidirect
3. C2×C3×#383C2semidirect
4. #74316C2direct (N × H)
5. C2×#4671C2semidirect
6. C2×#4672C2semidirect
15. C2×#876C22semidirect
17. C2×#874C22semidirect
18. C3×#383C22semidirect
20. #4672C22semidirect
21. #4671C22semidirect
27. C2×#383D3semidirect
38. C2×C3×C42D4semidirect
61. #383C2×D3semidirect
84. C3×C42C2×D4semidirect
106. C2×C42D12semidirect
156. C42C2×D12semidirect
194. C2×C3#2818semidirect
213. C3C2×#2818semidirect
216. C2#74316direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8742
C22×#8762
C2×C3×#3832
#743162
C2×#46712
C2×#46722
C2×#28183