Finite Group Databasebeta

← C23×#6249⋊C2 #72276414

Subgroups — C23×#6249⋊C2 #72276414

Order 3456.

3328 conjugacy classes of subgroups; only the 508 normal subgroups are shown.

normal subgroups only

This group has 3328 conjugacy classes of subgroups — too many to draw. Only its 508 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×SU(3,𝔽2)C211yesmaximal, characteristic, complemented
2C23×#1088⋊C2C211yesmaximal, characteristic
3C23×#6249C211yesmaximal, characteristic, complemented
4C22×#6249⋊C2C2128yesmaximal, direct product factor
5C23×((C3⋊D3)⋊C3)⋊C2—21no—
6C23×#1088—21no—
7C23×#1088C2211yescharacteristic
8C22×SU(3,𝔽2)—27no—
9C22×SU(3,𝔽2)C22114yescomplemented
10C22×#1088⋊C2—27no—
11C22×#1088⋊C2C22114yes—
12C22×#6249C22114yescomplemented
13C2×#6249⋊C2C221112yesdirect product factor
14C23×#272D411yescharacteristic
15C23×(C3⋊D3)⋊C3—41no—
16C22×((C3⋊D3)⋊C3)⋊C2—214no—
17C22×((C3⋊D3)⋊C3)⋊C2—47no—
18C22×#1088—214no—
19C22×#1088D417yes—
20C22×#1088C2317yes—
21C2×SU(3,𝔽2)—242no—
22C2×SU(3,𝔽2)C23128yescomplemented
23C2×#1088⋊C2—242no—
24C2×#1088⋊C2C23128yes—
25C2×#6249C23128yescomplemented
26#6249⋊C2C23164yesdirect product factor
27C23×#215—91nomaximal
28C23×D32—121no—
29C23×H(3,𝔽3)SD1611yescharacteristic, complemented
30C22×#272SD1617yescomplemented
31C22×#272C2×D417yes—
32C22×(C3⋊D3)⋊C3—414no—
33C2×((C3⋊D3)⋊C3)⋊C2—228no—
34C2×((C3⋊D3)⋊C3)⋊C2—442no—
35C2×#1088—228no—
36C2×#1088C2×D4121yes—
37C2×#1088C2417yes—
38SU(3,𝔽2)—228no—
39SU(3,𝔽2)C2418yescomplemented
40#1088⋊C2—228no—
41#1088⋊C2C2418yes—
42#6249C2418yescomplemented
43C23×C3×Q8—91no—
44C23×D12—91no—
45C22×#215—928no—
46C23×C3⋊C8—91no—
47C23×C3⋊D3—41no—
48C23×C3×D3—121no—
49C23×C3×D3—121no—
50C22×D32—1214no—
51C22×D32—1214no—
52C23×SD16—271noSylow
53C22×H(3,𝔽3)C2×SD1617yescomplemented
54C2×#272C2×SD16121yescomplemented
55C2×#272C22×D417yes—
56C2×(C3⋊D3)⋊C3—428no—
57((C3⋊D3)⋊C3)⋊C2—28no—
58((C3⋊D3)⋊C3)⋊C2—428no—
59#1088—28no—
60#1088C22×D417yes—
61#1088C2511yescharacteristic
62C24×D3—181no—
63C24×D3—361no—
64C22×C3×Q8—914no—
65C22×C3×Q8—187no—
66C23×C3×C4—91no—
67C23×C3×C4—181no—
68C22×D12—914no—
69C22×D12—187no—
70C2×#215—9112no—
71C22×C3⋊C8—914no—
72C23×C32—41no—
73C22×C3⋊D3—414no—
74C22×C3×D3—127no—
75C22×C3×D3—127no—
76C22×C3×D3—1214no—
77C2×D32—1228no—
78C2×D32—1284no—
79C23×Q8—91no—
80C23×D4—271no—
81C22×SD16—2728no—
82C23×C8—271no—
83C2×H(3,𝔽3)C22×SD1617yescomplemented
84#272C22×SD1617yescomplemented
85#272C23×D411yescharacteristic
86(C3⋊D3)⋊C3—48no—
87C24×C3—91no—
88C24×C3—361no—
89C23×D3—121no—
90C23×D3—121no—
91C23×D3—1814no—
92C23×D3—361no—
93C23×D3—3614no—
94C23×D3—367no—
95C23×D3—3614no—
96C2×C3×Q8—928no—
97C2×C3×Q8—1842no—
98C22×C3×C4—97no—
99C22×C3×C4—97no—
100C22×C3×C4—1814no—
101C2×D12—928no—
102C2×D12—1842no—
103#215—964no—
104C2×C3⋊C8—928no—
105C22×C32—47no—
106C2×C3⋊D3—428no—
107C2×C3×D3—1221no—
108C2×C3×D3—127no—
109C2×C3×D3—1228no—
110D32—128no—
111D32—1256no—
112C25—541no—
113C22×Q8—914no—
114C22×Q8—187no—
115C22×D4—2714no—
116C22×D4—547no—
117C23×C4—91no—
118C23×C4—181no—
119C2×SD16—27112no—
120C22×C8—2714no—
121H(3,𝔽3)C23×SD1611yescharacteristic, complemented, Sylow
122C23×C3—97no—
123C23×C3—97no—
124C23×C3—121no—
125C23×C3—3614no—
126C23×C3AΓL(1,𝔽9)11yescharacteristic
127C22×D3—127no—
128C22×D3—127no—
129C22×D3—1214no—
130C22×D3—1828no—
131C22×D3—3614no—
132C22×D3—3628no—
133C22×D3—3642no—
134C22×D3—3684no—
135C3×Q8—98no—
136C3×Q8—1828no—
137C2×C3×C4—921no—
138C2×C3×C4—97no—
139C2×C3×C4—1828no—
140D12—98no—
141D12—1828no—
142C3⋊C8—98no—
143C2×C32—47no—
144C3⋊D3—48no—
145C3×D3—127no—
146C3×D3—121no—
147C3×D3—128no—
148C24—91no—
149C24—361no—
150C24—5414no—
151C24—1087no—
152C2×Q8—928no—
153C2×Q8—1842no—
154C2×D4—2728no—
155C2×D4—5442no—
156C22×C4—97no—
157C22×C4—97no—
158C22×C4—1814no—
159SD16—2764no—
160C2×C8—2728no—
161C22×C3—97no—
162C22×C3—921no—
163C22×C3—127no—
164C22×C3—3628no—
165C22×C3C2×AΓL(1,𝔽9)17yes—
166C2×D3—127no—
167C2×D3—1221no—
168C2×D3—1228no—
169C2×D3—188no—
170C2×D3—3628no—
171C2×D3—368no—
172C2×D3—3628no—
173C2×D3—3656no—
174C3×C4—91no—
175C3×C4—97no—
176C3×C4—188no—
177C32—41no—
178C23—97no—
179C23—97no—
180C23—3614no—
181C23—5428no—
182C23—10842no—
183C23#6249⋊C211yescharacteristic, direct product factor
184Q8—98no—
185Q8—1828no—
186D4—278no—
187D4—5428no—
188C2×C4—97no—
189C2×C4—921no—
190C2×C4—1828no—
191C8—278no—
192C2×C3—91no—
193C2×C3—97no—
194C2×C3—127no—
195C2×C3—368no—
196C2×C3C22×AΓL(1,𝔽9)17yes—
197D3—121no—
198D3—127no—
199D3—128no—
200D3—368no—
201C22—97no—
202C22—921no—
203C22—3628no—
204C22—548no—
205C22—10828no—
206C22C2×#6249⋊C217yesdirect product factor
207C4—91no—
208C4—97no—
209C4—188no—
210C3—121no—
211C3C23×AΓL(1,𝔽9)11yescharacteristic
212C2—91no—
213C2—97no—
214C2—368no—
215C2C22×#6249⋊C217yesdirect product factor
subgroupG/H
center, hypercenterC23#6249⋊C2
commutator, cosocle#1088C25
FrattiniC3C23×AΓL(1,𝔽9)
FittingC23×H(3,𝔽3)SD16
socleC23×C3AΓL(1,𝔽9)
prime radicalC23×#1088⋊C2C2
hypocenterH(3,𝔽3)C23×SD16
LCSG ▷ #1088 ▷ #272 ▷ H(3,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×SU(3,𝔽2)C2semidirect
3. C23×#6249C2semidirect
4. C22×#6249⋊C2C2direct (N × H)
9. C22×SU(3,𝔽2)C22semidirect
12. C22×#6249C22semidirect
13. C2×#6249⋊C2C22direct (N × H)
22. C2×SU(3,𝔽2)C23semidirect
25. C2×#6249C23semidirect
26. #6249⋊C2C23direct (N × H)
29. C23×H(3,𝔽3)SD16semidirect
30. C22×#272SD16semidirect
39. SU(3,𝔽2)C24semidirect
42. #6249C24semidirect
53. C22×H(3,𝔽3)C2×SD16semidirect
54. C2×#272C2×SD16semidirect
83. C2×H(3,𝔽3)C22×SD16semidirect
84. #272C22×SD16semidirect
121. H(3,𝔽3)C23×SD16semidirect
183. C23#6249⋊C2direct (N × H)
206. C22C2×#6249⋊C2direct (N × H)
215. C2C22×#6249⋊C2direct (N × H)

Maximal subgroups

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

subgroupindex
C23×SU(3,𝔽2)2
C23×#1088⋊C22
C23×#62492
C22×#6249⋊C22
C23×#2159