Finite Group Databasebeta

← C2×C32×#986 #13181364

Subgroups — C2×C32×#986 #13181364

Order 1728.

1118 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C2×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C32×#986C214yesmaximal, direct product factor
3C2×C32×GL(2,𝔽3)C212yesmaximal, complemented
4C2×C3×#986C314yesmaximal, direct product factor
5C2×C32×AGL(1,ℤ/8)—31nomaximal
6C2×C32×D12—41nomaximal
7C32×#41⋊C3C2212yescomplemented
8C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
9C32×GL(2,𝔽3)C2214yescomplemented
10C2×C32×#41D311yescharacteristic, complemented
11C22×C32×D4—31no—
12C2×C3×#41⋊C3—24no—
13C2×C3×#41⋊C3C2×C314yescomplemented
14C3×#986C2×C3116yesdirect product factor
15C2×C3×GL(2,𝔽3)C2×C318yescomplemented
16C32×AGL(1,ℤ/8)—34no—
17C32×AGL(1,ℤ/8)—34no—
18C2×C32×SD16—32no—
19C2×C32×D8—32no—
20C2×C32×M16—31no—
21C22×C32×D3—42no—
22C2×C33×C4—41no—
23C32×D12—44no—
24C32×SL(2,𝔽3)C2311yescharacteristic, complemented
25C2×#986C3211yescharacteristic, direct product factor
26C2×C3×AGL(1,ℤ/8)—34no—
27C24×C32—61no—
28C32×#41—32no—
29C32×#41C2×D312yescomplemented
30C2×C32×Q8C2×D311yescharacteristic
31C2×C32×D4—31no—
32C2×C32×D4—32no—
33C2×C32×D4—34no—
34C2×C32×D4—62no—
35C2×C32×D4—61no—
36C22×C32×C4—31no—
37C2×C3×D12—44no—
38C3×#41⋊C3—28no—
39C3×#41⋊C3C22×C318yescomplemented
40C2×C3×SL(2,𝔽3)—24no—
41C2×C3×SL(2,𝔽3)C22×C314yescomplemented
42C3×GL(2,𝔽3)C22×C3116yescomplemented
43C32×SD16—34no—
44C32×SD16—34no—
45C32×D8—38no—
46C32×M16—34no—
47C2×C32×C8—32no—
48C22×C33—41no—
49C2×C32×D3—44no—
50C2×C32×D3—84no—
51C33×C4—42no—
52C2×C3×#41C3×D314yescomplemented
53C22×C3×D4—34no—
54C2×#41⋊C3—24no—
55C2×#41⋊C3C2×C3211yescharacteristic, complemented
56#986C2×C3214yesdirect product factor
57C2×GL(2,𝔽3)C2×C3212yescomplemented
58C3×AGL(1,ℤ/8)—316no—
59C3×AGL(1,ℤ/8)—316no—
60C2×C3×SD16—38no—
61C2×C3×D8—38no—
62C2×C3×M16—34no—
63C23×C32—31no—
64C23×C32—64no—
65C23×C32—62no—
66C23×C32—124no—
67C22×C3×D3—48no—
68C32×Q8—31no—
69C32×Q8C22×D311yescharacteristic
70C32×D4—32no—
71C32×D4—34no—
72C32×D4—61no—
73C32×D4—62no—
74C32×D4—64no—
75C2×C32×C4—31no—
76C2×C32×C4—32no—
77C2×C32×C4—32no—
78C2×C32×C4—44no—
79C2×C32×C4—84no—
80C2×C32×C4S411yescharacteristic
81C3×D12—416no—
82C3×SL(2,𝔽3)—24no—
83C3×SL(2,𝔽3)C23×C314yescomplemented
84C32×C8—34no—
85C2×AGL(1,ℤ/8)—31noSylow
86C2×C33—42no—
87C2×C33—41no—
88C32×D3—84no—
89C24×C3—64no—
90C3×#41—38no—
91C3×#41C2×C3×D318yescomplemented
92C2×C3×Q8C2×C3×D314yes—
93C2×C3×D4—34no—
94C2×C3×D4—38no—
95C2×C3×D4—316no—
96C2×C3×D4—68no—
97C2×C3×D4—64no—
98C22×C3×C4—34no—
99C2×D12—41no—
100#41⋊C3—28no—
101#41⋊C3C22×C3212yescomplemented
102C2×SL(2,𝔽3)—24no—
103C2×SL(2,𝔽3)C22×C3211yescharacteristic, complemented
104GL(2,𝔽3)C22×C3214yescomplemented
105C3×SD16—316no—
106C3×SD16—316no—
107C3×D8—332no—
108C3×M16—316no—
109C2×C3×C8—38no—
110C22×C32—32no—
111C22×C32—44no—
112C22×C32—62no—
113C22×C32—64no—
114C22×C32—84no—
115C22×C32—128no—
116C22×C32—124no—
117C22×C32C2×S411yescharacteristic
118C2×C3×D3—416no—
119C2×C3×D3—816no—
120C32×C4—31no—
121C32×C4—31no—
122C32×C4—48no—
123C32×C4—88no—
124C32×C4C2×S412yes—
125C2×#41C32×D311yescharacteristic, complemented
126C22×D4—31no—
127AGL(1,ℤ/8)—34no—
128AGL(1,ℤ/8)—34no—
129C2×SD16—32no—
130C2×D8—32no—
131C2×M16—31no—
132C33—41noSylow
133C23×C3—34no—
134C23×C3—616no—
135C23×C3—68no—
136C23×C3—1216no—
137C22×D3—42no—
138C3×Q8—34no—
139C3×Q8C22×C3×D314yes—
140C3×D4—38no—
141C3×D4—316no—
142C3×D4—64no—
143C3×D4—68no—
144C3×D4—616no—
145C2×C3×C4—34no—
146C2×C3×C4—38no—
147C2×C3×C4—38no—
148C2×C3×C4—41no—
149C2×C3×C4—84no—
150C2×C3×C4C3×S414yes—
151D12—44no—
152SL(2,𝔽3)—24no—
153SL(2,𝔽3)C23×C3211yescharacteristic, complemented
154C3×C8—316no—
155C2×C32—48no—
156C2×C32—44no—
157C2×C32—62no—
158C2×C32—88no—
159C2×C32—84no—
160C2×C32—124no—
161C2×C32#98612yesdirect product factor
162C2×C32C22×S411yescharacteristic
163C3×D3—816no—
164C24—61no—
165#41—32no—
166#41C2×C32×D312yescomplemented
167C2×Q8C2×C32×D311yescharacteristic
168C2×D4—31no—
169C2×D4—32no—
170C2×D4—34no—
171C2×D4—62no—
172C2×D4—61no—
173C22×C4—31no—
174SD16—34no—
175SD16—34no—
176D8—38no—
177M16—34no—
178C2×C8—32no—
179C22×C3—38no—
180C22×C3—41no—
181C22×C3—68no—
182C22×C3—616no—
183C22×C3—84no—
184C22×C3—1232no—
185C22×C3—1216no—
186C22×C3C2×C3×S414yes—
187C2×D3—44no—
188C2×D3—84no—
189C3×C4—34no—
190C3×C4—34no—
191C3×C4—42no—
192C3×C4—88no—
193C3×C4C2×C3×S418yes—
194C32—44no—
195C32—84no—
196C32C2×#98611yescharacteristic, direct product factor
197C23—31no—
198C23—64no—
199C23—62no—
200C23—124no—
201Q8—31no—
202Q8C22×C32×D311yescharacteristic
203D4—32no—
204D4—34no—
205D4—61no—
206D4—62no—
207D4—64no—
208C2×C4—31no—
209C2×C4—32no—
210C2×C4—32no—
211C2×C4C32×S411yescharacteristic
212C8—34no—
213C2×C3—42no—
214C2×C3—41no—
215C2×C3—68no—
216C2×C3—88no—
217C2×C3—84no—
218C2×C3—1216no—
219C2×C3C3×#98618yesdirect product factor
220C2×C3C22×C3×S414yes—
221D3—84no—
222C22—32no—
223C22—62no—
224C22—64no—
225C22—128no—
226C22—124no—
227C22C2×C32×S411yescharacteristic
228C4—31no—
229C4—31no—
230C4C2×C32×S412yes—
231C3—41no—
232C3—84no—
233C3C2×C3×#98614yesdirect product factor
234C2—62no—
235C2—124no—
236C2C32×#98612yesdirect product factor
237C2C22×C32×S411yescharacteristic
subgroupG/H
center, socleC22×C32C2×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C23×C32
FrattiniC2C22×C32×S4
FittingC2×C32×#41D3
hypercenterC2×C32×C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C32 ◁ C2×C32×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C32×#986C2direct (N × H)
3. C2×C32×GL(2,𝔽3)C2semidirect
4. C2×C3×#986C3direct (N × H)
7. C32×#41⋊C3C22semidirect
8. C2×C32×SL(2,𝔽3)C22semidirect
9. C32×GL(2,𝔽3)C22semidirect
10. C2×C32×#41D3semidirect
13. C2×C3×#41⋊C3C2×C3semidirect
14. C3×#986C2×C3direct (N × H)
15. C2×C3×GL(2,𝔽3)C2×C3semidirect
24. C32×SL(2,𝔽3)C23semidirect
25. C2×#986C32direct (N × H)
29. C32×#41C2×D3semidirect
39. C3×#41⋊C3C22×C3semidirect
41. C2×C3×SL(2,𝔽3)C22×C3semidirect
42. C3×GL(2,𝔽3)C22×C3semidirect
52. C2×C3×#41C3×D3semidirect
55. C2×#41⋊C3C2×C32semidirect
56. #986C2×C32direct (N × H)
57. C2×GL(2,𝔽3)C2×C32semidirect
83. C3×SL(2,𝔽3)C23×C3semidirect
91. C3×#41C2×C3×D3semidirect
101. #41⋊C3C22×C32semidirect
103. C2×SL(2,𝔽3)C22×C32semidirect
104. GL(2,𝔽3)C22×C32semidirect
125. C2×#41C32×D3semidirect
153. SL(2,𝔽3)C23×C32semidirect
161. C2×C32#986direct (N × H)
166. #41C2×C32×D3semidirect
196. C32C2×#986direct (N × H)
219. C2×C3C3×#986direct (N × H)
233. C3C2×C3×#986direct (N × H)
236. C2C32×#986direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C32×#9862
C2×C32×GL(2,𝔽3)2
C2×C3×#9863
C2×C32×AGL(1,ℤ/8)3
C2×C32×D124