Finite Group Databasebeta

← C32×#5244 #13162680

Subgroups — C32×#5244 #13162680

Order 1728.

1852 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1852 conjugacy classes of subgroups — too many to draw. Only its 258 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
1C33×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C32×#937C213yesmaximal, complemented
3C32×#834C213yesmaximal, complemented
4C32×B(4,𝔽2)—31nomaximal
5C3×#5244C314yesmaximal, direct product factor
6C23×C32×D3C2211yescharacteristic, complemented
7C33×#41—23no—
8C2×C33×D4—23no—
9C2×C33×D4C2213yes—
10C2×C32×#67—23no—
11C32×#217C2213yescomplemented
12C32×(C2×D3)⋊C4—23no—
13C32×H(5,𝔽2)—24no—
14C32×H(5,𝔽2)D311yescharacteristic, complemented
15C32×H(5,𝔽2)C2×C314yescomplemented
16C32×#120—33no—
17C3×#937C2×C3112yescomplemented
18C32×#99—33no—
19C3×#834C2×C3112yescomplemented
20C23×C33—21no—
21C23×C33D413yescomplemented
22C23×C33C2311yescharacteristic
23C22×C32×D3—23no—
24C22×C32×D3—42no—
25C33×Q8—21no—
26C33×D4—23no—
27C33×D4—26no—
28C2×C33×C4—23no—
29C2×C33×C4D413yescomplemented
30C32×#67—46no—
31C2×C32×Dic3—23no—
32C3×B(4,𝔽2)—34no—
33#5244C3211yescharacteristic, direct product factor
34C24×C32—31no—
35C23×C3×D3C22×C314yescomplemented
36C32×#41—224no—
37C32×#41—23no—
38C32×#41—212no—
39C2×C32×D4—224no—
40C2×C32×D4—212no—
41C2×C32×D4—23no—
42C2×C32×D4—212no—
43C2×C32×D4—63no—
44C2×C32×D4C2×D313yes—
45C2×C32×D4C22×C3112yes—
46C2×C3×#67—212no—
47C32×C22⋊C4—33no—
48C32×C22⋊C4—63no—
49C3×#217C22×C3112yescomplemented
50C3×(C2×D3)⋊C4—212no—
51C22×C33—23no—
52C22×C33—41no—
53C22×C33—43no—
54C22×C33—81no—
55C22×C33C2×D413yes—
56C2×C32×D3—26no—
57C2×C32×D3—43no—
58C2×C32×D3—41no—
59C33×C4—23no—
60C32×Dic3—43no—
61C3×H(5,𝔽2)—24no—
62C3×H(5,𝔽2)C3×D314yescomplemented
63C3×H(5,𝔽2)C2×C3211yescharacteristic, complemented
64C3×#120—312no—
65#937C2×C3213yescomplemented
66C3×#99—312no—
67#834C2×C3213yescomplemented
68C23×C32—28no—
69C23×C32—21no—
70C23×C32—24no—
71C23×C32—212no—
72C23×C32—24no—
73C23×C32—63no—
74C23×C32—122no—
75C23×C32#6713yescomplemented
76C23×C32C3×D4112yescomplemented
77C23×C32C22×D311yescharacteristic
78C23×C32C23×C314yes—
79C22×C3×D3—212no—
80C22×C3×D3—48no—
81C32×Q8—28no—
82C32×Q8—21no—
83C32×Q8—24no—
84C32×D4—224no—
85C32×D4—23no—
86C32×D4—212no—
87C32×D4—248no—
88C32×D4—26no—
89C32×D4—224no—
90C32×D4—126no—
91C2×C32×C4—224no—
92C2×C32×C4—23no—
93C2×C32×C4—212no—
94C2×C32×C4—212no—
95C2×C32×C4—63no—
96C2×C32×C4#6713yescomplemented
97C2×C32×C4C3×D4112yescomplemented
98C3×#67—424no—
99C2×C3×Dic3—212no—
100B(4,𝔽2)—31noSylow
101C2×C33—23no—
102C2×C33—43no—
103C2×C33#12011yescharacteristic
104C32×D3—42no—
105C24×C3—34no—
106C23×D3C22×C3211yescharacteristic, complemented
107C3×#41—224no—
108C3×#41—212no—
109C3×#41—23no—
110C2×C3×D4—224no—
111C2×C3×D4—212no—
112C2×C3×D4—23no—
113C2×C3×D4—212no—
114C2×C3×D4—612no—
115C2×C3×D4C2×C3×D3112yes—
116C2×C3×D4C22×C3213yes—
117C2×#67—23no—
118C3×C22⋊C4—312no—
119C3×C22⋊C4—612no—
120#217C22×C3213yescomplemented
121(C2×D3)⋊C4—23no—
122C22×C32—224no—
123C22×C32—23no—
124C22×C32—212no—
125C22×C32—212no—
126C22×C32—41no—
127C22×C32—43no—
128C22×C32—44no—
129C22×C32—412no—
130C22×C32—66no—
131C22×C32—84no—
132C22×C32—812no—
133C22×C32—88no—
134C22×C32—81no—
135C22×C32—84no—
136C22×C32—123no—
137C22×C32—121no—
138C22×C32C2×#6713yes—
139C22×C32C2×C3×D4112yes—
140C2×C3×D3—224no—
141C2×C3×D3—412no—
142C2×C3×D3—44no—
143C32×C4—224no—
144C32×C4—23no—
145C32×C4—212no—
146C32×C4—123no—
147C3×Dic3—412no—
148H(5,𝔽2)C32×D311yescharacteristic, complemented
149#120—33no—
150#99—33no—
151C33B(4,𝔽2)11yescharacteristic, complemented, Sylow
152C23×C3—28no—
153C23×C3—24no—
154C23×C3—21no—
155C23×C3—212no—
156C23×C3—24no—
157C23×C3—612no—
158C23×C3—128no—
159C23×C3C3×#67112yescomplemented
160C23×C3C32×D413yescomplemented
161C23×C3C22×C3×D314yes—
162C23×C3C23×C3211yescharacteristic
163C22×D3—23no—
164C22×D3—42no—
165C3×Q8—28no—
166C3×Q8—24no—
167C3×Q8—21no—
168C3×D4—224no—
169C3×D4—212no—
170C3×D4—23no—
171C3×D4—248no—
172C3×D4—224no—
173C3×D4—26no—
174C3×D4—1224no—
175C2×C3×C4—224no—
176C2×C3×C4—212no—
177C2×C3×C4—23no—
178C2×C3×C4—212no—
179C2×C3×C4—612no—
180C2×C3×C4C3×#67112yescomplemented
181C2×C3×C4C32×D413yescomplemented
182#67—46no—
183C2×Dic3—23no—
184C2×C32—23no—
185C2×C32—24no—
186C2×C32—212no—
187C2×C32—412no—
188C2×C32—424no—
189C2×C32—43no—
190C2×C32—412no—
191C2×C32—122no—
192C2×C32#95311yescharacteristic
193C2×C32C3×#12014yes—
194C3×D3—48no—
195C24—31no—
196#41—23no—
197C2×D4—23no—
198C2×D4—63no—
199C2×D4C2×C32×D313yes—
200C22⋊C4—33no—
201C22⋊C4—63no—
202C22×C3—224no—
203C22×C3—212no—
204C22×C3—23no—
205C22×C3—212no—
206C22×C3—41no—
207C22×C3—44no—
208C22×C3—43no—
209C22×C3—412no—
210C22×C3—624no—
211C22×C3—84no—
212C22×C3—812no—
213C22×C3—88no—
214C22×C3—81no—
215C22×C3—84no—
216C22×C3—1212no—
217C22×C3—124no—
218C22×C3C2×C3×#67112yes—
219C22×C3C2×C32×D413yes—
220C2×D3—26no—
221C2×D3—43no—
222C2×D3—41no—
223C3×C4—224no—
224C3×C4—212no—
225C3×C4—23no—
226C3×C4—1212no—
227Dic3—43no—
228C32—24no—
229C32#524411yescharacteristic, direct product factor
230C32C3×B(4,𝔽2)14yescomplemented
231C23—21no—
232C23—63no—
233C23—122no—
234C23C32×#6713yescomplemented
235C23C22×C32×D311yescharacteristic
236Q8—21no—
237D4—23no—
238D4—26no—
239D4—126no—
240C2×C4—23no—
241C2×C4—63no—
242C2×C4C32×#6713yescomplemented
243C2×C3—23no—
244C2×C3—212no—
245C2×C3—24no—
246C2×C3—412no—
247C2×C3—424no—
248C2×C3—43no—
249C2×C3—412no—
250C2×C3—128no—
251C2×C3C3×#95314yes—
252C2×C3C32×#12011yescharacteristic
253D3—42no—
254C22—23no—
255C22—41no—
256C22—43no—
257C22—66no—
258C22—81no—
259C22—123no—
260C22—121no—
261C22C2×C32×#6713yes—
262C4—23no—
263C4—123no—
264C3—24no—
265C3C3×#524414yesdirect product factor
266C3C32×B(4,𝔽2)11yescharacteristic, complemented
267C2—23no—
268C2—43no—
269C2—122no—
270C2C32×#95311yescharacteristic
subgroupG/H
centerC2×C32#953
commutator, cosocleC23×C3C23×C32
FrattiniC23C22×C32×D3
FittingC33×H(5,𝔽2)C2
socleC2×C33#120
hypercenterC32×H(5,𝔽2)D3
hypocenterC3C32×B(4,𝔽2)
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C23×C32 ◁ C32×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C33×H(5,𝔽2)C2semidirect
2. C32×#937C2semidirect
3. C32×#834C2semidirect
5. C3×#5244C3direct (N × H)
6. C23×C32×D3C22semidirect
11. C32×#217C22semidirect
14. C32×H(5,𝔽2)D3semidirect
15. C32×H(5,𝔽2)C2×C3semidirect
17. C3×#937C2×C3semidirect
19. C3×#834C2×C3semidirect
21. C23×C33D4semidirect
29. C2×C33×C4D4semidirect
33. #5244C32direct (N × H)
35. C23×C3×D3C22×C3semidirect
49. C3×#217C22×C3semidirect
62. C3×H(5,𝔽2)C3×D3semidirect
63. C3×H(5,𝔽2)C2×C32semidirect
65. #937C2×C32semidirect
67. #834C2×C32semidirect
75. C23×C32#67semidirect
76. C23×C32C3×D4semidirect
96. C2×C32×C4#67semidirect
97. C2×C32×C4C3×D4semidirect
106. C23×D3C22×C32semidirect
120. #217C22×C32semidirect
148. H(5,𝔽2)C32×D3semidirect
151. C33B(4,𝔽2)semidirect
159. C23×C3C3×#67semidirect
160. C23×C3C32×D4semidirect
180. C2×C3×C4C3×#67semidirect
181. C2×C3×C4C32×D4semidirect
229. C32#5244direct (N × H)
230. C32C3×B(4,𝔽2)semidirect
234. C23C32×#67semidirect
242. C2×C4C32×#67semidirect
265. C3C3×#5244direct (N × H)
266. C3C32×B(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C33×H(5,𝔽2)2
C32×#9372
C32×#8342
C32×B(4,𝔽2)3
C3×#52443