Finite Group Databasebeta

← C32×A4×#41 #13181701

Subgroups — C32×A4×#41 #13181701

Order 1728.

1930 conjugacy classes of subgroups; only the 578 normal subgroups are shown.

normal subgroups only

This group has 1930 conjugacy classes of subgroups — too many to draw. Only its 578 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
1C32×Q8×A4C211yesmaximal, characteristic, complemented
2C32×D4×A4C213yesmaximal, complemented
3C2×C32×C4×A4C213yesmaximal, complemented
4C22×C32×#41C311yesmaximal, characteristic, complemented
5C3×A4×#41C3112yesmaximal, direct product factor
6C22×C32×A4C2213yes—
7C33×#41—41nomaximal
8C32×C4×A4C2211yescharacteristic
9C32×C4×A4C2213yes—
10C2×C32×#41—33no—
11C2×C32×#41—33no—
12C2×C32×#41—31no—
13C2×C32×#41—31no—
14C22×C32×Q8C2×C311yescharacteristic, complemented
15C22×C32×D4C2×C313yescomplemented
16C23×C32×C4C2×C313yescomplemented
17C3×Q8×A4C2×C3112yescomplemented
18C3×D4×A4C2×C3136yescomplemented
19C2×C3×C4×A4C2×C3136yescomplemented
20C2×C32×A4—23no—
21C2×C32×A4C2311yescharacteristic
22C33×Q8—41no—
23C33×D4—43no—
24C2×C33×C4—43no—
25C22×C3×#41C3214yescomplemented
26A4×#41C3219yesdirect product factor
27C24×C32C22×C313yes—
28C22×C3×A4C22×C3136yes—
29C32×#41—36no—
30C32×#41—36no—
31C32×#41—32no—
32C32×#41—31no—
33C32×#41—33no—
34C32×#41—33no—
35C32×#41—412no—
36C32×#41A411yescharacteristic, direct product factor
37C2×C32×Q8—31no—
38C2×C32×Q8—33no—
39C2×C32×D4—33no—
40C2×C32×D4—33no—
41C2×C32×D4—36no—
42C22×C32×C4—33no—
43C22×C32×C4—33no—
44C22×C32×C4—33no—
45C22×C32×C4—33no—
46C22×C32×C4C22×C311yescharacteristic
47C22×C32×C4C22×C313yes—
48C3×C4×A4C22×C3112yes—
49C3×C4×A4C22×C3136yes—
50C22×C33—43no—
51C32×A4#4111yescharacteristic, direct product factor
52C33×C4—43no—
53C33×C4—41no—
54C2×C3×#41—312no—
55C2×C3×#41—312no—
56C2×C3×#41—34no—
57C2×C3×#41—34no—
58C22×C3×Q8C2×C3214yescomplemented
59C22×C3×D4C2×C32112yescomplemented
60C23×C3×C4C2×C32112yescomplemented
61Q8×A4C2×C3219yescomplemented
62D4×A4C2×C32127yescomplemented
63C2×C4×A4C2×C32127yescomplemented
64C23×C32—23no—
65C23×C32—33no—
66C23×C32—33no—
67C23×C32—63no—
68C23×C32C23×C311yescharacteristic
69C2×C3×A4—236no—
70C2×C3×A4C23×C3112yes—
71C32×Q8—32no—
72C32×Q8—33no—
73C32×Q8—412no—
74C32×Q8C2×A411yescharacteristic, complemented
75C32×D4—36no—
76C32×D4—36no—
77C32×D4—33no—
78C32×D4—436no—
79C32×D4C2×A413yescomplemented
80C2×C32×C4—36no—
81C2×C32×C4—33no—
82C2×C32×C4—33no—
83C2×C32×C4—33no—
84C2×C32×C4—31no—
85C2×C32×C4—33no—
86C2×C32×C4—31no—
87C2×C32×C4—33no—
88C2×C32×C4—436no—
89C2×C32×C4C2×A413yescomplemented
90C22×#41C3311yescharacteristic, complemented, Sylow
91C2×C33—41no—
92C2×C33—83no—
93C24×C3C22×C32112yes—
94C22×A4C22×C32127yes—
95C3×#41—324no—
96C3×#41—324no—
97C3×#41—38no—
98C3×#41—34no—
99C3×#41—312no—
100C3×#41—312no—
101C3×#41—49no—
102C3×#41C3×A414yesdirect product factor
103C2×C3×Q8—312no—
104C2×C3×Q8—34no—
105C2×C3×D4—312no—
106C2×C3×D4—312no—
107C2×C3×D4—324no—
108C22×C3×C4—312no—
109C22×C3×C4—312no—
110C22×C3×C4—312no—
111C22×C3×C4—312no—
112C22×C3×C4C22×C3214yes—
113C22×C3×C4C22×C32112yes—
114C4×A4C22×C3219yes—
115C4×A4C22×C32127yes—
116C22×C32—31no—
117C22×C32—31no—
118C22×C32—33no—
119C22×C32—436no—
120C22×C32—63no—
121C22×C32—63no—
122C22×C32—63no—
123C22×C32—63no—
124C22×C32C3×#4111yescharacteristic, complemented
125C22×C32C22×A413yes—
126C3×A4C3×#41112yesdirect product factor
127C32×C4—31no—
128C32×C4—33no—
129C32×C4—412no—
130C32×C4—436no—
131C32×C4C22×A413yes—
132C32×C4C22×A411yescharacteristic
133C2×#41—33no—
134C2×#41—33no—
135C2×#41—31no—
136C2×#41—31no—
137C22×Q8C2×C3311yescharacteristic, complemented
138C22×D4C2×C3313yescomplemented
139C23×C4C2×C3313yescomplemented
140C33—41noSylow
141C23×C3—212no—
142C23×C3—312no—
143C23×C3—312no—
144C23×C3—612no—
145C23×C3C23×C3214yes—
146C2×A4—227no—
147C2×A4C23×C3219yes—
148C3×Q8—38no—
149C3×Q8—312no—
150C3×Q8—49no—
151C3×Q8C2×C3×A414yescomplemented
152C3×D4—324no—
153C3×D4—324no—
154C3×D4—312no—
155C3×D4—427no—
156C3×D4C2×C3×A4112yescomplemented
157C2×C3×C4—324no—
158C2×C3×C4—312no—
159C2×C3×C4—34no—
160C2×C3×C4—312no—
161C2×C3×C4—312no—
162C2×C3×C4—312no—
163C2×C3×C4—34no—
164C2×C3×C4—312no—
165C2×C3×C4—427no—
166C2×C3×C4C2×C3×A4112yescomplemented
167C2×C32—23no—
168C2×C32—31no—
169C2×C32—31no—
170C2×C32—412no—
171C2×C32—63no—
172C2×C32—836no—
173C2×C32C23×A411yescharacteristic
174C24C22×C3313yes—
175#41—36no—
176#41—36no—
177#41—32no—
178#41—31no—
179#41—33no—
180#41—33no—
181#41C32×A411yescharacteristic, direct product factor
182C2×Q8—33no—
183C2×Q8—31no—
184C2×D4—33no—
185C2×D4—33no—
186C2×D4—36no—
187C22×C4—33no—
188C22×C4—33no—
189C22×C4—33no—
190C22×C4—33no—
191C22×C4C22×C3311yescharacteristic
192C22×C4C22×C3313yes—
193C22×C3—34no—
194C22×C3—34no—
195C22×C3—312no—
196C22×C3—427no—
197C22×C3—612no—
198C22×C3—612no—
199C22×C3—612no—
200C22×C3—612no—
201C22×C3C32×#4114yescomplemented
202C22×C3C22×C3×A4112yes—
203A4C32×#4119yesdirect product factor
204C3×C4—34no—
205C3×C4—312no—
206C3×C4—49no—
207C3×C4—427no—
208C3×C4C22×C3×A414yes—
209C3×C4C22×C3×A4112yes—
210C32—412no—
211C32A4×#4111yescharacteristic, direct product factor
212C23—23no—
213C23—33no—
214C23—33no—
215C23—63no—
216C23C23×C3311yescharacteristic
217Q8—32no—
218Q8—33no—
219Q8C2×C32×A411yescharacteristic, complemented
220D4—36no—
221D4—36no—
222D4—33no—
223D4C2×C32×A413yescomplemented
224C2×C4—36no—
225C2×C4—33no—
226C2×C4—31no—
227C2×C4—33no—
228C2×C4—33no—
229C2×C4—33no—
230C2×C4—31no—
231C2×C4—33no—
232C2×C4C2×C32×A413yescomplemented
233C2×C3—212no—
234C2×C3—34no—
235C2×C3—34no—
236C2×C3—49no—
237C2×C3—612no—
238C2×C3—827no—
239C2×C3C23×C3×A414yes—
240C22—31no—
241C22—31no—
242C22—33no—
243C22—63no—
244C22—63no—
245C22—63no—
246C22—63no—
247C22C33×#4111yescharacteristic, complemented
248C22C22×C32×A413yes—
249C4—31no—
250C4—33no—
251C4C22×C32×A411yescharacteristic
252C4C22×C32×A413yes—
253C3—49no—
254C3C3×A4×#4114yesdirect product factor
255C2—23no—
256C2—31no—
257C2—31no—
258C2—63no—
259C2C23×C32×A411yescharacteristic
subgroupG/H
centerC32×C4C22×A4
commutator, cosocleC23C23×C33
FrattiniC2C23×C32×A4
FittingC22×C32×#41C3
socleC23×C32C23×C3
hypercenterC32×#41A4
hypocenterC22C33×#41
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C32×C4 ◁ C32×#41

Complements

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

NH ≅ G/Ntype
1. C32×Q8×A4C2semidirect
2. C32×D4×A4C2semidirect
3. C2×C32×C4×A4C2semidirect
4. C22×C32×#41C3semidirect
5. C3×A4×#41C3direct (N × H)
14. C22×C32×Q8C2×C3semidirect
15. C22×C32×D4C2×C3semidirect
16. C23×C32×C4C2×C3semidirect
17. C3×Q8×A4C2×C3semidirect
18. C3×D4×A4C2×C3semidirect
19. C2×C3×C4×A4C2×C3semidirect
25. C22×C3×#41C32semidirect
26. A4×#41C32direct (N × H)
36. C32×#41A4direct (N × H)
51. C32×A4#41direct (N × H)
58. C22×C3×Q8C2×C32semidirect
59. C22×C3×D4C2×C32semidirect
60. C23×C3×C4C2×C32semidirect
61. Q8×A4C2×C32semidirect
62. D4×A4C2×C32semidirect
63. C2×C4×A4C2×C32semidirect
74. C32×Q8C2×A4semidirect
79. C32×D4C2×A4semidirect
89. C2×C32×C4C2×A4semidirect
90. C22×#41C33semidirect
102. C3×#41C3×A4direct (N × H)
124. C22×C32C3×#41semidirect
126. C3×A4C3×#41direct (N × H)
137. C22×Q8C2×C33semidirect
138. C22×D4C2×C33semidirect
139. C23×C4C2×C33semidirect
151. C3×Q8C2×C3×A4semidirect
156. C3×D4C2×C3×A4semidirect
166. C2×C3×C4C2×C3×A4semidirect
181. #41C32×A4direct (N × H)
201. C22×C3C32×#41semidirect
203. A4C32×#41direct (N × H)
211. C32A4×#41direct (N × H)
219. Q8C2×C32×A4semidirect
223. D4C2×C32×A4semidirect
232. C2×C4C2×C32×A4semidirect
247. C22C33×#41semidirect
254. C3C3×A4×#41direct (N × H)

Maximal subgroups

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

subgroupindex
C32×Q8×A42
C32×D4×A42
C2×C32×C4×A42
C22×C32×#413
C3×A4×#413
C33×#414