Finite Group Databasebeta

← C32×#5919 #13181361

Subgroups — C32×#5919 #13181361

Order 1728.

1074 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1074 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
1C32×Q8×A4C211yesmaximal, characteristic, complemented
2C32×C4⋊S4C213yesmaximal, complemented
3C32×C4×S4C213yesmaximal, complemented
4C3×#5919C314yesmaximal, direct product factor
5C32×#548—31nomaximal
6C2×C32×S4C2213yes—
7C32×#239—41nomaximal
8C32×C4×A4C2213yes—
9C32×SL(2,ℤ/4)C2211yescharacteristic
10C2×C32×#41—31no—
11C22×C32×Q8D311yescharacteristic, complemented
12C3×Q8×A4—24no—
13C3×Q8×A4C2×C314yescomplemented
14C3×C4⋊S4C2×C3112yescomplemented
15C3×C4×S4C2×C3112yescomplemented
16C32×#124—33no—
17C32×C22⋊Q8—33no—
18C32×#121—33no—
19C32×C4×Q8—31no—
20C32×C4×D4—33no—
21C2×C32×A4C2311yescharacteristic
22C32×S4—23no—
23C33×Q8—41no—
24C32×D12—43no—
25C32×C4×D3—43no—
26#5919C3211yescharacteristic, direct product factor
27C3×#548—34no—
28C2×C3×S4C22×C3112yes—
29C32×#41—63no—
30C32×#41—61no—
31C2×C32×Q8—31no—
32C2×C32×Q8—33no—
33C2×C32×D4—33no—
34C2×C32×D4—33no—
35C22×C32×C4—33no—
36C22×C32×C4C2×D313yes—
37C3×#239—44no—
38C3×C4×A4—212no—
39C3×C4×A4C22×C3112yes—
40C3×SL(2,ℤ/4)C22×C314yes—
41C32×C4⋊C4—33no—
42C32×C4⋊C4—33no—
43C32×C22⋊C4—33no—
44C32×C22⋊C4—36no—
45C32×C22⋊C4—31no—
46C32×C42—33no—
47C2×C32×D3—43no—
48C32×A4#4111yescharacteristic, complemented
49C33×C4—43no—
50C32×Dic3—41no—
51C2×C3×#41—34no—
52C22×C3×Q8C3×D314yescomplemented
53Q8×A4—24no—
54Q8×A4C2×C3211yescharacteristic, complemented
55C4⋊S4C2×C3213yescomplemented
56C4×S4C2×C3213yescomplemented
57C3×#124—312no—
58C3×C22⋊Q8—312no—
59C3×#121—312no—
60C3×C4×Q8—34no—
61C3×C4×D4—312no—
62C23×C32—33no—
63C23×C32C22×D311yescharacteristic
64C2×C3×A4—24no—
65C2×C3×A4C23×C314yes—
66C3×S4—212no—
67C32×Q8—33no—
68C32×Q8—44no—
69C32×Q8—61no—
70C32×Q8—84no—
71C32×Q8S411yescharacteristic, complemented
72C32×D4—63no—
73C32×D4—63no—
74C32×D4—63no—
75C32×D4—63no—
76C2×C32×C4—33no—
77C2×C32×C4—33no—
78C2×C32×C4—31no—
79C2×C32×C4—33no—
80C2×C32×C4—31no—
81C2×C32×C4—63no—
82C2×C32×C4—63no—
83C3×D12—412no—
84C3×C4×D3—412no—
85#548—31noSylow
86C2×C33—41no—
87C32×D3—83no—
88C2×S4C22×C3213yes—
89C3×#41—612no—
90C3×#41—64no—
91C2×C3×Q8—34no—
92C2×C3×Q8—312no—
93C2×C3×D4—312no—
94C2×C3×D4—312no—
95C22×C3×C4—312no—
96C22×C3×C4C2×C3×D3112yes—
97#239—41no—
98C4×A4—212no—
99C4×A4C22×C3213yes—
100SL(2,ℤ/4)C22×C3211yescharacteristic
101C3×C4⋊C4—312no—
102C3×C4⋊C4—312no—
103C3×C22⋊C4—312no—
104C3×C22⋊C4—324no—
105C3×C22⋊C4—34no—
106C3×C42—312no—
107C22×C32—31no—
108C22×C32—31no—
109C22×C32—63no—
110C22×C32—63no—
111C22×C32—63no—
112C22×C32#23911yescharacteristic, complemented
113C2×C3×D3—412no—
114C3×A4—24no—
115C3×A4C3×#4114yescomplemented
116C32×C4—33no—
117C32×C4—412no—
118C32×C4—61no—
119C32×C4—63no—
120C32×C4—61no—
121C32×C4—812no—
122C32×C4C2×S413yes—
123C3×Dic3—44no—
124C2×#41—31no—
125C22×Q8C32×D311yescharacteristic, complemented
126#124—33no—
127C22⋊Q8—33no—
128#121—33no—
129C4×Q8—31no—
130C4×D4—33no—
131C33—41noSylow
132C23×C3—312no—
133C23×C3C22×C3×D314yes—
134C2×A4—24no—
135C2×A4C23×C3211yescharacteristic
136S4—23no—
137C3×Q8—312no—
138C3×Q8—41no—
139C3×Q8—64no—
140C3×Q8—84no—
141C3×Q8C3×S414yescomplemented
142C3×D4—612no—
143C3×D4—612no—
144C3×D4—612no—
145C3×D4—612no—
146C2×C3×C4—312no—
147C2×C3×C4—312no—
148C2×C3×C4—34no—
149C2×C3×C4—312no—
150C2×C3×C4—34no—
151C2×C3×C4—612no—
152C2×C3×C4—612no—
153D12—43no—
154C4×D3—43no—
155C2×C32—31no—
156C2×C32—31no—
157C2×C32—44no—
158C2×C32—84no—
159C2×C32—123no—
160C2×C32C22×S411yescharacteristic
161C3×D3—812no—
162#41—63no—
163#41—61no—
164C2×Q8—31no—
165C2×Q8—33no—
166C2×D4—33no—
167C2×D4—33no—
168C22×C4—33no—
169C22×C4C2×C32×D313yes—
170C4⋊C4—33no—
171C4⋊C4—33no—
172C22⋊C4—33no—
173C22⋊C4—36no—
174C22⋊C4—31no—
175C42—33no—
176C22×C3—34no—
177C22×C3—34no—
178C22×C3—612no—
179C22×C3—612no—
180C22×C3—612no—
181C22×C3C3×#23914yescomplemented
182C2×D3—43no—
183A4—24no—
184A4C32×#4111yescharacteristic, complemented
185C3×C4—312no—
186C3×C4—43no—
187C3×C4—612no—
188C3×C4—64no—
189C3×C4—64no—
190C3×C4—812no—
191C3×C4C2×C3×S4112yes—
192Dic3—41no—
193C32—44no—
194C32—84no—
195C32#591911yescharacteristic, direct product factor
196C23—33no—
197C23C22×C32×D311yescharacteristic
198Q8—33no—
199Q8—61no—
200Q8C32×S411yescharacteristic, complemented
201D4—63no—
202D4—63no—
203D4—63no—
204D4—63no—
205C2×C4—33no—
206C2×C4—33no—
207C2×C4—31no—
208C2×C4—33no—
209C2×C4—31no—
210C2×C4—63no—
211C2×C4—63no—
212C2×C3—34no—
213C2×C3—34no—
214C2×C3—41no—
215C2×C3—84no—
216C2×C3—1212no—
217C2×C3C22×C3×S414yes—
218D3—83no—
219C22—31no—
220C22—31no—
221C22—63no—
222C22—63no—
223C22—63no—
224C22C32×#23911yescharacteristic, complemented
225C4—33no—
226C4—63no—
227C4—61no—
228C4—61no—
229C4C2×C32×S413yes—
230C3—41no—
231C3—84no—
232C3C3×#591914yesdirect product factor
233C2—31no—
234C2—31no—
235C2—123no—
236C2C22×C32×S411yescharacteristic
subgroupG/H
centerC2×C32C22×S4
commutator, cosocleC2×A4C23×C32
FrattiniC2C22×C32×S4
FittingC22×C32×Q8D3
socleC23×C32C22×D3
hypercenterC32×Q8S4
hypocenterA4C32×#41
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C32 ◁ C32×Q8

Complements

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

NH ≅ G/Ntype
1. C32×Q8×A4C2semidirect
2. C32×C4⋊S4C2semidirect
3. C32×C4×S4C2semidirect
4. C3×#5919C3direct (N × H)
11. C22×C32×Q8D3semidirect
13. C3×Q8×A4C2×C3semidirect
14. C3×C4⋊S4C2×C3semidirect
15. C3×C4×S4C2×C3semidirect
26. #5919C32direct (N × H)
48. C32×A4#41semidirect
52. C22×C3×Q8C3×D3semidirect
54. Q8×A4C2×C32semidirect
55. C4⋊S4C2×C32semidirect
56. C4×S4C2×C32semidirect
71. C32×Q8S4semidirect
112. C22×C32#239semidirect
115. C3×A4C3×#41semidirect
125. C22×Q8C32×D3semidirect
141. C3×Q8C3×S4semidirect
181. C22×C3C3×#239semidirect
184. A4C32×#41semidirect
195. C32#5919direct (N × H)
200. Q8C32×S4semidirect
224. C22C32×#239semidirect
232. C3C3×#5919direct (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×C4⋊S42
C32×C4×S42
C3×#59193
C32×#5483
C32×#2394