Finite Group Databasebeta

← C3×#10596205 #13179485

Subgroups — C3×#10596205 #13179485

Order 1728.

1688 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4⋊Q8C211yesmaximal, characteristic, complemented
2C3×#64156C214yesmaximal, complemented
3C3×#64104C214yesmaximal, complemented
4C3×#64101C214yesmaximal, complemented
5C3×C4×C4⋊(C3⋊D3)C212yesmaximal, complemented
6#10596205C311yesmaximal, characteristic, direct product factor
7C3×#5734—34nomaximal
8C2×C33×Q8C2212yes—
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C2×C3×C4⋊(C3⋊D3)C2214yes—
11C2×C3×C4×C3⋊D3C2214yes—
12C33×C4⋊C4C2214yes—
13C33×C42C2211yescharacteristic
14C3×#3752C2214yes—
15C3×#3752C2218yes—
16C3×C4⋊(C3⋊Dic3)C2212yes—
17C3×C4×C3⋊Dic3C2214yes—
18C32×C4⋊Q8—24no—
19C32×C4⋊Q8D314yescomplemented
20C32×C4⋊Q8C2×C311yescharacteristic, complemented
21#64156C2×C314yescomplemented
22#64104C2×C314yescomplemented
23#64101C2×C314yescomplemented
24C4×C4⋊(C3⋊D3)C2×C312yescomplemented
25C3×#947—316no—
26C3×#895—316no—
27C3×#892—316no—
28C3×C4×D12—38no—
29C22×C3×C3⋊D3C2314yes—
30C33×Q8—42no—
31C2×C33×C4C2314yes—
32C2×C33×C4C2311yescharacteristic
33C2×C33×C4C2312yes—
34C2×C3×C3⋊Dic3C2314yes—
35C3×C4⋊(C3⋊D3)—24no—
36C3×C4⋊(C3⋊D3)—44no—
37C3×C4×C3⋊D3—28no—
38C3×#555—91no—
39#5734—34no—
40C2×C32×Q8—28no—
41C2×C32×Q8C2×D318yes—
42C2×C32×Q8C22×C312yes—
43C2×C4⋊(C3⋊D3)C22×C312yes—
44C2×C4⋊(C3⋊D3)C22×C314yes—
45C2×C4×C3⋊D3C22×C314yes—
46C2×C3×D12—38no—
47C2×C3×D12—316no—
48C2×C3×C4×D3—316no—
49C32×C4⋊C4—216no—
50C32×C4⋊C4C2×D3116yes—
51C32×C4⋊C4C22×C314yes—
52C32×C42—24no—
53C32×C42C2×D314yes—
54C32×C42C22×C311yescharacteristic
55#3752C22×C314yes—
56#3752C22×C318yes—
57C4⋊(C3⋊Dic3)C22×C312yes—
58C4×C3⋊Dic3C22×C314yes—
59C3×(C2×D3)⋊C4—332no—
60C3×(C2×D3)⋊C4—316no—
61C3×C4⋊Dic3—38no—
62C3×C4×Dic3—316no—
63C22×C33C2411yescharacteristic
64C2×C3×C3⋊D3—24no—
65C2×C3×C3⋊D3—24no—
66C2×C3×C3⋊D3—24no—
67C33×C4—21no—
68C33×C4—24no—
69C33×C4#4114yes—
70C3×C3⋊Dic3—24no—
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8C3×D314yescomplemented
73C3×C4⋊Q8C3⋊D311yescharacteristic, complemented
74C3×#124—94no—
75C3×#121—94no—
76C3×C4×D4—92no—
77C3×#117—94no—
78#947—316no—
79#895—316no—
80#892—316no—
81C4×D12—38no—
82C22×C3⋊D3C23×C314yes—
83C22×C3×D3—316no—
84C32×Q8—48no—
85C32×Q8—42no—
86C32×Q8—416no—
87C2×C32×C4—28no—
88C2×C32×C4—24no—
89C2×C32×C4—216no—
90C2×C32×C4C22×D3116yes—
91C2×C32×C4C22×D314yes—
92C2×C32×C4C22×D318yes—
93C2×C32×C4C23×C314yes—
94C2×C32×C4C23×C311yescharacteristic
95C2×C32×C4C23×C312yes—
96C2×C3⋊Dic3C23×C314yes—
97C4⋊(C3⋊D3)—24no—
98C4⋊(C3⋊D3)—44no—
99C4×C3⋊D3—28no—
100C2×C3×Dic3—316no—
101C3×D12—616no—
102C3×D12—1216no—
103C3×C4×D3—632no—
104#555—91noSylow
105C2×C33C2×#4112yes—
106C2×C33H(5,𝔽2)11yescharacteristic
107C3×C3⋊D3—44no—
108C2×C3×Q8—28no—
109C2×C3×Q8C2×C3×D318yes—
110C2×C3×Q8C2×C3⋊D312yes—
111C2×C3×D4—92no—
112C2×C3×D4—94no—
113C22×C3×C4—94no—
114C2×D12—38no—
115C2×D12—316no—
116C2×C4×D3—316no—
117C3×C4⋊C4—216no—
118C3×C4⋊C4—92no—
119C3×C4⋊C4C2×C3×D3116yes—
120C3×C4⋊C4C2×C3⋊D314yes—
121C3×C22⋊C4—98no—
122C3×C22⋊C4—94no—
123C3×C42—24no—
124C3×C42—94no—
125C3×C42C2×C3×D314yes—
126C3×C42C2×C3⋊D311yescharacteristic
127(C2×D3)⋊C4—316no—
128(C2×D3)⋊C4—332no—
129C4⋊Dic3—38no—
130C4×Dic3—316no—
131C22×C32—24no—
132C22×C32C23×D314yes—
133C22×C32C24×C311yescharacteristic
134C2×C3⋊D3—24no—
135C2×C3⋊D3—24no—
136C2×C3⋊D3—24no—
137C2×C3×D3—616no—
138C2×C3×D3—616no—
139C2×C3×D3—616no—
140C32×C4—28no—
141C32×C4—24no—
142C32×C4—24no—
143C32×C4—21no—
144C32×C4—216no—
145C32×C4—216no—
146C32×C4—416no—
147C32×C4#239116yes—
148C32×C4C3×#4114yes—
149C3⋊Dic3—24no—
150C3×Dic3—616no—
151C4⋊Q8C3×C3⋊D311yescharacteristic, complemented
152#124—94no—
153#121—94no—
154C4×D4—92no—
155#117—94no—
156C33#55511yescharacteristic, complemented, Sylow
157C23×C3—94no—
158C22×D3—316no—
159C3×Q8—48no—
160C3×Q8—42no—
161C3×Q8—416no—
162C3×D4—184no—
163C3×D4—364no—
164C2×C3×C4—28no—
165C2×C3×C4—24no—
166C2×C3×C4—216no—
167C2×C3×C4—94no—
168C2×C3×C4—188no—
169C2×C3×C4C22×C3×D3116yes—
170C2×C3×C4C22×C3×D314yes—
171C2×C3×C4C22×C3×D318yes—
172C2×C3×C4C22×C3⋊D314yes—
173C2×C3×C4C22×C3⋊D311yescharacteristic
174C2×C3×C4C22×C3⋊D312yes—
175C2×Dic3—316no—
176D12—616no—
177D12—1216no—
178C4×D3—632no—
179C2×C32—24no—
180C2×C32—28no—
181C2×C32#100414yes—
182C2×C32C2×#23918yes—
183C2×C32C2×C3×#4112yes—
184C2×C32C3×H(5,𝔽2)11yescharacteristic
185C3⋊D3—44no—
186C3×D3—1216no—
187C2×Q8C2×C3×C3⋊D312yes—
188C2×D4—92no—
189C2×D4—94no—
190C22×C4—94no—
191C4⋊C4—92no—
192C4⋊C4C2×C3×C3⋊D314yes—
193C22⋊C4—98no—
194C22⋊C4—94no—
195C42—94no—
196C42C2×C3×C3⋊D311yescharacteristic
197C22×C3—24no—
198C22×C3—184no—
199C22×C3—184no—
200C22×C3—184no—
201C22×C3C23×C3×D314yes—
202C22×C3C23×C3⋊D311yescharacteristic
203C2×D3—616no—
204C2×D3—616no—
205C2×D3—616no—
206C3×C4—28no—
207C3×C4—216no—
208C3×C4—24no—
209C3×C4—24no—
210C3×C4—21no—
211C3×C4—216no—
212C3×C4—416no—
213C3×C4—184no—
214C3×C4C3×#239116yes—
215C3×C4#383214yes—
216Dic3—616no—
217C32—24no—
218C32#573414yescomplemented
219C32C3×#55511yescharacteristic, complemented
220C23—94no—
221Q8—42no—
222D4—184no—
223D4—364no—
224C2×C4—94no—
225C2×C4—188no—
226C2×C4C22×C3×C3⋊D314yes—
227C2×C4C22×C3×C3⋊D311yescharacteristic
228C2×C4C22×C3×C3⋊D312yes—
229C2×C3—24no—
230C2×C3—28no—
231C2×C3—364no—
232C2×C3C3×#100414yes—
233C2×C3C2×C3×#23918yes—
234C2×C3#6436111yescharacteristic
235C2×C3C2×#383212yes—
236D3—1216no—
237C22—184no—
238C22—184no—
239C22—184no—
240C22C23×C3×C3⋊D311yescharacteristic
241C4—24no—
242C4—21no—
243C4—184no—
244C4C3×#383214yes—
245C3—24no—
246C3C3×#573414yescomplemented
247C3#1059620511yescharacteristic, direct product factor
248C2—364no—
249C2C3×#6436111yescharacteristic
250C2C2×C3×#383212yes—
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4⋊Q8C2
socleC22×C33C24
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterC32C3×#555
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C33×C4⋊Q8C2semidirect
2. C3×#64156C2semidirect
3. C3×#64104C2semidirect
4. C3×#64101C2semidirect
5. C3×C4×C4⋊(C3⋊D3)C2semidirect
6. #10596205C3direct (N × H)
19. C32×C4⋊Q8D3semidirect
20. C32×C4⋊Q8C2×C3semidirect
21. #64156C2×C3semidirect
22. #64104C2×C3semidirect
23. #64101C2×C3semidirect
24. C4×C4⋊(C3⋊D3)C2×C3semidirect
72. C3×C4⋊Q8C3×D3semidirect
73. C3×C4⋊Q8C3⋊D3semidirect
151. C4⋊Q8C3×C3⋊D3semidirect
156. C33#555semidirect
218. C32#5734semidirect
219. C32C3×#555semidirect
246. C3C3×#5734semidirect
247. C3#10596205direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4⋊Q82
C3×#641562
C3×#641042
C3×#641012
C3×C4×C4⋊(C3⋊D3)2
#105962053
C3×#57343