Finite Group Databasebeta

← C3×#10592566 #13162906

Subgroups — C3×#10592566 #13162906

Order 1728.

804 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 194 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×C3×#3746C211yesmaximal, characteristic, complemented
2C2×C3×#3743C211yesmaximal, characteristic
3C3×#64106C212yesmaximal
4C33×#106C211yesmaximal, characteristic, complemented
5C3×#63637C212yesmaximal, complemented
6#10592566C311yesmaximal, characteristic, direct product factor
7C3×#4867—34nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3×#634C2212yes—
10C2×C33×C8C2211yescharacteristic
11C33×C4⋊C4C2212yes—
12C3×#3752—22no—
13C3×C4⋊(C3⋊Dic3)—22no—
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×#3746—22no—
16C3×#3743—22no—
17C2×#3746C2×C311yescharacteristic, complemented
18C2×#3743C2×C311yescharacteristic
19#64106C2×C312yes—
20C2×C3×Dic12—34no—
21C2×C3×ΓL(1,𝔽25)—34no—
22C3×#897—38no—
23C32×#106—24no—
24C32×#106D314yescomplemented
25C32×#106C2×C311yescharacteristic, complemented
26#63637C2×C312yescomplemented
27C3×#810—38no—
28C22×C3×C3⋊D3D411yescharacteristic
29C2×C33×C4—22no—
30C2×C33×C4C2311yescharacteristic
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3D411yescharacteristic
33C3×C4×C3⋊D3—21no—
34C3×C4×C3⋊D3—21no—
35C3×#634—22no—
36C3×#634—22no—
37C33×C8D412yes—
38C3×C3⋊(C3⋊C8)—21no—
39C3×#470—91no—
40#4867—34no—
41C2×C4×C3⋊D3C22×C311yescharacteristic
42C2×#634C22×C312yes—
43C2×C3×C4×D3—34no—
44C2×C3×Dic6—38no—
45C2×C32×C8—24no—
46C2×C32×C8C2×D314yes—
47C2×C32×C8C22×C311yescharacteristic
48C32×C4⋊C4—28no—
49C32×C4⋊C4C2×D318yes—
50C32×C4⋊C4C22×C312yes—
51#3752—22no—
52C4⋊(C3⋊Dic3)—22no—
53C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
54#3746—22no—
55#3743—22no—
56C3×(C2×D3)⋊C4—68no—
57C3×C4⋊Dic3—68no—
58C2×C3×C3⋊C8—34no—
59C3×Dic12—68no—
60C3×ΓL(1,𝔽25)—68no—
61C22×C33C2×D411yescharacteristic
62C2×C3×C3⋊D3—21no—
63C2×C3×C3⋊D3—22no—
64C33×C4—42no—
65C33×C4C2×D411yescharacteristic
66C33×C4#4111yescharacteristic
67C3×C3⋊Dic3—21no—
68C3×C3⋊Dic3—42no—
69C2×C3×Q16—91no—
70C2×C3×M16—91no—
71C3×C22⋊Q8—92no—
72C2×Dic12—34no—
73C2×ΓL(1,𝔽25)—34no—
74#897—38no—
75C3×#106—24no—
76C3×#106C3×D314yescomplemented
77C3×#106C3⋊D311yescharacteristic, complemented
78C3×#103—92no—
79#810—38no—
80C22×C3⋊D3C3×D411yescharacteristic
81C22×C3×D3—34no—
82C2×C32×C4—22no—
83C2×C32×C4—24no—
84C2×C32×C4—28no—
85C2×C32×C4—48no—
86C2×C32×C4C22×D314yes—
87C2×C32×C4C23×C311yescharacteristic
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3C3×D411yescharacteristic
90C4×C3⋊D3—21no—
91C4×C3⋊D3—21no—
92#634—22no—
93#634—22no—
94C2×C3×Dic3—34no—
95C2×C3×Dic3—68no—
96C3×C4×D3—64no—
97C3×C4×D3—64no—
98C3×Dic6—68no—
99C3×Dic6—68no—
100C32×C8—28no—
101C32×C8D1218yes—
102C32×C8C3×D412yes—
103C3⋊(C3⋊C8)—21no—
104C3×C3⋊C8—64no—
105#470—91noSylow
106C2×C33#12111yescharacteristic
107C2×C33#13712yes—
108C3×C3⋊D3—41no—
109C2×C3×Q8—92no—
110C22×C3×C4—91no—
111C2×C4×D3—34no—
112C2×Dic6—38no—
113C3×Q16—182no—
114C3×M16—182no—
115C2×C3×C8—24no—
116C2×C3×C8—91no—
117C2×C3×C8C2×C3×D314yes—
118C2×C3×C8C2×C3⋊D311yescharacteristic
119C3×C4⋊C4—28no—
120C3×C4⋊C4—182no—
121C3×C4⋊C4C2×C3×D318yes—
122C3×C4⋊C4C2×C3⋊D312yes—
123C3×C22⋊C4—182no—
124(C2×D3)⋊C4—68no—
125C4⋊Dic3—68no—
126C2×C3⋊C8—34no—
127Dic12—68no—
128ΓL(1,𝔽25)—68no—
129C22×C32—24no—
130C22×C32D3×D414yes—
131C22×C32C2×C3×D411yescharacteristic
132C2×C3⋊D3—21no—
133C2×C3⋊D3—22no—
134C2×C3×D3—64no—
135C2×C3×D3—68no—
136C32×C4—24no—
137C32×C4—24no—
138C32×C4—42no—
139C32×C4—48no—
140C32×C4—88no—
141C32×C4C2×D1214yes—
142C32×C4#23914yes—
143C32×C4C2×C3×D411yescharacteristic
144C32×C4C3×#4111yescharacteristic
145C3⋊Dic3—21no—
146C3⋊Dic3—42no—
147C3×Dic3—64no—
148C3×Dic3—128no—
149C2×Q16—91no—
150C2×M16—91no—
151C22⋊Q8—92no—
152#106C3×C3⋊D311yescharacteristic, complemented
153#103—92no—
154C33#47011yescharacteristic, complemented, Sylow
155C23×C3—91no—
156C22×D3—34no—
157C3×Q8—182no—
158C3×Q8—182no—
159C2×C3×C4—28no—
160C2×C3×C4—22no—
161C2×C3×C4—24no—
162C2×C3×C4—48no—
163C2×C3×C4—91no—
164C2×C3×C4—181no—
165C2×C3×C4—181no—
166C2×C3×C4—182no—
167C2×C3×C4C22×C3×D314yes—
168C2×C3×C4C22×C3⋊D311yescharacteristic
169C2×Dic3—34no—
170C2×Dic3—68no—
171C4×D3—64no—
172C4×D3—64no—
173Dic6—68no—
174Dic6—68no—
175C3×C8—28no—
176C3×C8—181no—
177C3×C8C3×D1218yes—
178C3×C8C4⋊(C3⋊D3)12yes—
179C3⋊C8—64no—
180C2×C32—24no—
181C2×C32—28no—
182C2×C32#89514yes—
183C2×C32#91518yes—
184C2×C32C3×#12111yescharacteristic
185C2×C32C3×#13712yes—
186C3⋊D3—41no—
187C3×D3—124no—
188C2×Q8—92no—
189C22×C4—91no—
190Q16—182no—
191M16—182no—
192C2×C8—91no—
193C2×C8C2×C3×C3⋊D311yescharacteristic
194C4⋊C4—182no—
195C4⋊C4C2×C3×C3⋊D312yes—
196C22⋊C4—182no—
197C22×C3—24no—
198C22×C3—181no—
199C22×C3—182no—
200C22×C3C3×D3×D414yes—
201C22×C3D4×C3⋊D311yescharacteristic
202C2×D3—64no—
203C2×D3—68no—
204C3×C4—24no—
205C3×C4—24no—
206C3×C4—48no—
207C3×C4—42no—
208C3×C4—88no—
209C3×C4—181no—
210C3×C4—362no—
211C3×C4C2×C3×D1214yes—
212C3×C4C3×#23914yes—
213C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
214C3×C4#383211yescharacteristic
215Dic3—64no—
216Dic3—128no—
217C32—24no—
218C32#486714yescomplemented
219C32C3×#47011yescharacteristic, complemented
220C23—91no—
221Q8—182no—
222Q8—182no—
223C2×C4—22no—
224C2×C4—91no—
225C2×C4—181no—
226C2×C4—181no—
227C2×C4—182no—
228C2×C4C22×C3×C3⋊D311yescharacteristic
229C8—181no—
230C8C3×C4⋊(C3⋊D3)12yes—
231C2×C3—24no—
232C2×C3—28no—
233C2×C3—361no—
234C2×C3C3×#89514yes—
235C2×C3C3×#91518yes—
236C2×C3#6410411yescharacteristic
237C2×C3#6412412yes—
238D3—124no—
239C22—181no—
240C22—182no—
241C22C3×D4×C3⋊D311yescharacteristic
242C4—42no—
243C4—181no—
244C4—362no—
245C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
246C4C3×#383211yescharacteristic
247C3—24no—
248C3C3×#486714yescomplemented
249C3#1059256611yescharacteristic, direct product factor
250C2—361no—
251C2C3×#6410411yescharacteristic
252C2C3×#6412412yes—
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#106C2
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC3×#106C3⋊D3
hypocenterC32C3×#470
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#106

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3746C2semidirect
4. C33×#106C2semidirect
5. C3×#63637C2semidirect
6. #10592566C3direct (N × H)
17. C2×#3746C2×C3semidirect
24. C32×#106D3semidirect
25. C32×#106C2×C3semidirect
26. #63637C2×C3semidirect
76. C3×#106C3×D3semidirect
77. C3×#106C3⋊D3semidirect
152. #106C3×C3⋊D3semidirect
154. C33#470semidirect
218. C32#4867semidirect
219. C32C3×#470semidirect
248. C3C3×#4867semidirect
249. C3#10592566direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37462
C2×C3×#37432
C3×#641062
C33×#1062
C3×#636372
#105925663
C3×#48673