Finite Group Databasebeta

← C3×#10592730 #13163069

Subgroups — C3×#10592730 #13163069

Order 1728.

928 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×Q8C211yesmaximal, characteristic
2C2×C3×#3756C212yesmaximal
3C3×#64078C211yesmaximal, characteristic
4C3×#63637C212yesmaximal
5C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
6#10592730C311yesmaximal, characteristic, direct product factor
7C3×#5031—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×#634C2212yes—
10C33×C4⋊C4—21no—
11C33×C4⋊C4C2211yescharacteristic
12C33×C42—21no—
13C33×C42C2211yescharacteristic
14C3×#3756—24no—
15C3×C4⋊(C3⋊Dic3)—21no—
16C3×C4⋊(C3⋊Dic3)—21no—
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18C32×C4×Q8—24no—
19C32×C4×Q8D314yes—
20C32×C4×Q8C2×C311yescharacteristic
21C2×#3756C2×C312yes—
22#64078C2×C311yescharacteristic
23C2×C3×#216—38no—
24C3×#869—34no—
25#63637C2×C312yes—
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic
27C3×#810—38no—
28C3×C4⋊(C3⋊C8)—34no—
29C33×Q8—21no—
30C33×Q8D412yes—
31C2×C33×C4—21no—
32C2×C33×C4—21no—
33C2×C33×C4D411yescharacteristic
34C2×C33×C4D411yescharacteristic
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—22no—
37C3×#634—22no—
38C3×#634—22no—
39C3×C3⋊(C3⋊C8)—22no—
40C3×#462—91no—
41#5031—34no—
42C2×C32×Q8—24no—
43C2×C32×Q8C2×D314yes—
44C2×C32×Q8C22×C311yescharacteristic
45C2×#634C22×C312yes—
46C2×C3×Dic6—38no—
47C32×C4⋊C4—28no—
48C32×C4⋊C4—24no—
49C32×C4⋊C4—24no—
50C32×C4⋊C4—21no—
51C32×C4⋊C4C2×D314yes—
52C32×C4⋊C4C22×C311yescharacteristic
53C32×C42—28no—
54C32×C42—24no—
55C32×C42—24no—
56C32×C42—21no—
57C32×C42C2×D314yes—
58C32×C42C22×C311yescharacteristic
59#3756—24no—
60C4⋊(C3⋊Dic3)—21no—
61C4⋊(C3⋊Dic3)—21no—
62C2×C3⋊(C3⋊C8)C22×C312yes—
63C3×#216—616no—
64C3×C4⋊Dic3—64no—
65C3×C4⋊Dic3—64no—
66C2×C3×C3⋊C8—38no—
67C22×C33C2×D411yescharacteristic
68C33×C4—22no—
69C33×C4—21no—
70C33×C4—41no—
71C33×C4Q1612yescomplemented
72C33×C4C2×D411yescharacteristic
73C33×C4#4111yescharacteristic
74C3×C3⋊Dic3—42no—
75C2×C3×Q16—92no—
76C3×C4⋊Q8—91no—
77C3×C4×Q8—24no—
78C3×C4×Q8C3×D314yes—
79C3×C4×Q8C3⋊D311yescharacteristic
80C2×#216—38no—
81#869—34no—
82C3×C4⋊C8—91no—
83C3×#103—92no—
84#810—38no—
85C4⋊(C3⋊C8)—34no—
86C32×Q8—28no—
87C32×Q8—24no—
88C32×Q8—21no—
89C32×Q8—28no—
90C32×Q8D1218yes—
91C32×Q8C3×D412yes—
92C2×C32×C4—28no—
93C2×C32×C4—24no—
94C2×C32×C4—21no—
95C2×C32×C4—28no—
96C2×C32×C4—24no—
97C2×C32×C4—21no—
98C2×C32×C4—24no—
99C2×C32×C4—24no—
100C2×C32×C4—24no—
101C2×C32×C4#6714yes—
102C2×C32×C4#6714yes—
103C2×C32×C4C3×D411yescharacteristic
104C2×C32×C4C3×D411yescharacteristic
105C2×C32×C4C22×D314yes—
106C2×C32×C4C23×C311yescharacteristic
107C2×C3⋊Dic3—22no—
108#634—22no—
109#634—22no—
110C2×C3×Dic3—68no—
111C3×Dic6—68no—
112C3×Dic6—68no—
113C3⋊(C3⋊C8)—22no—
114C3×C3⋊C8—68no—
115#462—91noSylow
116C2×C33#12111yescharacteristic
117C2×C33C2×Q1611yescharacteristic
118C2×C33#13711yescharacteristic
119C2×C3×Q8—24no—
120C2×C3×Q8—92no—
121C2×C3×Q8C2×C3×D314yes—
122C2×C3×Q8C2×C3⋊D311yescharacteristic
123C2×Dic6—38no—
124C3×Q16—184no—
125C2×C3×C8—92no—
126C3×C4⋊C4—28no—
127C3×C4⋊C4—24no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4—181no—
131C3×C4⋊C4—181no—
132C3×C4⋊C4C2×C3×D314yes—
133C3×C4⋊C4C2×C3⋊D311yescharacteristic
134C3×C42—28no—
135C3×C42—24no—
136C3×C42—21no—
137C3×C42—24no—
138C3×C42C2×C3×D314yes—
139C3×C42C2×C3⋊D311yescharacteristic
140#216—616no—
141C4⋊Dic3—64no—
142C4⋊Dic3—64no—
143C2×C3⋊C8—38no—
144C22×C32—24no—
145C22×C32C2×#6714yes—
146C22×C32C2×C3×D411yescharacteristic
147C32×C4—216no—
148C32×C4—28no—
149C32×C4—22no—
150C32×C4—21no—
151C32×C4—28no—
152C32×C4—24no—
153C32×C4—24no—
154C32×C4—24no—
155C32×C4—44no—
156C32×C4—48no—
157C32×C4—44no—
158C32×C4—41no—
159C32×C4#21618yescomplemented
160C32×C4C3×Q1612yescomplemented
161C32×C4C2×D1214yes—
162C32×C4#23514yes—
163C32×C4C2×C3×D411yescharacteristic
164C32×C4C3×#4111yescharacteristic
165C3⋊Dic3—42no—
166C3×Dic3—128no—
167C2×Q16—92no—
168C4⋊Q8—91no—
169C4×Q8C3×C3⋊D311yescharacteristic
170C4⋊C8—91no—
171#103—92no—
172C33#46211yescharacteristic, complemented, Sylow
173C3×Q8—28no—
174C3×Q8—24no—
175C3×Q8—21no—
176C3×Q8—28no—
177C3×Q8—182no—
178C3×Q8—182no—
179C3×Q8C3×D1218yes—
180C3×Q8C4⋊(C3⋊D3)12yes—
181C2×C3×C4—28no—
182C2×C3×C4—24no—
183C2×C3×C4—21no—
184C2×C3×C4—28no—
185C2×C3×C4—24no—
186C2×C3×C4—21no—
187C2×C3×C4—24no—
188C2×C3×C4—24no—
189C2×C3×C4—24no—
190C2×C3×C4—182no—
191C2×C3×C4C3×#6714yes—
192C2×C3×C4C3×#6714yes—
193C2×C3×C4#63811yescharacteristic
194C2×C3×C4#63811yescharacteristic
195C2×C3×C4C22×C3×D314yes—
196C2×C3×C4C22×C3⋊D311yescharacteristic
197C2×Dic3—68no—
198Dic6—68no—
199Dic6—68no—
200C3×C8—182no—
201C3⋊C8—68no—
202C2×C32—24no—
203C2×C32—24no—
204C2×C32—24no—
205C2×C32#93014yes—
206C2×C32C2×#21614yes—
207C2×C32#95114yes—
208C2×C32C3×#12111yescharacteristic
209C2×C32C2×C3×Q1611yescharacteristic
210C2×C32C3×#13711yescharacteristic
211C2×Q8—92no—
212C2×Q8C2×C3×C3⋊D311yescharacteristic
213Q16—184no—
214C2×C8—92no—
215C4⋊C4—21no—
216C4⋊C4—181no—
217C4⋊C4—181no—
218C4⋊C4C2×C3×C3⋊D311yescharacteristic
219C42—21no—
220C42C2×C3×C3⋊D311yescharacteristic
221C22×C3—24no—
222C22×C3C2×C3×#6714yes—
223C22×C3C2×#63811yescharacteristic
224C3×C4—216no—
225C3×C4—28no—
226C3×C4—24no—
227C3×C4—22no—
228C3×C4—21no—
229C3×C4—28no—
230C3×C4—24no—
231C3×C4—24no—
232C3×C4—44no—
233C3×C4—48no—
234C3×C4—44no—
235C3×C4—41no—
236C3×C4—362no—
237C3×C4C3×#21618yescomplemented
238C3×C4#375612yescomplemented
239C3×C4C2×C3×D1214yes—
240C3×C4C3×#23514yes—
241C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
242C3×C4#382811yescharacteristic
243Dic3—128no—
244C32—24no—
245C32#503114yescomplemented
246C32C3×#46211yescharacteristic, complemented
247Q8—21no—
248Q8—182no—
249Q8—182no—
250Q8C3×C4⋊(C3⋊D3)12yes—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—182no—
254C2×C4C3×#63811yescharacteristic
255C2×C4C3×#63811yescharacteristic
256C2×C4C22×C3×C3⋊D311yescharacteristic
257C8—182no—
258C2×C3—24no—
259C2×C3—24no—
260C2×C3—24no—
261C2×C3C3×#93014yes—
262C2×C3C2×C3×#21614yes—
263C2×C3C3×#95114yes—
264C2×C3#6413911yescharacteristic
265C2×C3C2×#375611yescharacteristic
266C2×C3#6416011yescharacteristic
267C22C2×C3×#63811yescharacteristic
268C4—22no—
269C4—21no—
270C4—41no—
271C4—362no—
272C4C3×#375612yescomplemented
273C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
274C4C3×#382811yescharacteristic
275C3—24no—
276C3C3×#503114yescomplemented
277C3#1059273011yescharacteristic, direct product factor
278C2C3×#6413911yescharacteristic
279C2C2×C3×#375611yescharacteristic
280C2C3×#6416011yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4×Q8C2
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4×Q8C3⋊D3
hypocenterC32C3×#462
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
6. #10592730C3direct (N × H)
71. C33×C4Q16semidirect
159. C32×C4#216semidirect
160. C32×C4C3×Q16semidirect
172. C33#462semidirect
237. C3×C4C3×#216semidirect
238. C3×C4#3756semidirect
245. C32#5031semidirect
246. C32C3×#462semidirect
272. C4C3×#3756semidirect
276. C3C3×#5031semidirect
277. C3#10592730direct (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
C2×C3×#37562
C3×#640782
C3×#636372
C3×C4⋊(C3⋊(C3⋊C8))2
#105927303
C3×#50313