Finite Group Databasebeta

← C3×#10592789 #13163130

Subgroups — C3×#10592789 #13163130

Order 1728.

920 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#3756C212yesmaximal
3C3×C4×#634C211yesmaximal, characteristic
4C3×#63662C212yesmaximal
5C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
6#10592789C311yesmaximal, characteristic, direct product factor
7C3×#5090—34nomaximal
8C2×C33×Q8C2212yes—
9C2×C3×#634C2211yescharacteristic
10C33×C4⋊C4—21no—
11C33×C4⋊C4—21no—
12C33×C42C2211yescharacteristic
13C3×#3756—24no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×#3750—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—
22C4×#634C2×C311yescharacteristic
23C2×C3×#216—38no—
24C3×C4×Dic6—34no—
25#63662C2×C312yes—
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic
27C3×#835—38no—
28C3×C4⋊(C3⋊C8)—34no—
29C33×Q8—22no—
30C33×Q8—22no—
31C2×C33×C4—22no—
32C2×C33×C4D411yescharacteristic
33C2×C33×C4D411yescharacteristic
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—21no—
37C3×#634—21no—
38C3×#634D412yes—
39C3×C3⋊(C3⋊C8)—22no—
40C3×#462—91no—
41#5090—34no—
42C2×C32×Q8—28no—
43C2×C32×Q8C2×D318yes—
44C2×C32×Q8C22×C312yes—
45C2×#634C22×C311yescharacteristic
46C2×C3×Dic6—34no—
47C32×C4⋊C4—28no—
48C32×C4⋊C4—28no—
49C32×C4⋊C4—24no—
50C32×C4⋊C4—24no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4—21no—
53C32×C42—24no—
54C32×C42C2×D314yes—
55C32×C42C22×C311yescharacteristic
56#3756—24no—
57C4⋊(C3⋊Dic3)C22×C311yescharacteristic
58#3750—21no—
59C4×C3⋊Dic3—21no—
60C2×C3⋊(C3⋊C8)C22×C312yes—
61C3×#216—616no—
62C3×C4⋊Dic3—34no—
63C3×Dic3⋊C4—64no—
64C3×C4×Dic3—64no—
65C2×C3×C3⋊C8—38no—
66C22×C33C2×D411yescharacteristic
67C33×C4—21no—
68C33×C4—42no—
69C33×C4Q1612yescomplemented
70C33×C4C2×D411yescharacteristic
71C33×C4#4111yescharacteristic
72C3×C3⋊Dic3—22no—
73C3×C3⋊Dic3—41no—
74C2×C3×Q16—92no—
75C3×C4⋊Q8—24no—
76C3×C4⋊Q8C3×D314yes—
77C3×C4⋊Q8C3⋊D311yescharacteristic
78C3×C4×Q8—91no—
79C2×#216—38no—
80C4×Dic6—34no—
81C3×C4⋊C8—91no—
82C3×#103—92no—
83#835—38no—
84C4⋊(C3⋊C8)—34no—
85C32×Q8—22no—
86C32×Q8—22no—
87C32×Q8—28no—
88C32×Q8—28no—
89C32×Q8—48no—
90C32×Q8—48no—
91C2×C32×C4—216no—
92C2×C32×C4—28no—
93C2×C32×C4—22no—
94C2×C32×C4—24no—
95C2×C32×C4—24no—
96C2×C32×C4—24no—
97C2×C32×C4#6714yes—
98C2×C32×C4#6714yes—
99C2×C32×C4C3×D411yescharacteristic
100C2×C32×C4C3×D411yescharacteristic
101C2×C32×C4C22×D314yes—
102C2×C32×C4C23×C311yescharacteristic
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—21no—
105#634—21no—
106#634C3×D412yes—
107C2×C3×Dic3—64no—
108C2×C3×Dic3—64no—
109C3×Dic6—38no—
110C3×Dic6—64no—
111C3⋊(C3⋊C8)—22no—
112C3×C3⋊C8—68no—
113#462—91noSylow
114C2×C33#12111yescharacteristic
115C2×C33C2×Q1611yescharacteristic
116C2×C33#13711yescharacteristic
117C2×C3×Q8—28no—
118C2×C3×Q8—91no—
119C2×C3×Q8C2×C3×D318yes—
120C2×C3×Q8C2×C3⋊D312yes—
121C2×Dic6—34no—
122C3×Q16—184no—
123C2×C3×C8—92no—
124C3×C4⋊C4—28no—
125C3×C4⋊C4—28no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4—24no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—91no—
131C3×C4⋊C4—181no—
132C3×C42—24no—
133C3×C42—181no—
134C3×C42C2×C3×D314yes—
135C3×C42C2×C3⋊D311yescharacteristic
136#216—616no—
137C4⋊Dic3—34no—
138Dic3⋊C4—64no—
139C4×Dic3—64no—
140C2×C3⋊C8—38no—
141C22×C32—24no—
142C22×C32C2×#6714yes—
143C22×C32C2×C3×D411yescharacteristic
144C32×C4—28no—
145C32×C4—24no—
146C32×C4—21no—
147C32×C4—28no—
148C32×C4—24no—
149C32×C4—24no—
150C32×C4—416no—
151C32×C4—48no—
152C32×C4—42no—
153C32×C4#21618yescomplemented
154C32×C4C3×Q1612yescomplemented
155C32×C4D3×D414yes—
156C32×C4#23714yes—
157C32×C4C2×C3×D411yescharacteristic
158C32×C4C3×#4111yescharacteristic
159C3⋊Dic3—22no—
160C3⋊Dic3—41no—
161C3×Dic3—68no—
162C3×Dic3—124no—
163C2×Q16—92no—
164C4⋊Q8C3×C3⋊D311yescharacteristic
165C4×Q8—91no—
166C4⋊C8—91no—
167#103—92no—
168C33#46211yescharacteristic, complemented, Sylow
169C3×Q8—28no—
170C3×Q8—28no—
171C3×Q8—22no—
172C3×Q8—22no—
173C3×Q8—48no—
174C3×Q8—48no—
175C3×Q8—92no—
176C3×Q8—181no—
177C2×C3×C4—216no—
178C2×C3×C4—28no—
179C2×C3×C4—22no—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—24no—
183C2×C3×C4—181no—
184C2×C3×C4—181no—
185C2×C3×C4C3×#6714yes—
186C2×C3×C4C3×#6714yes—
187C2×C3×C4#63811yescharacteristic
188C2×C3×C4#63811yescharacteristic
189C2×C3×C4C22×C3×D314yes—
190C2×C3×C4C22×C3⋊D311yescharacteristic
191C2×Dic3—64no—
192C2×Dic3—64no—
193Dic6—38no—
194Dic6—64no—
195C3×C8—182no—
196C3⋊C8—68no—
197C2×C32—24no—
198C2×C32—24no—
199C2×C32—24no—
200C2×C32#93814yes—
201C2×C32#94214yes—
202C2×C32C2×#21614yes—
203C2×C32C3×#12111yescharacteristic
204C2×C32C2×C3×Q1611yescharacteristic
205C2×C32C3×#13711yescharacteristic
206C2×Q8—91no—
207C2×Q8C2×C3×C3⋊D312yes—
208Q16—184no—
209C2×C8—92no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—91no—
213C4⋊C4—181no—
214C42—181no—
215C42C2×C3×C3⋊D311yescharacteristic
216C22×C3—24no—
217C22×C3C2×C3×#6714yes—
218C22×C3C2×#63811yescharacteristic
219C3×C4—28no—
220C3×C4—24no—
221C3×C4—21no—
222C3×C4—28no—
223C3×C4—24no—
224C3×C4—24no—
225C3×C4—416no—
226C3×C4—48no—
227C3×C4—42no—
228C3×C4—182no—
229C3×C4—361no—
230C3×C4C3×#21618yescomplemented
231C3×C4#375612yescomplemented
232C3×C4C3×D3×D414yes—
233C3×C4C3×#23714yes—
234C3×C4D4×C3⋊D311yescharacteristic
235C3×C4#383011yescharacteristic
236Dic3—68no—
237Dic3—124no—
238C32—24no—
239C32#509014yescomplemented
240C32C3×#46211yescharacteristic, complemented
241Q8—22no—
242Q8—22no—
243Q8—92no—
244Q8—181no—
245C2×C4—22no—
246C2×C4—181no—
247C2×C4—181no—
248C2×C4C3×#63811yescharacteristic
249C2×C4C3×#63811yescharacteristic
250C2×C4C22×C3×C3⋊D311yescharacteristic
251C8—182no—
252C2×C3—24no—
253C2×C3—24no—
254C2×C3—24no—
255C2×C3C3×#93814yes—
256C2×C3C3×#94214yes—
257C2×C3C2×C3×#21614yes—
258C2×C3#6414711yescharacteristic
259C2×C3#6415111yescharacteristic
260C2×C3C2×#375611yescharacteristic
261C22C2×C3×#63811yescharacteristic
262C4—21no—
263C4—42no—
264C4—182no—
265C4—361no—
266C4C3×#375612yescomplemented
267C4C3×D4×C3⋊D311yescharacteristic
268C4C3×#383011yescharacteristic
269C3—24no—
270C3C3×#509014yescomplemented
271C3#1059278911yescharacteristic, direct product factor
272C2C3×#6414711yescharacteristic
273C2C3×#6415111yescharacteristic
274C2C2×C3×#375611yescharacteristic
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. #10592789C3direct (N × H)
69. C33×C4Q16semidirect
153. C32×C4#216semidirect
154. C32×C4C3×Q16semidirect
168. C33#462semidirect
230. C3×C4C3×#216semidirect
231. C3×C4#3756semidirect
239. C32#5090semidirect
240. C32C3×#462semidirect
266. C4C3×#3756semidirect
270. C3C3×#5090semidirect
271. C3#10592789direct (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×C4×#6342
C3×#636622
C3×C4⋊(C3⋊(C3⋊C8))2
#105927893
C3×#50903