Finite Group Databasebeta

← C3×#10592793 #13163132

Subgroups — C3×#10592793 #13163132

Order 1728.

848 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4⋊Q8C211yesmaximal, characteristic
2C3×C4×#634C211yesmaximal, characteristic
3C3×#63637C212yesmaximal
4C3×#63635C212yesmaximal
5C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
6#10592793C311yesmaximal, characteristic, direct product factor
7C3×#5094—34nomaximal
8C2×C33×Q8—21no—
9C2×C3×#634C2211yescharacteristic
10C33×C4⋊C4—21no—
11C33×C4⋊C4C2212yes—
12C33×C42C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×#3750—21no—
15C3×C4×C3⋊Dic3—21no—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C32×C4⋊Q8—24no—
18C32×C4⋊Q8D314yes—
19C32×C4⋊Q8C2×C311yescharacteristic
20C4×#634C2×C311yescharacteristic
21C3×C4×Dic6—34no—
22#63637C2×C312yes—
23#63635C2×C312yes—
24C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic
25C3×#810—38no—
26C3×#808—38no—
27C3×C4⋊(C3⋊C8)—34no—
28C33×Q8—42no—
29C2×C33×C4—22no—
30C2×C33×C4D411yescharacteristic
31C2×C33×C4D411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C3×#634—21no—
36C3×#634Q812yescomplemented
37C3×C3⋊(C3⋊C8)—22no—
38C3×#475—91no—
39#5094—34no—
40C2×C32×Q8—28no—
41C2×C32×Q8—24no—
42C2×C32×Q8—21no—
43C2×#634C22×C311yescharacteristic
44C2×C3×Dic6—34no—
45C32×C4⋊C4—28no—
46C32×C4⋊C4—28no—
47C32×C4⋊C4—24no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4C2×D318yes—
50C32×C4⋊C4C22×C312yes—
51C32×C42—24no—
52C32×C42C2×D314yes—
53C32×C42C22×C311yescharacteristic
54C4⋊(C3⋊Dic3)C22×C311yescharacteristic
55#3750—21no—
56C4×C3⋊Dic3—21no—
57C2×C3⋊(C3⋊C8)C22×C312yes—
58C3×C4⋊Dic3—34no—
59C3×Dic3⋊C4—64no—
60C3×C4×Dic3—64no—
61C2×C3×C3⋊C8—38no—
62C22×C33C2×D411yescharacteristic
63C33×C4—21no—
64C33×C4—42no—
65C33×C4SD1612yes—
66C33×C4C2×Q811yescharacteristic
67C33×C4#4111yescharacteristic
68C3×C3⋊Dic3—22no—
69C3×C3⋊Dic3—41no—
70C3×C4⋊Q8—24no—
71C3×C4⋊Q8C3×D314yes—
72C3×C4⋊Q8C3⋊D311yescharacteristic
73C3×C4×Q8—91no—
74C4×Dic6—34no—
75C3×#106—92no—
76C3×C4⋊C8—91no—
77C3×#103—92no—
78#810—38no—
79#808—38no—
80C4⋊(C3⋊C8)—34no—
81C32×Q8—416no—
82C32×Q8—48no—
83C32×Q8—42no—
84C2×C32×C4—216no—
85C2×C32×C4—28no—
86C2×C32×C4—22no—
87C2×C32×C4—24no—
88C2×C32×C4—24no—
89C2×C32×C4—24no—
90C2×C32×C4#6714yes—
91C2×C32×C4#6714yes—
92C2×C32×C4C3×D411yescharacteristic
93C2×C32×C4C3×D411yescharacteristic
94C2×C32×C4C22×D314yes—
95C2×C32×C4C23×C311yescharacteristic
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—21no—
98#634—21no—
99#634C3×Q812yescomplemented
100C2×C3×Dic3—64no—
101C2×C3×Dic3—64no—
102C3×Dic6—38no—
103C3×Dic6—64no—
104C3⋊(C3⋊C8)—22no—
105C3×C3⋊C8—68no—
106#475—91noSylow
107C2×C33C22⋊Q811yescharacteristic
108C2×C33C2×SD1611yescharacteristic
109C2×C33#13711yescharacteristic
110C2×C3×Q8—28no—
111C2×C3×Q8—24no—
112C2×C3×Q8—21no—
113C2×C3×Q8—91no—
114C2×Dic6—34no—
115C2×C3×C8—92no—
116C3×C4⋊C4—28no—
117C3×C4⋊C4—24no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—28no—
120C3×C4⋊C4—91no—
121C3×C4⋊C4—181no—
122C3×C4⋊C4C2×C3×D318yes—
123C3×C4⋊C4C2×C3⋊D312yes—
124C3×C42—24no—
125C3×C42—181no—
126C3×C42C2×C3×D314yes—
127C3×C42C2×C3⋊D311yescharacteristic
128C4⋊Dic3—34no—
129Dic3⋊C4—64no—
130C4×Dic3—64no—
131C2×C3⋊C8—38no—
132C22×C32—24no—
133C22×C32C2×#6714yes—
134C22×C32C2×C3×D411yescharacteristic
135C32×C4—28no—
136C32×C4—24no—
137C32×C4—21no—
138C32×C4—28no—
139C32×C4—24no—
140C32×C4—24no—
141C32×C4—416no—
142C32×C4—48no—
143C32×C4—42no—
144C32×C4#21418yes—
145C32×C4C3×SD1612yes—
146C32×C4D3×Q814yes—
147C32×C4#23914yes—
148C32×C4C2×C3×Q811yescharacteristic
149C32×C4C3×#4111yescharacteristic
150C3⋊Dic3—22no—
151C3⋊Dic3—41no—
152C3×Dic3—68no—
153C3×Dic3—124no—
154C4⋊Q8C3×C3⋊D311yescharacteristic
155C4×Q8—91no—
156#106—92no—
157C4⋊C8—91no—
158#103—92no—
159C33#47511yescharacteristic, complemented, Sylow
160C3×Q8—416no—
161C3×Q8—48no—
162C3×Q8—42no—
163C3×Q8—92no—
164C3×Q8—181no—
165C2×C3×C4—216no—
166C2×C3×C4—28no—
167C2×C3×C4—22no—
168C2×C3×C4—24no—
169C2×C3×C4—24no—
170C2×C3×C4—24no—
171C2×C3×C4—181no—
172C2×C3×C4—181no—
173C2×C3×C4C3×#6714yes—
174C2×C3×C4C3×#6714yes—
175C2×C3×C4#63811yescharacteristic
176C2×C3×C4#63811yescharacteristic
177C2×C3×C4C22×C3×D314yes—
178C2×C3×C4C22×C3⋊D311yescharacteristic
179C2×Dic3—64no—
180C2×Dic3—64no—
181Dic6—38no—
182Dic6—64no—
183C3×C8—182no—
184C3⋊C8—68no—
185C2×C32—24no—
186C2×C32—24no—
187C2×C32—24no—
188C2×C32C2×#21414yes—
189C2×C32#94214yes—
190C2×C32#94614yes—
191C2×C32C3×C22⋊Q811yescharacteristic
192C2×C32C2×C3×SD1611yescharacteristic
193C2×C32C3×#13711yescharacteristic
194C2×Q8—21no—
195C2×Q8—91no—
196C2×C8—92no—
197C4⋊C4—21no—
198C4⋊C4—91no—
199C4⋊C4—181no—
200C4⋊C4C2×C3×C3⋊D312yes—
201C42—181no—
202C42C2×C3×C3⋊D311yescharacteristic
203C22×C3—24no—
204C22×C3C2×C3×#6714yes—
205C22×C3C2×#63811yescharacteristic
206C3×C4—28no—
207C3×C4—24no—
208C3×C4—21no—
209C3×C4—28no—
210C3×C4—24no—
211C3×C4—24no—
212C3×C4—416no—
213C3×C4—48no—
214C3×C4—42no—
215C3×C4—182no—
216C3×C4—361no—
217C3×C4C3×#21418yes—
218C3×C4#375412yes—
219C3×C4C3×D3×Q814yes—
220C3×C4C3×#23914yes—
221C3×C4Q8×C3⋊D311yescharacteristic
222C3×C4#383211yescharacteristic
223Dic3—68no—
224Dic3—124no—
225C32—24no—
226C32#509414yescomplemented
227C32C3×#47511yescharacteristic, complemented
228Q8—42no—
229Q8—92no—
230Q8—181no—
231C2×C4—22no—
232C2×C4—181no—
233C2×C4—181no—
234C2×C4C3×#63811yescharacteristic
235C2×C4C3×#63811yescharacteristic
236C2×C4C22×C3×C3⋊D311yescharacteristic
237C8—182no—
238C2×C3—24no—
239C2×C3—24no—
240C2×C3—24no—
241C2×C3C2×C3×#21414yes—
242C2×C3C3×#94214yes—
243C2×C3C3×#94614yes—
244C2×C3C2×#375411yescharacteristic
245C2×C3#6415111yescharacteristic
246C2×C3#6415511yescharacteristic
247C22C2×C3×#63811yescharacteristic
248C4—21no—
249C4—42no—
250C4—182no—
251C4—361no—
252C4C3×#375412yes—
253C4C3×Q8×C3⋊D311yescharacteristic
254C4C3×#383211yescharacteristic
255C3—24no—
256C3C3×#509414yescomplemented
257C3#1059279311yescharacteristic, direct product factor
258C2C2×C3×#375411yescharacteristic
259C2C3×#6415111yescharacteristic
260C2C3×#6415511yescharacteristic
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×#475
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. #10592793C3direct (N × H)
36. C3×#634Q8semidirect
99. #634C3×Q8semidirect
159. C33#475semidirect
226. C32#5094semidirect
227. C32C3×#475semidirect
256. C3C3×#5094semidirect
257. C3#10592793direct (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×C4×#6342
C3×#636372
C3×#636352
C3×C4⋊(C3⋊(C3⋊C8))2
#105927933
C3×#50943