Finite Group Databasebeta

← C3×#10592674 #13163007

Subgroups — C3×#10592674 #13163007

Order 1728.

1484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C4⋊C4C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C2×C3×#3750C214yesmaximal
4C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
5#10592674C311yesmaximal, characteristic, direct product factor
6C3×#4975—34nomaximal
7C22×C33×C4C2212yes—
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2212yes—
11C33×C4⋊C4—22no—
12C3×C4⋊(C3⋊Dic3)—22no—
13C3×#3750—28no—
14C3×C4×C3⋊Dic3C414yescomplemented
15C2×C32×C4⋊C4—24no—
16C2×C32×C4⋊C4D314yes—
17C2×C32×C4⋊C4C2×C311yescharacteristic
18C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
19C2×#3750C2×C314yes—
20C2×C4×C3⋊Dic3C2×C311yescharacteristic
21C2×C3×C4⋊Dic3—34no—
22C2×C3×Dic3⋊C4—316no—
23C2×C3×C4×Dic3—34no—
24C23×C33C2311yescharacteristic
25C2×C33×C4—22no—
26C2×C33×C4—24no—
27C2×C33×C4C2×C412yes—
28C2×C33×C4D412yes—
29C2×C33×C4Q812yes—
30C2×C3×C3⋊Dic3—24no—
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3C2×C414yes—
33C2×C3×C3⋊Dic3D414yes—
34C2×C3×C3⋊Dic3Q814yes—
35C3×#384—91no—
36#4975—34no—
37C22×C32×C4—24no—
38C22×C32×C4—28no—
39C22×C32×C4C2×D318yes—
40C22×C32×C4C2×D314yes—
41C22×C32×C4C22×C312yes—
42C22×C32×C4C22×C311yescharacteristic
43C22×C3⋊Dic3C22×C312yes—
44C22×C3⋊Dic3C22×C312yes—
45C22×C3×Dic3—38no—
46C22×C3×Dic3—38no—
47C32×C4⋊C4—216no—
48C32×C4⋊C4—28no—
49C32×C4⋊C4—22no—
50C4⋊(C3⋊Dic3)—22no—
51#3750—28no—
52C4×C3⋊Dic3C3×C414yescomplemented
53C3×C4⋊Dic3—68no—
54C3×Dic3⋊C4—632no—
55C3×C4×Dic3—316no—
56C22×C33C22×C411yescharacteristic
57C22×C33C2×D412yes—
58C22×C33C2×D411yescharacteristic
59C22×C33C2×Q811yescharacteristic
60C22×C33C2×Q812yes—
61C33×C4—42no—
62C33×C4C4⋊C414yescomplemented
63C3×C3⋊Dic3—42no—
64C3×C3⋊Dic3C4⋊C418yescomplemented
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—94no—
67C2×C3×C4⋊C4—91no—
68C2×C3×C4⋊C4C3×D314yes—
69C2×C3×C4⋊C4C3⋊D311yescharacteristic
70C2×C3×C42—91no—
71C2×C4⋊Dic3—34no—
72C2×Dic3⋊C4—316no—
73C2×C4×Dic3—34no—
74C23×C32—24no—
75C23×C32C22×D314yes—
76C23×C32C23×C311yescharacteristic
77C2×C32×C4—216no—
78C2×C32×C4—28no—
79C2×C32×C4—24no—
80C2×C32×C4—22no—
81C2×C32×C4—28no—
82C2×C32×C4—28no—
83C2×C32×C4—28no—
84C2×C32×C4—216no—
85C2×C32×C4—416no—
86C2×C32×C4Dic618yes—
87C2×C32×C4C4×D318yes—
88C2×C32×C4#6718yes—
89C2×C32×C4C2×C3×C412yes—
90C2×C32×C4C3×D412yes—
91C2×C32×C4C3×Q812yes—
92C2×C3⋊Dic3—24no—
93C2×C3⋊Dic3—22no—
94C2×C3⋊Dic3C2×C3×C414yes—
95C2×C3⋊Dic3C3×D414yes—
96C2×C3⋊Dic3C3×Q814yes—
97C2×C3×Dic3—316no—
98C2×C3×Dic3—316no—
99C2×C3×Dic3—316no—
100C2×C3×Dic3—616no—
101C2×C3×Dic3—68no—
102#384—91noSylow
103C2×C33C2×C4⋊C412yes—
104C2×C33C2×C4⋊C411yescharacteristic
105C2×C33#12711yescharacteristic
106C2×C33C4⋊Q812yes—
107C2×C33C4⋊Q811yescharacteristic
108C22×C3×C4—24no—
109C22×C3×C4—28no—
110C22×C3×C4—92no—
111C22×C3×C4—92no—
112C22×C3×C4C2×C3×D318yes—
113C22×C3×C4C2×C3×D314yes—
114C22×C3×C4C2×C3⋊D312yes—
115C22×C3×C4C2×C3⋊D311yescharacteristic
116C22×Dic3—38no—
117C22×Dic3—38no—
118C3×C4⋊C4—216no—
119C3×C4⋊C4—28no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—188no—
122C3×C4⋊C4—182no—
123C3×C42—94no—
124C4⋊Dic3—68no—
125Dic3⋊C4—632no—
126C4×Dic3—316no—
127C22×C32—24no—
128C22×C32—24no—
129C22×C32—28no—
130C22×C32—28no—
131C22×C32—24no—
132C22×C32C2×Dic614yes—
133C22×C32C2×C4×D314yes—
134C22×C32D3×D418yes—
135C22×C32D3×Q818yes—
136C22×C32C2×#6714yes—
137C22×C32C22×C3×C411yescharacteristic
138C22×C32C2×C3×D412yes—
139C22×C32C2×C3×D411yescharacteristic
140C22×C32C2×C3×Q811yescharacteristic
141C22×C32C2×C3×Q812yes—
142C32×C4—216no—
143C32×C4—416no—
144C32×C4—48no—
145C32×C4—42no—
146C32×C4Dic3⋊C4116yescomplemented
147C32×C4C3×C4⋊C414yescomplemented
148C3⋊Dic3—42no—
149C3⋊Dic3C3×C4⋊C418yescomplemented
150C3×Dic3—332no—
151C3×Dic3—128no—
152C2×C4⋊C4—94no—
153C2×C4⋊C4—91no—
154C2×C4⋊C4C3×C3⋊D311yescharacteristic
155C2×C42—91no—
156C33#38411yescharacteristic, complemented, Sylow
157C23×C3—24no—
158C23×C3C22×C3×D314yes—
159C23×C3C22×C3⋊D311yescharacteristic
160C2×C3×C4—216no—
161C2×C3×C4—216no—
162C2×C3×C4—28no—
163C2×C3×C4—24no—
164C2×C3×C4—22no—
165C2×C3×C4—28no—
166C2×C3×C4—28no—
167C2×C3×C4—28no—
168C2×C3×C4—416no—
169C2×C3×C4—94no—
170C2×C3×C4—94no—
171C2×C3×C4—94no—
172C2×C3×C4—184no—
173C2×C3×C4—182no—
174C2×C3×C4C3×Dic618yes—
175C2×C3×C4C3×C4×D318yes—
176C2×C3×C4C3×#6718yes—
177C2×C3×C4#63412yes—
178C2×C3×C4C4×C3⋊D312yes—
179C2×C3×C4#63812yes—
180C2×Dic3—316no—
181C2×Dic3—316no—
182C2×Dic3—316no—
183C2×Dic3—616no—
184C2×Dic3—68no—
185C2×C32—28no—
186C2×C32—28no—
187C2×C32—24no—
188C2×C32—24no—
189C2×C32—24no—
190C2×C32#88818yes—
191C2×C32D3×C4⋊C418yes—
192C2×C32C2×Dic3⋊C414yes—
193C2×C32#94014yes—
194C2×C32#94414yes—
195C2×C32C2×C3×C4⋊C412yes—
196C2×C32C2×C3×C4⋊C411yescharacteristic
197C2×C32C3×#12711yescharacteristic
198C2×C32C3×C4⋊Q812yes—
199C2×C32C3×C4⋊Q811yescharacteristic
200C22×C4—92no—
201C22×C4—92no—
202C22×C4C2×C3×C3⋊D312yes—
203C22×C4C2×C3×C3⋊D311yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—188no—
206C4⋊C4—182no—
207C42—94no—
208C22×C3—24no—
209C22×C3—24no—
210C22×C3—28no—
211C22×C3—28no—
212C22×C3—24no—
213C22×C3C2×C3×Dic614yes—
214C22×C3C2×C3×C4×D314yes—
215C22×C3C3×D3×D418yes—
216C22×C3C3×D3×Q818yes—
217C22×C3C2×C3×#6714yes—
218C22×C3C2×#63411yescharacteristic
219C22×C3C2×C4×C3⋊D311yescharacteristic
220C22×C3D4×C3⋊D312yes—
221C22×C3Q8×C3⋊D312yes—
222C22×C3C2×#63811yescharacteristic
223C3×C4—216no—
224C3×C4—416no—
225C3×C4—48no—
226C3×C4—42no—
227C3×C4—98no—
228C3×C4—362no—
229C3×C4C3×Dic3⋊C4116yescomplemented
230C3×C4#375014yescomplemented
231Dic3—332no—
232Dic3—128no—
233C32—24no—
234C32#497514yescomplemented
235C32C3×#38411yescharacteristic, complemented
236C23C22×C3×C3⋊D311yescharacteristic
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—94no—
240C2×C4—94no—
241C2×C4—94no—
242C2×C4—184no—
243C2×C4—182no—
244C2×C4C3×#63412yes—
245C2×C4C3×C4×C3⋊D312yes—
246C2×C4C3×#63812yes—
247C2×C3—28no—
248C2×C3—28no—
249C2×C3—24no—
250C2×C3—24no—
251C2×C3—24no—
252C2×C3C3×#88818yes—
253C2×C3C3×D3×C4⋊C418yes—
254C2×C3C2×C3×Dic3⋊C414yes—
255C2×C3C3×#94014yes—
256C2×C3C3×#94414yes—
257C2×C3#6409712yes—
258C2×C3C4⋊C4×C3⋊D312yes—
259C2×C3C2×#375011yescharacteristic
260C2×C3#6414911yescharacteristic
261C2×C3#6415311yescharacteristic
262C22C2×C3×#63411yescharacteristic
263C22C2×C3×C4×C3⋊D311yescharacteristic
264C22C3×D4×C3⋊D312yes—
265C22C3×Q8×C3⋊D312yes—
266C22C2×C3×#63811yescharacteristic
267C4—42no—
268C4—98no—
269C4—362no—
270C4C3×#375014yescomplemented
271C3—24no—
272C3C3×#497514yescomplemented
273C3#1059267411yescharacteristic, direct product factor
274C2C3×#6409712yes—
275C2C3×C4⋊C4×C3⋊D312yes—
276C2C2×C3×#375011yescharacteristic
277C2C3×#6414911yescharacteristic
278C2C3×#6415311yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C4⋊C4C2
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterC32C3×#384
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #10592674C3direct (N × H)
14. C3×C4×C3⋊Dic3C4semidirect
52. C4×C3⋊Dic3C3×C4semidirect
62. C33×C4C4⋊C4semidirect
64. C3×C3⋊Dic3C4⋊C4semidirect
146. C32×C4Dic3⋊C4semidirect
147. C32×C4C3×C4⋊C4semidirect
149. C3⋊Dic3C3×C4⋊C4semidirect
156. C33#384semidirect
229. C3×C4C3×Dic3⋊C4semidirect
230. C3×C4#3750semidirect
234. C32#4975semidirect
235. C32C3×#384semidirect
270. C4C3×#3750semidirect
272. C3C3×#4975semidirect
273. C3#10592674direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×C4⋊C42
C2×C3×C4⋊(C3⋊Dic3)2
C2×C3×#37502
C2×C3×C4×C3⋊Dic32
#105926743
C3×#49753