Finite Group Databasebeta

← C3×#10592672 #13163005

Subgroups — C3×#10592672 #13163005

Order 1728.

1484 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C2×C33×C4⋊C4C211yesmaximal, characteristic
3C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
4C3×#63658C214yesmaximal
5#10592672C311yesmaximal, characteristic, direct product factor
6C3×#4973—34nomaximal
7C22×C33×C4C2212yes—
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2212yes—
11C2×C3×#634—22no—
12C2×C3×#634—22no—
13C2×C3×#634C414yescomplemented
14C33×C4⋊C4—22no—
15C3×C4×C3⋊Dic3—22no—
16C22×#634C2×C311yescharacteristic
17C22×C3×Dic6—34no—
18C2×C32×C4⋊C4—24no—
19C2×C32×C4⋊C4D314yes—
20C2×C32×C4⋊C4C2×C311yescharacteristic
21C2×C4×C3⋊Dic3C2×C311yescharacteristic
22C2×C3×C4×Dic3—34no—
23#63658C2×C314yes—
24C3×C3⋊#95—316no—
25C23×C33C2311yescharacteristic
26C2×C33×C4—22no—
27C2×C33×C4—24no—
28C2×C33×C4C2×C412yes—
29C2×C33×C4D412yes—
30C2×C33×C4D412yes—
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3C2×C414yes—
36C2×C3×C3⋊Dic3Q814yescomplemented
37C3×#634—28no—
38C3×#391—91no—
39#4973—34no—
40C22×C32×C4—24no—
41C22×C32×C4—28no—
42C22×C32×C4C2×D318yes—
43C22×C32×C4C2×D314yes—
44C22×C32×C4C22×C312yes—
45C22×C32×C4C22×C311yescharacteristic
46C22×C3⋊Dic3C22×C312yes—
47C22×C3⋊Dic3C22×C312yes—
48C2×#634—22no—
49C2×#634—22no—
50C2×#634C3×C414yescomplemented
51C22×C3×Dic3—38no—
52C22×C3×Dic3—38no—
53C2×C3×Dic6—316no—
54C2×C3×Dic6—68no—
55C2×C3×Dic6—68no—
56C32×C4⋊C4—216no—
57C32×C4⋊C4—28no—
58C32×C4⋊C4—22no—
59C4×C3⋊Dic3—22no—
60C3×C4×Dic3—68no—
61C22×C33C22×C411yescharacteristic
62C22×C33C2×D411yescharacteristic
63C22×C33C2×D411yescharacteristic
64C22×C33C2×Q812yes—
65C22×C33#4112yes—
66C33×C4—42no—
67C33×C4C22⋊C414yes—
68C3×C3⋊Dic3—24no—
69C3×C3⋊Dic3—24no—
70C22×C3×Q8—91no—
71C22×Dic6—34no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4C3×D314yes—
74C2×C3×C4⋊C4C3⋊D311yescharacteristic
75C2×C3×C42—91no—
76C2×C4×Dic3—34no—
77C3×#95—94no—
78C3⋊#95—316no—
79C23×C32—24no—
80C23×C32C22×D314yes—
81C23×C32C23×C311yescharacteristic
82C2×C32×C4—216no—
83C2×C32×C4—28no—
84C2×C32×C4—22no—
85C2×C32×C4—24no—
86C2×C32×C4—28no—
87C2×C32×C4—28no—
88C2×C32×C4—28no—
89C2×C32×C4—216no—
90C2×C32×C4—416no—
91C2×C32×C4C4×D318yes—
92C2×C32×C4D1218yes—
93C2×C32×C4#6718yes—
94C2×C32×C4C2×C3×C412yes—
95C2×C32×C4C3×D412yes—
96C2×C32×C4C3×D412yes—
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3—22no—
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3C2×C3×C414yes—
102C2×C3⋊Dic3C3×Q814yescomplemented
103#634—28no—
104C2×C3×Dic3—316no—
105C2×C3×Dic3—316no—
106C2×C3×Dic3—68no—
107C2×C3×Dic3—68no—
108C2×C3×Dic3—68no—
109C2×C3×Dic3—68no—
110C3×Dic6—632no—
111#391—91noSylow
112C2×C33C2×C22⋊C411yescharacteristic
113C2×C33C4×Q812yes—
114C2×C33C22⋊Q812yes—
115C2×C33#12411yescharacteristic
116C2×C33C4⋊Q811yescharacteristic
117C2×C3×Q8—94no—
118C2×C3×Q8—182no—
119C2×C3×Q8—182no—
120C22×C3×C4—24no—
121C22×C3×C4—28no—
122C22×C3×C4—92no—
123C22×C3×C4—92no—
124C22×C3×C4C2×C3×D318yes—
125C22×C3×C4C2×C3×D314yes—
126C22×C3×C4C2×C3⋊D312yes—
127C22×C3×C4C2×C3⋊D311yescharacteristic
128C22×Dic3—38no—
129C22×Dic3—38no—
130C2×Dic6—316no—
131C2×Dic6—68no—
132C2×Dic6—68no—
133C3×C4⋊C4—216no—
134C3×C4⋊C4—28no—
135C3×C4⋊C4—22no—
136C3×C42—182no—
137C4×Dic3—68no—
138C22×C32—24no—
139C22×C32—24no—
140C22×C32—28no—
141C22×C32—28no—
142C22×C32—24no—
143C22×C32C2×C4×D314yes—
144C22×C32C2×D1214yes—
145C22×C32#23718yes—
146C22×C32D3×Q818yes—
147C22×C32C2×#6714yes—
148C22×C32C22×C3×C411yescharacteristic
149C22×C32C2×C3×D411yescharacteristic
150C22×C32C2×C3×D411yescharacteristic
151C22×C32C2×C3×Q812yes—
152C22×C32C3×#4112yes—
153C32×C4—216no—
154C32×C4—416no—
155C32×C4—48no—
156C32×C4—42no—
157C32×C4(C2×D3)⋊C4116yes—
158C32×C4C3×C22⋊C414yes—
159C3⋊Dic3—24no—
160C3⋊Dic3—24no—
161C3×Dic3—616no—
162C3×Dic3—616no—
163C22×Q8—91no—
164C2×C4⋊C4C3×C3⋊D311yescharacteristic
165C2×C42—91no—
166#95—94no—
167C33#39111yescharacteristic, complemented, Sylow
168C23×C3—24no—
169C23×C3C22×C3×D314yes—
170C23×C3C22×C3⋊D311yescharacteristic
171C3×Q8—188no—
172C2×C3×C4—216no—
173C2×C3×C4—216no—
174C2×C3×C4—28no—
175C2×C3×C4—24no—
176C2×C3×C4—22no—
177C2×C3×C4—28no—
178C2×C3×C4—28no—
179C2×C3×C4—28no—
180C2×C3×C4—416no—
181C2×C3×C4—94no—
182C2×C3×C4—94no—
183C2×C3×C4—182no—
184C2×C3×C4—182no—
185C2×C3×C4—182no—
186C2×C3×C4—182no—
187C2×C3×C4C3×C4×D318yes—
188C2×C3×C4C3×D1218yes—
189C2×C3×C4C3×#6718yes—
190C2×C3×C4C4×C3⋊D312yes—
191C2×C3×C4C4⋊(C3⋊D3)12yes—
192C2×C3×C4#63812yes—
193C2×Dic3—316no—
194C2×Dic3—316no—
195C2×Dic3—68no—
196C2×Dic3—68no—
197C2×Dic3—68no—
198C2×Dic3—68no—
199Dic6—632no—
200C2×C32—28no—
201C2×C32—28no—
202C2×C32—24no—
203C2×C32—24no—
204C2×C32—24no—
205C2×C32#88718yes—
206C2×C32#89718yes—
207C2×C32C2×(C2×D3)⋊C414yes—
208C2×C32#93614yes—
209C2×C32#94414yes—
210C2×C32C2×C3×C22⋊C411yescharacteristic
211C2×C32C3×C4×Q812yes—
212C2×C32C3×C22⋊Q812yes—
213C2×C32C3×#12411yescharacteristic
214C2×C32C3×C4⋊Q811yescharacteristic
215C2×Q8—94no—
216C2×Q8—182no—
217C2×Q8—182no—
218C22×C4—92no—
219C22×C4—92no—
220C22×C4C2×C3×C3⋊D312yes—
221C22×C4C2×C3×C3⋊D311yescharacteristic
222C4⋊C4—22no—
223C42—182no—
224C22×C3—24no—
225C22×C3—24no—
226C22×C3—28no—
227C22×C3—28no—
228C22×C3—24no—
229C22×C3C2×C3×C4×D314yes—
230C22×C3C2×C3×D1214yes—
231C22×C3C3×#23718yes—
232C22×C3C3×D3×Q818yes—
233C22×C3C2×C3×#6714yes—
234C22×C3C2×C4×C3⋊D311yescharacteristic
235C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
236C22×C3#383012yes—
237C22×C3Q8×C3⋊D312yes—
238C22×C3C2×#63811yescharacteristic
239C3×C4—216no—
240C3×C4—416no—
241C3×C4—48no—
242C3×C4—42no—
243C3×C4—184no—
244C3×C4—184no—
245C3×C4C3×(C2×D3)⋊C4116yes—
246C3×C4#375214yes—
247Dic3—616no—
248Dic3—616no—
249C32—24no—
250C32#497314yescomplemented
251C32C3×#39111yescharacteristic, complemented
252C23C22×C3×C3⋊D311yescharacteristic
253Q8—188no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—94no—
257C2×C4—94no—
258C2×C4—182no—
259C2×C4—182no—
260C2×C4—182no—
261C2×C4—182no—
262C2×C4C3×C4×C3⋊D312yes—
263C2×C4C3×C4⋊(C3⋊D3)12yes—
264C2×C4C3×#63812yes—
265C2×C3—28no—
266C2×C3—28no—
267C2×C3—24no—
268C2×C3—24no—
269C2×C3—24no—
270C2×C3C3×#88718yes—
271C2×C3C3×#89718yes—
272C2×C3C2×C3×(C2×D3)⋊C414yes—
273C2×C3C3×#93614yes—
274C2×C3C3×#94414yes—
275C2×C3#6409612yes—
276C2×C3#6410612yes—
277C2×C3C2×#375211yescharacteristic
278C2×C3#6414511yescharacteristic
279C2×C3#6415311yescharacteristic
280C22C2×C3×C4×C3⋊D311yescharacteristic
281C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
282C22C3×#383012yes—
283C22C3×Q8×C3⋊D312yes—
284C22C2×C3×#63811yescharacteristic
285C4—42no—
286C4—184no—
287C4—184no—
288C4C3×#375214yes—
289C3—24no—
290C3C3×#497314yescomplemented
291C3#1059267211yescharacteristic, direct product factor
292C2C3×#6409612yes—
293C2C3×#6410612yes—
294C2C2×C3×#375211yescharacteristic
295C2C3×#6414511yescharacteristic
296C2C3×#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×#391
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. #10592672C3direct (N × H)
13. C2×C3×#634C4semidirect
36. C2×C3×C3⋊Dic3Q8semidirect
50. C2×#634C3×C4semidirect
102. C2×C3⋊Dic3C3×Q8semidirect
167. C33#391semidirect
250. C32#4973semidirect
251. C32C3×#391semidirect
290. C3C3×#4973semidirect
291. C3#10592672direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C2×C33×C4⋊C42
C2×C3×C4×C3⋊Dic32
C3×#636582
#105926723
C3×#49733