Finite Group Databasebeta

← C3×#10592628 #13162965

Subgroups — C3×#10592628 #13162965

Order 1728.

1548 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C2×C33×C42C211yesmaximal, characteristic
3C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
4C3×#63658C214yesmaximal
5#10592628C311yesmaximal, characteristic, direct product factor
6C3×#4929—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×C42—22no—
15C3×C4⋊(C3⋊Dic3)—22no—
16C22×#634C2×C311yescharacteristic
17C22×C3×Dic6—34no—
18C2×C32×C42—24no—
19C2×C32×C42D314yes—
20C2×C32×C42C2×C311yescharacteristic
21C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
22C2×C3×C4⋊Dic3—34no—
23#63658C2×C314yes—
24C3×C3⋊#95—316no—
25C23×C33C2311yescharacteristic
26C2×C33×C4—22no—
27C2×C33×C4—22no—
28C2×C33×C4C2×C412yes—
29C2×C33×C4D412yes—
30C2×C33×C4D412yes—
31C2×C33×C4Q814yescomplemented
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3—24no—
36C2×C3×C3⋊Dic3C2×C414yes—
37C3×#634—28no—
38C3×#391—91no—
39#4929—34no—
40C22×C32×C4—28no—
41C22×C32×C4—24no—
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×C42—216no—
57C32×C42—28no—
58C32×C42—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—24no—
67C33×C4C22⋊C414yes—
68C3×C3⋊Dic3—24no—
69C3×C3⋊Dic3—42no—
70C22×C3×Q8—91no—
71C22×Dic6—34no—
72C2×C3×C4⋊C4—91no—
73C2×C3×C42—24no—
74C2×C3×C42C3×D314yes—
75C2×C3×C42C3⋊D311yescharacteristic
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—216no—
84C2×C32×C4—28no—
85C2×C32×C4—28no—
86C2×C32×C4—22no—
87C2×C32×C4—22no—
88C2×C32×C4—216no—
89C2×C32×C4—28no—
90C2×C32×C4—28no—
91C2×C32×C4—28no—
92C2×C32×C4Dic6116yescomplemented
93C2×C32×C4C4×D318yes—
94C2×C32×C4D1218yes—
95C2×C32×C4#6718yes—
96C2×C32×C4C2×C3×C412yes—
97C2×C32×C4C3×D412yes—
98C2×C32×C4C3×D412yes—
99C2×C32×C4C3×Q814yescomplemented
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—24no—
104C2×C3⋊Dic3C2×C3×C414yes—
105#634—28no—
106C2×C3×Dic3—316no—
107C2×C3×Dic3—68no—
108C2×C3×Dic3—68no—
109C2×C3×Dic3—68no—
110C2×C3×Dic3—616no—
111C3×Dic6—632no—
112#391—91noSylow
113C2×C33C2×C22⋊C411yescharacteristic
114C2×C33C4×Q812yes—
115C2×C33C22⋊Q812yes—
116C2×C33#12411yescharacteristic
117C2×C33C4⋊Q811yescharacteristic
118C2×C3×Q8—94no—
119C2×C3×Q8—182no—
120C2×C3×Q8—182no—
121C22×C3×C4—28no—
122C22×C3×C4—24no—
123C22×C3×C4—92no—
124C22×C3×C4—92no—
125C22×C3×C4C2×C3×D318yes—
126C22×C3×C4C2×C3×D314yes—
127C22×C3×C4C2×C3⋊D312yes—
128C22×C3×C4C2×C3⋊D311yescharacteristic
129C22×Dic3—38no—
130C22×Dic3—38no—
131C2×Dic6—316no—
132C2×Dic6—68no—
133C2×Dic6—68no—
134C3×C4⋊C4—182no—
135C3×C42—216no—
136C3×C42—28no—
137C3×C42—22no—
138C4⋊Dic3—68no—
139C22×C32—28no—
140C22×C32—24no—
141C22×C32—24no—
142C22×C32—28no—
143C22×C32—24no—
144C22×C32C2×Dic618yes—
145C22×C32C2×C4×D314yes—
146C22×C32C2×D1214yes—
147C22×C32#23518yes—
148C22×C32C2×#6714yes—
149C22×C32C22×C3×C411yescharacteristic
150C22×C32C2×C3×D411yescharacteristic
151C22×C32C2×C3×D411yescharacteristic
152C22×C32C2×C3×Q812yes—
153C22×C32C3×#4112yes—
154C32×C4—232no—
155C32×C4—216no—
156C32×C4—24no—
157C32×C4—216no—
158C32×C4(C2×D3)⋊C4116yes—
159C32×C4C3×C22⋊C414yes—
160C3⋊Dic3—24no—
161C3⋊Dic3—42no—
162C3×Dic3—616no—
163C3×Dic3—128no—
164C22×Q8—91no—
165C2×C4⋊C4—91no—
166C2×C42C3×C3⋊D311yescharacteristic
167#95—94no—
168C33#39111yescharacteristic, complemented, Sylow
169C23×C3—24no—
170C23×C3C22×C3×D314yes—
171C23×C3C22×C3⋊D311yescharacteristic
172C3×Q8—188no—
173C2×C3×C4—216no—
174C2×C3×C4—216no—
175C2×C3×C4—28no—
176C2×C3×C4—28no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—216no—
180C2×C3×C4—28no—
181C2×C3×C4—28no—
182C2×C3×C4—28no—
183C2×C3×C4—94no—
184C2×C3×C4—182no—
185C2×C3×C4—182no—
186C2×C3×C4—182no—
187C2×C3×C4—184no—
188C2×C3×C4C3×Dic6116yescomplemented
189C2×C3×C4C3×C4×D318yes—
190C2×C3×C4C3×D1218yes—
191C2×C3×C4C3×#6718yes—
192C2×C3×C4#63414yescomplemented
193C2×C3×C4C4×C3⋊D312yes—
194C2×C3×C4C4⋊(C3⋊D3)12yes—
195C2×C3×C4#63812yes—
196C2×Dic3—316no—
197C2×Dic3—68no—
198C2×Dic3—68no—
199C2×Dic3—68no—
200C2×Dic3—616no—
201Dic6—632no—
202C2×C32—28no—
203C2×C32—24no—
204C2×C32—28no—
205C2×C32—24no—
206C2×C32—24no—
207C2×C32C4×Dic618yes—
208C2×C32#86914yes—
209C2×C32#87514yes—
210C2×C32#92418yes—
211C2×C32C2×(C2×D3)⋊C414yes—
212C2×C32C2×C3×C22⋊C411yescharacteristic
213C2×C32C3×C4×Q812yes—
214C2×C32C3×C22⋊Q812yes—
215C2×C32C3×#12411yescharacteristic
216C2×C32C3×C4⋊Q811yescharacteristic
217C2×Q8—94no—
218C2×Q8—182no—
219C2×Q8—182no—
220C22×C4—92no—
221C22×C4—92no—
222C22×C4C2×C3×C3⋊D312yes—
223C22×C4C2×C3×C3⋊D311yescharacteristic
224C4⋊C4—182no—
225C42—22no—
226C22×C3—28no—
227C22×C3—24no—
228C22×C3—24no—
229C22×C3—28no—
230C22×C3—24no—
231C22×C3C2×C3×Dic618yes—
232C22×C3C2×C3×C4×D314yes—
233C22×C3C2×C3×D1214yes—
234C22×C3C3×#23518yes—
235C22×C3C2×C3×#6714yes—
236C22×C3C2×#63412yes—
237C22×C3C2×C4×C3⋊D311yescharacteristic
238C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
239C22×C3#382812yes—
240C22×C3C2×#63811yescharacteristic
241C3×C4—232no—
242C3×C4—216no—
243C3×C4—24no—
244C3×C4—216no—
245C3×C4—184no—
246C3×C4—362no—
247C3×C4C3×(C2×D3)⋊C4116yes—
248C3×C4#375214yes—
249Dic3—616no—
250Dic3—128no—
251C32—24no—
252C32#492914yescomplemented
253C32C3×#39111yescharacteristic, complemented
254C23C22×C3×C3⋊D311yescharacteristic
255Q8—188no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—94no—
259C2×C4—182no—
260C2×C4—182no—
261C2×C4—182no—
262C2×C4—184no—
263C2×C4C3×#63414yescomplemented
264C2×C4C3×C4×C3⋊D312yes—
265C2×C4C3×C4⋊(C3⋊D3)12yes—
266C2×C4C3×#63812yes—
267C2×C3—28no—
268C2×C3—24no—
269C2×C3—24no—
270C2×C3—28no—
271C2×C3—24no—
272C2×C3C3×C4×Dic618yes—
273C2×C3C3×#86914yes—
274C2×C3C3×#87514yes—
275C2×C3C3×#92418yes—
276C2×C3C2×C3×(C2×D3)⋊C414yes—
277C2×C3C4×#63412yes—
278C2×C3#6407811yescharacteristic
279C2×C3#6408411yescharacteristic
280C2×C3#6413312yes—
281C2×C3C2×#375211yescharacteristic
282C22C2×C3×#63412yes—
283C22C2×C3×C4×C3⋊D311yescharacteristic
284C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
285C22C3×#382812yes—
286C22C2×C3×#63811yescharacteristic
287C4—24no—
288C4—184no—
289C4—362no—
290C4C3×#375214yes—
291C3—24no—
292C3C3×#492914yescomplemented
293C3#1059262811yescharacteristic, direct product factor
294C2C3×C4×#63412yes—
295C2C3×#6407811yescharacteristic
296C2C3×#6408411yescharacteristic
297C2C3×#6413312yes—
298C2C2×C3×#375211yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C42C2
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC2×C3×C42C3⋊D3
hypocenterC32C3×#391
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C2×C3×C42

Complements

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

NH ≅ G/Ntype
5. #10592628C3direct (N × H)
13. C2×C3×#634C4semidirect
31. C2×C33×C4Q8semidirect
50. C2×#634C3×C4semidirect
92. C2×C32×C4Dic6semidirect
99. C2×C32×C4C3×Q8semidirect
168. C33#391semidirect
188. C2×C3×C4C3×Dic6semidirect
192. C2×C3×C4#634semidirect
252. C32#4929semidirect
253. C32C3×#391semidirect
263. C2×C4C3×#634semidirect
292. C3C3×#4929semidirect
293. C3#10592628direct (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×C422
C2×C3×C4⋊(C3⋊Dic3)2
C3×#636582
#105926283
C3×#49293