Finite Group Databasebeta

← C3×#10592928 #13163266

Subgroups — C3×#10592928 #13163266

Order 1728.

1740 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C33×Q8C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
4C3×#63658C214yesmaximal
5#10592928C311yesmaximal, characteristic, direct product factor
6C3×#5229—34nomaximal
7C2×C33×Q8—22no—
8C2×C33×Q8—22no—
9C2×C33×Q8C414yescomplemented
10C22×C33×C4C2212yes—
11C22×C33×C4C2211yescharacteristic
12C22×C3×C3⋊Dic3C2212yes—
13C22×C3×C3⋊Dic3C2212yes—
14C3×C4⋊(C3⋊Dic3)—22no—
15C3×C4×C3⋊Dic3—22no—
16C22×C32×Q8—24no—
17C22×C32×Q8D314yes—
18C22×C32×Q8C2×C311yescharacteristic
19C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
20C2×C4×C3⋊Dic3C2×C311yescharacteristic
21C2×C3×C4⋊Dic3—34no—
22C2×C3×C4×Dic3—34no—
23#63658C2×C314yes—
24C3×C3⋊#95—316no—
25C23×C33C2311yescharacteristic
26C33×Q8—28no—
27C2×C33×C4—22no—
28C2×C33×C4—22no—
29C2×C33×C4C2×C414yes—
30C2×C33×C4C2×C412yes—
31C2×C33×C4D412yes—
32C2×C33×C4D412yes—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—24no—
37C2×C3×C3⋊Dic3Q814yescomplemented
38C3×#391—91no—
39#5229—34no—
40C2×C32×Q8—216no—
41C2×C32×Q8—216no—
42C2×C32×Q8—216no—
43C2×C32×Q8—28no—
44C2×C32×Q8—28no—
45C2×C32×Q8—22no—
46C2×C32×Q8—22no—
47C2×C32×Q8Dic3116yescomplemented
48C2×C32×Q8C3×C414yescomplemented
49C22×C32×C4—28no—
50C22×C32×C4—24no—
51C22×C32×C4C2×D318yes—
52C22×C32×C4C2×D314yes—
53C22×C32×C4C22×C312yes—
54C22×C32×C4C22×C311yescharacteristic
55C22×C3⋊Dic3C22×C312yes—
56C22×C3⋊Dic3C22×C312yes—
57C22×C3×Dic3—38no—
58C22×C3×Dic3—38no—
59C4⋊(C3⋊Dic3)—22no—
60C4×C3⋊Dic3—22no—
61C3×C4⋊Dic3—68no—
62C3×C4×Dic3—68no—
63C22×C33C22×C411yescharacteristic
64C22×C33C2×D411yescharacteristic
65C22×C33C2×D411yescharacteristic
66C22×C33C2×Q812yes—
67C22×C33#4112yes—
68C33×C4—24no—
69C33×C4C22⋊C414yes—
70C3×C3⋊Dic3—24no—
71C3×C3⋊Dic3—42no—
72C22×C3×Q8—24no—
73C22×C3×Q8C3×D314yes—
74C22×C3×Q8C3⋊D311yescharacteristic
75C2×C3×C4⋊C4—91no—
76C2×C3×C42—91no—
77C2×C4⋊Dic3—34no—
78C2×C4×Dic3—34no—
79C3×#95—94no—
80C3⋊#95—316no—
81C23×C32—24no—
82C23×C32C22×D314yes—
83C23×C32C23×C311yescharacteristic
84C32×Q8—264no—
85C32×Q8—232no—
86C32×Q8—28no—
87C2×C32×C4—216no—
88C2×C32×C4—216no—
89C2×C32×C4—28no—
90C2×C32×C4—28no—
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4—216no—
94C2×C32×C4—28no—
95C2×C32×C4—28no—
96C2×C32×C4—28no—
97C2×C32×C4C2×Dic3116yes—
98C2×C32×C4C2×Dic318yes—
99C2×C32×C4#6718yes—
100C2×C32×C4#6718yes—
101C2×C32×C4C2×C3×C414yes—
102C2×C32×C4C2×C3×C412yes—
103C2×C32×C4C3×D412yes—
104C2×C32×C4C3×D412yes—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—24no—
109C2×C3⋊Dic3C3×Q814yescomplemented
110C2×C3×Dic3—316no—
111C2×C3×Dic3—68no—
112C2×C3×Dic3—68no—
113C2×C3×Dic3—616no—
114C2×C3×Dic3—68no—
115#391—91noSylow
116C2×C33C2×C22⋊C411yescharacteristic
117C2×C33C4×Q812yes—
118C2×C33C22⋊Q812yes—
119C2×C33#12411yescharacteristic
120C2×C33C4⋊Q811yescharacteristic
121C2×C3×Q8—216no—
122C2×C3×Q8—216no—
123C2×C3×Q8—28no—
124C2×C3×Q8—28no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—216no—
128C2×C3×Q8C3×Dic3116yescomplemented
129C2×C3×Q8C3⋊Dic314yescomplemented
130C22×C3×C4—28no—
131C22×C3×C4—24no—
132C22×C3×C4—92no—
133C22×C3×C4—92no—
134C22×C3×C4C2×C3×D318yes—
135C22×C3×C4C2×C3×D314yes—
136C22×C3×C4C2×C3⋊D312yes—
137C22×C3×C4C2×C3⋊D311yescharacteristic
138C22×Dic3—38no—
139C22×Dic3—38no—
140C3×C4⋊C4—182no—
141C3×C42—182no—
142C4⋊Dic3—68no—
143C4×Dic3—68no—
144C22×C32—28no—
145C22×C32—28no—
146C22×C32—24no—
147C22×C32—24no—
148C22×C32—24no—
149C22×C32D3×Q818yes—
150C22×C32#23918yes—
151C22×C32C22×Dic314yes—
152C22×C32C2×#6714yes—
153C22×C32C2×#6714yes—
154C22×C32C22×C3×C411yescharacteristic
155C22×C32C2×C3×D411yescharacteristic
156C22×C32C2×C3×D411yescharacteristic
157C22×C32C2×C3×Q812yes—
158C22×C32C3×#4112yes—
159C32×C4—232no—
160C32×C4—216no—
161C32×C4—24no—
162C32×C4—216no—
163C32×C4#217116yes—
164C32×C4C3×C22⋊C414yes—
165C3⋊Dic3—24no—
166C3⋊Dic3—42no—
167C3×Dic3—616no—
168C3×Dic3—128no—
169C22×Q8C3×C3⋊D311yescharacteristic
170C2×C4⋊C4—91no—
171C2×C42—91no—
172#95—94no—
173C33#39111yescharacteristic, complemented, Sylow
174C23×C3—24no—
175C23×C3C22×C3×D314yes—
176C23×C3C22×C3⋊D311yescharacteristic
177C3×Q8—264no—
178C3×Q8—232no—
179C3×Q8—28no—
180C2×C3×C4—216no—
181C2×C3×C4—216no—
182C2×C3×C4—28no—
183C2×C3×C4—28no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—216no—
187C2×C3×C4—28no—
188C2×C3×C4—28no—
189C2×C3×C4—28no—
190C2×C3×C4—94no—
191C2×C3×C4—182no—
192C2×C3×C4—182no—
193C2×C3×C4—184no—
194C2×C3×C4—182no—
195C2×C3×C4C2×C3×Dic3116yes—
196C2×C3×C4C2×C3×Dic318yes—
197C2×C3×C4C3×#6718yes—
198C2×C3×C4C3×#6718yes—
199C2×C3×C4C2×C3⋊Dic314yes—
200C2×C3×C4C2×C3⋊Dic312yes—
201C2×C3×C4#63812yes—
202C2×C3×C4#63812yes—
203C2×Dic3—316no—
204C2×Dic3—68no—
205C2×Dic3—68no—
206C2×Dic3—616no—
207C2×Dic3—68no—
208C2×C32—24no—
209C2×C32—28no—
210C2×C32—28no—
211C2×C32—24no—
212C2×C32—24no—
213C2×C32#94414yes—
214C2×C32Q8×Dic318yes—
215C2×C32#94618yes—
216C2×C32#94714yes—
217C2×C32C2×#21714yes—
218C2×C32C2×C3×C22⋊C411yescharacteristic
219C2×C32C3×C4×Q812yes—
220C2×C32C3×C22⋊Q812yes—
221C2×C32C3×#12411yescharacteristic
222C2×C32C3×C4⋊Q811yescharacteristic
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8C3×C3⋊Dic314yescomplemented
226C22×C4—92no—
227C22×C4—92no—
228C22×C4C2×C3×C3⋊D312yes—
229C22×C4C2×C3×C3⋊D311yescharacteristic
230C4⋊C4—182no—
231C42—182no—
232C22×C3—28no—
233C22×C3—28no—
234C22×C3—24no—
235C22×C3—24no—
236C22×C3—24no—
237C22×C3C3×D3×Q818yes—
238C22×C3C3×#23918yes—
239C22×C3C22×C3×Dic314yes—
240C22×C3C2×C3×#6714yes—
241C22×C3C2×C3×#6714yes—
242C22×C3Q8×C3⋊D312yes—
243C22×C3#383212yes—
244C22×C3C22×C3⋊Dic311yescharacteristic
245C22×C3C2×#63811yescharacteristic
246C22×C3C2×#63811yescharacteristic
247C3×C4—232no—
248C3×C4—216no—
249C3×C4—24no—
250C3×C4—216no—
251C3×C4—184no—
252C3×C4—362no—
253C3×C4C3×#217116yes—
254C3×C4#375714yes—
255Dic3—616no—
256Dic3—128no—
257C32—24no—
258C32#522914yescomplemented
259C32C3×#39111yescharacteristic, complemented
260C23C22×C3×C3⋊D311yescharacteristic
261Q8—28no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—94no—
265C2×C4—182no—
266C2×C4—182no—
267C2×C4—184no—
268C2×C4—182no—
269C2×C4C2×C3×C3⋊Dic314yes—
270C2×C4C2×C3×C3⋊Dic312yes—
271C2×C4C3×#63812yes—
272C2×C4C3×#63812yes—
273C2×C3—24no—
274C2×C3—28no—
275C2×C3—28no—
276C2×C3—24no—
277C2×C3—24no—
278C2×C3C3×#94414yes—
279C2×C3C3×Q8×Dic318yes—
280C2×C3C3×#94618yes—
281C2×C3C3×#94714yes—
282C2×C3C2×C3×#21714yes—
283C2×C3#6415311yescharacteristic
284C2×C3Q8×C3⋊Dic312yes—
285C2×C3#6415512yes—
286C2×C3#6415611yescharacteristic
287C2×C3C2×#375711yescharacteristic
288C22C3×Q8×C3⋊D312yes—
289C22C3×#383212yes—
290C22C22×C3×C3⋊Dic311yescharacteristic
291C22C2×C3×#63811yescharacteristic
292C22C2×C3×#63811yescharacteristic
293C4—24no—
294C4—184no—
295C4—362no—
296C4C3×#375714yes—
297C3—24no—
298C3C3×#522914yescomplemented
299C3#1059292811yescharacteristic, direct product factor
300C2C3×#6415311yescharacteristic
301C2C3×Q8×C3⋊Dic312yes—
302C2C3×#6415512yes—
303C2C3×#6415611yescharacteristic
304C2C2×C3×#375711yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC22×C33×Q8C2
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×Q8C3⋊D3
hypocenterC32C3×#391
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
5. #10592928C3direct (N × H)
9. C2×C33×Q8C4semidirect
37. C2×C3×C3⋊Dic3Q8semidirect
47. C2×C32×Q8Dic3semidirect
48. C2×C32×Q8C3×C4semidirect
109. C2×C3⋊Dic3C3×Q8semidirect
128. C2×C3×Q8C3×Dic3semidirect
129. C2×C3×Q8C3⋊Dic3semidirect
173. C33#391semidirect
225. C2×Q8C3×C3⋊Dic3semidirect
258. C32#5229semidirect
259. C32C3×#391semidirect
298. C3C3×#5229semidirect
299. C3#10592928direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C33×Q82
C2×C3×C4⋊(C3⋊Dic3)2
C2×C3×C4×C3⋊Dic32
C3×#636582
#105929283
C3×#52293