Finite Group Databasebeta

← C3×#10592355 #13162693

Subgroups — C3×#10592355 #13162693

Order 1728.

1396 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C212yesmaximal
3C2×C3×#3750C211yesmaximal, characteristic
4C33×#95C211yesmaximal, characteristic
5C3×#63658C212yesmaximal
6#10592355C311yesmaximal, characteristic, direct product factor
7C3×#4656—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C33×C4C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C2×C3×#634—22no—
14C2×C3×#634—24no—
15C3×C4⋊(C3⋊Dic3)—24no—
16C3×#3750—22no—
17C22×#634C2×C311yescharacteristic
18C22×C3×Dic6—34no—
19C2×C4⋊(C3⋊Dic3)C2×C312yes—
20C2×#3750C2×C311yescharacteristic
21C2×C3×C4⋊Dic3—38no—
22C2×C3×Dic3⋊C4—34no—
23C32×#95—24no—
24C32×#95D314yes—
25C32×#95C2×C311yescharacteristic
26#63658C2×C312yes—
27C3×C3⋊#95—38no—
28C23×C33C2311yescharacteristic
29C2×C33×C4—22no—
30C2×C33×C4—22no—
31C2×C33×C4—22no—
32C2×C33×C4D412yes—
33C2×C33×C4Q814yescomplemented
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3—21no—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3D414yes—
41C2×C3×C3⋊Dic3Q812yescomplemented
42C3×#634—28no—
43C3×#395—91no—
44#4656—34no—
45C22×C32×C4—28no—
46C22×C32×C4—24no—
47C22×C32×C4C2×D318yes—
48C22×C32×C4C2×D314yes—
49C22×C32×C4C22×C311yescharacteristic
50C22×C32×C4C22×C312yes—
51C22×C3⋊Dic3C22×C311yescharacteristic
52C22×C3⋊Dic3C22×C312yes—
53C22×C3⋊Dic3C22×C311yescharacteristic
54C2×#634—22no—
55C2×#634—24no—
56C22×C3×Dic3—34no—
57C22×C3×Dic3—38no—
58C22×C3×Dic3—34no—
59C2×C3×Dic6—616no—
60C2×C3×Dic6—68no—
61C4⋊(C3⋊Dic3)—24no—
62#3750—22no—
63C3×C4⋊Dic3—616no—
64C3×Dic3⋊C4—68no—
65C22×C33C2×D411yescharacteristic
66C22×C33C2×D412yes—
67C22×C33C2×Q812yes—
68C22×C33C2×Q811yescharacteristic
69C22×C33#4111yescharacteristic
70C33×C4—22no—
71C33×C4—24no—
72C3×C3⋊Dic3—24no—
73C3×C3⋊Dic3—22no—
74C3×C3⋊Dic3—41no—
75C22×C3×Q8—91no—
76C22×Dic6—34no—
77C2×C3×C4⋊C4—91no—
78C2×C3×C4⋊C4—92no—
79C2×C4⋊Dic3—38no—
80C2×Dic3⋊C4—34no—
81C3×#95—24no—
82C3×#95—92no—
83C3×#95C3×D314yes—
84C3×#95C3⋊D311yescharacteristic
85C3⋊#95—38no—
86C23×C32—24no—
87C23×C32C22×D314yes—
88C23×C32C23×C311yescharacteristic
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4—216no—
93C2×C32×C4—28no—
94C2×C32×C4—28no—
95C2×C32×C4—28no—
96C2×C32×C4—28no—
97C2×C32×C4—48no—
98C2×C32×C4—48no—
99C2×C32×C4—48no—
100C2×C32×C4Dic6116yescomplemented
101C2×C32×C4D1218yes—
102C2×C32×C4C3×D412yes—
103C2×C32×C4C3×Q814yescomplemented
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3C3×D414yes—
111C2×C3⋊Dic3C3×Q812yescomplemented
112#634—28no—
113C2×C3×Dic3—316no—
114C2×C3×Dic3—38no—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—68no—
117C2×C3×Dic3—68no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3—64no—
120C2×C3×Dic3—68no—
121C3×Dic6—632no—
122#395—91noSylow
123C2×C33#12011yescharacteristic
124C2×C33C22⋊Q812yes—
125C2×C33C22⋊Q811yescharacteristic
126C2×C33C4⋊Q811yescharacteristic
127C2×C33C4⋊Q812yes—
128C2×C3×Q8—184no—
129C2×C3×Q8—182no—
130C22×C3×C4—28no—
131C22×C3×C4—24no—
132C22×C3×C4—91no—
133C22×C3×C4—92no—
134C22×C3×C4—91no—
135C22×C3×C4C2×C3×D318yes—
136C22×C3×C4C2×C3×D314yes—
137C22×C3×C4C2×C3⋊D312yes—
138C22×C3×C4C2×C3⋊D311yescharacteristic
139C22×Dic3—34no—
140C22×Dic3—38no—
141C22×Dic3—34no—
142C2×Dic6—616no—
143C2×Dic6—68no—
144C3×C4⋊C4—184no—
145C3×C4⋊C4—182no—
146C4⋊Dic3—616no—
147Dic3⋊C4—68no—
148C22×C32—28no—
149C22×C32—24no—
150C22×C32—28no—
151C22×C32—24no—
152C22×C32—24no—
153C22×C32C2×Dic618yes—
154C22×C32C2×D1214yes—
155C22×C32D3×D418yes—
156C22×C32#23714yes—
157C22×C32D3×Q814yes—
158C22×C32C2×C3×D411yescharacteristic
159C22×C32C2×C3×D412yes—
160C22×C32C2×C3×Q812yes—
161C22×C32C2×C3×Q811yescharacteristic
162C22×C32C3×#4111yescharacteristic
163C32×C4—22no—
164C32×C4—24no—
165C32×C4—216no—
166C32×C4—28no—
167C32×C4—416no—
168C32×C4—48no—
169C3⋊Dic3—24no—
170C3⋊Dic3—22no—
171C3⋊Dic3—41no—
172C3×Dic3—616no—
173C3×Dic3—68no—
174C3×Dic3—124no—
175C22×Q8—91no—
176C2×C4⋊C4—91no—
177C2×C4⋊C4—92no—
178#95—92no—
179#95C3×C3⋊D311yescharacteristic
180C33#39511yescharacteristic, complemented, Sylow
181C23×C3—24no—
182C23×C3C22×C3×D314yes—
183C23×C3C22×C3⋊D311yescharacteristic
184C3×Q8—188no—
185C2×C3×C4—28no—
186C2×C3×C4—28no—
187C2×C3×C4—28no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—216no—
192C2×C3×C4—28no—
193C2×C3×C4—48no—
194C2×C3×C4—48no—
195C2×C3×C4—48no—
196C2×C3×C4—94no—
197C2×C3×C4—92no—
198C2×C3×C4—181no—
199C2×C3×C4—182no—
200C2×C3×C4—182no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4—182no—
204C2×C3×C4C3×Dic6116yescomplemented
205C2×C3×C4C3×D1218yes—
206C2×C3×C4#63414yescomplemented
207C2×C3×C4C4⋊(C3⋊D3)12yes—
208C2×Dic3—316no—
209C2×Dic3—38no—
210C2×Dic3—64no—
211C2×Dic3—68no—
212C2×Dic3—68no—
213C2×Dic3—64no—
214C2×Dic3—64no—
215C2×Dic3—68no—
216Dic6—632no—
217C2×C32—28no—
218C2×C32—24no—
219C2×C32—28no—
220C2×C32—24no—
221C2×C32—24no—
222C2×C32#86914yes—
223C2×C32#87818yes—
224C2×C32#88214yes—
225C2×C32#88818yes—
226C2×C32#89714yes—
227C2×C32C3×#12011yescharacteristic
228C2×C32C3×C22⋊Q812yes—
229C2×C32C3×C22⋊Q811yescharacteristic
230C2×C32C3×C4⋊Q812yes—
231C2×C32C3×C4⋊Q811yescharacteristic
232C2×Q8—184no—
233C2×Q8—182no—
234C22×C4—91no—
235C22×C4—92no—
236C22×C4—91no—
237C22×C4C2×C3×C3⋊D312yes—
238C22×C4C2×C3×C3⋊D311yescharacteristic
239C4⋊C4—184no—
240C4⋊C4—182no—
241C22×C3—28no—
242C22×C3—24no—
243C22×C3—28no—
244C22×C3—24no—
245C22×C3—24no—
246C22×C3C2×C3×Dic618yes—
247C22×C3C2×C3×D1214yes—
248C22×C3C3×D3×D418yes—
249C22×C3C3×#23714yes—
250C22×C3C3×D3×Q814yes—
251C22×C3C2×#63412yes—
252C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
253C22×C3D4×C3⋊D312yes—
254C22×C3#383011yescharacteristic
255C22×C3Q8×C3⋊D311yescharacteristic
256C3×C4—216no—
257C3×C4—28no—
258C3×C4—24no—
259C3×C4—22no—
260C3×C4—416no—
261C3×C4—48no—
262C3×C4—184no—
263C3×C4—182no—
264C3×C4—361no—
265Dic3—616no—
266Dic3—68no—
267Dic3—124no—
268C32—24no—
269C32#465614yescomplemented
270C32C3×#39511yescharacteristic, complemented
271C23C22×C3×C3⋊D311yescharacteristic
272Q8—188no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—94no—
277C2×C4—92no—
278C2×C4—181no—
279C2×C4—182no—
280C2×C4—182no—
281C2×C4—181no—
282C2×C4—181no—
283C2×C4—182no—
284C2×C4C3×#63414yescomplemented
285C2×C4C3×C4⋊(C3⋊D3)12yes—
286C2×C3—24no—
287C2×C3—28no—
288C2×C3—24no—
289C2×C3—28no—
290C2×C3—24no—
291C2×C3C3×#86914yes—
292C2×C3C3×#87818yes—
293C2×C3C3×#88214yes—
294C2×C3C3×#88818yes—
295C2×C3C3×#89714yes—
296C2×C3#6407811yescharacteristic
297C2×C3#6408712yes—
298C2×C3#6409111yescharacteristic
299C2×C3#6409712yes—
300C2×C3#6410611yescharacteristic
301C22C2×C3×#63412yes—
302C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
303C22C3×D4×C3⋊D312yes—
304C22C3×#383011yescharacteristic
305C22C3×Q8×C3⋊D311yescharacteristic
306C4—24no—
307C4—22no—
308C4—184no—
309C4—182no—
310C4—361no—
311C3—24no—
312C3C3×#465614yescomplemented
313C3#1059235511yescharacteristic, direct product factor
314C2C3×#6407811yescharacteristic
315C2C3×#6408712yes—
316C2C3×#6409111yescharacteristic
317C2C3×#6409712yes—
318C2C3×#6410611yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutator, cosocleC23×C32C23×C3
FrattiniC23C22×C3×C3⋊D3
FittingC33×#95C2
socle, prime radicalC23×C33C23
hypercenterC3×#95C3⋊D3
hypocenterC32C3×#395
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
6. #10592355C3direct (N × H)
33. C2×C33×C4Q8semidirect
41. C2×C3×C3⋊Dic3Q8semidirect
100. C2×C32×C4Dic6semidirect
103. C2×C32×C4C3×Q8semidirect
111. C2×C3⋊Dic3C3×Q8semidirect
180. C33#395semidirect
204. C2×C3×C4C3×Dic6semidirect
206. C2×C3×C4#634semidirect
269. C32#4656semidirect
270. C32C3×#395semidirect
284. C2×C4C3×#634semidirect
312. C3C3×#4656semidirect
313. C3#10592355direct (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×C3×C4⋊(C3⋊Dic3)2
C2×C3×#37502
C33×#952
C3×#636582
#105923553
C3×#46563