Finite Group Databasebeta

← C2×C3×#63459 #13152406

Subgroups — C2×C3×#63459 #13152406

Order 1728.

1058 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D9C211yesmaximal, characteristic
2C22×C3×Dic18C211yesmaximal, characteristic, complemented
3C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×(C2×D9)⋊C4C212yesmaximal
5C2×C3×C4⋊Dic9C212yesmaximal, complemented
6C3×#63459C218yesmaximal, direct product factor
7C2×C3×#897—31nomaximal
8C2×#63459C311yesmaximal, characteristic, direct product factor
9C23×C3×D9C2211yescharacteristic
10C22×C3×C4×C9C2212yes—
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3×Dic9C2212yes—
13C22×C3×Dic9C2211yescharacteristic
14C2×C3×C4×D9—24no—
15C2×C3×C4×D9C2214yes—
16C2×C3×Dic18—24no—
17C2×C3×Dic18C2214yescomplemented
18C3×C9×C4⋊C4C2214yescomplemented
19C3×(C2×D9)⋊C4C2218yes—
20C3×C4⋊Dic9C2218yescomplemented
21C22×C3×C4×D3—31no—
22C22×C3×Dic6—31no—
23C2×C32×C4⋊C4D311yescharacteristic
24C2×C3×(C2×D3)⋊C4—32no—
25C2×C3×C4⋊Dic3—32no—
26C3×#897—38no—
27C22×C4×D9C2×C311yescharacteristic
28C22×Dic18C2×C311yescharacteristic, complemented
29C2×C9×C4⋊C4—21no—
30C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
31C2×(C2×D9)⋊C4C2×C312yes—
32C2×C4⋊Dic9C2×C312yescomplemented
33#63459C2×C318yesdirect product factor
34C23×C3×C9C2311yescharacteristic
35C22×C3×D9—22no—
36C22×C3×D9—22no—
37C22×C3×D9Q814yescomplemented
38C22×C3×D9C2312yes—
39C2×C3×C4×C9—24no—
40C2×C3×C4×C9D414yes—
41C2×C3×C4×C9C2314yes—
42C2×C3×C4×C9C2312yes—
43C2×C3×Dic9—24no—
44C2×C3×Dic9—22no—
45C2×C3×Dic9C2314yes—
46C2×C3×Dic9C2312yes—
47C3×C4×D9—28no—
48C3×Dic18—28no—
49C2×C3×C22⋊Q8—91no—
50C2×#897—31no—
51C23×C3×D3—31no—
52C22×C32×C4C2×D312yes—
53C22×C32×C4C2×D311yescharacteristic
54C22×C3×Dic3—31no—
55C22×C3×Dic3—32no—
56C2×C3×C4×D3—34no—
57C2×C3×C4×D3—64no—
58C2×C3×Dic6—34no—
59C2×C3×Dic6—64no—
60C23×D9C22×C311yescharacteristic
61C32×C4⋊C4C2×D314yes—
62C3×(C2×D3)⋊C4—38no—
63C3×C4⋊Dic3—38no—
64C22×C4×C9—21no—
65C22×C4×C9—22no—
66C22×C4×C9C22×C312yes—
67C22×C4×C9C22×C311yescharacteristic
68C22×Dic9C22×C312yes—
69C22×Dic9C22×C311yescharacteristic
70C2×C4×D9—24no—
71C2×C4×D9C22×C314yes—
72C2×Dic18—24no—
73C2×Dic18C22×C314yescomplemented
74C9×C4⋊C4—24no—
75C9×C4⋊C4C22×C314yescomplemented
76(C2×D9)⋊C4C22×C318yes—
77C4⋊Dic9C22×C318yescomplemented
78C22×C3×C9C2×D412yes—
79C22×C3×C9C2×Q812yes—
80C22×C3×C9#4112yes—
81C22×C3×C9C2411yescharacteristic
82C2×C3×D9—28no—
83C2×C3×D9—22no—
84C2×C3×D9—22no—
85C2×C3×D9C2×Q814yescomplemented
86C3×C4×C9—24no—
87C3×C4×C9C2×D414yes—
88C3×Dic9—24no—
89C3×Dic9—22no—
90C22×C3×Q8—91no—
91C23×C3×C4—91no—
92C22×C4×D3—31no—
93C22×Dic6—31no—
94C3×C22⋊Q8—98no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—92no—
97C2×C3×C4⋊C4D911yescharacteristic, complemented
98C2×C3×C4⋊C4C3×D311yescharacteristic
99C2×C3×C22⋊C4—92no—
100C2×(C2×D3)⋊C4—32no—
101C2×C4⋊Dic3—32no—
102#897—38no—
103C23×C32C22×D311yescharacteristic
104C22×C3×D3—34no—
105C22×C3×D3—32no—
106C22×C3×D3—62no—
107C22×C3×D3—62no—
108C2×C32×C4—24no—
109C2×C32×C4D1214yes—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D312yes—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—34no—
114C2×C3×Dic3—64no—
115C2×C3×Dic3—62no—
116C3×C4×D3—68no—
117C3×Dic6—68no—
118C23×C9—21no—
119C23×C9C23×C311yescharacteristic
120C22×D9—22no—
121C22×D9—22no—
122C22×D9C3×Q814yescomplemented
123C22×D9C23×C312yes—
124C2×C4×C9—24no—
125C2×C4×C9—24no—
126C2×C4×C9—22no—
127C2×C4×C9—24no—
128C2×C4×C9—44no—
129C2×C4×C9C3×D414yes—
130C2×C4×C9C23×C314yes—
131C2×C4×C9C23×C312yes—
132C2×Dic9—24no—
133C2×Dic9—22no—
134C2×Dic9C23×C312yes—
135C2×Dic9C23×C314yes—
136C4×D9—28no—
137Dic18—28no—
138C2×C22⋊Q8—91noSylow
139C2×C3×C9C22⋊Q814yescomplemented
140C2×C3×C9C22×D411yescharacteristic
141C2×C3×C9C22×Q811yescharacteristic
142C2×C3×C9C2×#4111yescharacteristic
143C3×D9—24no—
144C24×C3—91no—
145C23×D3—31no—
146C2×C3×Q8—94no—
147C2×C3×Q8—184no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—92no—
151C22×C3×C4—91no—
152C22×C3×C4—94no—
153C22×C3×C4—184no—
154C22×C3×C4C2×D912yes—
155C22×C3×C4C2×D911yescharacteristic
156C22×C3×C4C2×C3×D312yes—
157C22×C3×C4C2×C3×D311yescharacteristic
158C22×Dic3—31no—
159C22×Dic3—32no—
160C2×C4×D3—34no—
161C2×C4×D3—64no—
162C2×Dic6—34no—
163C2×Dic6—64no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—98no—
166C3×C4⋊C4C2×D914yescomplemented
167C3×C4⋊C4C2×C3×D314yes—
168C3×C22⋊C4—98no—
169(C2×D3)⋊C4—38no—
170C4⋊Dic3—38no—
171C22×C32C2×D1212yes—
172C22×C32#23712yes—
173C22×C32D3×Q812yes—
174C22×C32C23×D311yescharacteristic
175C2×C3×D3—34no—
176C2×C3×D3—68no—
177C2×C3×D3—62no—
178C2×C3×D3—62no—
179C32×C4—24no—
180C32×C4C2×D1214yes—
181C3×Dic3—64no—
182C3×Dic3—62no—
183C22×C9—21no—
184C22×C9—22no—
185C22×C9—22no—
186C22×C9—22no—
187C22×C9C2×C3×D412yes—
188C22×C9C2×C3×Q812yes—
189C22×C9C3×#4112yes—
190C22×C9C24×C311yescharacteristic
191C2×D9—28no—
192C2×D9—22no—
193C2×D9—22no—
194C2×D9C2×C3×Q814yescomplemented
195C4×C9—24no—
196C4×C9—24no—
197C4×C9—44no—
198C4×C9C2×C3×D414yes—
199Dic9—24no—
200Dic9—22no—
201C22×Q8—91no—
202C23×C4—91no—
203C22⋊Q8—98no—
204C2×C4⋊C4—92no—
205C2×C4⋊C4C3×D911yescharacteristic, complemented
206C2×C22⋊C4—92no—
207C3×C9C2×C22⋊Q811yescharacteristic, complemented, Sylow
208C23×C3—21no—
209C23×C3—94no—
210C23×C3—92no—
211C23×C3—182no—
212C23×C3—182no—
213C23×C3C22×D911yescharacteristic
214C23×C3C22×C3×D311yescharacteristic
215C22×D3—32no—
216C22×D3—34no—
217C22×D3—62no—
218C22×D3—62no—
219C3×Q8—188no—
220C2×C3×C4—24no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—22no—
224C2×C3×C4—24no—
225C2×C3×C4—44no—
226C2×C3×C4—94no—
227C2×C3×C4—92no—
228C2×C3×C4—182no—
229C2×C3×C4—184no—
230C2×C3×C4—188no—
231C2×C3×C4D3614yes—
232C2×C3×C4C22×D914yes—
233C2×C3×C4C22×D912yes—
234C2×C3×C4C3×D1214yes—
235C2×C3×C4C22×C3×D314yes—
236C2×C3×C4C22×C3×D312yes—
237C2×Dic3—32no—
238C2×Dic3—34no—
239C2×Dic3—64no—
240C2×Dic3—62no—
241C4×D3—68no—
242Dic6—68no—
243C2×C32#89714yes—
244C2×C32C22×D1211yescharacteristic
245C2×C32C2×#23711yescharacteristic
246C2×C32C2×D3×Q811yescharacteristic
247C3×D3—64no—
248C2×C9—24no—
249C2×C9—21no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9C3×C22⋊Q814yescomplemented
253C2×C9C22×C3×D411yescharacteristic
254C2×C9C22×C3×Q811yescharacteristic
255C2×C9C2×C3×#4111yescharacteristic
256D9—24no—
257C24—91no—
258C2×Q8—94no—
259C2×Q8—184no—
260C22×C4—92no—
261C22×C4—91no—
262C22×C4—94no—
263C22×C4—184no—
264C22×C4C2×C3×D912yes—
265C22×C4C2×C3×D911yescharacteristic
266C4⋊C4—98no—
267C4⋊C4C2×C3×D914yescomplemented
268C22⋊C4—98no—
269C22×C3—21no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—94no—
274C22×C3—182no—
275C22×C3—182no—
276C22×C3—188no—
277C22×C3C2×D3612yes—
278C22×C3#369912yes—
279C22×C3Q8×D912yes—
280C22×C3C23×D911yescharacteristic
281C22×C3C2×C3×D1212yes—
282C22×C3C3×#23712yes—
283C22×C3C3×D3×Q812yes—
284C22×C3C23×C3×D311yescharacteristic
285C2×D3—34no—
286C2×D3—68no—
287C2×D3—62no—
288C2×D3—62no—
289C3×C4—24no—
290C3×C4—24no—
291C3×C4—24no—
292C3×C4—44no—
293C3×C4—182no—
294C3×C4—184no—
295C3×C4C2×D3614yes—
296C3×C4C2×C3×D1214yes—
297Dic3—64no—
298Dic3—62no—
299C32C2×#89711yescharacteristic
300C9—21no—
301C9C2×C3×C22⋊Q811yescharacteristic, complemented
302C23—92no—
303C23—94no—
304C23—182no—
305C23—182no—
306C23C22×C3×D911yescharacteristic
307Q8—188no—
308C2×C4—24no—
309C2×C4—94no—
310C2×C4—92no—
311C2×C4—182no—
312C2×C4—184no—
313C2×C4—188no—
314C2×C4C3×D3614yes—
315C2×C4C22×C3×D914yes—
316C2×C4C22×C3×D912yes—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—24no—
321C2×C3—184no—
322C2×C3#6345914yesdirect product factor
323C2×C3C22×D3611yescharacteristic
324C2×C3C2×#369911yescharacteristic
325C2×C3C2×Q8×D911yescharacteristic
326C2×C3C3×#89714yes—
327C2×C3C22×C3×D1211yescharacteristic
328C2×C3C2×C3×#23711yescharacteristic
329C2×C3C2×C3×D3×Q811yescharacteristic
330D3—64no—
331C22—94no—
332C22—182no—
333C22—182no—
334C22—188no—
335C22C2×C3×D3612yes—
336C22C3×#369912yes—
337C22C3×Q8×D912yes—
338C22C23×C3×D911yescharacteristic
339C4—24no—
340C4—182no—
341C4—184no—
342C4C2×C3×D3614yes—
343C3—21no—
344C3C2×#6345911yescharacteristic, direct product factor
345C3C2×C3×#89711yescharacteristic
346C2—184no—
347C2C3×#6345914yesdirect product factor
348C2C22×C3×D3611yescharacteristic
349C2C2×C3×#369911yescharacteristic
350C2C2×C3×Q8×D911yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC23×C32C22×D3
prime radicalC23×C3×D9C22
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C2×C3×C22⋊Q8
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×Dic18C2semidirect
3. C2×C3×C9×C4⋊C4C2semidirect
5. C2×C3×C4⋊Dic9C2semidirect
6. C3×#63459C2direct (N × H)
8. C2×#63459C3direct (N × H)
17. C2×C3×Dic18C22semidirect
18. C3×C9×C4⋊C4C22semidirect
20. C3×C4⋊Dic9C22semidirect
28. C22×Dic18C2×C3semidirect
30. C2×C9×C4⋊C4C2×C3semidirect
32. C2×C4⋊Dic9C2×C3semidirect
33. #63459C2×C3direct (N × H)
37. C22×C3×D9Q8semidirect
73. C2×Dic18C22×C3semidirect
75. C9×C4⋊C4C22×C3semidirect
77. C4⋊Dic9C22×C3semidirect
85. C2×C3×D9C2×Q8semidirect
97. C2×C3×C4⋊C4D9semidirect
122. C22×D9C3×Q8semidirect
139. C2×C3×C9C22⋊Q8semidirect
166. C3×C4⋊C4C2×D9semidirect
194. C2×D9C2×C3×Q8semidirect
205. C2×C4⋊C4C3×D9semidirect
207. C3×C9C2×C22⋊Q8semidirect
252. C2×C9C3×C22⋊Q8semidirect
267. C4⋊C4C2×C3×D9semidirect
301. C9C2×C3×C22⋊Q8semidirect
322. C2×C3#63459direct (N × H)
344. C3C2×#63459direct (N × H)
347. C2C3×#63459direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D92
C22×C3×Dic182
C2×C3×C9×C4⋊C42
C2×C3×(C2×D9)⋊C42
C2×C3×C4⋊Dic92
C3×#634592
C2×C3×#8973
C2×#634593