Finite Group Databasebeta

← C2×#63450⋊C3 #13152449

Subgroups — C2×#63450⋊C3 #13152449

Order 1728.

916 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×3−1+2⋊Q8C212yesmaximal
2C2×C4⋊C4×3−1+2C211yesmaximal, characteristic
3C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic
4C2×(3−1+2⋊C4)⋊C4C212yesmaximal
5C2×C4×3−1+2⋊C4C211yesmaximal, characteristic
6#63450⋊C3C218yesmaximal, direct product factor
7C2×C3×#888—31nomaximal
8C2×#63450C311yesmaximal, characteristic, complemented
9C22×C4×3−1+2C2212yes—
10C22×C4×3−1+2C2211yescharacteristic
11C22×3−1+2⋊C4C2212yes—
12C22×3−1+2⋊C4C2211yescharacteristic
13C22×3−1+2⋊C4C2211yescharacteristic
14C2×3−1+2⋊Q8—28no—
15C2×3−1+2⋊Q8C2218yes—
16C4⋊C4×3−1+2C2214yes—
17C4⋊(3−1+2⋊C4)C2214yes—
18(3−1+2⋊C4)⋊C4C2218yes—
19C4×3−1+2⋊C4C2214yes—
20C22×C3×Dic6—32no—
21C2×C32×C4⋊C4D311yescharacteristic
22C2×C3×C4⋊Dic3—31no—
23C2×C3×Dic3⋊C4—32no—
24C2×C3×C4×Dic3—31no—
25C3×#888—38no—
26C22×Dic18C2×C312yes—
27C2×C9×C4⋊C4—21no—
28C2×C9×C4⋊C4C2×C311yescharacteristic
29C2×C4⋊Dic9C2×C311yescharacteristic
30C2×Dic9⋊C4C2×C312yes—
31C2×C4×Dic9C2×C311yescharacteristic
32#63450C2×C318yescomplemented
33C23×3−1+2C2311yescharacteristic
34C2×C4×3−1+2—24no—
35C2×C4×3−1+2Q814yescomplemented
36C2×C4×3−1+2C2314yes—
37C2×C4×3−1+2C2312yes—
38C2×3−1+2⋊C4—24no—
39C2×3−1+2⋊C4D414yes—
40C2×3−1+2⋊C4Q814yescomplemented
41C2×3−1+2⋊C4C2314yes—
42C2×3−1+2⋊C4C2312yes—
43C2×3−1+2⋊C4C2312yes—
443−1+2⋊Q8—216no—
45C2×C3×C4⋊Q8—91no—
46C2×#888—31no—
47C22×C32×C4C2×D312yes—
48C22×C32×C4C2×D311yescharacteristic
49C22×C3×Dic3—31no—
50C22×C3×Dic3—31no—
51C22×C3×Dic3—32no—
52C2×C3×Dic6—38no—
53C2×C3×Dic6—68no—
54C32×C4⋊C4C2×D314yes—
55C3×C4⋊Dic3—34no—
56C3×Dic3⋊C4—38no—
57C3×C4×Dic3—34no—
58C22×C4×C9—21no—
59C22×C4×C9—22no—
60C22×C4×C9C22×C312yes—
61C22×C4×C9C22×C311yescharacteristic
62C22×Dic9C22×C312yes—
63C22×Dic9C22×C311yescharacteristic
64C22×Dic9C22×C311yescharacteristic
65C2×Dic18—28no—
66C2×Dic18C22×C318yes—
67C9×C4⋊C4—24no—
68C9×C4⋊C4C22×C314yes—
69C4⋊Dic9C22×C314yes—
70Dic9⋊C4C22×C318yes—
71C4×Dic9C22×C314yes—
72C22×3−1+2C2×D412yes—
73C22×3−1+2C2×Q812yes—
74C22×3−1+2C2×Q812yes—
75C22×3−1+2C2411yescharacteristic
76C4×3−1+2—24no—
77C4×3−1+2C2×Q814yescomplemented
783−1+2⋊C4—24no—
793−1+2⋊C4C2×D414yes—
803−1+2⋊C4C2×Q814yescomplemented
81C22×C3×Q8—92no—
82C22×Dic6—32no—
83C3×C4⋊Q8—98no—
84C2×C3×C4⋊C4—31no—
85C2×C3×C4⋊C4—92no—
86C2×C3×C4⋊C4—91no—
87C2×C3×C4⋊C4C3×D311yescharacteristic
88C2×C3×C42—91no—
89C2×C4⋊Dic3—31no—
90C2×Dic3⋊C4—32no—
91C2×C4×Dic3—31no—
92#888—38no—
93C23×C32C22×D311yescharacteristic
94C2×C32×C4—24no—
95C2×C32×C4Dic614yes—
96C2×C32×C4C22×D314yes—
97C2×C32×C4C22×D312yes—
98C2×C3×Dic3—34no—
99C2×C3×Dic3—34no—
100C2×C3×Dic3—32no—
101C2×C3×Dic3—32no—
102C2×C3×Dic3—34no—
103C2×C3×Dic3—64no—
104C3×Dic6—616no—
105C23×C9—21no—
106C23×C9C23×C311yescharacteristic
107C2×C4×C9—24no—
108C2×C4×C9—24no—
109C2×C4×C9—22no—
110C2×C4×C9—24no—
111C2×C4×C9—44no—
112C2×C4×C9C3×Q814yescomplemented
113C2×C4×C9C23×C314yes—
114C2×C4×C9C23×C312yes—
115C2×Dic9—24no—
116C2×Dic9C3×D414yes—
117C2×Dic9C3×Q814yescomplemented
118C2×Dic9C23×C314yes—
119C2×Dic9C23×C312yes—
120C2×Dic9C23×C312yes—
121Dic18—216no—
122C2×C4⋊Q8—91noSylow
123C2×3−1+2C4⋊Q814yescomplemented
124C2×3−1+2C22×D411yescharacteristic
125C2×3−1+2C22×Q811yescharacteristic
126C2×3−1+2C22×Q811yescharacteristic
127C2×C3×Q8—98no—
128C2×C3×Q8—188no—
129C22×C3×C4—31no—
130C22×C3×C4—32no—
131C22×C3×C4—91no—
132C22×C3×C4—91no—
133C22×C3×C4—92no—
134C22×C3×C4C2×C3×D312yes—
135C22×C3×C4C2×C3×D311yescharacteristic
136C22×Dic3—31no—
137C22×Dic3—31no—
138C22×Dic3—32no—
139C2×Dic6—38no—
140C2×Dic6—68no—
141C3×C4⋊C4—34no—
142C3×C4⋊C4—98no—
143C3×C4⋊C4—94no—
144C3×C4⋊C4C2×C3×D314yes—
145C3×C42—94no—
146C4⋊Dic3—34no—
147Dic3⋊C4—38no—
148C4×Dic3—34no—
149C22×C32C2×Dic612yes—
150C22×C32D3×D412yes—
151C22×C32D3×Q812yes—
152C22×C32C23×D311yescharacteristic
153C32×C4—24no—
154C32×C4C2×Dic614yes—
155C3×Dic3—34no—
156C3×Dic3—34no—
157C3×Dic3—64no—
158C22×C9—21no—
159C22×C9—22no—
160C22×C9—22no—
161C22×C9—22no—
162C22×C9C2×C3×D412yes—
163C22×C9C2×C3×Q812yes—
164C22×C9C2×C3×Q812yes—
165C22×C9C24×C311yescharacteristic
166C4×C9—24no—
167C4×C9—24no—
168C4×C9—44no—
169C4×C9C2×C3×Q814yescomplemented
170Dic9—24no—
171Dic9C2×C3×D414yes—
172Dic9C2×C3×Q814yescomplemented
173C22×Q8—92no—
174C4⋊Q8—98no—
175C2×C4⋊C4—92no—
176C2×C4⋊C4—91no—
177C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic
178C2×C42—91no—
1793−1+2C2×C4⋊Q811yescharacteristic, complemented, Sylow
180C23×C3—31no—
181C23×C3C22×C3×D311yescharacteristic
182C3×Q8—1816no—
183C2×C3×C4—24no—
184C2×C3×C4—34no—
185C2×C3×C4—32no—
186C2×C3×C4—34no—
187C2×C3×C4—64no—
188C2×C3×C4—92no—
189C2×C3×C4—92no—
190C2×C3×C4—94no—
191C2×C3×C4—94no—
192C2×C3×C4—94no—
193C2×C3×C4—184no—
194C2×C3×C4C3×Dic614yes—
195C2×C3×C4C22×C3×D314yes—
196C2×C3×C4C22×C3×D312yes—
197C2×Dic3—32no—
198C2×Dic3—32no—
199C2×Dic3—34no—
200C2×Dic3—34no—
201C2×Dic3—34no—
202C2×Dic3—64no—
203Dic6—616no—
204C2×C32#88814yes—
205C2×C32C22×Dic611yescharacteristic
206C2×C32C2×D3×D411yescharacteristic
207C2×C32C2×D3×Q811yescharacteristic
208C2×C9—24no—
209C2×C9—21no—
210C2×C9—21no—
211C2×C9—21no—
212C2×C9C3×C4⋊Q814yescomplemented
213C2×C9C22×C3×D411yescharacteristic
214C2×C9C22×C3×Q811yescharacteristic
215C2×C9C22×C3×Q811yescharacteristic
216C2×Q8—98no—
217C2×Q8—188no—
218C22×C4—91no—
219C22×C4—91no—
220C22×C4—92no—
221C22×C4C2×AGL(1,ℤ/9)12yes—
222C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
223C4⋊C4—98no—
224C4⋊C4—94no—
225C4⋊C4C2×AGL(1,ℤ/9)14yes—
226C42—94no—
227C22×C3—31no—
228C22×C3—32no—
229C22×C3—32no—
230C22×C3—32no—
231C22×C3C2×C3×Dic612yes—
232C22×C3C3×D3×D412yes—
233C22×C3C3×D3×Q812yes—
234C22×C3C23×C3×D311yescharacteristic
235C3×C4—24no—
236C3×C4—34no—
237C3×C4—64no—
238C3×C4—94no—
239C3×C4—94no—
240C3×C4—184no—
241C3×C4C2×C3×Dic614yes—
242Dic3—34no—
243Dic3—34no—
244Dic3—64no—
245C32C2×#88811yescharacteristic
246C9—21no—
247C9C2×C3×C4⋊Q811yescharacteristic, complemented
248C23C22×AGL(1,ℤ/9)11yescharacteristic
249Q8—1816no—
250C2×C4—24no—
251C2×C4—92no—
252C2×C4—92no—
253C2×C4—94no—
254C2×C4—94no—
255C2×C4—94no—
256C2×C4—184no—
257C2×C43−1+2⋊Q814yescomplemented
258C2×C4C22×AGL(1,ℤ/9)14yes—
259C2×C4C22×AGL(1,ℤ/9)12yes—
260C2×C3—34no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—31no—
264C2×C3C3×#88814yes—
265C2×C3C22×C3×Dic611yescharacteristic
266C2×C3C2×C3×D3×D411yescharacteristic
267C2×C3C2×C3×D3×Q811yescharacteristic
268C22C2×3−1+2⋊Q812yes—
269C22D4×AGL(1,ℤ/9)12yes—
270C22Q8×AGL(1,ℤ/9)12yes—
271C22C23×AGL(1,ℤ/9)11yescharacteristic
272C4—24no—
273C4—94no—
274C4—94no—
275C4—184no—
276C4C2×3−1+2⋊Q814yescomplemented
277C3—31no—
278C3C2×C3×#88811yescharacteristic
279C2#63450⋊C314yesdirect product factor
280C2C22×3−1+2⋊Q811yescharacteristic
281C2C2×D4×AGL(1,ℤ/9)11yescharacteristic
282C2C2×Q8×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C4⋊C4×3−1+2C2
socleC23×C3C22×C3×D3
prime radicalC23×C32C22×D3
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C2×C3×C4⋊Q8
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #63450⋊C3C2direct (N × H)
8. C2×#63450C3semidirect
32. #63450C2×C3semidirect
35. C2×C4×3−1+2Q8semidirect
40. C2×3−1+2⋊C4Q8semidirect
77. C4×3−1+2C2×Q8semidirect
80. 3−1+2⋊C4C2×Q8semidirect
112. C2×C4×C9C3×Q8semidirect
117. C2×Dic9C3×Q8semidirect
123. C2×3−1+2C4⋊Q8semidirect
169. C4×C9C2×C3×Q8semidirect
172. Dic9C2×C3×Q8semidirect
179. 3−1+2C2×C4⋊Q8semidirect
212. C2×C9C3×C4⋊Q8semidirect
247. C9C2×C3×C4⋊Q8semidirect
257. C2×C43−1+2⋊Q8semidirect
276. C4C2×3−1+2⋊Q8semidirect
279. C2#63450⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×3−1+2⋊Q82
C2×C4⋊C4×3−1+22
C2×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
C2×C4×3−1+2⋊C42
#63450⋊C32
C2×C3×#8883
C2×#634503