Finite Group Databasebeta

← C2×#63439⋊C3 #13152371

Subgroups — C2×#63439⋊C3 #13152371

Order 1728.

1056 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×3−1+2⋊C4C211yesmaximal, characteristic
2C2×C22⋊C4×3−1+2C211yesmaximal, characteristic
3C2×#3676⋊C3C211yesmaximal, characteristic
4C2×(3−1+2⋊C4)⋊C4C212yesmaximal, complemented
5C2×C4×3−1+2⋊C4C212yesmaximal, complemented
6#63439⋊C3C218yesmaximal, direct product factor
7C2×C3×#877—31nomaximal
8C2×#63439C311yesmaximal, characteristic, complemented
9C24×3−1+2C2211yescharacteristic
10C22×C4×3−1+2C2212yes—
11C22×3−1+2⋊C4C418yescomplemented
12C22×3−1+2⋊C4C2212yes—
13C22×3−1+2⋊C4C2211yescharacteristic
14C22×3−1+2⋊C4C2211yescharacteristic
15C22×3−1+2⋊C4C2214yes—
16C22⋊C4×3−1+2C2214yes—
17#3676⋊C3C2214yes—
18(3−1+2⋊C4)⋊C4C2218yescomplemented
19C4×3−1+2⋊C4C2218yescomplemented
20C23×C3×Dic3—31no—
21C2×C32×C22⋊C4D311yescharacteristic
22C2×C3×#217—31no—
23C2×C3×Dic3⋊C4—32no—
24C2×C3×C4×Dic3—32no—
25C3×#877—38no—
26C23×Dic9C2×C311yescharacteristic
27C2×C9×C22⋊C4—21no—
28C2×C9×C22⋊C4C2×C311yescharacteristic
29C2×#3676C2×C311yescharacteristic
30C2×Dic9⋊C4C2×C312yescomplemented
31C2×C4×Dic9C2×C312yescomplemented
32#63439C2×C318yescomplemented
33C23×3−1+2—24no—
34C23×3−1+2C2×C414yes—
35C23×3−1+2C2312yes—
36C23×3−1+2C2311yescharacteristic
37C2×C4×3−1+2—24no—
38C2×C4×3−1+2C2314yes—
39C2×3−1+2⋊C4—24no—
40C2×3−1+2⋊C4C2×C414yes—
41C2×3−1+2⋊C4C2×C414yes—
42C2×3−1+2⋊C4C2×C4116yescomplemented
43C2×3−1+2⋊C4C2314yes—
44C2×3−1+2⋊C4C2312yes—
45C2×3−1+2⋊C4C2312yes—
46C2×C3×#117—91no—
47C2×#877—31no—
48C24×C32C2×D311yescharacteristic
49C22×C32×C4C2×D312yes—
50C22×C3×Dic3—38no—
51C22×C3×Dic3—34no—
52C22×C3×Dic3—31no—
53C22×C3×Dic3—31no—
54C22×C3×Dic3—32no—
55C24×C9—21no—
56C24×C9C22×C311yescharacteristic
57C32×C22⋊C4C2×D314yes—
58C3×#217—34no—
59C3×Dic3⋊C4—38no—
60C3×C4×Dic3—38no—
61C22×C4×C9—22no—
62C22×C4×C9C22×C312yes—
63C22×Dic9C3×C418yescomplemented
64C22×Dic9C22×C312yes—
65C22×Dic9C22×C311yescharacteristic
66C22×Dic9C22×C311yescharacteristic
67C22×Dic9C22×C314yes—
68C9×C22⋊C4—24no—
69C9×C22⋊C4C22×C314yes—
70#3676C22×C314yes—
71Dic9⋊C4C22×C318yescomplemented
72C4×Dic9C22×C318yescomplemented
73C22×3−1+2—28no—
74C22×3−1+2—24no—
75C22×3−1+2C22×C414yes—
76C22×3−1+2C22×C412yes—
77C22×3−1+2#4114yes—
78C22×3−1+2C2411yescharacteristic
79C4×3−1+2—24no—
803−1+2⋊C4—24no—
813−1+2⋊C4C22×C414yes—
823−1+2⋊C4C22×C414yes—
83C23×C3×C4—91no—
84C23×Dic3—31no—
85C3×#117—98no—
86C2×C3×C4⋊C4—92no—
87C2×C3×C22⋊C4—31no—
88C2×C3×C22⋊C4—91no—
89C2×C3×C22⋊C4C3×D311yescharacteristic
90C2×C3×C42—92no—
91C2×#217—31no—
92C2×Dic3⋊C4—32no—
93C2×C4×Dic3—32no—
94#877—38no—
95C23×C32—24no—
96C23×C32C4×D314yes—
97C23×C32C22×D312yes—
98C23×C32C22×D311yescharacteristic
99C2×C32×C4—24no—
100C2×C32×C4C22×D314yes—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—34no—
103C2×C3×Dic3—316no—
104C2×C3×Dic3—32no—
105C2×C3×Dic3—32no—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—64no—
108C23×C9—24no—
109C23×C9—24no—
110C23×C9—21no—
111C23×C9—22no—
112C23×C9—44no—
113C23×C9C2×C3×C414yes—
114C23×C9C23×C312yes—
115C23×C9C23×C311yescharacteristic
116C2×C4×C9—24no—
117C2×C4×C9—24no—
118C2×C4×C9—44no—
119C2×C4×C9C23×C314yes—
120C2×Dic9—24no—
121C2×Dic9C2×C3×C414yes—
122C2×Dic9C2×C3×C414yes—
123C2×Dic9C2×C3×C4116yescomplemented
124C2×Dic9C23×C314yes—
125C2×Dic9C23×C312yes—
126C2×Dic9C23×C312yes—
127C2×#117—91noSylow
128C2×3−1+2—24no—
129C2×3−1+2#11714yescomplemented
130C2×3−1+2C23×C411yescharacteristic
131C2×3−1+2C2×#4112yes—
132C24×C3—31no—
133C24×C3C2×C3×D311yescharacteristic
134C22×C3×C4—32no—
135C22×C3×C4—91no—
136C22×C3×C4—91no—
137C22×C3×C4—94no—
138C22×C3×C4—98no—
139C22×C3×C4—92no—
140C22×C3×C4C2×C3×D312yes—
141C22×Dic3—38no—
142C22×Dic3—34no—
143C22×Dic3—31no—
144C22×Dic3—31no—
145C22×Dic3—32no—
146C3×C4⋊C4—98no—
147C3×C22⋊C4—34no—
148C3×C22⋊C4—94no—
149C3×C22⋊C4C2×C3×D314yes—
150C3×C42—98no—
151#217—34no—
152Dic3⋊C4—38no—
153C4×Dic3—38no—
154C22×C32—28no—
155C22×C32—24no—
156C22×C32C2×C4×D314yes—
157C22×C32C2×C4×D312yes—
158C22×C32#23714yes—
159C22×C32C23×D311yescharacteristic
160C32×C4—24no—
161C3×Dic3—34no—
162C3×Dic3—34no—
163C3×Dic3—64no—
164C22×C9—28no—
165C22×C9—24no—
166C22×C9—21no—
167C22×C9—22no—
168C22×C9—24no—
169C22×C9—24no—
170C22×C9—48no—
171C22×C9—44no—
172C22×C9C22×C3×C414yes—
173C22×C9C22×C3×C412yes—
174C22×C9C3×#4114yes—
175C22×C9C24×C311yescharacteristic
176C4×C9—24no—
177C4×C9—44no—
178Dic9—24no—
179Dic9C22×C3×C414yes—
180Dic9C22×C3×C414yes—
181C23×C4—91no—
182#117—98no—
183C2×C4⋊C4—92no—
184C2×C22⋊C4—91no—
185C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic
186C2×C42—92no—
1873−1+2C2×#11711yescharacteristic, complemented, Sylow
188C23×C3—24no—
189C23×C3—34no—
190C23×C3—32no—
191C23×C3—31no—
192C23×C3—64no—
193C23×C3C3×C4×D314yes—
194C23×C3C22×C3×D312yes—
195C23×C3C22×C3×D311yescharacteristic
196C2×C3×C4—24no—
197C2×C3×C4—34no—
198C2×C3×C4—64no—
199C2×C3×C4—92no—
200C2×C3×C4—92no—
201C2×C3×C4—94no—
202C2×C3×C4—94no—
203C2×C3×C4—916no—
204C2×C3×C4—94no—
205C2×C3×C4—184no—
206C2×C3×C4C22×C3×D314yes—
207C2×Dic3—32no—
208C2×Dic3—32no—
209C2×Dic3—34no—
210C2×Dic3—34no—
211C2×Dic3—316no—
212C2×Dic3—34no—
213C2×Dic3—64no—
214C2×C32—24no—
215C2×C32#87714yes—
216C2×C32C22×C4×D311yescharacteristic
217C2×C32C2×#23712yes—
218C2×C9—24no—
219C2×C9—24no—
220C2×C9—21no—
221C2×C9—22no—
222C2×C9—44no—
223C2×C9C3×#11714yescomplemented
224C2×C9C23×C3×C411yescharacteristic
225C2×C9C2×C3×#4112yes—
226C24C2×AGL(1,ℤ/9)11yescharacteristic
227C22×C4—98no—
228C22×C4—94no—
229C22×C4—91no—
230C22×C4—91no—
231C22×C4—92no—
232C22×C4C2×AGL(1,ℤ/9)12yes—
233C4⋊C4—98no—
234C22⋊C4—94no—
235C22⋊C4C2×AGL(1,ℤ/9)14yes—
236C42—98no—
237C22×C3—28no—
238C22×C3—24no—
239C22×C3—31no—
240C22×C3—32no—
241C22×C3—34no—
242C22×C3—34no—
243C22×C3—64no—
244C22×C3—68no—
245C22×C3C2×C3×C4×D314yes—
246C22×C3C2×C3×C4×D312yes—
247C22×C3C3×#23714yes—
248C22×C3C23×C3×D311yescharacteristic
249C3×C4—24no—
250C3×C4—64no—
251C3×C4—94no—
252C3×C4—94no—
253C3×C4—184no—
254Dic3—34no—
255Dic3—34no—
256Dic3—64no—
257C32C2×#87711yescharacteristic
258C9—21no—
259C9C2×C3×#11711yescharacteristic, complemented
260C23—24no—
261C23C4×AGL(1,ℤ/9)14yes—
262C23C22×AGL(1,ℤ/9)12yes—
263C23C22×AGL(1,ℤ/9)11yescharacteristic
264C2×C4—24no—
265C2×C4—92no—
266C2×C4—92no—
267C2×C4—916no—
268C2×C4—94no—
269C2×C4—94no—
270C2×C4—94no—
271C2×C4—184no—
272C2×C4C22×AGL(1,ℤ/9)14yes—
273C2×C3—24no—
274C2×C3—31no—
275C2×C3—32no—
276C2×C3—34no—
277C2×C3—64no—
278C2×C3C3×#87714yes—
279C2×C3C22×C3×C4×D311yescharacteristic
280C2×C3C2×C3×#23712yes—
281C22—24no—
282C22—28no—
283C22C2×C4×AGL(1,ℤ/9)14yes—
284C22C2×C4×AGL(1,ℤ/9)12yes—
285C22#3699⋊C314yes—
286C22C23×AGL(1,ℤ/9)11yescharacteristic
287C4—24no—
288C4—94no—
289C4—94no—
290C4—184no—
291C3—31no—
292C3C2×C3×#87711yescharacteristic
293C2—24no—
294C2#63439⋊C314yesdirect product factor
295C2C22×C4×AGL(1,ℤ/9)11yescharacteristic
296C2C2×#3699⋊C312yes—
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutatorC2×C9C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC2×C22⋊C4×3−1+2C2
socleC23×C3C22×C3×D3
prime radicalC24×C32C2×D3
cosocleC22×C9C24×C3
hypercenterC2×C22⋊C4AGL(1,ℤ/9)
hypocenterC9C2×C3×#117
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×(3−1+2⋊C4)⋊C4C2semidirect
5. C2×C4×3−1+2⋊C4C2semidirect
6. #63439⋊C3C2direct (N × H)
8. C2×#63439C3semidirect
11. C22×3−1+2⋊C4C4semidirect
18. (3−1+2⋊C4)⋊C4C22semidirect
19. C4×3−1+2⋊C4C22semidirect
30. C2×Dic9⋊C4C2×C3semidirect
31. C2×C4×Dic9C2×C3semidirect
32. #63439C2×C3semidirect
42. C2×3−1+2⋊C4C2×C4semidirect
63. C22×Dic9C3×C4semidirect
71. Dic9⋊C4C22×C3semidirect
72. C4×Dic9C22×C3semidirect
123. C2×Dic9C2×C3×C4semidirect
129. C2×3−1+2#117semidirect
187. 3−1+2C2×#117semidirect
223. C2×C9C3×#117semidirect
259. C9C2×C3×#117semidirect
294. C2#63439⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×3−1+2⋊C42
C2×C22⋊C4×3−1+22
C2×#3676⋊C32
C2×(3−1+2⋊C4)⋊C42
C2×C4×3−1+2⋊C42
#63439⋊C32
C2×C3×#8773
C2×#634393