Finite Group Databasebeta

← C2×#63460⋊C3 #13152463

Subgroups — C2×#63460⋊C3 #13152463

Order 1728.

784 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×3−1+2C211yesmaximal, characteristic, complemented
2C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
3C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
4C2×(C2×AGL(1,ℤ/9))⋊C4C211yesmaximal, characteristic
5C2×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
6C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
7C2×C4×3−1+2⋊C4C211yesmaximal, characteristic, complemented
8#63460⋊C3C218yesmaximal, direct product factor
9C2×C3×#898—31nomaximal
10C2×#63460C311yesmaximal, characteristic, complemented
11C23×AGL(1,ℤ/9)C2211yescharacteristic
12C22×C4×3−1+2C2211yescharacteristic
13C22×C4×3−1+2C2211yescharacteristic
14C22×C4×3−1+2C2211yescharacteristic
15C22×3−1+2⋊C4C2211yescharacteristic
16C22×3−1+2⋊C4C2211yescharacteristic
17C22×3−1+2⋊C4C2211yescharacteristic
18C4⋊C4×3−1+2C2214yescomplemented
19(C2×AGL(1,ℤ/9))⋊C4C2214yes—
20(C2×AGL(1,ℤ/9))⋊C4C2214yes—
21(C2×AGL(1,ℤ/9))⋊C4C2214yes—
22C4⋊(3−1+2⋊C4)C2214yescomplemented
23(3−1+2⋊C4)⋊C4C2214yescomplemented
24C4×3−1+2⋊C4C2214yescomplemented
25C2×C32×C4⋊C4D311yescharacteristic
26C2×C3×(C2×D3)⋊C4—31no—
27C2×C3×(C2×D3)⋊C4—31no—
28C2×C3×(C2×D3)⋊C4—31no—
29C2×C3×C4⋊Dic3—31no—
30C2×C3×Dic3⋊C4—31no—
31C2×C3×C4×Dic3—31no—
32C3×#898—38no—
33C2×C9×C4⋊C4—21no—
34C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
35C2×(C2×D9)⋊C4C2×C311yescharacteristic
36C2×(C2×D9)⋊C4C2×C311yescharacteristic
37C2×(C2×D9)⋊C4C2×C311yescharacteristic
38C2×C4⋊Dic9C2×C311yescharacteristic, complemented
39C2×Dic9⋊C4C2×C311yescharacteristic, complemented
40C2×C4×Dic9C2×C311yescharacteristic, complemented
41#63460C2×C318yescomplemented
42C23×3−1+2C2311yescharacteristic
43C22×AGL(1,ℤ/9)—22no—
44C22×AGL(1,ℤ/9)—22no—
45C22×AGL(1,ℤ/9)—22no—
46C22×AGL(1,ℤ/9)C2312yes—
47C2×C4×3−1+2—22no—
48C2×C4×3−1+2—22no—
49C2×C4×3−1+2—22no—
50C2×C4×3−1+2C2312yes—
51C2×C4×3−1+2C2312yes—
52C2×C4×3−1+2C2312yes—
53C2×3−1+2⋊C4—22no—
54C2×3−1+2⋊C4—22no—
55C2×3−1+2⋊C4—22no—
56C2×3−1+2⋊C4C2312yes—
57C2×3−1+2⋊C4C2312yes—
58C2×3−1+2⋊C4C2312yes—
59C2×C3×#126—91no—
60C2×#898—31no—
61C23×C3×D3—31no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C2×D311yescharacteristic
65C22×C3×Dic3—31no—
66C22×C3×Dic3—31no—
67C22×C3×Dic3—31no—
68C23×D9C22×C311yescharacteristic
69C32×C4⋊C4C2×D314yes—
70C3×(C2×D3)⋊C4—34no—
71C3×(C2×D3)⋊C4—34no—
72C3×(C2×D3)⋊C4—34no—
73C3×C4⋊Dic3—34no—
74C3×Dic3⋊C4—34no—
75C3×C4×Dic3—34no—
76C22×C4×C9—21no—
77C22×C4×C9—21no—
78C22×C4×C9—21no—
79C22×C4×C9C22×C311yescharacteristic
80C22×C4×C9C22×C311yescharacteristic
81C22×C4×C9C22×C311yescharacteristic
82C22×Dic9C22×C311yescharacteristic
83C22×Dic9C22×C311yescharacteristic
84C22×Dic9C22×C311yescharacteristic
85C9×C4⋊C4—24no—
86C9×C4⋊C4C22×C314yescomplemented
87(C2×D9)⋊C4C22×C314yes—
88(C2×D9)⋊C4C22×C314yes—
89(C2×D9)⋊C4C22×C314yes—
90C4⋊Dic9C22×C314yescomplemented
91Dic9⋊C4C22×C314yescomplemented
92C4×Dic9C22×C314yescomplemented
93C22×3−1+2#4112yes—
94C22×3−1+2#4112yes—
95C22×3−1+2#4112yes—
96C22×3−1+2C2411yescharacteristic
97C2×AGL(1,ℤ/9)—22no—
98C2×AGL(1,ℤ/9)—22no—
99C2×AGL(1,ℤ/9)—22no—
100C2×AGL(1,ℤ/9)—44no—
101C4×3−1+2—22no—
102C4×3−1+2—22no—
103C4×3−1+2—22no—
1043−1+2⋊C4—22no—
1053−1+2⋊C4—22no—
1063−1+2⋊C4—22no—
107C3×#126—98no—
108C2×C3×C4⋊C4—31no—
109C2×C3×C4⋊C4—91no—
110C2×C3×C4⋊C4—91no—
111C2×C3×C4⋊C4C3×D311yescharacteristic
112C2×C3×C22⋊C4—91no—
113C2×C3×C22⋊C4—91no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C42—91no—
116C2×(C2×D3)⋊C4—31no—
117C2×(C2×D3)⋊C4—31no—
118C2×(C2×D3)⋊C4—31no—
119C2×C4⋊Dic3—31no—
120C2×Dic3⋊C4—31no—
121C2×C4×Dic3—31no—
122#898—38no—
123C23×C32C22×D311yescharacteristic
124C22×C3×D3—32no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C22×C3×D3—62no—
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4C22×D312yes—
132C2×C32×C4C22×D312yes—
133C2×C32×C4C22×D312yes—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—62no—
140C23×C9—21no—
141C23×C9C23×C311yescharacteristic
142C22×D9—22no—
143C22×D9—22no—
144C22×D9—22no—
145C22×D9C23×C312yes—
146C2×C4×C9—24no—
147C2×C4×C9—22no—
148C2×C4×C9—22no—
149C2×C4×C9—22no—
150C2×C4×C9—22no—
151C2×C4×C9—22no—
152C2×C4×C9—22no—
153C2×C4×C9—42no—
154C2×C4×C9—42no—
155C2×C4×C9C23×C312yes—
156C2×C4×C9C23×C312yes—
157C2×C4×C9C23×C312yes—
158C2×Dic9—22no—
159C2×Dic9—22no—
160C2×Dic9—22no—
161C2×Dic9C23×C312yes—
162C2×Dic9C23×C312yes—
163C2×Dic9C23×C312yes—
164C2×#126—91noSylow
165C2×3−1+2#12614yescomplemented
166C2×3−1+2C2×#4111yescharacteristic
167C2×3−1+2C2×#4111yescharacteristic
168C2×3−1+2C2×#4111yescharacteristic
169AGL(1,ℤ/9)—42no—
170C24×C3—91no—
171C23×D3—31no—
172C22×C3×C4—31no—
173C22×C3×C4—31no—
174C22×C3×C4—31no—
175C22×C3×C4—91no—
176C22×C3×C4—91no—
177C22×C3×C4—91no—
178C22×C3×C4C2×C3×D311yescharacteristic
179C22×C3×C4C2×C3×D311yescharacteristic
180C22×C3×C4C2×C3×D311yescharacteristic
181C22×Dic3—31no—
182C22×Dic3—31no—
183C22×Dic3—31no—
184C3×C4⋊C4—34no—
185C3×C4⋊C4—94no—
186C3×C4⋊C4—94no—
187C3×C4⋊C4C2×C3×D314yes—
188C3×C22⋊C4—94no—
189C3×C22⋊C4—94no—
190C3×C22⋊C4—94no—
191C3×C42—94no—
192(C2×D3)⋊C4—34no—
193(C2×D3)⋊C4—34no—
194(C2×D3)⋊C4—34no—
195C4⋊Dic3—34no—
196Dic3⋊C4—34no—
197C4×Dic3—34no—
198C22×C32#23512yes—
199C22×C32#23712yes—
200C22×C32#23912yes—
201C22×C32C23×D311yescharacteristic
202C2×C3×D3—62no—
203C2×C3×D3—62no—
204C2×C3×D3—62no—
205C2×C3×D3—124no—
206C32×C4—22no—
207C32×C4—22no—
208C32×C4—22no—
209C3×Dic3—62no—
210C3×Dic3—62no—
211C3×Dic3—62no—
212C22×C9—21no—
213C22×C9—22no—
214C22×C9—22no—
215C22×C9—22no—
216C22×C9C3×#4112yes—
217C22×C9C3×#4112yes—
218C22×C9C3×#4112yes—
219C22×C9C24×C311yescharacteristic
220C2×D9—22no—
221C2×D9—22no—
222C2×D9—22no—
223C2×D9—44no—
224C4×C9—24no—
225C4×C9—22no—
226C4×C9—22no—
227C4×C9—22no—
228C4×C9—42no—
229C4×C9—42no—
230Dic9—22no—
231Dic9—22no—
232Dic9—22no—
233#126—98no—
234C2×C4⋊C4—91no—
235C2×C4⋊C4—91no—
236C2×C4⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
237C2×C22⋊C4—91no—
238C2×C22⋊C4—91no—
239C2×C22⋊C4—91no—
240C2×C42—91no—
2413−1+2C2×#12611yescharacteristic, complemented, Sylow
242C23×C3—31no—
243C23×C3—92no—
244C23×C3—182no—
245C23×C3—182no—
246C23×C3—182no—
247C23×C3C22×C3×D311yescharacteristic
248C22×D3—32no—
249C22×D3—62no—
250C22×D3—62no—
251C22×D3—62no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—32no—
256C2×C3×C4—32no—
257C2×C3×C4—32no—
258C2×C3×C4—62no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—92no—
262C2×C3×C4—92no—
263C2×C3×C4—92no—
264C2×C3×C4—182no—
265C2×C3×C4—182no—
266C2×C3×C4—182no—
267C2×C3×C4C22×C3×D312yes—
268C2×C3×C4C22×C3×D312yes—
269C2×C3×C4C22×C3×D312yes—
270C2×Dic3—32no—
271C2×Dic3—32no—
272C2×Dic3—32no—
273C2×Dic3—62no—
274C2×Dic3—62no—
275C2×Dic3—62no—
276C2×C32#89814yes—
277C2×C32C2×#23511yescharacteristic
278C2×C32C2×#23711yescharacteristic
279C2×C32C2×#23911yescharacteristic
280C3×D3—122no—
281C2×C9—24no—
282C2×C9—21no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9C3×#12614yescomplemented
286C2×C9C2×C3×#4111yescharacteristic
287C2×C9C2×C3×#4111yescharacteristic
288C2×C9C2×C3×#4111yescharacteristic
289D9—42no—
290C24—91no—
291C22×C4—91no—
292C22×C4—91no—
293C22×C4—91no—
294C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
295C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
296C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
297C4⋊C4—94no—
298C4⋊C4—94no—
299C4⋊C4C2×AGL(1,ℤ/9)14yescomplemented
300C22⋊C4—94no—
301C22⋊C4—94no—
302C22⋊C4—94no—
303C42—94no—
304C22×C3—31no—
305C22×C3—32no—
306C22×C3—32no—
307C22×C3—32no—
308C22×C3—182no—
309C22×C3—182no—
310C22×C3—182no—
311C22×C3—364no—
312C22×C3C3×#23512yes—
313C22×C3C3×#23712yes—
314C22×C3C3×#23912yes—
315C22×C3C23×C3×D311yescharacteristic
316C2×D3—62no—
317C2×D3—62no—
318C2×D3—62no—
319C2×D3—124no—
320C3×C4—22no—
321C3×C4—22no—
322C3×C4—22no—
323C3×C4—62no—
324C3×C4—62no—
325C3×C4—62no—
326C3×C4—182no—
327C3×C4—182no—
328C3×C4—182no—
329Dic3—62no—
330Dic3—62no—
331Dic3—62no—
332C32C2×#89811yescharacteristic
333C9—21no—
334C9C2×C3×#12611yescharacteristic, complemented
335C23—92no—
336C23—182no—
337C23—182no—
338C23—182no—
339C23C22×AGL(1,ℤ/9)11yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—92no—
344C2×C4—92no—
345C2×C4—92no—
346C2×C4—182no—
347C2×C4—182no—
348C2×C4—182no—
349C2×C4C22×AGL(1,ℤ/9)12yes—
350C2×C4C22×AGL(1,ℤ/9)12yes—
351C2×C4C22×AGL(1,ℤ/9)12yes—
352C2×C3—34no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—362no—
357C2×C3C3×#89814yes—
358C2×C3C2×C3×#23511yescharacteristic
359C2×C3C2×C3×#23711yescharacteristic
360C2×C3C2×C3×#23911yescharacteristic
361D3—122no—
362C22—182no—
363C22—182no—
364C22—182no—
365C22—364no—
366C22#3697⋊C312yes—
367C22#3699⋊C312yes—
368C22#8872412yes—
369C22C23×AGL(1,ℤ/9)11yescharacteristic
370C4—22no—
371C4—22no—
372C4—22no—
373C4—182no—
374C4—182no—
375C4—182no—
376C3—31no—
377C3C2×C3×#89811yescharacteristic
378C2—362no—
379C2#63460⋊C314yesdirect product factor
380C2C2×#3697⋊C311yescharacteristic
381C2C2×#3699⋊C311yescharacteristic
382C2C2×#8872411yescharacteristic
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×AGL(1,ℤ/9)C22
hypercenterC2×C4⋊C4AGL(1,ℤ/9)
hypocenterC9C2×C3×#126
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×3−1+2C2semidirect
5. C2×C4⋊(3−1+2⋊C4)C2semidirect
6. C2×(3−1+2⋊C4)⋊C4C2semidirect
7. C2×C4×3−1+2⋊C4C2semidirect
8. #63460⋊C3C2direct (N × H)
10. C2×#63460C3semidirect
18. C4⋊C4×3−1+2C22semidirect
22. C4⋊(3−1+2⋊C4)C22semidirect
23. (3−1+2⋊C4)⋊C4C22semidirect
24. C4×3−1+2⋊C4C22semidirect
34. C2×C9×C4⋊C4C2×C3semidirect
38. C2×C4⋊Dic9C2×C3semidirect
39. C2×Dic9⋊C4C2×C3semidirect
40. C2×C4×Dic9C2×C3semidirect
41. #63460C2×C3semidirect
86. C9×C4⋊C4C22×C3semidirect
90. C4⋊Dic9C22×C3semidirect
91. Dic9⋊C4C22×C3semidirect
92. C4×Dic9C22×C3semidirect
165. C2×3−1+2#126semidirect
236. C2×C4⋊C4AGL(1,ℤ/9)semidirect
241. 3−1+2C2×#126semidirect
285. C2×C9C3×#126semidirect
299. C4⋊C4C2×AGL(1,ℤ/9)semidirect
334. C9C2×C3×#126semidirect
379. C2#63460⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×3−1+22
C2×(C2×AGL(1,ℤ/9))⋊C42
C2×C4⋊(3−1+2⋊C4)2
C2×(3−1+2⋊C4)⋊C42
C2×C4×3−1+2⋊C42
#63460⋊C32
C2×C3×#8983
C2×#634603