Finite Group Databasebeta

← C2×C3×#5697 #11901482

Subgroups — C2×C3×#5697 #11901482

Order 1152.

2633 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#125C211yesmaximal, characteristic, complemented
2C3×#5697C2116yesmaximal, direct product factor
3C2×C3×#895C214yesmaximal, complemented
4C2×C3×#895C212yesmaximal, complemented
5C2×C3×#894C214yesmaximal, complemented
6C2×C3×#893C212yesmaximal, complemented
7C2×C3×#874C211yesmaximal, characteristic, complemented
8C2×C3×#872C211yesmaximal, characteristic, complemented
9C2×C3×#540—31nomaximal
10C2×#5697C311yesmaximal, characteristic, direct product factor
11C22×C3×D12C2212yes—
12C22×C3×D12C2214yes—
13C22×C3×D12C2214yes—
14C22×C3×C4×D3C2211yescharacteristic
15C22×C3×C4×D3C2214yes—
16C32×#125C2218yescomplemented
17C2×C32×C4⋊C4C2212yes—
18C2×C32×C4⋊C4C2214yes—
19C2×C32×C42C2211yescharacteristic
20C2×C3×(C2×D3)⋊C4C2214yes—
21C2×C3×(C2×D3)⋊C4C2214yes—
22C2×C3×(C2×D3)⋊C4C2212yes—
23C2×C3×Dic3⋊C4C2212yes—
24C2×C3×C4×Dic3C2211yescharacteristic
25C3×#895C22132yescomplemented
26C3×#895C22116yescomplemented
27C3×#894C22132yescomplemented
28C3×#893C22116yescomplemented
29C3×#874C2218yescomplemented
30C3×#872C2218yescomplemented
31C3×#540—316no—
32C2×C3×#127—31no—
33C2×C3×#125—21no—
34C2×C3×#125D311yescharacteristic, complemented
35C2×C3×#125C2×C311yescharacteristic, complemented
36C2×C3×#123—34no—
37C2×C3×#121—34no—
38C2×C3×#121—32no—
39C2×C3×C4×D4—32no—
40C2×C3×#117—31no—
41#5697C2×C3116yesdirect product factor
42C2×#895C2×C314yescomplemented
43C2×#895C2×C312yescomplemented
44C2×#894C2×C314yescomplemented
45C2×#893C2×C312yescomplemented
46C2×#874C2×C311yescharacteristic, complemented
47C2×#872C2×C311yescharacteristic, complemented
48C23×C3×D3C2311yescharacteristic
49C23×C3×D3C2314yes—
50C22×C32×C4C2312yes—
51C22×C32×C4C2314yes—
52C22×C32×C4C2311yescharacteristic
53C22×C3×Dic3C2312yes—
54C22×C3×Dic3C2311yescharacteristic
55C2×C3×D12—28no—
56C2×C3×D12—48no—
57C2×C3×D12—48no—
58C2×C3×D12C2318yes—
59C2×C3×D12C23116yes—
60C2×C3×D12C23116yes—
61C2×C3×C4×D3—24no—
62C2×C3×C4×D3—48no—
63C2×C3×C4×D3C2314yes—
64C2×C3×C4×D3C23116yes—
65C32×C4⋊C4C2318yes—
66C32×C4⋊C4C23116yes—
67C32×C42C2314yes—
68C3×(C2×D3)⋊C4C23116yes—
69C3×(C2×D3)⋊C4C23116yes—
70C3×(C2×D3)⋊C4C2318yes—
71C3×Dic3⋊C4C2318yes—
72C3×C4×Dic3C2314yes—
73C2×#540—31noSylow
74C22×C3×D4—32no—
75C22×C3×D4—34no—
76C22×C3×D4—34no—
77C23×C3×C4—31no—
78C23×C3×C4—34no—
79C22×D12C22×C312yes—
80C22×D12C22×C314yes—
81C22×D12C22×C314yes—
82C22×C4×D3C22×C311yescharacteristic
83C22×C4×D3C22×C314yes—
84C3×#127—38no—
85C3×#125—28no—
86C3×#125C2×D318yescomplemented
87C3×#125C22×C318yescomplemented
88C3×#123—332no—
89C3×#121—332no—
90C3×#121—316no—
91C3×C4×D4—316no—
92C3×#117—38no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—24no—
95C2×C3×C4⋊C4—32no—
96C2×C3×C4⋊C4C2×D312yes—
97C2×C3×C4⋊C4C2×D314yes—
98C2×C3×C4⋊C4C22×C312yes—
99C2×C3×C4⋊C4C22×C314yes—
100C2×C3×C22⋊C4—32no—
101C2×C3×C22⋊C4—34no—
102C2×C3×C22⋊C4—34no—
103C2×C3×C42—21no—
104C2×C3×C42—31no—
105C2×C3×C42C2×D311yescharacteristic
106C2×C3×C42C22×C311yescharacteristic
107C2×(C2×D3)⋊C4C22×C312yes—
108C2×(C2×D3)⋊C4C22×C314yes—
109C2×(C2×D3)⋊C4C22×C314yes—
110C2×Dic3⋊C4C22×C312yes—
111C2×C4×Dic3C22×C311yescharacteristic
112#895C22×C3132yescomplemented
113#895C22×C3116yescomplemented
114#894C22×C3132yescomplemented
115#893C22×C3116yescomplemented
116#874C22×C318yescomplemented
117#872C22×C318yescomplemented
118C23×C32C2411yescharacteristic
119C22×C3×D3—22no—
120C22×C3×D3—28no—
121C22×C3×D3—28no—
122C22×C3×D3—24no—
123C22×C3×D3—28no—
124C22×C3×D3C2412yes—
125C22×C3×D3C2418yes—
126C2×C32×C4—28no—
127C2×C32×C4—24no—
128C2×C32×C4#4114yes—
129C2×C32×C4C2414yes—
130C2×C32×C4C2418yes—
131C2×C32×C4C2412yes—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3C2414yes—
135C2×C3×Dic3C2412yes—
136C3×D12—216no—
137C3×D12—416no—
138C3×D12—416no—
139C3×C4×D3—28no—
140C3×C4×D3—416no—
141#540—316no—
142C2×#127—31no—
143C2×#125C3×D311yescharacteristic, complemented
144C2×#123—34no—
145C2×#121—34no—
146C2×#121—32no—
147C2×C4×D4—32no—
148C2×#117—31no—
149C24×C3—31no—
150C24×C3—34no—
151C23×D3C23×C311yescharacteristic
152C23×D3C23×C314yes—
153C2×C3×D4—38no—
154C2×C3×D4—316no—
155C2×C3×D4—316no—
156C2×C3×D4—68no—
157C2×C3×D4—128no—
158C2×C3×D4—128no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4—32no—
163C22×C3×C4—31no—
164C22×C3×C4—34no—
165C22×C3×C4—316no—
166C22×C3×C4—64no—
167C22×C3×C4—128no—
168C22×C3×C4C22×D312yes—
169C22×C3×C4C22×D314yes—
170C22×C3×C4C22×D311yescharacteristic
171C22×C3×C4C23×C312yes—
172C22×C3×C4C23×C314yes—
173C22×C3×C4C23×C311yescharacteristic
174C22×Dic3C23×C312yes—
175C22×Dic3C23×C311yescharacteristic
176C2×D12—28no—
177C2×D12—48no—
178C2×D12—48no—
179C2×D12C23×C318yes—
180C2×D12C23×C3116yes—
181C2×D12C23×C3116yes—
182C2×C4×D3—24no—
183C2×C4×D3—48no—
184C2×C4×D3C23×C314yes—
185C2×C4×D3C23×C3116yes—
186C3×C4⋊C4—28no—
187C3×C4⋊C4—216no—
188C3×C4⋊C4—38no—
189C3×C4⋊C4C22×D318yes—
190C3×C4⋊C4C22×D3116yes—
191C3×C4⋊C4C23×C318yes—
192C3×C4⋊C4C23×C3116yes—
193C3×C22⋊C4—38no—
194C3×C22⋊C4—316no—
195C3×C22⋊C4—316no—
196C3×C42—24no—
197C3×C42—34no—
198C3×C42C22×D314yes—
199C3×C42C23×C314yes—
200(C2×D3)⋊C4C23×C318yes—
201(C2×D3)⋊C4C23×C3116yes—
202(C2×D3)⋊C4C23×C3116yes—
203Dic3⋊C4C23×C318yes—
204C4×Dic3C23×C314yes—
205C22×C32C2×#4112yes—
206C22×C32H(5,𝔽2)14yes—
207C22×C32C2511yescharacteristic
208C2×C3×D3—22no—
209C2×C3×D3—28no—
210C2×C3×D3—28no—
211C2×C3×D3—24no—
212C2×C3×D3—28no—
213C2×C3×D3—44no—
214C2×C3×D3—416no—
215C32×C4—28no—
216C32×C4—24no—
217C32×C4C2×#4114yes—
218C3×Dic3—22no—
219C3×Dic3—24no—
220C22×D4—32no—
221C22×D4—34no—
222C22×D4—34no—
223C23×C4—31no—
224C23×C4—34no—
225#127—38no—
226#125C2×C3×D318yescomplemented
227#123—332no—
228#121—332no—
229#121—316no—
230C4×D4—316no—
231#117—38no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4C2×C3×D312yes—
234C2×C4⋊C4C2×C3×D314yes—
235C2×C22⋊C4—32no—
236C2×C22⋊C4—34no—
237C2×C22⋊C4—34no—
238C2×C42—31no—
239C2×C42C2×C3×D311yescharacteristic
240C23×C3—21no—
241C23×C3—32no—
242C23×C3—38no—
243C23×C3—62no—
244C23×C3—68no—
245C23×C3—68no—
246C23×C3—64no—
247C23×C3—68no—
248C23×C3C23×D311yescharacteristic
249C23×C3C24×C311yescharacteristic
250C22×D3—22no—
251C22×D3—28no—
252C22×D3—28no—
253C22×D3—24no—
254C22×D3—28no—
255C22×D3C24×C312yes—
256C22×D3C24×C318yes—
257C3×D4—616no—
258C3×D4—1216no—
259C3×D4—1216no—
260C2×C3×C4—28no—
261C2×C3×C4—24no—
262C2×C3×C4—28no—
263C2×C3×C4—24no—
264C2×C3×C4—22no—
265C2×C3×C4—24no—
266C2×C3×C4—28no—
267C2×C3×C4—24no—
268C2×C3×C4—34no—
269C2×C3×C4—32no—
270C2×C3×C4—48no—
271C2×C3×C4—44no—
272C2×C3×C4—62no—
273C2×C3×C4—64no—
274C2×C3×C4—68no—
275C2×C3×C4—1216no—
276C2×C3×C4#23914yes—
277C2×C3×C4C3×#4114yes—
278C2×C3×C4C23×D314yes—
279C2×C3×C4C23×D318yes—
280C2×C3×C4C23×D312yes—
281C2×C3×C4C24×C314yes—
282C2×C3×C4C24×C318yes—
283C2×C3×C4C24×C312yes—
284C2×Dic3—22no—
285C2×Dic3—24no—
286C2×Dic3C24×C314yes—
287C2×Dic3C24×C312yes—
288D12—216no—
289D12—416no—
290D12—416no—
291C4×D3—28no—
292C4×D3—416no—
293C2×C32#54014yescomplemented
294C2×C32C22×#4111yescharacteristic
295C2×C32C2×H(5,𝔽2)12yes—
296C3×D3—42no—
297C3×D3—48no—
298C24—31no—
299C24—34no—
300C2×D4—38no—
301C2×D4—316no—
302C2×D4—316no—
303C2×D4—68no—
304C2×D4—128no—
305C2×D4—128no—
306C22×C4—32no—
307C22×C4—31no—
308C22×C4—34no—
309C22×C4—316no—
310C22×C4—64no—
311C22×C4—128no—
312C22×C4C22×C3×D312yes—
313C22×C4C22×C3×D314yes—
314C22×C4C22×C3×D311yescharacteristic
315C4⋊C4—38no—
316C4⋊C4C22×C3×D318yes—
317C4⋊C4C22×C3×D3116yes—
318C22⋊C4—38no—
319C22⋊C4—316no—
320C22⋊C4—316no—
321C42—34no—
322C42C22×C3×D314yes—
323C22×C3—22no—
324C22×C3—24no—
325C22×C3—21no—
326C22×C3—62no—
327C22×C3—68no—
328C22×C3—68no—
329C22×C3—64no—
330C22×C3—68no—
331C22×C3—124no—
332C22×C3—1216no—
333C22×C3C2×#23912yes—
334C22×C3#100914yes—
335C22×C3C2×C3×#4112yes—
336C22×C3C3×H(5,𝔽2)14yes—
337C22×C3C24×D311yescharacteristic
338C22×C3C25×C311yescharacteristic
339C2×D3—22no—
340C2×D3—28no—
341C2×D3—28no—
342C2×D3—24no—
343C2×D3—28no—
344C2×D3—44no—
345C2×D3—416no—
346C3×C4—28no—
347C3×C4—24no—
348C3×C4—28no—
349C3×C4—24no—
350C3×C4—24no—
351C3×C4—48no—
352C3×C4—44no—
353C3×C4—62no—
354C3×C4—64no—
355C3×C4C2×#23914yes—
356C3×C4C2×C3×#4114yes—
357Dic3—22no—
358Dic3—24no—
359C32C2×#54011yescharacteristic, complemented, Sylow
360C23—32no—
361C23—38no—
362C23—62no—
363C23—68no—
364C23—68no—
365C23—64no—
366C23—68no—
367C23C23×C3×D311yescharacteristic
368D4—616no—
369D4—1216no—
370D4—1216no—
371C2×C4—28no—
372C2×C4—24no—
373C2×C4—34no—
374C2×C4—32no—
375C2×C4—62no—
376C2×C4—64no—
377C2×C4—68no—
378C2×C4—1216no—
379C2×C4C3×#23914yes—
380C2×C4C23×C3×D314yes—
381C2×C4C23×C3×D318yes—
382C2×C4C23×C3×D312yes—
383C2×C3—21no—
384C2×C3—22no—
385C2×C3—24no—
386C2×C3—122no—
387C2×C3—128no—
388C2×C3#569714yesdirect product factor
389C2×C3C3×#54014yescomplemented
390C2×C3C22×#23911yescharacteristic
391C2×C3C2×#100912yes—
392C2×C3C22×C3×#4111yescharacteristic
393C2×C3C2×C3×H(5,𝔽2)12yes—
394D3—42no—
395D3—48no—
396C22—62no—
397C22—68no—
398C22—68no—
399C22—64no—
400C22—68no—
401C22—124no—
402C22—1216no—
403C22C2×C3×#23912yes—
404C22C3×#100914yes—
405C22C24×C3×D311yescharacteristic
406C4—28no—
407C4—24no—
408C4—62no—
409C4—64no—
410C4C2×C3×#23914yes—
411C3—21no—
412C3C2×#569711yescharacteristic, direct product factor
413C3C2×C3×#54011yescharacteristic, complemented
414C2—122no—
415C2—128no—
416C2C3×#569714yesdirect product factor
417C2C22×C3×#23911yescharacteristic
418C2C2×C3×#100912yes—
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×#125C2
socleC23×C32C24
hypercenterC2×C3×#125D3
hypocenterC3C2×C3×#540
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×#125

Complements

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

NH ≅ G/Ntype
1. C2×C32×#125C2semidirect
2. C3×#5697C2direct (N × H)
3. C2×C3×#895C2semidirect
4. C2×C3×#895C2semidirect
5. C2×C3×#894C2semidirect
6. C2×C3×#893C2semidirect
7. C2×C3×#874C2semidirect
8. C2×C3×#872C2semidirect
10. C2×#5697C3direct (N × H)
16. C32×#125C22semidirect
25. C3×#895C22semidirect
26. C3×#895C22semidirect
27. C3×#894C22semidirect
28. C3×#893C22semidirect
29. C3×#874C22semidirect
30. C3×#872C22semidirect
34. C2×C3×#125D3semidirect
35. C2×C3×#125C2×C3semidirect
41. #5697C2×C3direct (N × H)
42. C2×#895C2×C3semidirect
43. C2×#895C2×C3semidirect
44. C2×#894C2×C3semidirect
45. C2×#893C2×C3semidirect
46. C2×#874C2×C3semidirect
47. C2×#872C2×C3semidirect
86. C3×#125C2×D3semidirect
87. C3×#125C22×C3semidirect
112. #895C22×C3semidirect
113. #895C22×C3semidirect
114. #894C22×C3semidirect
115. #893C22×C3semidirect
116. #874C22×C3semidirect
117. #872C22×C3semidirect
143. C2×#125C3×D3semidirect
226. #125C2×C3×D3semidirect
293. C2×C32#540semidirect
359. C32C2×#540semidirect
388. C2×C3#5697direct (N × H)
389. C2×C3C3×#540semidirect
412. C3C2×#5697direct (N × H)
413. C3C2×C3×#540semidirect
416. C2C3×#5697direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1252
C3×#56972
C2×C3×#8952
C2×C3×#8942
C2×C3×#8932
C2×C3×#8742
C2×C3×#8722
C2×C3×#5403
C2×#56973