Finite Group Databasebeta

← C2×C3×#5585 #11903812

Subgroups — C2×C3×#5585 #11903812

Order 1152.

2629 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#947C212yesmaximal, complemented
3C3×#5585C2116yesmaximal, direct product factor
4C2×C3×#894C214yesmaximal, complemented
5C2×C3×#893C212yesmaximal, complemented
6C2×C3×#875C212yesmaximal, complemented
7C2×C3×#874C211yesmaximal, characteristic
8C2×C3×C4×D12C212yesmaximal, complemented
9C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
10C2×C3×#559—31nomaximal
11C2×#5585C311yesmaximal, characteristic, direct product factor
12C22×C32×Q8C2211yescharacteristic
13C22×C3×D12C2212yes—
14C22×C3×D12C2212yes—
15C22×C3×D12C2212yes—
16C22×C3×C4×D3C2214yes—
17C22×C3×Dic6C2211yescharacteristic
18C32×C4×Q8C2218yescomplemented
19C2×C32×C4⋊C4C2211yescharacteristic
20C2×C32×C4⋊C4C2212yes—
21C2×C32×C42C2211yescharacteristic
22C2×C32×C42C2212yes—
23C3×#947C22116yescomplemented
24C2×C3×(C2×D3)⋊C4C2214yes—
25C2×C3×(C2×D3)⋊C4C2214yes—
26C2×C3×(C2×D3)⋊C4C2214yes—
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C2×C3×Dic3⋊C4C2212yes—
29C2×C3×C4×Dic3C2212yes—
30C3×#894C22132yescomplemented
31C3×#893C22116yescomplemented
32C3×#875C22116yescomplemented
33C3×#874C2218yes—
34C3×C4×D12C22116yescomplemented
35C3×C4×Dic6C2218yescomplemented
36C3×#559—316no—
37C2×C3×#127—31no—
38C2×C3×#124—32no—
39C2×C3×#124—32no—
40C2×C3×#123—34no—
41C2×C3×C4×Q8—21no—
42C2×C3×C4×Q8—31no—
43C2×C3×C4×Q8D311yescharacteristic, complemented
44C2×C3×C4×Q8C2×C311yescharacteristic, complemented
45C2×C3×C4×D4—32no—
46C2×C3×C4×D4—32no—
47C2×#947C2×C312yescomplemented
48#5585C2×C3116yesdirect product factor
49C2×#894C2×C314yescomplemented
50C2×#893C2×C312yescomplemented
51C2×#875C2×C312yescomplemented
52C2×#874C2×C311yescharacteristic
53C2×C4×D12C2×C312yescomplemented
54C2×C4×Dic6C2×C311yescharacteristic, complemented
55C23×C3×D3C2314yes—
56C2×C32×Q8—24no—
57C2×C32×Q8C2314yes—
58C22×C32×C4C2311yescharacteristic
59C22×C32×C4C2312yes—
60C22×C32×C4C2312yes—
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2311yescharacteristic
63C22×C3×Dic3C2312yes—
64C22×C3×Dic3C2312yes—
65C2×C3×D12—28no—
66C2×C3×D12—28no—
67C2×C3×D12—44no—
68C2×C3×D12C2318yes—
69C2×C3×D12C2318yes—
70C2×C3×D12C2318yes—
71C2×C3×C4×D3—48no—
72C2×C3×C4×D3C23116yes—
73C2×C3×Dic6—24no—
74C2×C3×Dic6C2314yes—
75C32×C4⋊C4C2314yes—
76C32×C4⋊C4C2318yes—
77C32×C42C2314yes—
78C32×C42C2318yes—
79C3×(C2×D3)⋊C4C23116yes—
80C3×(C2×D3)⋊C4C23116yes—
81C3×(C2×D3)⋊C4C23116yes—
82C3×C4⋊Dic3C2314yes—
83C3×Dic3⋊C4C2318yes—
84C3×C4×Dic3C2318yes—
85C2×#559—31noSylow
86C22×C3×Q8—21no—
87C22×C3×Q8—31no—
88C22×C3×Q8C2×D311yescharacteristic
89C22×C3×Q8C22×C311yescharacteristic
90C22×C3×D4—32no—
91C22×C3×D4—32no—
92C22×C3×D4—32no—
93C23×C3×C4—34no—
94C22×D12C22×C312yes—
95C22×D12C22×C312yes—
96C22×D12C22×C312yes—
97C22×C4×D3C22×C314yes—
98C22×Dic6C22×C311yescharacteristic
99C3×#127—38no—
100C3×#124—316no—
101C3×#124—316no—
102C3×#123—332no—
103C3×C4×Q8—28no—
104C3×C4×Q8—38no—
105C3×C4×Q8C2×D318yescomplemented
106C3×C4×Q8C22×C318yescomplemented
107C3×C4×D4—316no—
108C3×C4×D4—316no—
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4—22no—
111C2×C3×C4⋊C4—32no—
112C2×C3×C4⋊C4—31no—
113C2×C3×C4⋊C4C2×D311yescharacteristic
114C2×C3×C4⋊C4C2×D312yes—
115C2×C3×C4⋊C4C22×C311yescharacteristic
116C2×C3×C4⋊C4C22×C312yes—
117C2×C3×C22⋊C4—34no—
118C2×C3×C22⋊C4—34no—
119C2×C3×C22⋊C4—34no—
120C2×C3×C42—22no—
121C2×C3×C42—21no—
122C2×C3×C42—32no—
123C2×C3×C42C2×D312yes—
124C2×C3×C42C2×D311yescharacteristic
125C2×C3×C42C22×C311yescharacteristic
126C2×C3×C42C22×C312yes—
127#947C22×C3116yescomplemented
128C2×(C2×D3)⋊C4C22×C314yes—
129C2×(C2×D3)⋊C4C22×C314yes—
130C2×(C2×D3)⋊C4C22×C314yes—
131C2×C4⋊Dic3C22×C311yescharacteristic
132C2×Dic3⋊C4C22×C312yes—
133C2×C4×Dic3C22×C312yes—
134#894C22×C3132yescomplemented
135#893C22×C3116yescomplemented
136#875C22×C3116yescomplemented
137#874C22×C318yes—
138C4×D12C22×C3116yescomplemented
139C4×Dic6C22×C318yescomplemented
140C23×C32C2411yescharacteristic
141C22×C3×D3—28no—
142C22×C3×D3—28no—
143C22×C3×D3—28no—
144C22×C3×D3C2418yes—
145C32×Q8—28no—
146C2×C32×C4—24no—
147C2×C32×C4—24no—
148C2×C32×C4—22no—
149C2×C32×C4#4114yes—
150C2×C32×C4#4114yes—
151C2×C32×C4C2412yes—
152C2×C32×C4C2414yes—
153C2×C32×C4C2414yes—
154C2×C32×C4C2412yes—
155C2×C32×C4C2412yes—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3C2414yes—
159C2×C3×Dic3C2414yes—
160C3×D12—216no—
161C3×D12—216no—
162C3×D12—48no—
163C3×C4×D3—416no—
164C3×Dic6—28no—
165#559—316no—
166C2×#127—31no—
167C2×#124—32no—
168C2×#124—32no—
169C2×#123—34no—
170C2×C4×Q8—31no—
171C2×C4×Q8C3×D311yescharacteristic, complemented
172C2×C4×D4—32no—
173C2×C4×D4—32no—
174C24×C3—34no—
175C23×D3C23×C314yes—
176C2×C3×Q8—28no—
177C2×C3×Q8—24no—
178C2×C3×Q8—24no—
179C2×C3×Q8—24no—
180C2×C3×Q8—34no—
181C2×C3×Q8—64no—
182C2×C3×Q8C22×D314yes—
183C2×C3×Q8C23×C314yes—
184C2×C3×D4—38no—
185C2×C3×D4—38no—
186C2×C3×D4—38no—
187C2×C3×D4—68no—
188C2×C3×D4—68no—
189C2×C3×D4—124no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4—22no—
194C22×C3×C4—21no—
195C22×C3×C4—32no—
196C22×C3×C4—32no—
197C22×C3×C4—316no—
198C22×C3×C4—128no—
199C22×C3×C4C22×D312yes—
200C22×C3×C4C22×D312yes—
201C22×C3×C4C22×D311yescharacteristic
202C22×C3×C4C22×D311yescharacteristic
203C22×C3×C4C22×D311yescharacteristic
204C22×C3×C4C23×C311yescharacteristic
205C22×C3×C4C23×C312yes—
206C22×C3×C4C23×C312yes—
207C22×C3×C4C23×C311yescharacteristic
208C22×C3×C4C23×C311yescharacteristic
209C22×Dic3C23×C312yes—
210C22×Dic3C23×C312yes—
211C2×D12—28no—
212C2×D12—28no—
213C2×D12—44no—
214C2×D12C23×C318yes—
215C2×D12C23×C318yes—
216C2×D12C23×C318yes—
217C2×C4×D3—48no—
218C2×C4×D3C23×C3116yes—
219C2×Dic6—24no—
220C2×Dic6C23×C314yes—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—28no—
223C3×C4⋊C4—38no—
224C3×C4⋊C4—34no—
225C3×C4⋊C4C22×D314yes—
226C3×C4⋊C4C22×D318yes—
227C3×C4⋊C4C23×C314yes—
228C3×C4⋊C4C23×C318yes—
229C3×C22⋊C4—316no—
230C3×C22⋊C4—316no—
231C3×C22⋊C4—316no—
232C3×C42—28no—
233C3×C42—24no—
234C3×C42—38no—
235C3×C42C22×D318yes—
236C3×C42C22×D314yes—
237C3×C42C23×C314yes—
238C3×C42C23×C318yes—
239(C2×D3)⋊C4C23×C3116yes—
240(C2×D3)⋊C4C23×C3116yes—
241(C2×D3)⋊C4C23×C3116yes—
242C4⋊Dic3C23×C314yes—
243Dic3⋊C4C23×C318yes—
244C4×Dic3C23×C318yes—
245C22×C32C2×#4112yes—
246C22×C32C2×#4112yes—
247C22×C32H(5,𝔽2)12yes—
248C22×C32C2511yescharacteristic
249C2×C3×D3—28no—
250C2×C3×D3—28no—
251C2×C3×D3—28no—
252C2×C3×D3—416no—
253C32×C4—24no—
254C32×C4—24no—
255C32×C4—22no—
256C32×C4C2×#4114yes—
257C32×C4C2×#4114yes—
258C3×Dic3—24no—
259C3×Dic3—24no—
260C22×Q8—31no—
261C22×Q8C2×C3×D311yescharacteristic
262C22×D4—32no—
263C22×D4—32no—
264C22×D4—32no—
265C23×C4—34no—
266#127—38no—
267#124—316no—
268#124—316no—
269#123—332no—
270C4×Q8—38no—
271C4×Q8C2×C3×D318yescomplemented
272C4×D4—316no—
273C4×D4—316no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—32no—
276C2×C4⋊C4C2×C3×D311yescharacteristic
277C2×C4⋊C4C2×C3×D312yes—
278C2×C22⋊C4—34no—
279C2×C22⋊C4—34no—
280C2×C22⋊C4—34no—
281C2×C42—32no—
282C2×C42C2×C3×D312yes—
283C2×C42C2×C3×D311yescharacteristic
284C23×C3—21no—
285C23×C3—38no—
286C23×C3—68no—
287C23×C3—68no—
288C23×C3—68no—
289C23×C3C23×D311yescharacteristic
290C23×C3C24×C311yescharacteristic
291C22×D3—28no—
292C22×D3—28no—
293C22×D3—28no—
294C22×D3C24×C318yes—
295C3×Q8—216no—
296C3×Q8—28no—
297C3×Q8—28no—
298C3×Q8—68no—
299C3×D4—616no—
300C3×D4—616no—
301C3×D4—128no—
302C2×C3×C4—28no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—34no—
317C2×C3×C4—34no—
318C2×C3×C4—44no—
319C2×C3×C4—42no—
320C2×C3×C4—64no—
321C2×C3×C4—64no—
322C2×C3×C4—1216no—
323C2×C3×C4#23514yes—
324C2×C3×C4#23914yes—
325C2×C3×C4C3×#4114yes—
326C2×C3×C4C3×#4114yes—
327C2×C3×C4C23×D314yes—
328C2×C3×C4C23×D314yes—
329C2×C3×C4C23×D312yes—
330C2×C3×C4C23×D312yes—
331C2×C3×C4C23×D312yes—
332C2×C3×C4C24×C312yes—
333C2×C3×C4C24×C314yes—
334C2×C3×C4C24×C314yes—
335C2×C3×C4C24×C312yes—
336C2×C3×C4C24×C312yes—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3C24×C314yes—
340C2×Dic3C24×C314yes—
341D12—216no—
342D12—216no—
343D12—48no—
344C4×D3—416no—
345Dic6—28no—
346C2×C32#55914yescomplemented
347C2×C32C22×#4111yescharacteristic
348C2×C32C22×#4111yescharacteristic
349C2×C32C2×H(5,𝔽2)11yescharacteristic
350C3×D3—48no—
351C24—34no—
352C2×Q8—24no—
353C2×Q8—34no—
354C2×Q8—64no—
355C2×Q8C22×C3×D314yes—
356C2×D4—38no—
357C2×D4—38no—
358C2×D4—38no—
359C2×D4—68no—
360C2×D4—68no—
361C2×D4—124no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—316no—
365C22×C4—128no—
366C22×C4C22×C3×D312yes—
367C22×C4C22×C3×D312yes—
368C22×C4C22×C3×D311yescharacteristic
369C22×C4C22×C3×D311yescharacteristic
370C22×C4C22×C3×D311yescharacteristic
371C4⋊C4—34no—
372C4⋊C4—38no—
373C4⋊C4C22×C3×D314yes—
374C4⋊C4C22×C3×D318yes—
375C22⋊C4—316no—
376C22⋊C4—316no—
377C22⋊C4—316no—
378C42—38no—
379C42C22×C3×D318yes—
380C42C22×C3×D314yes—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—21no—
385C22×C3—68no—
386C22×C3—68no—
387C22×C3—68no—
388C22×C3—1216no—
389C22×C3C2×#23512yes—
390C22×C3C2×#23912yes—
391C22×C3#100912yes—
392C22×C3C2×C3×#4112yes—
393C22×C3C2×C3×#4112yes—
394C22×C3C3×H(5,𝔽2)12yes—
395C22×C3C24×D311yescharacteristic
396C22×C3C25×C311yescharacteristic
397C2×D3—28no—
398C2×D3—28no—
399C2×D3—28no—
400C2×D3—416no—
401C3×C4—28no—
402C3×C4—24no—
403C3×C4—24no—
404C3×C4—22no—
405C3×C4—24no—
406C3×C4—24no—
407C3×C4—22no—
408C3×C4—24no—
409C3×C4—24no—
410C3×C4—44no—
411C3×C4—42no—
412C3×C4—64no—
413C3×C4—64no—
414C3×C4C2×#23514yes—
415C3×C4C2×#23914yes—
416C3×C4C2×C3×#4114yes—
417C3×C4C2×C3×#4114yes—
418Dic3—24no—
419Dic3—24no—
420C32C2×#55911yescharacteristic, complemented, Sylow
421C23—38no—
422C23—68no—
423C23—68no—
424C23—68no—
425C23C23×C3×D311yescharacteristic
426Q8—28no—
427Q8—68no—
428D4—616no—
429D4—616no—
430D4—128no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—34no—
435C2×C4—34no—
436C2×C4—64no—
437C2×C4—64no—
438C2×C4—1216no—
439C2×C4C3×#23514yes—
440C2×C4C3×#23914yes—
441C2×C4C23×C3×D314yes—
442C2×C4C23×C3×D314yes—
443C2×C4C23×C3×D312yes—
444C2×C4C23×C3×D312yes—
445C2×C4C23×C3×D312yes—
446C2×C3—24no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—128no—
451C2×C3#558514yesdirect product factor
452C2×C3C3×#55914yescomplemented
453C2×C3C22×#23511yescharacteristic
454C2×C3C22×#23911yescharacteristic
455C2×C3C2×#100911yescharacteristic
456C2×C3C22×C3×#4111yescharacteristic
457C2×C3C22×C3×#4111yescharacteristic
458C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
459D3—48no—
460C22—68no—
461C22—68no—
462C22—68no—
463C22—1216no—
464C22C2×C3×#23512yes—
465C22C2×C3×#23912yes—
466C22C3×#100912yes—
467C22C24×C3×D311yescharacteristic
468C4—24no—
469C4—24no—
470C4—22no—
471C4—64no—
472C4—64no—
473C4C2×C3×#23514yes—
474C4C2×C3×#23914yes—
475C3—21no—
476C3C2×#558511yescharacteristic, direct product factor
477C3C2×C3×#55911yescharacteristic, complemented
478C2—128no—
479C2C3×#558514yesdirect product factor
480C2C22×C3×#23511yescharacteristic
481C2C22×C3×#23911yescharacteristic
482C2C2×C3×#100911yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC2×C32×C4×Q8C2
socleC23×C32C24
prime radicalC2×C3×#874C2
hypercenterC2×C3×C4×Q8D3
hypocenterC3C2×C3×#559
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×Q8C2semidirect
2. C2×C3×#947C2semidirect
3. C3×#5585C2direct (N × H)
4. C2×C3×#894C2semidirect
5. C2×C3×#893C2semidirect
6. C2×C3×#875C2semidirect
8. C2×C3×C4×D12C2semidirect
9. C2×C3×C4×Dic6C2semidirect
11. C2×#5585C3direct (N × H)
18. C32×C4×Q8C22semidirect
23. C3×#947C22semidirect
30. C3×#894C22semidirect
31. C3×#893C22semidirect
32. C3×#875C22semidirect
34. C3×C4×D12C22semidirect
35. C3×C4×Dic6C22semidirect
43. C2×C3×C4×Q8D3semidirect
44. C2×C3×C4×Q8C2×C3semidirect
47. C2×#947C2×C3semidirect
48. #5585C2×C3direct (N × H)
49. C2×#894C2×C3semidirect
50. C2×#893C2×C3semidirect
51. C2×#875C2×C3semidirect
53. C2×C4×D12C2×C3semidirect
54. C2×C4×Dic6C2×C3semidirect
105. C3×C4×Q8C2×D3semidirect
106. C3×C4×Q8C22×C3semidirect
127. #947C22×C3semidirect
134. #894C22×C3semidirect
135. #893C22×C3semidirect
136. #875C22×C3semidirect
138. C4×D12C22×C3semidirect
139. C4×Dic6C22×C3semidirect
171. C2×C4×Q8C3×D3semidirect
271. C4×Q8C2×C3×D3semidirect
346. C2×C32#559semidirect
420. C32C2×#559semidirect
451. C2×C3#5585direct (N × H)
452. C2×C3C3×#559semidirect
476. C3C2×#5585direct (N × H)
477. C3C2×C3×#559semidirect
479. C2C3×#5585direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C4×Q82
C2×C3×#9472
C3×#55852
C2×C3×#8942
C2×C3×#8932
C2×C3×#8752
C2×C3×#8742
C2×C3×C4×D122
C2×C3×C4×Dic62
C2×C3×#5593
C2×#55853