Finite Group Databasebeta

← C2×C3×#4817 #11860052

Subgroups — C2×C3×#4817 #11860052

Order 1152.

1169 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×Dic3C211yesmaximal, characteristic, complemented
2C22×C3×#215C211yesmaximal, characteristic
3C2×C3×#893C211yesmaximal, characteristic
4C2×C32×#103C211yesmaximal, characteristic, complemented
5C2×C3×#821C211yesmaximal, characteristic
6C2×C3×#815C211yesmaximal, characteristic, complemented
7C2×C3×#807C211yesmaximal, characteristic, complemented
8C3×#4817C218yesmaximal, direct product factor
9C2×C3×#440—31nomaximal
10C2×#4817C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C3×D12C2211yescharacteristic
13C22×C3×C4×D3—21no—
14C22×C32×C8C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C3×Q8×Dic3—22no—
17C3×Q8×Dic3C2214yescomplemented
18C2×C3×#215C418yescomplemented
19C2×C3×#215C2214yes—
20C2×C3×(C2×D3)⋊C4—21no—
21C2×C3×C4⋊Dic3—21no—
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C2×C3×C4×Dic3—21no—
24C2×C3×C4×Dic3C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#893—22no—
27C3×#893C2214yes—
28C32×#103C2214yescomplemented
29C3×#821C2214yes—
30C3×#815C2214yescomplemented
31C3×#807C2214yescomplemented
32C22×C3×SD16—31no—
33C2×C3×C4×Q8—31no—
34C2×C3×C4×D4—31no—
35C2×Q8×Dic3C2×C311yescharacteristic, complemented
36C22×#215C2×C311yescharacteristic
37C2×#893C2×C311yescharacteristic
38C3×#440—38no—
39C2×C3×#107—31no—
40C2×C3×#103—21no—
41C2×C3×#103D311yescharacteristic, complemented
42C2×C3×#103C2×C311yescharacteristic, complemented
43C2×C3×#102—31no—
44C2×C3×#97—31no—
45C2×#821C2×C311yescharacteristic
46C2×#815C2×C311yescharacteristic, complemented
47C2×#807C2×C311yescharacteristic, complemented
48#4817C2×C318yesdirect product factor
49C23×C3×D3—21no—
50C2×C32×Q8—21no—
51C2×C32×Q8—22no—
52C2×C32×Q8C2×C414yescomplemented
53C2×C32×Q8C2312yes—
54C22×C32×C4—21no—
55C22×C32×C4—21no—
56C22×C32×C4C2311yescharacteristic
57C22×C3×Dic3—21no—
58C22×C3×Dic3D411yescharacteristic
59C22×C3×Dic3D411yescharacteristic
60C2×C3×D12—21no—
61C2×C3×D12—22no—
62C2×C3×D12C2×C414yes—
63C2×C3×D12C2312yes—
64C2×C3×C4×D3—24no—
65C2×C3×C4×D3—44no—
66C2×C32×C8—22no—
67C2×C32×C8C2312yes—
68C32×C4⋊C4—21no—
69C32×C4⋊C4C2312yes—
70C3×#215C2×C4116yescomplemented
71C3×(C2×D3)⋊C4—24no—
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3—24no—
74C3×C4⋊Dic3C2312yes—
75C3×C4×Dic3—24no—
76C3×C4×Dic3D412yes—
77C3×C4×Dic3C2312yes—
78C2×C3×C3⋊C8C2×C414yescomplemented
79C2×C3×C3⋊C8C2312yes—
80C2×#440—31noSylow
81C22×C3×Q8—21no—
82C22×C3×Q8C2×D311yescharacteristic
83C22×C3×Q8C22×C311yescharacteristic
84C22×C3×D4—31no—
85C23×C3×C4—61no—
86C22×D12C22×C311yescharacteristic
87C22×C4×D3—21no—
88C2×C3×SD16—38no—
89C2×C3×SD16—34no—
90C22×C3×C8—21no—
91C22×C3×C8—31no—
92C22×C3×C8C2×D311yescharacteristic
93C22×C3×C8C22×C311yescharacteristic
94C3×C4×Q8—34no—
95C3×C4×Q8—62no—
96C3×C4×D4—34no—
97C3×C4×D4—62no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—31no—
100C2×C3×C4⋊C4—61no—
101C2×C3×C4⋊C4C2×D311yescharacteristic
102C2×C3×C4⋊C4C22×C311yescharacteristic
103C2×C3×C22⋊C4—61no—
104C2×C3×C42—31no—
105C2×C3×C42—61no—
106Q8×Dic3—22no—
107Q8×Dic3C22×C314yescomplemented
108C2×#215C3×C418yescomplemented
109C2×#215C22×C314yes—
110C2×(C2×D3)⋊C4—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3C22×C311yescharacteristic
113C2×C4×Dic3—21no—
114C2×C4×Dic3C22×C311yescharacteristic
115C22×C3⋊C8C22×C311yescharacteristic
116#893—22no—
117#893C22×C314yes—
118C3×#107—34no—
119C3×#103—24no—
120C3×#103C2×D314yescomplemented
121C3×#103C22×C314yescomplemented
122C3×#102—34no—
123C3×#97—34no—
124#821C22×C314yes—
125#815C22×C314yescomplemented
126#807C22×C314yescomplemented
127C23×C32C2×D411yescharacteristic
128C22×C3×D3—24no—
129C22×C3×D3—22no—
130C22×C3×D3—42no—
131C22×C3×D3—42no—
132C32×Q8—24no—
133C32×Q8—22no—
134C32×Q8C22×C414yescomplemented
135C2×C32×C4—24no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—42no—
139C2×C32×C4C22×C412yes—
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4#4112yes—
142C2×C32×C4C2411yescharacteristic
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—42no—
147C2×C3×Dic3C2×D412yes—
148C2×C3×Dic3C2×D412yes—
149C3×D12—24no—
150C3×D12—22no—
151C3×D12C22×C414yes—
152C3×C4×D3—48no—
153C32×C8—22no—
154C3×C3⋊C8C22×C414yescomplemented
155C22×SD16—31no—
156C2×C4×Q8—31no—
157C2×C4×D4—31no—
158#440—38no—
159C2×#107—31no—
160C2×#103C3×D311yescharacteristic, complemented
161C2×#102—31no—
162C2×#97—31no—
163C24×C3—61no—
164C23×D3—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—22no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—24no—
170C2×C3×Q8—22no—
171C2×C3×Q8—41no—
172C2×C3×Q8—42no—
173C2×C3×Q8C4×D314yescomplemented
174C2×C3×Q8C2×C3×C414yescomplemented
175C2×C3×Q8C22×D312yes—
176C2×C3×Q8C23×C312yes—
177C2×C3×D4—34no—
178C2×C3×D4—32no—
179C2×C3×D4—61no—
180C2×C3×D4—62no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—41no—
189C22×C3×C4—41no—
190C22×C3×C4—64no—
191C22×C3×C4—61no—
192C22×C3×C4—124no—
193C22×C3×C4C22×D311yescharacteristic
194C22×C3×C4C23×C311yescharacteristic
195C22×Dic3—21no—
196C22×Dic3C3×D411yescharacteristic
197C22×Dic3C3×D411yescharacteristic
198C2×D12—21no—
199C2×D12—22no—
200C2×D12C2×C3×C414yes—
201C2×D12C23×C312yes—
202C2×C4×D3—24no—
203C2×C4×D3—44no—
204C3×SD16—316no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—34no—
209C2×C3×C8—32no—
210C2×C3×C8—42no—
211C2×C3×C8C22×D312yes—
212C2×C3×C8C23×C312yes—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—32no—
217C3×C4⋊C4—41no—
218C3×C4⋊C4—64no—
219C3×C4⋊C4—61no—
220C3×C4⋊C4C22×D312yes—
221C3×C4⋊C4C23×C312yes—
222C3×C22⋊C4—64no—
223C3×C42—32no—
224C3×C42—32no—
225C3×C42—64no—
226#215C2×C3×C4116yescomplemented
227(C2×D3)⋊C4—24no—
228C4⋊Dic3—21no—
229C4⋊Dic3—24no—
230C4⋊Dic3C23×C312yes—
231C4×Dic3—24no—
232C4×Dic3C3×D412yes—
233C4×Dic3C23×C312yes—
234C2×C3⋊C8C2×C3×C414yescomplemented
235C2×C3⋊C8C23×C312yes—
236C22×C32C4×D412yes—
237C22×C32AGL(1,ℤ/8)12yes—
238C22×C32#13712yes—
239C22×C32C22×D411yescharacteristic
240C2×C3×D3—24no—
241C2×C3×D3—42no—
242C2×C3×D3—42no—
243C2×C3×D3—48no—
244C32×C4—24no—
245C32×C4—42no—
246C32×C4C4×D412yes—
247C32×C4C23×C411yescharacteristic
248C32×C4C2×#4111yescharacteristic
249C3×Dic3—22no—
250C3×Dic3—22no—
251C3×Dic3—42no—
252C22×Q8C2×C3×D311yescharacteristic
253C22×D4—31no—
254C23×C4—61no—
255C2×SD16—38no—
256C2×SD16—34no—
257C22×C8—31no—
258C22×C8C2×C3×D311yescharacteristic
259C4×Q8—34no—
260C4×Q8—62no—
261C4×D4—34no—
262C4×D4—62no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4C2×C3×D311yescharacteristic
266C2×C22⋊C4—61no—
267C2×C42—31no—
268C2×C42—61no—
269#107—34no—
270#103C2×C3×D314yescomplemented
271#102—34no—
272#97—34no—
273C23×C3—21no—
274C23×C3—64no—
275C23×C3—62no—
276C23×C3—122no—
277C23×C3—122no—
278C23×C3D3×D411yescharacteristic
279C23×C3C2×C3×D411yescharacteristic
280C22×D3—24no—
281C22×D3—22no—
282C22×D3—42no—
283C22×D3—42no—
284C3×Q8—24no—
285C3×Q8—24no—
286C3×Q8—22no—
287C3×Q8—22no—
288C3×Q8—24no—
289C3×Q8—44no—
290C3×Q8—42no—
291C3×Q8C2×C4×D314yescomplemented
292C3×Q8C22×C3×C414yescomplemented
293C3×D4—34no—
294C3×D4—64no—
295C3×D4—62no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—21no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—32no—
307C2×C3×C4—32no—
308C2×C3×C4—44no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—82no—
317C2×C3×C4—128no—
318C2×C3×C4—122no—
319C2×C3×C4C2×C4×D312yes—
320C2×C3×C4D3×D411yescharacteristic
321C2×C3×C4#23712yes—
322C2×C3×C4C22×C3×C412yes—
323C2×C3×C4C2×C3×D411yescharacteristic
324C2×C3×C4C3×#4112yes—
325C2×C3×C4C23×D311yescharacteristic
326C2×C3×C4C24×C311yescharacteristic
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—42no—
331C2×Dic3C2×C3×D412yes—
332C2×Dic3C2×C3×D412yes—
333D12—24no—
334D12—22no—
335D12C22×C3×C414yes—
336C4×D3—48no—
337C3×C8—22no—
338C3×C8—22no—
339C3×C8—34no—
340C3×C8—42no—
341C3⋊C8C22×C3×C414yescomplemented
342C2×C32#44014yescomplemented
343C2×C32C2×C4×D411yescharacteristic
344C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
345C2×C32C2×#13711yescharacteristic
346C3×D3—44no—
347C24—61no—
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8C3×C4×D314yescomplemented
351C2×Q8C22×C3×D312yes—
352C2×D4—32no—
353C2×D4—34no—
354C2×D4—61no—
355C2×D4—62no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—31no—
359C22×C4—31no—
360C22×C4—64no—
361C22×C4—61no—
362C22×C4—124no—
363C22×C4C22×C3×D311yescharacteristic
364SD16—316no—
365C2×C8—22no—
366C2×C8—32no—
367C2×C8—34no—
368C2×C8C22×C3×D312yes—
369C4⋊C4—21no—
370C4⋊C4—32no—
371C4⋊C4—64no—
372C4⋊C4—61no—
373C4⋊C4C22×C3×D312yes—
374C22⋊C4—64no—
375C42—32no—
376C42—32no—
377C42—64no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—21no—
382C22×C3—64no—
383C22×C3—122no—
384C22×C3—122no—
385C22×C3—128no—
386C22×C3#67⋊C412yes—
387C22×C3#91412yes—
388C22×C3#91812yes—
389C22×C3C3×C4×D412yes—
390C22×C3C3×AGL(1,ℤ/8)12yes—
391C22×C3C3×#13712yes—
392C22×C3C2×D3×D411yescharacteristic
393C22×C3C22×C3×D411yescharacteristic
394C2×D3—24no—
395C2×D3—42no—
396C2×D3—42no—
397C2×D3—48no—
398C3×C4—24no—
399C3×C4—24no—
400C3×C4—22no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—44no—
404C3×C4—42no—
405C3×C4—42no—
406C3×C4—62no—
407C3×C4—62no—
408C3×C4—82no—
409C3×C4—122no—
410C3×C4#67⋊C412yes—
411C3×C4C3×C4×D412yes—
412C3×C4C22×C4×D311yescharacteristic
413C3×C4C2×#23711yescharacteristic
414C3×C4C23×C3×C411yescharacteristic
415C3×C4C2×C3×#4111yescharacteristic
416Dic3—22no—
417Dic3—22no—
418Dic3—42no—
419C32C2×#44011yescharacteristic, complemented, Sylow
420C23—64no—
421C23—62no—
422C23—122no—
423C23—122no—
424C23C3×D3×D411yescharacteristic
425Q8—24no—
426Q8—22no—
427Q8C2×C3×C4×D314yescomplemented
428D4—34no—
429D4—64no—
430D4—62no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—32no—
435C2×C4—32no—
436C2×C4—42no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—128no—
441C2×C4—122no—
442C2×C4C2×C3×C4×D312yes—
443C2×C4C3×D3×D411yescharacteristic
444C2×C4C3×#23712yes—
445C2×C4C23×C3×D311yescharacteristic
446C8—22no—
447C8—34no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—24no—
451C2×C3—21no—
452C2×C3—124no—
453C2×C3#481714yesdirect product factor
454C2×C3C3×#44014yescomplemented
455C2×C3C2×#67⋊C411yescharacteristic
456C2×C3C2×#91411yescharacteristic
457C2×C3C2×#91811yescharacteristic
458C2×C3C2×C3×C4×D411yescharacteristic
459C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
460C2×C3C2×C3×#13711yescharacteristic
461D3—44no—
462C22—64no—
463C22—122no—
464C22—122no—
465C22—128no—
466C22C3×#67⋊C412yes—
467C22C3×#91412yes—
468C22C3×#91812yes—
469C22C2×C3×D3×D411yescharacteristic
470C4—24no—
471C4—42no—
472C4—62no—
473C4—62no—
474C4—122no—
475C4C3×#67⋊C412yes—
476C4C22×C3×C4×D311yescharacteristic
477C4C2×C3×#23711yescharacteristic
478C3—21no—
479C3C2×#481711yescharacteristic, direct product factor
480C3C2×C3×#44011yescharacteristic, complemented
481C2—124no—
482C2C3×#481714yesdirect product factor
483C2C2×C3×#67⋊C411yescharacteristic
484C2C2×C3×#91411yescharacteristic
485C2C2×C3×#91811yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#103C2
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
cosocleC2×C3×C4C24×C3
hypercenterC2×C3×#103D3
hypocenterC3C2×C3×#440
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#103

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×Dic3C2semidirect
4. C2×C32×#103C2semidirect
6. C2×C3×#815C2semidirect
7. C2×C3×#807C2semidirect
8. C3×#4817C2direct (N × H)
10. C2×#4817C3direct (N × H)
17. C3×Q8×Dic3C22semidirect
18. C2×C3×#215C4semidirect
28. C32×#103C22semidirect
30. C3×#815C22semidirect
31. C3×#807C22semidirect
35. C2×Q8×Dic3C2×C3semidirect
41. C2×C3×#103D3semidirect
42. C2×C3×#103C2×C3semidirect
46. C2×#815C2×C3semidirect
47. C2×#807C2×C3semidirect
48. #4817C2×C3direct (N × H)
52. C2×C32×Q8C2×C4semidirect
70. C3×#215C2×C4semidirect
78. C2×C3×C3⋊C8C2×C4semidirect
107. Q8×Dic3C22×C3semidirect
108. C2×#215C3×C4semidirect
120. C3×#103C2×D3semidirect
121. C3×#103C22×C3semidirect
125. #815C22×C3semidirect
126. #807C22×C3semidirect
134. C32×Q8C22×C4semidirect
154. C3×C3⋊C8C22×C4semidirect
160. C2×#103C3×D3semidirect
173. C2×C3×Q8C4×D3semidirect
174. C2×C3×Q8C2×C3×C4semidirect
226. #215C2×C3×C4semidirect
234. C2×C3⋊C8C2×C3×C4semidirect
270. #103C2×C3×D3semidirect
291. C3×Q8C2×C4×D3semidirect
292. C3×Q8C22×C3×C4semidirect
341. C3⋊C8C22×C3×C4semidirect
342. C2×C32#440semidirect
350. C2×Q8C3×C4×D3semidirect
419. C32C2×#440semidirect
427. Q8C2×C3×C4×D3semidirect
453. C2×C3#4817direct (N × H)
454. C2×C3C3×#440semidirect
479. C3C2×#4817direct (N × H)
480. C3C2×C3×#440semidirect
482. C2C3×#4817direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×Dic32
C22×C3×#2152
C2×C3×#8932
C2×C32×#1032
C2×C3×#8212
C2×C3×#8152
C2×C3×#8072
C3×#48172
C2×C3×#4403
C2×#48173