Finite Group Databasebeta

← C2×C3×#4785 #11860141

Subgroups — C2×C3×#4785 #11860141

Order 1152.

1317 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Dic3C211yesmaximal, characteristic, complemented
2C22×C3×#213C211yesmaximal, characteristic
3C2×C3×#893C211yesmaximal, characteristic, complemented
4C2×C32×#102C211yesmaximal, characteristic, complemented
5C2×C3×#821C211yesmaximal, characteristic, complemented
6C2×C3×#815C211yesmaximal, characteristic, complemented
7C2×C3×#808C211yesmaximal, characteristic, complemented
8C3×#4785C218yesmaximal, direct product factor
9C2×C3×#442—31nomaximal
10C2×#4785C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C23×C3×Dic3—21no—
13C22×C3×D12C2211yescharacteristic
14C22×C3×C4×D3—21no—
15C22×C32×C8C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C3×#217—21no—
18C3×D4×Dic3—22no—
19C3×D4×Dic3C2214yescomplemented
20C2×C3×#213C418yescomplemented
21C2×C3×#213C2214yes—
22C2×C3×(C2×D3)⋊C4—21no—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×C4×Dic3C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#893—22no—
27C3×#893C2214yescomplemented
28C32×#102C2214yescomplemented
29C3×#821C2214yescomplemented
30C3×#815C2214yescomplemented
31C3×#808C2214yescomplemented
32C22×C3×D8—31no—
33C2×C3×C4×D4—31no—
34C2×C3×C4×D4—31no—
35C2×D4×Dic3C2×C311yescharacteristic, complemented
36C22×#213C2×C311yescharacteristic
37C2×#893C2×C311yescharacteristic, complemented
38C3×#442—38no—
39C2×C3×#106—31no—
40C2×C3×#102—21no—
41C2×C3×#102—31no—
42C2×C3×#102D311yescharacteristic, complemented
43C2×C3×#102C2×C311yescharacteristic, complemented
44C2×C3×#97—31no—
45C2×#821C2×C311yescharacteristic, complemented
46C2×#815C2×C311yescharacteristic, complemented
47C2×#808C2×C311yescharacteristic, complemented
48#4785C2×C318yesdirect product factor
49C24×C32—21no—
50C23×C3×D3—21no—
51C2×C32×D4—21no—
52C2×C32×D4—22no—
53C2×C32×D4C2×C414yescomplemented
54C2×C32×D4C2312yes—
55C22×C32×C4—21no—
56C22×C32×C4C2311yescharacteristic
57C22×C3×Dic3—21no—
58C22×C3×Dic3—24no—
59C22×C3×Dic3—44no—
60C22×C3×Dic3D411yescharacteristic
61C22×C3×Dic3D411yescharacteristic
62C2×C3×D12—21no—
63C2×C3×D12—22no—
64C2×C3×D12C2×C414yescomplemented
65C2×C3×D12C2312yes—
66C2×C3×C4×D3—24no—
67C2×C3×C4×D3—44no—
68C2×C32×C8—22no—
69C2×C32×C8C2312yes—
70C32×C4⋊C4—21no—
71C32×C4⋊C4C2312yes—
72C3×#217—24no—
73C3×#213C2×C4116yescomplemented
74C3×(C2×D3)⋊C4—24no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3C2312yes—
77C3×C4×Dic3D412yes—
78C3×C4×Dic3C2312yes—
79C2×C3×C3⋊C8C2×C414yescomplemented
80C2×C3×C3⋊C8C2312yes—
81C2×#442—31noSylow
82C22×C3×D4—21no—
83C22×C3×D4—31no—
84C22×C3×D4C2×D311yescharacteristic
85C22×C3×D4C22×C311yescharacteristic
86C23×C3×C4—61no—
87C23×C3×C4—61no—
88C23×Dic3—21no—
89C22×D12C22×C311yescharacteristic
90C22×C4×D3—21no—
91C2×C3×D8—38no—
92C2×C3×D8—34no—
93C22×C3×C8—21no—
94C22×C3×C8—31no—
95C22×C3×C8C2×D311yescharacteristic
96C22×C3×C8C22×C311yescharacteristic
97C3×C4×D4—34no—
98C3×C4×D4—34no—
99C3×C4×D4—62no—
100C3×C4×D4—62no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—31no—
103C2×C3×C4⋊C4C2×D311yescharacteristic
104C2×C3×C4⋊C4C22×C311yescharacteristic
105C2×C3×C22⋊C4—61no—
106C2×C3×C22⋊C4—61no—
107C2×C3×C42—31no—
108C2×#217—21no—
109D4×Dic3—22no—
110D4×Dic3C22×C314yescomplemented
111C2×#213C3×C418yescomplemented
112C2×#213C22×C314yes—
113C2×(C2×D3)⋊C4—21no—
114C2×C4⋊Dic3C22×C311yescharacteristic
115C2×C4×Dic3C22×C311yescharacteristic
116C22×C3⋊C8C22×C311yescharacteristic
117#893—22no—
118#893C22×C314yescomplemented
119C3×#106—34no—
120C3×#102—24no—
121C3×#102—34no—
122C3×#102C2×D314yescomplemented
123C3×#102C22×C314yescomplemented
124C3×#97—34no—
125#821C22×C314yescomplemented
126#815C22×C314yescomplemented
127#808C22×C314yescomplemented
128C23×C32—24no—
129C23×C32—22no—
130C23×C32—42no—
131C23×C32—42no—
132C23×C32C2×D411yescharacteristic
133C22×C3×D3—24no—
134C22×C3×D3—22no—
135C22×C3×D3—42no—
136C22×C3×D3—42no—
137C32×D4—24no—
138C32×D4—22no—
139C32×D4C22×C414yescomplemented
140C2×C32×C4—22no—
141C2×C32×C4—42no—
142C2×C32×C4C22×C412yes—
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4#4112yes—
145C2×C32×C4C2411yescharacteristic
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—48no—
150C2×C3×Dic3—42no—
151C2×C3×Dic3C2×D412yes—
152C2×C3×Dic3C2×D412yes—
153C3×D12—24no—
154C3×D12—22no—
155C3×D12C22×C414yescomplemented
156C3×C4×D3—48no—
157C32×C8—22no—
158C3×C3⋊C8C22×C414yescomplemented
159C22×D8—31no—
160C2×C4×D4—31no—
161C2×C4×D4—31no—
162#442—38no—
163C2×#106—31no—
164C2×#102—31no—
165C2×#102C3×D311yescharacteristic, complemented
166C2×#97—31no—
167C24×C3—21no—
168C24×C3—21no—
169C24×C3—41no—
170C24×C3—61no—
171C23×D3—21no—
172C2×C3×D4—21no—
173C2×C3×D4—22no—
174C2×C3×D4—21no—
175C2×C3×D4—22no—
176C2×C3×D4—24no—
177C2×C3×D4—22no—
178C2×C3×D4—34no—
179C2×C3×D4—32no—
180C2×C3×D4—41no—
181C2×C3×D4—42no—
182C2×C3×D4—61no—
183C2×C3×D4—62no—
184C2×C3×D4C4×D314yescomplemented
185C2×C3×D4C2×C3×C414yescomplemented
186C2×C3×D4C22×D312yes—
187C2×C3×D4C23×C312yes—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—31no—
192C22×C3×C4—31no—
193C22×C3×C4—41no—
194C22×C3×C4—64no—
195C22×C3×C4—64no—
196C22×C3×C4—61no—
197C22×C3×C4—124no—
198C22×C3×C4—124no—
199C22×C3×C4C22×D311yescharacteristic
200C22×C3×C4C23×C311yescharacteristic
201C22×Dic3—21no—
202C22×Dic3—24no—
203C22×Dic3—44no—
204C22×Dic3C3×D411yescharacteristic
205C22×Dic3C3×D411yescharacteristic
206C2×D12—21no—
207C2×D12—22no—
208C2×D12C2×C3×C414yescomplemented
209C2×D12C23×C312yes—
210C2×C4×D3—24no—
211C2×C4×D3—44no—
212C3×D8—316no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—34no—
217C2×C3×C8—32no—
218C2×C3×C8—42no—
219C2×C3×C8C22×D312yes—
220C2×C3×C8C23×C312yes—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—32no—
225C3×C4⋊C4—41no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4C22×D312yes—
228C3×C4⋊C4C23×C312yes—
229C3×C22⋊C4—64no—
230C3×C22⋊C4—64no—
231C3×C42—32no—
232C3×C42—32no—
233#217—24no—
234#213C2×C3×C4116yescomplemented
235(C2×D3)⋊C4—24no—
236C4⋊Dic3—21no—
237C4⋊Dic3C23×C312yes—
238C4×Dic3C3×D412yes—
239C4×Dic3C23×C312yes—
240C2×C3⋊C8C2×C3×C414yescomplemented
241C2×C3⋊C8C23×C312yes—
242C22×C32—24no—
243C22×C32—42no—
244C22×C32—42no—
245C22×C32—48no—
246C22×C32C4×D412yes—
247C22×C32AGL(1,ℤ/8)12yes—
248C22×C32AGL(1,ℤ/8)12yes—
249C22×C32C22×D411yescharacteristic
250C2×C3×D3—24no—
251C2×C3×D3—42no—
252C2×C3×D3—42no—
253C2×C3×D3—48no—
254C32×C4—42no—
255C32×C4C4×D412yes—
256C32×C4C23×C411yescharacteristic
257C32×C4C2×#4111yescharacteristic
258C3×Dic3—22no—
259C3×Dic3—22no—
260C3×Dic3—42no—
261C22×D4—31no—
262C22×D4C2×C3×D311yescharacteristic
263C23×C4—61no—
264C23×C4—61no—
265C2×D8—38no—
266C2×D8—34no—
267C22×C8—31no—
268C22×C8C2×C3×D311yescharacteristic
269C4×D4—34no—
270C4×D4—34no—
271C4×D4—62no—
272C4×D4—62no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4C2×C3×D311yescharacteristic
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C42—31no—
278#106—34no—
279#102—34no—
280#102C2×C3×D314yescomplemented
281#97—34no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—21no—
287C23×C3—44no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—64no—
294C23×C3—62no—
295C23×C3—82no—
296C23×C3—82no—
297C23×C3—122no—
298C23×C3—122no—
299C23×C3D3×D411yescharacteristic
300C23×C3C2×C3×D411yescharacteristic
301C22×D3—24no—
302C22×D3—22no—
303C22×D3—42no—
304C22×D3—42no—
305C3×D4—24no—
306C3×D4—24no—
307C3×D4—22no—
308C3×D4—22no—
309C3×D4—24no—
310C3×D4—34no—
311C3×D4—44no—
312C3×D4—42no—
313C3×D4—64no—
314C3×D4—62no—
315C3×D4C2×C4×D314yescomplemented
316C3×D4C22×C3×C414yescomplemented
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—32no—
324C2×C3×C4—32no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—82no—
332C2×C3×C4—128no—
333C2×C3×C4—128no—
334C2×C3×C4—122no—
335C2×C3×C4C2×C4×D312yes—
336C2×C3×C4D3×D411yescharacteristic
337C2×C3×C4#23712yes—
338C2×C3×C4C22×C3×C412yes—
339C2×C3×C4C2×C3×D411yescharacteristic
340C2×C3×C4C3×#4112yes—
341C2×C3×C4C23×D311yescharacteristic
342C2×C3×C4C24×C311yescharacteristic
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—48no—
347C2×Dic3—42no—
348C2×Dic3C2×C3×D412yes—
349C2×Dic3C2×C3×D412yes—
350D12—24no—
351D12—22no—
352D12C22×C3×C414yescomplemented
353C4×D3—48no—
354C3×C8—22no—
355C3×C8—22no—
356C3×C8—34no—
357C3×C8—42no—
358C3⋊C8C22×C3×C414yescomplemented
359C2×C32—44no—
360C2×C32#44214yescomplemented
361C2×C32C2×C4×D411yescharacteristic
362C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
363C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
364C3×D3—44no—
365C24—21no—
366C24—61no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—32no—
370C2×D4—34no—
371C2×D4—61no—
372C2×D4—62no—
373C2×D4C3×C4×D314yescomplemented
374C2×D4C22×C3×D312yes—
375C22×C4—21no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—64no—
379C22×C4—64no—
380C22×C4—61no—
381C22×C4—124no—
382C22×C4—124no—
383C22×C4C22×C3×D311yescharacteristic
384D8—316no—
385C2×C8—22no—
386C2×C8—32no—
387C2×C8—34no—
388C2×C8C22×C3×D312yes—
389C4⋊C4—21no—
390C4⋊C4—32no—
391C4⋊C4—61no—
392C4⋊C4C22×C3×D312yes—
393C22⋊C4—64no—
394C22⋊C4—64no—
395C42—32no—
396C42—32no—
397C22×C3—24no—
398C22×C3—24no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—21no—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—48no—
409C22×C3—48no—
410C22×C3—64no—
411C22×C3—82no—
412C22×C3—82no—
413C22×C3—88no—
414C22×C3—122no—
415C22×C3—122no—
416C22×C3—128no—
417C22×C3#67⋊C412yes—
418C22×C3#91112yes—
419C22×C3#91412yes—
420C22×C3C3×C4×D412yes—
421C22×C3C3×AGL(1,ℤ/8)12yes—
422C22×C3C3×AGL(1,ℤ/8)12yes—
423C22×C3C2×D3×D411yescharacteristic
424C22×C3C22×C3×D411yescharacteristic
425C2×D3—24no—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—48no—
429C3×C4—22no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—62no—
435C3×C4—62no—
436C3×C4—82no—
437C3×C4—122no—
438C3×C4#67⋊C412yes—
439C3×C4C3×C4×D412yes—
440C3×C4C22×C4×D311yescharacteristic
441C3×C4C2×#23711yescharacteristic
442C3×C4C23×C3×C411yescharacteristic
443C3×C4C2×C3×#4111yescharacteristic
444Dic3—22no—
445Dic3—22no—
446Dic3—42no—
447C32C2×#44211yescharacteristic, complemented, Sylow
448C23—24no—
449C23—22no—
450C23—42no—
451C23—42no—
452C23—64no—
453C23—62no—
454C23—122no—
455C23—122no—
456C23C3×D3×D411yescharacteristic
457D4—24no—
458D4—22no—
459D4—34no—
460D4—64no—
461D4—62no—
462D4C2×C3×C4×D314yescomplemented
463C2×C4—22no—
464C2×C4—32no—
465C2×C4—32no—
466C2×C4—42no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—128no—
471C2×C4—128no—
472C2×C4—122no—
473C2×C4C2×C3×C4×D312yes—
474C2×C4C3×D3×D411yescharacteristic
475C2×C4C3×#23712yes—
476C2×C4C23×C3×D311yescharacteristic
477C8—22no—
478C8—34no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—24no—
482C2×C3—21no—
483C2×C3—44no—
484C2×C3—44no—
485C2×C3—84no—
486C2×C3—124no—
487C2×C3#478514yesdirect product factor
488C2×C3C3×#44214yescomplemented
489C2×C3C2×#67⋊C411yescharacteristic
490C2×C3C2×#91111yescharacteristic
491C2×C3C2×#91411yescharacteristic
492C2×C3C2×C3×C4×D411yescharacteristic
493C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
494C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
495D3—44no—
496C22—24no—
497C22—42no—
498C22—42no—
499C22—48no—
500C22—64no—
501C22—122no—
502C22—122no—
503C22—128no—
504C22C3×#67⋊C412yes—
505C22C3×#91112yes—
506C22C3×#91412yes—
507C22C2×C3×D3×D411yescharacteristic
508C4—42no—
509C4—62no—
510C4—62no—
511C4—122no—
512C4C3×#67⋊C412yes—
513C4C22×C3×C4×D311yescharacteristic
514C4C2×C3×#23711yescharacteristic
515C3—21no—
516C3C2×#478511yescharacteristic, direct product factor
517C3C2×C3×#44211yescharacteristic, complemented
518C2—44no—
519C2—124no—
520C2C3×#478514yesdirect product factor
521C2C2×C3×#67⋊C411yescharacteristic
522C2C2×C3×#91111yescharacteristic
523C2C2×C3×#91411yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#102C2
socleC23×C32C2×D4
prime radicalC22×C3×#213C2
cosocleC2×C3×C4C24×C3
hypercenterC2×C3×#102D3
hypocenterC3C2×C3×#442
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#102

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Dic3C2semidirect
3. C2×C3×#893C2semidirect
4. C2×C32×#102C2semidirect
5. C2×C3×#821C2semidirect
6. C2×C3×#815C2semidirect
7. C2×C3×#808C2semidirect
8. C3×#4785C2direct (N × H)
10. C2×#4785C3direct (N × H)
19. C3×D4×Dic3C22semidirect
20. C2×C3×#213C4semidirect
27. C3×#893C22semidirect
28. C32×#102C22semidirect
29. C3×#821C22semidirect
30. C3×#815C22semidirect
31. C3×#808C22semidirect
35. C2×D4×Dic3C2×C3semidirect
37. C2×#893C2×C3semidirect
42. C2×C3×#102D3semidirect
43. C2×C3×#102C2×C3semidirect
45. C2×#821C2×C3semidirect
46. C2×#815C2×C3semidirect
47. C2×#808C2×C3semidirect
48. #4785C2×C3direct (N × H)
53. C2×C32×D4C2×C4semidirect
64. C2×C3×D12C2×C4semidirect
73. C3×#213C2×C4semidirect
79. C2×C3×C3⋊C8C2×C4semidirect
110. D4×Dic3C22×C3semidirect
111. C2×#213C3×C4semidirect
118. #893C22×C3semidirect
122. C3×#102C2×D3semidirect
123. C3×#102C22×C3semidirect
125. #821C22×C3semidirect
126. #815C22×C3semidirect
127. #808C22×C3semidirect
139. C32×D4C22×C4semidirect
155. C3×D12C22×C4semidirect
158. C3×C3⋊C8C22×C4semidirect
165. C2×#102C3×D3semidirect
184. C2×C3×D4C4×D3semidirect
185. C2×C3×D4C2×C3×C4semidirect
208. C2×D12C2×C3×C4semidirect
234. #213C2×C3×C4semidirect
240. C2×C3⋊C8C2×C3×C4semidirect
280. #102C2×C3×D3semidirect
315. C3×D4C2×C4×D3semidirect
316. C3×D4C22×C3×C4semidirect
352. D12C22×C3×C4semidirect
358. C3⋊C8C22×C3×C4semidirect
360. C2×C32#442semidirect
373. C2×D4C3×C4×D3semidirect
447. C32C2×#442semidirect
462. D4C2×C3×C4×D3semidirect
487. C2×C3#4785direct (N × H)
488. C2×C3C3×#442semidirect
516. C3C2×#4785direct (N × H)
517. C3C2×C3×#442semidirect
520. C2C3×#4785direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Dic32
C22×C3×#2132
C2×C3×#8932
C2×C32×#1022
C2×C3×#8212
C2×C3×#8152
C2×C3×#8082
C3×#47852
C2×C3×#4423
C2×#47853