Finite Group Databasebeta

← C2×C3×C4×#214 #11859859

Subgroups — C2×C3×C4×#214 #11859859

Order 1152.

1317 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic
2C22×C3×#214C211yesmaximal, characteristic
3C2×C3×C4×Dic6C211yesmaximal, characteristic, complemented
4C2×C3×#832C211yesmaximal, characteristic
5C3×C4×#214C218yesmaximal, direct product factor
6C2×C3×#810C211yesmaximal, characteristic, complemented
7C2×C3×#808C211yesmaximal, characteristic, complemented
8C2×C3×C4×C3⋊C8C211yesmaximal, characteristic, complemented
9C2×C3×C4×SD16—31nomaximal
10C2×C4×#214C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C23×C32×C4—21no—
13C22×C3×Dic6C2211yescharacteristic
14C32×C4×D4—22no—
15C32×C4×D4C2214yes—
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C32×C22⋊C4—21no—
18C2×C32×C42C2211yescharacteristic
19C2×C3×#214C418yesdirect product factor
20C2×C3×#214C2214yes—
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C2×C3×Dic3⋊C4—21no—
23C2×C3×C4×Dic3—21no—
24C22×C3×C3⋊C8C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×C4×Dic6—22no—
27C3×C4×Dic6C2214yescomplemented
28C3×#832C2214yes—
29C3×#810C2214yescomplemented
30C3×#808C2214yescomplemented
31C3×C4×C3⋊C8C2214yescomplemented
32C22×C3×SD16—31no—
33C2×C3×C4×Q8—31no—
34C2×C3×C4×D4—21no—
35C2×C3×C4×D4D311yescharacteristic
36C2×C3×C4×D4C2×C311yescharacteristic
37C22×#214C2×C311yescharacteristic
38C2×C4×Dic6C2×C311yescharacteristic, complemented
39C3×C4×SD16—38no—
40C2×C3×#106—31no—
41C2×C3×#103—31no—
42C2×C3×#102—31no—
43C2×C3×C4×C8—31no—
44C2×#832C2×C311yescharacteristic
45C4×#214C2×C318yesdirect product factor
46C2×#810C2×C311yescharacteristic, complemented
47C2×#808C2×C311yescharacteristic, complemented
48C2×C4×C3⋊C8C2×C311yescharacteristic, complemented
49C24×C32—21no—
50C2×C32×D4—21no—
51C2×C32×D4—22no—
52C2×C32×D4C2×C414yes—
53C2×C32×D4C2312yes—
54C22×C32×C4—21no—
55C22×C32×C4—24no—
56C22×C32×C4—44no—
57C22×C32×C4D411yescharacteristic
58C22×C32×C4D411yescharacteristic
59C22×C32×C4C2311yescharacteristic
60C22×C3×Dic3—21no—
61C22×C3×Dic3—21no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—22no—
64C2×C3×Dic6C2×C414yescomplemented
65C2×C3×Dic6C2312yes—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2312yes—
68C32×C22⋊C4—24no—
69C32×C42D412yes—
70C32×C42C2312yes—
71C3×#214C2×C4116yesdirect product factor
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3C2312yes—
74C3×Dic3⋊C4—24no—
75C3×C4×Dic3—24no—
76C2×C3×C3⋊C8—22no—
77C2×C3×C3⋊C8C2×C414yescomplemented
78C2×C3×C3⋊C8C2312yes—
79C2×C3×C3⋊C8C2312yes—
80C2×C4×SD16—31noSylow
81C22×C3×Q8—31no—
82C22×C3×D4—21no—
83C22×C3×D4C2×D311yescharacteristic
84C22×C3×D4C22×C311yescharacteristic
85C23×C3×C4—22no—
86C23×C3×C4—21no—
87C23×C3×C4—21no—
88C22×Dic6C22×C311yescharacteristic
89C2×C3×SD16—38no—
90C2×C3×SD16—34no—
91C22×C3×C8—31no—
92C22×C3×C8—31no—
93C3×C4×Q8—34no—
94C3×C4×Q8—62no—
95C3×C4×D4—24no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—24no—
99C3×C4×D4C2×D314yes—
100C3×C4×D4C22×C314yes—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—31no—
103C2×C3×C4⋊C4—61no—
104C2×C3×C4⋊C4C2×D311yescharacteristic
105C2×C3×C4⋊C4C22×C311yescharacteristic
106C2×C3×C22⋊C4—22no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C42—21no—
110C2×C3×C42—61no—
111C2×C3×C42C2×D311yescharacteristic
112C2×C3×C42C22×C311yescharacteristic
113C2×#214C3×C418yesdirect product factor
114C2×#214C22×C314yes—
115C2×C4⋊Dic3C22×C311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—21no—
118C22×C3⋊C8C22×C311yescharacteristic
119C22×C3⋊C8C22×C311yescharacteristic
120C4×Dic6—22no—
121C4×Dic6C22×C314yescomplemented
122C3×#106—34no—
123C3×#103—34no—
124C3×#102—34no—
125C3×C4×C8—34no—
126#832C22×C314yes—
127#810C22×C314yescomplemented
128#808C22×C314yescomplemented
129C4×C3⋊C8C22×C314yescomplemented
130C23×C32—24no—
131C23×C32—22no—
132C23×C32—42no—
133C23×C32—42no—
134C23×C32C2×D411yescharacteristic
135C32×D4—24no—
136C32×D4—22no—
137C32×D4C22×C414yes—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—42no—
141C2×C32×C4—48no—
142C2×C32×C4SD1614yescomplemented
143C2×C32×C4C22×C412yes—
144C2×C32×C4C2×D412yes—
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×D412yes—
147C2×C32×C4#4112yes—
148C2×C32×C4C2411yescharacteristic
149C2×C3×Dic3—24no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—42no—
153C3×Dic6—24no—
154C3×Dic6—22no—
155C3×Dic6C22×C414yescomplemented
156C3×C3⋊C8—22no—
157C3×C3⋊C8C22×C414yescomplemented
158C22×SD16—31no—
159C2×C4×Q8—31no—
160C2×C4×D4C3×D311yescharacteristic
161C4×SD16—38no—
162C2×#106—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×C4×C8—31no—
166C24×C3—22no—
167C24×C3—21no—
168C24×C3—21no—
169C2×C3×Q8—34no—
170C2×C3×Q8—32no—
171C2×C3×Q8—62no—
172C2×C3×Q8—61no—
173C2×C3×D4—22no—
174C2×C3×D4—24no—
175C2×C3×D4—22no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—24no—
180C2×C3×D4—22no—
181C2×C3×D4C4×D314yes—
182C2×C3×D4C2×C3×C414yes—
183C2×C3×D4C22×D312yes—
184C2×C3×D4C23×C312yes—
185C22×C3×C4—22no—
186C22×C3×C4—28no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—24no—
190C22×C3×C4—24no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—48no—
195C22×C3×C4—44no—
196C22×C3×C4—44no—
197C22×C3×C4—61no—
198C22×C3×C4—61no—
199C22×C3×C4#6711yescharacteristic
200C22×C3×C4#6711yescharacteristic
201C22×C3×C4C3×D411yescharacteristic
202C22×C3×C4C3×D411yescharacteristic
203C22×C3×C4C22×D311yescharacteristic
204C22×C3×C4C23×C311yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C2×Dic6—21no—
208C2×Dic6—22no—
209C2×Dic6C2×C3×C414yescomplemented
210C2×Dic6C23×C312yes—
211C3×SD16—316no—
212C2×C3×C8—34no—
213C2×C3×C8—32no—
214C2×C3×C8—32no—
215C2×C3×C8—62no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—32no—
221C3×C4⋊C4—64no—
222C3×C4⋊C4—61no—
223C3×C4⋊C4C22×D312yes—
224C3×C4⋊C4C23×C312yes—
225C3×C22⋊C4—28no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C42—22no—
229C3×C42—22no—
230C3×C42—64no—
231C3×C42#6712yes—
232C3×C42C3×D412yes—
233C3×C42C22×D312yes—
234C3×C42C23×C312yes—
235#214C2×C3×C4116yesdirect product factor
236C4⋊Dic3—21no—
237C4⋊Dic3C23×C312yes—
238Dic3⋊C4—24no—
239C4×Dic3—24no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8C2×C3×C414yescomplemented
242C2×C3⋊C8C23×C312yes—
243C2×C3⋊C8C23×C312yes—
244C22×C32—24no—
245C22×C32—42no—
246C22×C32—42no—
247C22×C32—48no—
248C22×C32C4×D412yes—
249C22×C32C2×SD1612yes—
250C22×C32#13512yes—
251C22×C32C22×D411yescharacteristic
252C32×C4—22no—
253C32×C4—42no—
254C32×C4C4×D412yes—
255C32×C4C2×SD1614yescomplemented
256C32×C4C23×C411yescharacteristic
257C32×C4C2×#4111yescharacteristic
258C3×Dic3—24no—
259C3×Dic3—42no—
260C22×Q8—31no—
261C22×D4C2×C3×D311yescharacteristic
262C23×C4—21no—
263C2×SD16—38no—
264C2×SD16—34no—
265C22×C8—31no—
266C22×C8—31no—
267C4×Q8—34no—
268C4×Q8—62no—
269C4×D4—22no—
270C4×D4C2×C3×D314yes—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4C2×C3×D311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C42—61no—
276C2×C42C2×C3×D311yescharacteristic
277#106—34no—
278#103—34no—
279#102—34no—
280C4×C8—34no—
281C23×C3—24no—
282C23×C3—28no—
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—21no—
288C23×C3—44no—
289C23×C3—44no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3C2×#6711yescharacteristic
295C23×C3C2×C3×D411yescharacteristic
296C3×Q8—34no—
297C3×Q8—64no—
298C3×Q8—62no—
299C3×D4—28no—
300C3×D4—24no—
301C3×D4—24no—
302C3×D4—24no—
303C3×D4—22no—
304C3×D4—22no—
305C3×D4—24no—
306C3×D4C2×C4×D314yes—
307C3×D4C22×C3×C414yes—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—21no—
314C2×C3×C4—24no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—42no—
321C2×C3×C4—416no—
322C2×C3×C4—44no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—48no—
326C2×C3×C4—48no—
327C2×C3×C4—64no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—122no—
331C2×C3×C4#21414yesdirect product factor
332C2×C3×C4C3×SD1614yescomplemented
333C2×C3×C4C2×C4×D312yes—
334C2×C3×C4#23512yes—
335C2×C3×C4C2×#6712yes—
336C2×C3×C4C2×#6711yescharacteristic
337C2×C3×C4C2×#6712yes—
338C2×C3×C4C22×C3×C412yes—
339C2×C3×C4C2×C3×D412yes—
340C2×C3×C4C2×C3×D411yescharacteristic
341C2×C3×C4C2×C3×D412yes—
342C2×C3×C4C3×#4112yes—
343C2×C3×C4C23×D311yescharacteristic
344C2×C3×C4C24×C311yescharacteristic
345C2×Dic3—24no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—42no—
349Dic6—24no—
350Dic6—22no—
351Dic6C22×C3×C414yescomplemented
352C3×C8—34no—
353C3×C8—62no—
354C3⋊C8—22no—
355C3⋊C8C22×C3×C414yescomplemented
356C2×C32—44no—
357C2×C32C4×SD1614yescomplemented
358C2×C32C2×C4×D411yescharacteristic
359C2×C32C22×SD1611yescharacteristic
360C2×C32C2×#13511yescharacteristic
361C24—21no—
362C2×Q8—32no—
363C2×Q8—34no—
364C2×Q8—62no—
365C2×Q8—61no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4C3×C4×D314yes—
369C2×D4C22×C3×D312yes—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—44no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4C3×#6711yescharacteristic
376C22×C4C3×#6711yescharacteristic
377C22×C4C22×C3×D311yescharacteristic
378SD16—316no—
379C2×C8—32no—
380C2×C8—34no—
381C2×C8—32no—
382C2×C8—62no—
383C4⋊C4—21no—
384C4⋊C4—32no—
385C4⋊C4—61no—
386C4⋊C4—64no—
387C4⋊C4C22×C3×D312yes—
388C22⋊C4—24no—
389C42—64no—
390C42C3×#6712yes—
391C42C22×C3×D312yes—
392C22×C3—28no—
393C22×C3—24no—
394C22×C3—24no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—21no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—416no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—48no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—48no—
408C22×C3C4×#6712yes—
409C22×C3C2×#21412yes—
410C22×C3#95012yes—
411C22×C3C3×C4×D412yes—
412C22×C3C2×C3×SD1612yes—
413C22×C3C3×#13512yes—
414C22×C3C22×#6711yescharacteristic
415C22×C3C22×C3×D411yescharacteristic
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—24no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—42no—
423C3×C4—44no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—64no—
427C3×C4—122no—
428C3×C4C4×#6712yes—
429C3×C4C2×#21414yesdirect product factor
430C3×C4C3×C4×D412yes—
431C3×C4C2×C3×SD1614yescomplemented
432C3×C4C22×C4×D311yescharacteristic
433C3×C4C2×#23511yescharacteristic
434C3×C4C23×C3×C411yescharacteristic
435C3×C4C2×C3×#4111yescharacteristic
436Dic3—24no—
437Dic3—42no—
438C32C2×C4×SD1611yescharacteristic, complemented, Sylow
439C23—24no—
440C23—22no—
441C23—42no—
442C23—42no—
443C23C2×C3×#6711yescharacteristic
444Q8—34no—
445Q8—64no—
446Q8—62no—
447D4—24no—
448D4—22no—
449D4C2×C3×C4×D314yes—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—48no—
454C2×C4—64no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—122no—
458C2×C4C3×#21414yesdirect product factor
459C2×C4C2×C3×C4×D312yes—
460C2×C4C3×#23512yes—
461C2×C4C2×C3×#6712yes—
462C2×C4C2×C3×#6711yescharacteristic
463C2×C4C2×C3×#6712yes—
464C2×C4C23×C3×D311yescharacteristic
465C8—34no—
466C8—62no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—24no—
471C2×C3—48no—
472C2×C3—44no—
473C2×C3—44no—
474C2×C3C4×#21414yesdirect product factor
475C2×C3C3×C4×SD1614yescomplemented
476C2×C3C2×C4×#6711yescharacteristic
477C2×C3C22×#21411yescharacteristic
478C2×C3C2×#95011yescharacteristic
479C2×C3C2×C3×C4×D411yescharacteristic
480C2×C3C22×C3×SD1611yescharacteristic
481C2×C3C2×C3×#13511yescharacteristic
482C22—24no—
483C22—42no—
484C22—42no—
485C22—48no—
486C22C3×C4×#6712yes—
487C22C2×C3×#21412yes—
488C22C3×#95012yes—
489C22C22×C3×#6711yescharacteristic
490C4—22no—
491C4—42no—
492C4—64no—
493C4—122no—
494C4C3×C4×#6712yes—
495C4C2×C3×#21414yesdirect product factor
496C4C22×C3×C4×D311yescharacteristic
497C4C2×C3×#23511yescharacteristic
498C3—21no—
499C3C2×C4×#21411yescharacteristic, direct product factor
500C3C2×C3×C4×SD1611yescharacteristic, complemented
501C2—44no—
502C2C3×C4×#21414yesdirect product factor
503C2C2×C3×C4×#6711yescharacteristic
504C2C22×C3×#21411yescharacteristic
505C2C2×C3×#95011yescharacteristic
subgroupG/H
centerC22×C3×C4#67
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C4×D4C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
cosocleC2×C3×C4C24×C3
hypercenterC2×C3×C4×D4D3
hypocenterC3C2×C3×C4×SD16
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C42 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4×Dic6C2semidirect
5. C3×C4×#214C2direct (N × H)
6. C2×C3×#810C2semidirect
7. C2×C3×#808C2semidirect
8. C2×C3×C4×C3⋊C8C2semidirect
10. C2×C4×#214C3direct (N × H)
19. C2×C3×#214C4direct (N × H)
27. C3×C4×Dic6C22semidirect
29. C3×#810C22semidirect
30. C3×#808C22semidirect
31. C3×C4×C3⋊C8C22semidirect
38. C2×C4×Dic6C2×C3semidirect
45. C4×#214C2×C3direct (N × H)
46. C2×#810C2×C3semidirect
47. C2×#808C2×C3semidirect
48. C2×C4×C3⋊C8C2×C3semidirect
64. C2×C3×Dic6C2×C4semidirect
71. C3×#214C2×C4direct (N × H)
77. C2×C3×C3⋊C8C2×C4semidirect
113. C2×#214C3×C4direct (N × H)
121. C4×Dic6C22×C3semidirect
127. #810C22×C3semidirect
128. #808C22×C3semidirect
129. C4×C3⋊C8C22×C3semidirect
142. C2×C32×C4SD16semidirect
155. C3×Dic6C22×C4semidirect
157. C3×C3⋊C8C22×C4semidirect
209. C2×Dic6C2×C3×C4semidirect
235. #214C2×C3×C4direct (N × H)
241. C2×C3⋊C8C2×C3×C4semidirect
255. C32×C4C2×SD16semidirect
331. C2×C3×C4#214direct (N × H)
332. C2×C3×C4C3×SD16semidirect
351. Dic6C22×C3×C4semidirect
355. C3⋊C8C22×C3×C4semidirect
357. C2×C32C4×SD16semidirect
429. C3×C4C2×#214direct (N × H)
431. C3×C4C2×C3×SD16semidirect
438. C32C2×C4×SD16semidirect
458. C2×C4C3×#214direct (N × H)
474. C2×C3C4×#214direct (N × H)
475. C2×C3C3×C4×SD16semidirect
495. C4C2×C3×#214direct (N × H)
499. C3C2×C4×#214direct (N × H)
500. C3C2×C3×C4×SD16semidirect
502. C2C3×C4×#214direct (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×D42
C22×C3×#2142
C2×C3×C4×Dic62
C2×C3×#8322
C3×C4×#2142
C2×C3×#8102
C2×C3×#8082
C2×C3×C4×C3⋊C82
C2×C3×C4×SD163
C2×C4×#2143