Finite Group Databasebeta

← C2×C3×#4864 #11866675

Subgroups — C2×C3×#4864 #11866675

Order 1152.

1193 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#204C211yesmaximal, characteristic, complemented
2C22×C3×D3×C8C211yesmaximal, characteristic, complemented
3C2×C3×#897C211yesmaximal, characteristic, complemented
4C2×C3×#895C211yesmaximal, characteristic
5C2×C32×#106C211yesmaximal, characteristic, complemented
6C2×C3×#810C211yesmaximal, characteristic, complemented
7C2×C3×#809C211yesmaximal, characteristic, complemented
8C3×#4864C218yesmaximal, direct product factor
9C2×C3×#465—31nomaximal
10C2×#4864C311yesmaximal, characteristic, direct product factor
11C22×C3×D12—21no—
12C22×C3×D12C2211yescharacteristic
13C22×C3×C4×D3C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C22×C32×C8C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C3×(C2×D3)⋊C4—21no—
19C2×C3×(C2×D3)⋊C4—21no—
20C2×C3×C4⋊Dic3—21no—
21C22×C3×C3⋊C8C2211yescharacteristic
22C2×C3×#204—24no—
23C2×C3×#204C2214yescomplemented
24C2×C3×D3×C8—24no—
25C2×C3×D3×C8C2214yescomplemented
26C3×#897—22no—
27C3×#897C2214yescomplemented
28C3×#895—22no—
29C3×#895C2214yes—
30C32×#106C2214yescomplemented
31C3×#810C2214yescomplemented
32C3×#809C2214yescomplemented
33C22×C3×SD16—31no—
34C23×C3×C8—31no—
35C2×C3×C22⋊Q8—31no—
36C2×C3×#121—31no—
37C22×#204C2×C311yescharacteristic, complemented
38C22×D3×C8C2×C311yescharacteristic, complemented
39C2×#897C2×C311yescharacteristic, complemented
40C2×#895C2×C311yescharacteristic
41C3×#465—38no—
42C2×C3×#106—21no—
43C2×C3×#106D311yescharacteristic, complemented
44C2×C3×#106C2×C311yescharacteristic, complemented
45C2×C3×#103—31no—
46C2×C3×#102—31no—
47C2×#810C2×C311yescharacteristic, complemented
48C2×#809C2×C311yescharacteristic, complemented
49#4864C2×C318yesdirect product factor
50C23×C3×D3—21no—
51C23×C3×D3D411yescharacteristic
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4C2311yescharacteristic
55C22×C3×Dic3—21no—
56C22×C3×Dic3D411yescharacteristic
57C2×C3×D12—22no—
58C2×C3×D12—22no—
59C2×C3×D12—24no—
60C2×C3×D12—21no—
61C2×C3×D12—44no—
62C2×C3×D12C2312yes—
63C2×C3×C4×D3—22no—
64C2×C3×C4×D3—22no—
65C2×C3×C4×D3D412yes—
66C2×C3×C4×D3C2312yes—
67C2×C3×Dic6—22no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2312yes—
71C2×C32×C8D414yescomplemented
72C2×C32×C8C2312yes—
73C32×C4⋊C4—21no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4C2312yes—
76C32×C4⋊C4C2312yes—
77C3×(C2×D3)⋊C4—24no—
78C3×(C2×D3)⋊C4—24no—
79C3×C4⋊Dic3—24no—
80C2×C3×C3⋊C8—22no—
81C2×C3×C3⋊C8C2312yes—
82C3×#204—28no—
83C3×D3×C8—28no—
84C2×#465—31noSylow
85C22×C3×Q8—31no—
86C22×C3×D4—31no—
87C22×C3×D4—61no—
88C23×C3×C4—31no—
89C22×D12—21no—
90C22×D12C22×C311yescharacteristic
91C22×C4×D3C22×C311yescharacteristic
92C22×Dic6C22×C311yescharacteristic
93C2×C3×SD16—34no—
94C2×C3×SD16—64no—
95C22×C3×C8—21no—
96C22×C3×C8—31no—
97C22×C3×C8—34no—
98C22×C3×C8—64no—
99C22×C3×C8C2×D311yescharacteristic
100C22×C3×C8C22×C311yescharacteristic
101C3×C22⋊Q8—34no—
102C3×C22⋊Q8—62no—
103C3×#121—34no—
104C3×#121—62no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—61no—
108C2×C3×C4⋊C4C2×D311yescharacteristic
109C2×C3×C4⋊C4C2×D311yescharacteristic
110C2×C3×C4⋊C4C22×C311yescharacteristic
111C2×C3×C4⋊C4C22×C311yescharacteristic
112C2×C3×C22⋊C4—61no—
113C2×C3×C22⋊C4—61no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×C4⋊Dic3—21no—
117C22×C3⋊C8C22×C311yescharacteristic
118C2×#204—24no—
119C2×#204C22×C314yescomplemented
120C2×D3×C8—24no—
121C2×D3×C8C22×C314yescomplemented
122#897—22no—
123#897C22×C314yescomplemented
124#895—22no—
125#895C22×C314yes—
126C3×#106—24no—
127C3×#106C2×D314yescomplemented
128C3×#106C22×C314yescomplemented
129C3×#103—34no—
130C3×#102—34no—
131#810C22×C314yescomplemented
132#809C22×C314yescomplemented
133C23×C32C2×D411yescharacteristic
134C22×C3×D3—22no—
135C22×C3×D3—22no—
136C22×C3×D3—22no—
137C22×C3×D3—42no—
138C22×C3×D3—42no—
139C22×C3×D3—42no—
140C22×C3×D3SD1614yescomplemented
141C22×C3×D3C2×D412yes—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—42no—
145C2×C32×C4—42no—
146C2×C32×C4C2×D412yes—
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4#4112yes—
149C2×C32×C4C2411yescharacteristic
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—42no—
153C2×C3×Dic3C2×D412yes—
154C3×D12—22no—
155C3×D12—22no—
156C3×D12—48no—
157C3×D12—42no—
158C3×C4×D3—24no—
159C3×C4×D3—22no—
160C3×C4×D3—22no—
161C3×Dic6—22no—
162C3×Dic6—22no—
163C3×Dic6—42no—
164C32×C8C2×D414yescomplemented
165C3×C3⋊C8—22no—
166C22×SD16—31no—
167C23×C8—31no—
168C2×C22⋊Q8—31no—
169C2×#121—31no—
170#465—38no—
171C2×#106C3×D311yescharacteristic, complemented
172C2×#103—31no—
173C2×#102—31no—
174C24×C3—31no—
175C24×C3—61no—
176C23×D3—21no—
177C23×D3C3×D411yescharacteristic
178C2×C3×Q8—32no—
179C2×C3×Q8—61no—
180C2×C3×Q8—62no—
181C2×C3×Q8—62no—
182C2×C3×D4—32no—
183C2×C3×D4—64no—
184C2×C3×D4—62no—
185C2×C3×D4—61no—
186C2×C3×D4—62no—
187C2×C3×D4—124no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—31no—
194C22×C3×C4—32no—
195C22×C3×C4—32no—
196C22×C3×C4—41no—
197C22×C3×C4—41no—
198C22×C3×C4—62no—
199C22×C3×C4—62no—
200C22×C3×C4—61no—
201C22×C3×C4C22×D311yescharacteristic
202C22×C3×C4C23×C311yescharacteristic
203C22×Dic3—21no—
204C22×Dic3C3×D411yescharacteristic
205C2×D12—22no—
206C2×D12—22no—
207C2×D12—24no—
208C2×D12—21no—
209C2×D12—44no—
210C2×D12C23×C312yes—
211C2×C4×D3—22no—
212C2×C4×D3—22no—
213C2×C4×D3C3×D412yes—
214C2×C4×D3C23×C312yes—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6—21no—
218C2×Dic6C23×C312yes—
219C3×SD16—68no—
220C2×C3×C8—24no—
221C2×C3×C8—22no—
222C2×C3×C8—32no—
223C2×C3×C8—62no—
224C2×C3×C8—68no—
225C2×C3×C8D1214yescomplemented
226C2×C3×C8C3×D414yescomplemented
227C2×C3×C8C22×D312yes—
228C2×C3×C8C23×C312yes—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—41no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4—64no—
238C3×C4⋊C4C22×D312yes—
239C3×C4⋊C4C22×D312yes—
240C3×C4⋊C4C23×C312yes—
241C3×C4⋊C4C23×C312yes—
242C3×C22⋊C4—64no—
243C3×C22⋊C4—64no—
244(C2×D3)⋊C4—24no—
245(C2×D3)⋊C4—24no—
246C4⋊Dic3—24no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8C23×C312yes—
249#204—28no—
250D3×C8—28no—
251C22×C32#12112yes—
252C22×C32C2×SD1612yes—
253C22×C32#13512yes—
254C22×C32C22×D411yescharacteristic
255C2×C3×D3—22no—
256C2×C3×D3—22no—
257C2×C3×D3—28no—
258C2×C3×D3—42no—
259C2×C3×D3—42no—
260C2×C3×D3—42no—
261C2×C3×D3—84no—
262C2×C3×D3C2×SD1614yescomplemented
263C32×C4—42no—
264C32×C4—42no—
265C32×C4#12112yes—
266C32×C4C22×D411yescharacteristic
267C32×C4C2×#4111yescharacteristic
268C3×Dic3—22no—
269C3×Dic3—42no—
270C22×Q8—31no—
271C22×D4—31no—
272C22×D4—61no—
273C23×C4—31no—
274C2×SD16—34no—
275C2×SD16—64no—
276C22×C8—31no—
277C22×C8—34no—
278C22×C8—64no—
279C22×C8C2×C3×D311yescharacteristic
280C22⋊Q8—34no—
281C22⋊Q8—62no—
282#121—34no—
283#121—62no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4C2×C3×D311yescharacteristic
286C2×C4⋊C4C2×C3×D311yescharacteristic
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289#106C2×C3×D314yescomplemented
290#103—34no—
291#102—34no—
292C23×C3—21no—
293C23×C3—34no—
294C23×C3—32no—
295C23×C3—62no—
296C23×C3—62no—
297C23×C3—62no—
298C23×C3—122no—
299C23×C3—122no—
300C23×C3—122no—
301C23×C3D3×D411yescharacteristic
302C23×C3C2×C3×D411yescharacteristic
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3C3×SD1614yescomplemented
310C22×D3C2×C3×D412yes—
311C3×Q8—62no—
312C3×Q8—62no—
313C3×Q8—122no—
314C3×D4—62no—
315C3×D4—62no—
316C3×D4—128no—
317C3×D4—122no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—32no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—62no—
334C2×C3×C4—64no—
335C2×C3×C4—62no—
336C2×C3×C4—62no—
337C2×C3×C4—62no—
338C2×C3×C4—82no—
339C2×C3×C4—82no—
340C2×C3×C4—122no—
341C2×C3×C4C2×D1212yes—
342C2×C3×C4D3×D411yescharacteristic
343C2×C3×C4#23912yes—
344C2×C3×C4C2×C3×D412yes—
345C2×C3×C4C2×C3×D411yescharacteristic
346C2×C3×C4C3×#4112yes—
347C2×C3×C4C23×D311yescharacteristic
348C2×C3×C4C24×C311yescharacteristic
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—42no—
352C2×Dic3C2×C3×D412yes—
353D12—22no—
354D12—22no—
355D12—48no—
356D12—42no—
357C4×D3—24no—
358C4×D3—22no—
359C4×D3—22no—
360Dic6—22no—
361Dic6—22no—
362Dic6—42no—
363C3×C8—24no—
364C3×C8—62no—
365C3×C8C2×D1214yescomplemented
366C3×C8C2×C3×D414yescomplemented
367C3⋊C8—22no—
368C2×C32#46514yescomplemented
369C2×C32C2×#12111yescharacteristic
370C2×C32C22×SD1611yescharacteristic
371C2×C32C2×#13511yescharacteristic
372C3×D3—24no—
373C3×D3—82no—
374C24—31no—
375C24—61no—
376C2×Q8—32no—
377C2×Q8—61no—
378C2×Q8—62no—
379C2×Q8—62no—
380C2×D4—32no—
381C2×D4—64no—
382C2×D4—62no—
383C2×D4—61no—
384C2×D4—62no—
385C2×D4—124no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—31no—
389C22×C4—32no—
390C22×C4—32no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—61no—
394C22×C4C22×C3×D311yescharacteristic
395SD16—68no—
396C2×C8—32no—
397C2×C8—62no—
398C2×C8—68no—
399C2×C8C3×D1214yescomplemented
400C2×C8C22×C3×D312yes—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—64no—
404C4⋊C4C22×C3×D312yes—
405C4⋊C4C22×C3×D312yes—
406C22⋊C4—64no—
407C22⋊C4—64no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—21no—
412C22×C3—34no—
413C22×C3—68no—
414C22×C3—62no—
415C22×C3—62no—
416C22×C3—122no—
417C22×C3—122no—
418C22×C3—122no—
419C22×C3—244no—
420C22×C3#89512yes—
421C22×C3D3×SD1612yes—
422C22×C3#91612yes—
423C22×C3C3×#12112yes—
424C22×C3C2×C3×SD1612yes—
425C22×C3C3×#13512yes—
426C22×C3C2×D3×D411yescharacteristic
427C22×C3C22×C3×D411yescharacteristic
428C2×D3—22no—
429C2×D3—28no—
430C2×D3—22no—
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—84no—
435C2×D3C2×C3×SD1614yescomplemented
436C3×C4—22no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—62no—
444C3×C4—82no—
445C3×C4—82no—
446C3×C4—122no—
447C3×C4#89512yes—
448C3×C4C3×#12112yes—
449C3×C4C22×D1211yescharacteristic
450C3×C4C2×#23911yescharacteristic
451C3×C4C22×C3×D411yescharacteristic
452C3×C4C2×C3×#4111yescharacteristic
453Dic3—22no—
454Dic3—42no—
455C32C2×#46511yescharacteristic, complemented, Sylow
456C23—34no—
457C23—32no—
458C23—62no—
459C23—62no—
460C23—62no—
461C23—122no—
462C23—122no—
463C23—122no—
464C23C3×D3×D411yescharacteristic
465Q8—62no—
466Q8—62no—
467Q8—122no—
468D4—62no—
469D4—62no—
470D4—128no—
471D4—122no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—32no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—62no—
478C2×C4—64no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—122no—
483C2×C4C2×C3×D1212yes—
484C2×C4C3×D3×D411yescharacteristic
485C2×C4C3×#23912yes—
486C2×C4C23×C3×D311yescharacteristic
487C8—62no—
488C8C2×C3×D1214yescomplemented
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—24no—
493C2×C3—64no—
494C2×C3—242no—
495C2×C3#486414yesdirect product factor
496C2×C3C3×#46514yescomplemented
497C2×C3C2×#89511yescharacteristic
498C2×C3C2×D3×SD1611yescharacteristic
499C2×C3C2×#91611yescharacteristic
500C2×C3C2×C3×#12111yescharacteristic
501C2×C3C22×C3×SD1611yescharacteristic
502C2×C3C2×C3×#13511yescharacteristic
503D3—24no—
504D3—82no—
505C22—34no—
506C22—62no—
507C22—62no—
508C22—68no—
509C22—122no—
510C22—122no—
511C22—122no—
512C22—244no—
513C22C3×#89512yes—
514C22C3×D3×SD1612yes—
515C22C3×#91612yes—
516C22C2×C3×D3×D411yescharacteristic
517C4—42no—
518C4—42no—
519C4—62no—
520C4—122no—
521C4C3×#89512yes—
522C4C22×C3×D1211yescharacteristic
523C4C2×C3×#23911yescharacteristic
524C3—21no—
525C3C2×#486411yescharacteristic, direct product factor
526C3C2×C3×#46511yescharacteristic, complemented
527C2—64no—
528C2—242no—
529C2C3×#486414yesdirect product factor
530C2C2×C3×#89511yescharacteristic
531C2C2×C3×D3×SD1611yescharacteristic
532C2C2×C3×#91611yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#106C2
socleC23×C32C2×D4
prime radicalC2×C3×#895C2
hypercenterC2×C3×#106D3
hypocenterC3C2×C3×#465
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#106

Complements

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

NH ≅ G/Ntype
1. C22×C3×#204C2semidirect
2. C22×C3×D3×C8C2semidirect
3. C2×C3×#897C2semidirect
5. C2×C32×#106C2semidirect
6. C2×C3×#810C2semidirect
7. C2×C3×#809C2semidirect
8. C3×#4864C2direct (N × H)
10. C2×#4864C3direct (N × H)
23. C2×C3×#204C22semidirect
25. C2×C3×D3×C8C22semidirect
27. C3×#897C22semidirect
30. C32×#106C22semidirect
31. C3×#810C22semidirect
32. C3×#809C22semidirect
37. C22×#204C2×C3semidirect
38. C22×D3×C8C2×C3semidirect
39. C2×#897C2×C3semidirect
43. C2×C3×#106D3semidirect
44. C2×C3×#106C2×C3semidirect
47. C2×#810C2×C3semidirect
48. C2×#809C2×C3semidirect
49. #4864C2×C3direct (N × H)
71. C2×C32×C8D4semidirect
119. C2×#204C22×C3semidirect
121. C2×D3×C8C22×C3semidirect
123. #897C22×C3semidirect
127. C3×#106C2×D3semidirect
128. C3×#106C22×C3semidirect
131. #810C22×C3semidirect
132. #809C22×C3semidirect
140. C22×C3×D3SD16semidirect
164. C32×C8C2×D4semidirect
171. C2×#106C3×D3semidirect
225. C2×C3×C8D12semidirect
226. C2×C3×C8C3×D4semidirect
262. C2×C3×D3C2×SD16semidirect
289. #106C2×C3×D3semidirect
309. C22×D3C3×SD16semidirect
365. C3×C8C2×D12semidirect
366. C3×C8C2×C3×D4semidirect
368. C2×C32#465semidirect
399. C2×C8C3×D12semidirect
435. C2×D3C2×C3×SD16semidirect
455. C32C2×#465semidirect
488. C8C2×C3×D12semidirect
495. C2×C3#4864direct (N × H)
496. C2×C3C3×#465semidirect
525. C3C2×#4864direct (N × H)
526. C3C2×C3×#465semidirect
529. C2C3×#4864direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2042
C22×C3×D3×C82
C2×C3×#8972
C2×C3×#8952
C2×C32×#1062
C2×C3×#8102
C2×C3×#8092
C3×#48642
C2×C3×#4653
C2×#48643