Finite Group Databasebeta

← C2×C3×#4468 #11773709

Subgroups — C2×C3×#4468 #11773709

Order 1152.

1013 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
2C2×C32×C22⋊C8C211yesmaximal, characteristic, complemented
3C2×C3×#830C211yesmaximal, characteristic, complemented
4C3×#4468C214yesmaximal, direct product factor
5C2×C3×#323—31nomaximal
6C2×#4468C311yesmaximal, characteristic, direct product factor
7C24×C3×D3C411yescharacteristic
8C23×C32×C4C2211yescharacteristic
9C23×C3×Dic3C411yescharacteristic
10C22×C32×C8—21no—
11C2×C3×(C2×D3)⋊C4—21no—
12C2×C3×(C2×D3)⋊C4—21no—
13C2×C3×(C2×D3)⋊C4—21no—
14C2×C3×(C2×D3)⋊C4—21no—
15C2×C3×(C2×D3)⋊C4C412yes—
16C2×C3×(C2×D3)⋊C4C2212yes—
17C22×C3×C3⋊C8—21no—
18C32×C22⋊C8—21no—
19C32×C22⋊C8C2212yescomplemented
20C3×#830—21no—
21C3×#830C2212yescomplemented
22C22×C3×C22⋊C4—31no—
23C22×(C2×D3)⋊C4C2×C311yescharacteristic
24C2×C3×C22⋊C8—21no—
25C2×C3×C22⋊C8—31no—
26C2×C3×C22⋊C8D311yescharacteristic, complemented
27C2×C3×C22⋊C8C2×C311yescharacteristic, complemented
28C2×#830C2×C311yescharacteristic, complemented
29C3×#323—34no—
30#4468C2×C314yesdirect product factor
31C24×C32C2×C411yescharacteristic
32C23×C3×D3—21no—
33C23×C3×D3—21no—
34C23×C3×D3—22no—
35C23×C3×D3—42no—
36C23×C3×D3—42no—
37C23×C3×D3C814yescomplemented
38C23×C3×D3C2×C412yes—
39C22×C32×C4—42no—
40C22×C32×C4C2×C411yescharacteristic
41C22×C32×C4D411yescharacteristic
42C22×C32×C4D411yescharacteristic
43C22×C32×C4D411yescharacteristic
44C22×C32×C4D411yescharacteristic
45C22×C32×C4C2311yescharacteristic
46C22×C3×Dic3—21no—
47C22×C3×Dic3—21no—
48C22×C3×Dic3—42no—
49C22×C3×Dic3C2×C412yes—
50C2×C32×C8—22no—
51C2×C32×C8—42no—
52C3×(C2×D3)⋊C4—22no—
53C3×(C2×D3)⋊C4—22no—
54C3×(C2×D3)⋊C4—22no—
55C3×(C2×D3)⋊C4—22no—
56C2×C3×C3⋊C8—22no—
57C2×C3×C3⋊C8—42no—
58C2×#323—31noSylow
59C25×C3—31no—
60C24×D3C3×C411yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4—31no—
63C23×C3×C4C2×D311yescharacteristic
64C23×C3×C4C22×C311yescharacteristic
65C23×Dic3C3×C411yescharacteristic
66C22×C3×C8—22no—
67C22×C3×C8—21no—
68C22×C3×C8—21no—
69C22×C3×C8—61no—
70C2×C3×C22⋊C4—32no—
71C2×C3×C22⋊C4—32no—
72C2×C3×C22⋊C4—61no—
73C2×C3×C22⋊C4—61no—
74C2×C3×C22⋊C4—61no—
75C2×C3×C22⋊C4—61no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4C3×C412yes—
81C2×(C2×D3)⋊C4C22×C312yes—
82C22×C3⋊C8—21no—
83C3×C22⋊C8—22no—
84C3×C22⋊C8—21no—
85C3×C22⋊C8—21no—
86C3×C22⋊C8—22no—
87C3×C22⋊C8—32no—
88C3×C22⋊C8—61no—
89C3×C22⋊C8C2×D312yescomplemented
90C3×C22⋊C8C22×C312yescomplemented
91#830—21no—
92#830C22×C312yescomplemented
93C23×C32—22no—
94C23×C32—22no—
95C23×C32C22⋊C411yescharacteristic
96C23×C32C22⋊C411yescharacteristic
97C23×C32C2×C812yes—
98C23×C32M1612yes—
99C23×C32C22×C411yescharacteristic
100C22×C3×D3—24no—
101C22×C3×D3—24no—
102C22×C3×D3—22no—
103C22×C3×D3—22no—
104C22×C3×D3—48no—
105C22×C3×D3—42no—
106C22×C3×D3—42no—
107C22×C3×D3—42no—
108C22×C3×D3—42no—
109C22×C3×D3—48no—
110C22×C3×D3—42no—
111C22×C3×D3—42no—
112C22×C3×D3C2×C814yescomplemented
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C32×C4C22⋊C411yescharacteristic
120C2×C32×C4C22⋊C411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C3×Dic3—22no—
124C2×C3×Dic3—42no—
125C2×C3×Dic3—42no—
126C2×C3×Dic3—42no—
127C32×C8—42no—
128C3×C3⋊C8—42no—
129C22×C22⋊C4—31no—
130C2×C22⋊C8—31no—
131C2×C22⋊C8C3×D311yescharacteristic, complemented
132#323—34no—
133C24×C3—21no—
134C24×C3—34no—
135C24×C3—32no—
136C24×C3—62no—
137C24×C3—61no—
138C24×C3—61no—
139C24×C3—122no—
140C24×C3—122no—
141C24×C3C4×D311yescharacteristic
142C24×C3C2×C3×C411yescharacteristic
143C23×D3—21no—
144C23×D3—21no—
145C23×D3—22no—
146C23×D3—42no—
147C23×D3—42no—
148C23×D3C3×C814yescomplemented
149C23×D3C2×C3×C412yes—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—32no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—44no—
160C22×C3×C4—61no—
161C22×C3×C4—61no—
162C22×C3×C4—122no—
163C22×C3×C4C4×D311yescharacteristic
164C22×C3×C4D1211yescharacteristic
165C22×C3×C4D1211yescharacteristic
166C22×C3×C4#6711yescharacteristic
167C22×C3×C4#6711yescharacteristic
168C22×C3×C4C2×C3×C411yescharacteristic
169C22×C3×C4C3×D411yescharacteristic
170C22×C3×C4C3×D411yescharacteristic
171C22×C3×C4C3×D411yescharacteristic
172C22×C3×C4C3×D411yescharacteristic
173C22×C3×C4C22×D311yescharacteristic
174C22×C3×C4C23×C311yescharacteristic
175C22×Dic3—21no—
176C22×Dic3—21no—
177C22×Dic3—42no—
178C22×Dic3C2×C3×C412yes—
179C2×C3×C8—24no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—44no—
183C2×C3×C8—42no—
184C2×C3×C8—42no—
185C2×C3×C8—62no—
186C2×C3×C8—122no—
187C3×C22⋊C4—62no—
188C3×C22⋊C4—62no—
189C3×C22⋊C4—62no—
190C3×C22⋊C4—62no—
191(C2×D3)⋊C4—22no—
192(C2×D3)⋊C4—22no—
193(C2×D3)⋊C4—22no—
194(C2×D3)⋊C4—22no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—42no—
197C22×C32—24no—
198C22×C32—24no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32C22⋊C812yes—
204C22×C32C22⋊C812yes—
205C22×C32#9912yes—
206C22×C32#10012yes—
207C22×C32C2×C22⋊C411yescharacteristic
208C22×C32C22×C811yescharacteristic
209C22×C32C2×M1611yescharacteristic
210C2×C3×D3—24no—
211C2×C3×D3—48no—
212C2×C3×D3—44no—
213C2×C3×D3—48no—
214C2×C3×D3—42no—
215C2×C3×D3—42no—
216C2×C3×D3—44no—
217C32×C4—21no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—21no—
221C3×Dic3—42no—
222C25—31no—
223C23×C4—31no—
224C23×C4C2×C3×D311yescharacteristic
225C22×C8—21no—
226C22×C8—61no—
227C2×C22⋊C4—32no—
228C2×C22⋊C4—32no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—61no—
231C2×C22⋊C4—61no—
232C2×C22⋊C4—61no—
233C22⋊C8—21no—
234C22⋊C8—32no—
235C22⋊C8—61no—
236C22⋊C8C2×C3×D312yescomplemented
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3—34no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—64no—
250C23×C3—64no—
251C23×C3—62no—
252C23×C3—62no—
253C23×C3—122no—
254C23×C3—122no—
255C23×C3—128no—
256C23×C3—122no—
257C23×C3—122no—
258C23×C3—128no—
259C23×C3—122no—
260C23×C3—122no—
261C23×C3D3×C812yes—
262C23×C3ΓL(1,𝔽25)12yes—
263C23×C3(C2×D3)⋊C411yescharacteristic
264C23×C3(C2×D3)⋊C411yescharacteristic
265C23×C3C3×C22⋊C411yescharacteristic
266C23×C3C3×C22⋊C411yescharacteristic
267C23×C3C2×C3×C812yes—
268C23×C3C3×M1612yes—
269C23×C3C2×C4×D311yescharacteristic
270C23×C3C22×C3×C411yescharacteristic
271C22×D3—24no—
272C22×D3—24no—
273C22×D3—22no—
274C22×D3—22no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—48no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3—48no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3C2×C3×C814yescomplemented
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—62no—
307C2×C3×C4—122no—
308C2×C3×C4—122no—
309C2×C3×C4—122no—
310C2×C3×C4(C2×D3)⋊C411yescharacteristic
311C2×C3×C4(C2×D3)⋊C411yescharacteristic
312C2×C3×C4C3×C22⋊C411yescharacteristic
313C2×C3×C4C3×C22⋊C411yescharacteristic
314C2×C3×C4C2×D1211yescharacteristic
315C2×C3×C4C2×#6711yescharacteristic
316C2×C3×C4C2×C3×D411yescharacteristic
317C2×C3×C4C2×C3×D411yescharacteristic
318C2×Dic3—22no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C3×C8—44no—
323C3×C8—42no—
324C3×C8—42no—
325C3×C8—122no—
326C3⋊C8—42no—
327C2×C32—22no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32#32314yescomplemented
331C2×C32C2×C22⋊C811yescharacteristic
332C2×C32C2×#9911yescharacteristic
333C2×C32C2×#10011yescharacteristic
334C3×D3—44no—
335C24—34no—
336C24—32no—
337C24—62no—
338C24—61no—
339C24—61no—
340C24—122no—
341C24—122no—
342C24C3×C4×D311yescharacteristic
343C22×C4—32no—
344C22×C4—42no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—122no—
348C22×C4C3×C4×D311yescharacteristic
349C22×C4C3×D1211yescharacteristic
350C22×C4C3×D1211yescharacteristic
351C22×C4C3×#6711yescharacteristic
352C22×C4C3×#6711yescharacteristic
353C22×C4C22×C3×D311yescharacteristic
354C2×C8—22no—
355C2×C8—42no—
356C2×C8—62no—
357C2×C8—122no—
358C22⋊C4—62no—
359C22⋊C4—62no—
360C22⋊C4—62no—
361C22⋊C4—62no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—24no—
367C22×C3—24no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—24no—
373C22×C3—24no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3—44no—
384C22×C3—44no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—64no—
388C22×C3—124no—
389C22×C3—128no—
390C22×C3—124no—
391C22×C3—122no—
392C22×C3—122no—
393C22×C3—128no—
394C22×C3#80612yes—
395C22×C3(C2×D3)⋊C812yes—
396C22×C3(C2×D3)⋊C812yes—
397C22×C3#82312yes—
398C22×C3C3×C22⋊C812yes—
399C22×C3C3×C22⋊C812yes—
400C22×C3C3×#9912yes—
401C22×C3C3×#10012yes—
402C22×C3C2×D3×C811yescharacteristic
403C22×C3C2×ΓL(1,𝔽25)11yescharacteristic
404C22×C3C2×(C2×D3)⋊C411yescharacteristic
405C22×C3C2×C3×C22⋊C411yescharacteristic
406C22×C3C22×C3×C811yescharacteristic
407C22×C3C2×C3×M1611yescharacteristic
408C2×D3—24no—
409C2×D3—48no—
410C2×D3—44no—
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—48no—
414C2×D3—44no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—122no—
428Dic3—42no—
429C32C2×#32311yescharacteristic, complemented, Sylow
430C23—22no—
431C23—22no—
432C23—34no—
433C23—62no—
434C23—62no—
435C23—64no—
436C23—64no—
437C23—128no—
438C23—122no—
439C23—122no—
440C23—128no—
441C23—122no—
442C23—122no—
443C23—122no—
444C23—122no—
445C23C3×D3×C812yes—
446C23C3×ΓL(1,𝔽25)12yes—
447C23C3×(C2×D3)⋊C411yescharacteristic
448C23C3×(C2×D3)⋊C411yescharacteristic
449C23C2×C3×C4×D311yescharacteristic
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—62no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4C3×(C2×D3)⋊C411yescharacteristic
461C2×C4C3×(C2×D3)⋊C411yescharacteristic
462C2×C4C2×C3×D1211yescharacteristic
463C2×C4C2×C3×#6711yescharacteristic
464C8—42no—
465C8—122no—
466C2×C3—22no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—22no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—24no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—41no—
477C2×C3—41no—
478C2×C3—42no—
479C2×C3—124no—
480C2×C3#446814yesdirect product factor
481C2×C3C3×#32314yescomplemented
482C2×C3C2×#80611yescharacteristic
483C2×C3C2×(C2×D3)⋊C811yescharacteristic
484C2×C3C2×#82311yescharacteristic
485C2×C3C2×C3×C22⋊C811yescharacteristic
486C2×C3C2×C3×#9911yescharacteristic
487C2×C3C2×C3×#10011yescharacteristic
488D3—44no—
489C22—24no—
490C22—24no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—64no—
496C22—124no—
497C22—124no—
498C22—128no—
499C22—128no—
500C22—122no—
501C22—122no—
502C22C3×#80612yes—
503C22C3×(C2×D3)⋊C812yes—
504C22C3×(C2×D3)⋊C812yes—
505C22C3×#82312yes—
506C22C2×C3×D3×C811yescharacteristic
507C22C2×C3×ΓL(1,𝔽25)11yescharacteristic
508C22C2×C3×(C2×D3)⋊C411yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—122no—
514C3—21no—
515C3C2×#446811yescharacteristic, direct product factor
516C3C2×C3×#32311yescharacteristic, complemented
517C2—21no—
518C2—21no—
519C2—22no—
520C2—124no—
521C2C3×#446814yesdirect product factor
522C2C2×C3×#80611yescharacteristic
523C2C2×C3×(C2×D3)⋊C811yescharacteristic
524C2C2×C3×#82311yescharacteristic
subgroupG/H
centerC23×C3(C2×D3)⋊C4
commutatorC22×C3C22×C3×C8
FrattiniC22×C4C22×C3×D3
FittingC2×C32×C22⋊C8C2
socleC23×C32C22⋊C4
prime radicalC24×C3×D3C4
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×C22⋊C8D3
hypocenterC3C2×C3×#323
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C32×C22⋊C8C2semidirect
3. C2×C3×#830C2semidirect
4. C3×#4468C2direct (N × H)
6. C2×#4468C3direct (N × H)
19. C32×C22⋊C8C22semidirect
21. C3×#830C22semidirect
26. C2×C3×C22⋊C8D3semidirect
27. C2×C3×C22⋊C8C2×C3semidirect
28. C2×#830C2×C3semidirect
30. #4468C2×C3direct (N × H)
37. C23×C3×D3C8semidirect
89. C3×C22⋊C8C2×D3semidirect
90. C3×C22⋊C8C22×C3semidirect
92. #830C22×C3semidirect
112. C22×C3×D3C2×C8semidirect
131. C2×C22⋊C8C3×D3semidirect
148. C23×D3C3×C8semidirect
236. C22⋊C8C2×C3×D3semidirect
283. C22×D3C2×C3×C8semidirect
330. C2×C32#323semidirect
429. C32C2×#323semidirect
480. C2×C3#4468direct (N × H)
481. C2×C3C3×#323semidirect
515. C3C2×#4468direct (N × H)
516. C3C2×C3×#323semidirect
521. C2C3×#4468direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×(C2×D3)⋊C42
C2×C32×C22⋊C82
C2×C3×#8302
C3×#44682
C2×C3×#3233
C2×#44683