Finite Group Databasebeta

← C2×C3×#4811 #11869586

Subgroups — C2×C3×#4811 #11869586

Order 1152.

1001 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#946C211yesmaximal, characteristic
2C2×C3×D3×C4⋊C4C211yesmaximal, characteristic
3C2×C32×#103C211yesmaximal, characteristic, complemented
4C2×C3×#835C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D3)⋊C8C211yesmaximal, characteristic
6C2×C3×#818C211yesmaximal, characteristic, complemented
7C2×C3×#807C211yesmaximal, characteristic, complemented
8C3×#4811C218yesmaximal, direct product factor
9C2×C3×#484—31nomaximal
10C2×#4811C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C3×C4×D3—21no—
13C22×C3×C4×D3C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C3×#946—22no—
17C3×#946C2214yes—
18C2×C3×(C2×D3)⋊C4—21no—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×Dic3⋊C4—21no—
22C2×C3×Dic3⋊C4—21no—
23C22×C3×C3⋊C8C2211yescharacteristic
24C3×D3×C4⋊C4—22no—
25C3×D3×C4⋊C4C2214yes—
26C32×#103C2214yescomplemented
27C3×#835C2214yescomplemented
28C3×(C2×D3)⋊C8C2214yes—
29C3×#818C2214yescomplemented
30C3×#807C2214yescomplemented
31C2×C3×C22⋊Q8—31no—
32C22×C3×C4⋊C4—31no—
33C2×#946C2×C311yescharacteristic
34C2×D3×C4⋊C4C2×C311yescharacteristic
35C3×#484—38no—
36C2×C3×#107—31no—
37C2×C3×#107—31no—
38C2×C3×#103—21no—
39C2×C3×#103—31no—
40C2×C3×#103D311yescharacteristic, complemented
41C2×C3×#103C2×C311yescharacteristic, complemented
42C2×C3×C22⋊C8—31no—
43C2×#835C2×C311yescharacteristic, complemented
44C2×(C2×D3)⋊C8C2×C311yescharacteristic
45C2×#818C2×C311yescharacteristic, complemented
46C2×#807C2×C311yescharacteristic, complemented
47#4811C2×C318yesdirect product factor
48C23×C3×D3D411yescharacteristic
49C2×C32×Q8—21no—
50C2×C32×Q8—22no—
51C2×C32×Q8—22no—
52C2×C32×Q8C2312yes—
53C22×C32×C4—21no—
54C22×C32×C4—21no—
55C22×C32×C4C2311yescharacteristic
56C22×C3×Dic3—21no—
57C22×C3×Dic3—21no—
58C22×C3×Dic3D411yescharacteristic
59C2×C3×C4×D3—22no—
60C2×C3×C4×D3—22no—
61C2×C3×C4×D3—24no—
62C2×C3×C4×D3—44no—
63C2×C3×C4×D3D412yes—
64C2×C3×C4×D3C2312yes—
65C2×C32×C8—22no—
66C2×C32×C8C2312yes—
67C32×C4⋊C4—21no—
68C32×C4⋊C4C2312yes—
69C3×(C2×D3)⋊C4—24no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3—21no—
72C3×C4⋊Dic3C2312yes—
73C3×C4⋊Dic3C2312yes—
74C3×Dic3⋊C4—24no—
75C3×Dic3⋊C4—24no—
76C2×C3×C3⋊C8—22no—
77C2×C3×C3⋊C8C2312yes—
78C2×#484—31noSylow
79C22×C3×Q8—21no—
80C22×C3×Q8C2×D311yescharacteristic
81C22×C3×Q8C22×C311yescharacteristic
82C23×C3×C4—31no—
83C23×C3×C4—61no—
84C22×C4×D3—21no—
85C22×C4×D3C22×C311yescharacteristic
86C22×C3×C8—21no—
87C22×C3×C8—31no—
88C22×C3×C8C2×D311yescharacteristic
89C22×C3×C8C22×C311yescharacteristic
90C3×C22⋊Q8—34no—
91C3×C22⋊Q8—62no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—31no—
94C2×C3×C4⋊C4—31no—
95C2×C3×C4⋊C4—34no—
96C2×C3×C4⋊C4—62no—
97C2×C3×C4⋊C4—61no—
98C2×C3×C4⋊C4—61no—
99C2×C3×C4⋊C4C2×D311yescharacteristic
100C2×C3×C4⋊C4C22×C311yescharacteristic
101C2×C3×C22⋊C4—61no—
102#946—22no—
103#946C22×C314yes—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3C22×C311yescharacteristic
106C2×C4⋊Dic3C22×C311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C22×C3⋊C8C22×C311yescharacteristic
110D3×C4⋊C4—22no—
111D3×C4⋊C4C22×C314yes—
112C3×#107—34no—
113C3×#107—34no—
114C3×#103—24no—
115C3×#103—34no—
116C3×#103C2×D314yescomplemented
117C3×#103C22×C314yescomplemented
118C3×C22⋊C8—34no—
119#835C22×C314yescomplemented
120(C2×D3)⋊C8C22×C314yes—
121#818C22×C314yescomplemented
122#807C22×C314yescomplemented
123C23×C32C2×D411yescharacteristic
124C22×C3×D3—22no—
125C22×C3×D3—22no—
126C22×C3×D3Q1614yes—
127C22×C3×D3C2×D412yes—
128C32×Q8—22no—
129C32×Q8—22no—
130C32×Q8—42no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—42no—
134C2×C32×C4—42no—
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4#4112yes—
137C2×C32×C4#4112yes—
138C2×C32×C4C2411yescharacteristic
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—42no—
143C2×C3×Dic3—42no—
144C2×C3×Dic3C2×D412yes—
145C3×C4×D3—24no—
146C3×C4×D3—22no—
147C3×C4×D3—22no—
148C3×C4×D3—48no—
149C32×C8—22no—
150C3×C3⋊C8—22no—
151C2×C22⋊Q8—31no—
152C22×C4⋊C4—31no—
153#484—38no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#103—31no—
157C2×#103C3×D311yescharacteristic, complemented
158C2×C22⋊C8—31no—
159C24×C3—31no—
160C23×D3C3×D411yescharacteristic
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—24no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—41no—
170C2×C3×Q8—42no—
171C2×C3×Q8C22×D312yes—
172C2×C3×Q8C23×C312yes—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—31no—
179C22×C3×C4—32no—
180C22×C3×C4—32no—
181C22×C3×C4—41no—
182C22×C3×C4—41no—
183C22×C3×C4—64no—
184C22×C3×C4—62no—
185C22×C3×C4—62no—
186C22×C3×C4—61no—
187C22×C3×C4—61no—
188C22×C3×C4—124no—
189C22×C3×C4C22×D311yescharacteristic
190C22×C3×C4C23×C311yescharacteristic
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3C3×D411yescharacteristic
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—24no—
197C2×C4×D3—44no—
198C2×C4×D3C3×D412yes—
199C2×C4×D3C23×C312yes—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—32no—
204C2×C3×C8—42no—
205C2×C3×C8—62no—
206C2×C3×C8C22×D312yes—
207C2×C3×C8C23×C312yes—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—32no—
212C3×C4⋊C4—32no—
213C3×C4⋊C4—41no—
214C3×C4⋊C4—64no—
215C3×C4⋊C4—64no—
216C3×C4⋊C4—61no—
217C3×C4⋊C4—61no—
218C3×C4⋊C4C22×D312yes—
219C3×C4⋊C4C23×C312yes—
220C3×C22⋊C4—64no—
221(C2×D3)⋊C4—24no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3C23×C312yes—
225C4⋊Dic3C23×C312yes—
226Dic3⋊C4—24no—
227Dic3⋊C4—24no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8C23×C312yes—
230C22×C32#12312yes—
231C22×C32C2×Q1612yes—
232C22×C32AGL(1,ℤ/8)12yes—
233C22×C32C22×D411yescharacteristic
234C2×C3×D3—22no—
235C2×C3×D3—22no—
236C2×C3×D3—28no—
237C2×C3×D3C2×Q1614yes—
238C32×C4—42no—
239C32×C4—42no—
240C32×C4#12312yes—
241C32×C4C2×#4111yescharacteristic
242C32×C4C2×#4111yescharacteristic
243C3×Dic3—22no—
244C3×Dic3—42no—
245C3×Dic3—42no—
246C22×Q8C2×C3×D311yescharacteristic
247C23×C4—31no—
248C23×C4—61no—
249C22×C8—31no—
250C22×C8C2×C3×D311yescharacteristic
251C22⋊Q8—34no—
252C22⋊Q8—62no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—34no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4C2×C3×D311yescharacteristic
260C2×C22⋊C4—61no—
261#107—34no—
262#107—34no—
263#103—34no—
264#103C2×C3×D314yescomplemented
265C22⋊C8—34no—
266C23×C3—21no—
267C23×C3—34no—
268C23×C3—32no—
269C23×C3—62no—
270C23×C3—62no—
271C23×C3D3×D411yescharacteristic
272C23×C3C2×C3×D411yescharacteristic
273C22×D3—22no—
274C22×D3—22no—
275C22×D3C3×Q1614yes—
276C22×D3C2×C3×D412yes—
277C3×Q8—24no—
278C3×Q8—22no—
279C3×Q8—22no—
280C3×Q8—22no—
281C3×Q8—22no—
282C3×Q8—42no—
283C3×Q8—44no—
284C3×Q8—42no—
285C3×Q8—42no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—21no—
293C2×C3×C4—22no—
294C2×C3×C4—32no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—44no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—62no—
303C2×C3×C4—64no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4—82no—
309C2×C3×C4—128no—
310C2×C3×C4—122no—
311C2×C3×C4—122no—
312C2×C3×C4#23512yes—
313C2×C3×C4D3×D411yescharacteristic
314C2×C3×C4#23712yes—
315C2×C3×C4C2×C3×D411yescharacteristic
316C2×C3×C4C3×#4112yes—
317C2×C3×C4C3×#4112yes—
318C2×C3×C4C23×D311yescharacteristic
319C2×C3×C4C24×C311yescharacteristic
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3C2×C3×D412yes—
326C4×D3—24no—
327C4×D3—22no—
328C4×D3—22no—
329C4×D3—48no—
330C3×C8—22no—
331C3×C8—22no—
332C3×C8—42no—
333C3×C8—62no—
334C3⋊C8—22no—
335C2×C32#48414yescomplemented
336C2×C32C2×#12311yescharacteristic
337C2×C32C22×Q1611yescharacteristic
338C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
339C3×D3—24no—
340C24—31no—
341C2×Q8—21no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8C22×C3×D312yes—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—31no—
348C22×C4—32no—
349C22×C4—32no—
350C22×C4—64no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—124no—
356C22×C4C22×C3×D311yescharacteristic
357C2×C8—22no—
358C2×C8—32no—
359C2×C8—62no—
360C2×C8C22×C3×D312yes—
361C4⋊C4—21no—
362C4⋊C4—32no—
363C4⋊C4—32no—
364C4⋊C4—64no—
365C4⋊C4—64no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4C22×C3×D312yes—
369C22⋊C4—64no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—21no—
374C22×C3—34no—
375C22×C3—62no—
376C22×C3—62no—
377C22×C3—68no—
378C22×C3#88312yes—
379C22×C3#91412yes—
380C22×C3D3×Q1612yes—
381C22×C3C3×#12312yes—
382C22×C3C2×C3×Q1612yes—
383C22×C3C3×AGL(1,ℤ/8)12yes—
384C22×C3C2×D3×D411yescharacteristic
385C22×C3C22×C3×D411yescharacteristic
386C2×D3—22no—
387C2×D3—22no—
388C2×D3—28no—
389C2×D3C2×C3×Q1614yes—
390C3×C4—22no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—44no—
394C3×C4—42no—
395C3×C4—42no—
396C3×C4—42no—
397C3×C4—42no—
398C3×C4—62no—
399C3×C4—82no—
400C3×C4—122no—
401C3×C4—122no—
402C3×C4#88312yes—
403C3×C4C3×#12312yes—
404C3×C4C2×#23511yescharacteristic
405C3×C4C2×#23711yescharacteristic
406C3×C4C2×C3×#4111yescharacteristic
407C3×C4C2×C3×#4111yescharacteristic
408Dic3—22no—
409Dic3—42no—
410Dic3—42no—
411C32C2×#48411yescharacteristic, complemented, Sylow
412C23—34no—
413C23—32no—
414C23—62no—
415C23—62no—
416C23C3×D3×D411yescharacteristic
417Q8—22no—
418Q8—22no—
419Q8—42no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—32no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—62no—
426C2×C4—64no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—128no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4C3×#23512yes—
435C2×C4C3×D3×D411yescharacteristic
436C2×C4C3×#23712yes—
437C2×C4C23×C3×D311yescharacteristic
438C8—22no—
439C8—62no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—24no—
444C2×C3—64no—
445C2×C3#481114yesdirect product factor
446C2×C3C3×#48414yescomplemented
447C2×C3C2×#88311yescharacteristic
448C2×C3C2×#91411yescharacteristic
449C2×C3C2×D3×Q1611yescharacteristic
450C2×C3C2×C3×#12311yescharacteristic
451C2×C3C22×C3×Q1611yescharacteristic
452C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
453D3—24no—
454C22—34no—
455C22—62no—
456C22—62no—
457C22—68no—
458C22C3×#88312yes—
459C22C3×#91412yes—
460C22C3×D3×Q1612yes—
461C22C2×C3×D3×D411yescharacteristic
462C4—42no—
463C4—42no—
464C4—62no—
465C4—122no—
466C4—122no—
467C4C3×#88312yes—
468C4C2×C3×#23511yescharacteristic
469C4C2×C3×#23711yescharacteristic
470C3—21no—
471C3C2×#481111yescharacteristic, direct product factor
472C3C2×C3×#48411yescharacteristic, complemented
473C2—64no—
474C2C3×#481114yesdirect product factor
475C2C2×C3×#88311yescharacteristic
476C2C2×C3×#91411yescharacteristic
477C2C2×C3×D3×Q1611yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#103C2
socleC23×C32C2×D4
prime radicalC23×C3×D3D4
hypercenterC2×C3×#103D3
hypocenterC3C2×C3×#484
LCSG ▷ C2×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
3. C2×C32×#103C2semidirect
4. C2×C3×#835C2semidirect
6. C2×C3×#818C2semidirect
7. C2×C3×#807C2semidirect
8. C3×#4811C2direct (N × H)
10. C2×#4811C3direct (N × H)
26. C32×#103C22semidirect
27. C3×#835C22semidirect
29. C3×#818C22semidirect
30. C3×#807C22semidirect
40. C2×C3×#103D3semidirect
41. C2×C3×#103C2×C3semidirect
43. C2×#835C2×C3semidirect
45. C2×#818C2×C3semidirect
46. C2×#807C2×C3semidirect
47. #4811C2×C3direct (N × H)
116. C3×#103C2×D3semidirect
117. C3×#103C22×C3semidirect
119. #835C22×C3semidirect
121. #818C22×C3semidirect
122. #807C22×C3semidirect
157. C2×#103C3×D3semidirect
264. #103C2×C3×D3semidirect
335. C2×C32#484semidirect
411. C32C2×#484semidirect
445. C2×C3#4811direct (N × H)
446. C2×C3C3×#484semidirect
471. C3C2×#4811direct (N × H)
472. C3C2×C3×#484semidirect
474. C2C3×#4811direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9462
C2×C3×D3×C4⋊C42
C2×C32×#1032
C2×C3×#8352
C2×C3×(C2×D3)⋊C82
C2×C3×#8182
C2×C3×#8072
C3×#48112
C2×C3×#4843
C2×#48113