Finite Group Databasebeta

← C2×C3×#4764 #11873565

Subgroups — C2×C3×#4764 #11873565

Order 1152.

1077 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#940C211yesmaximal, characteristic
2C2×C3×#890C211yesmaximal, characteristic, complemented
3C2×C32×#102C211yesmaximal, characteristic, complemented
4C2×C3×#832C211yesmaximal, characteristic, complemented
5C2×C3×#821C211yesmaximal, characteristic, complemented
6C2×C3×#815C211yesmaximal, characteristic, complemented
7C2×C3×#809C211yesmaximal, characteristic, complemented
8C3×#4764C218yesmaximal, direct product factor
9C2×C3×#490—31nomaximal
10C2×#4764C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C22×C3×#67—21no—
13C22×C3×#67—21no—
14C22×C3×D12C2211yescharacteristic
15C22×C32×C8C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C3×#940—22no—
18C3×#940C2214yes—
19C2×C3×C4⋊Dic3—21no—
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×Dic3⋊C4—21no—
22C2×C3×C4×Dic3C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24C3×#890—22no—
25C3×#890C2214yescomplemented
26C32×#102C2214yescomplemented
27C3×#832C2214yescomplemented
28C3×#821C2214yescomplemented
29C3×#815C2214yescomplemented
30C3×#809C2214yescomplemented
31C2×C3×#127—31no—
32C2×C3×#125—31no—
33C2×#940C2×C311yescharacteristic
34C2×#890C2×C311yescharacteristic, complemented
35C3×#490—38no—
36C2×C3×#102—21no—
37C2×C3×#102—31no—
38C2×C3×#102—31no—
39C2×C3×#102—31no—
40C2×C3×#102D311yescharacteristic, complemented
41C2×C3×#102C2×C311yescharacteristic, complemented
42C2×C3×#97—31no—
43C2×#832C2×C311yescharacteristic, complemented
44C2×#821C2×C311yescharacteristic, complemented
45C2×#815C2×C311yescharacteristic, complemented
46C2×#809C2×C311yescharacteristic, complemented
47#4764C2×C318yesdirect product factor
48C24×C32—21no—
49C23×C3×D3—21no—
50C2×C32×D4—21no—
51C2×C32×D4—22no—
52C2×C32×D4—22no—
53C2×C32×D4C2312yes—
54C22×C32×C4—21no—
55C22×C32×C4C2311yescharacteristic
56C2×C3×#67—24no—
57C2×C3×#67—24no—
58C2×C3×#67—44no—
59C2×C3×#67—44no—
60C22×C3×Dic3—21no—
61C22×C3×Dic3D411yescharacteristic
62C22×C3×Dic3D411yescharacteristic
63C2×C3×D12—22no—
64C2×C3×D12—22no—
65C2×C3×D12—21no—
66C2×C3×D12C2312yes—
67C2×C32×C8—22no—
68C2×C32×C8C2312yes—
69C32×C4⋊C4—21no—
70C32×C4⋊C4C2312yes—
71C3×C4⋊Dic3—21no—
72C3×C4⋊Dic3—24no—
73C3×C4⋊Dic3C2312yes—
74C3×Dic3⋊C4—24no—
75C3×C4×Dic3D412yes—
76C3×C4×Dic3C2312yes—
77C2×C3×C3⋊C8—22no—
78C2×C3×C3⋊C8C2312yes—
79C2×#490—31noSylow
80C22×C3×D4—21no—
81C22×C3×D4—31no—
82C22×C3×D4—61no—
83C22×C3×D4—61no—
84C22×C3×D4C2×D311yescharacteristic
85C22×C3×D4C22×C311yescharacteristic
86C22×#67—21no—
87C22×#67—21no—
88C22×D12C22×C311yescharacteristic
89C22×C3×C8—21no—
90C22×C3×C8—31no—
91C22×C3×C8C2×D311yescharacteristic
92C22×C3×C8C22×C311yescharacteristic
93C3×#127—34no—
94C3×#127—62no—
95C3×#125—34no—
96C3×#125—62no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—31no—
99C2×C3×C4⋊C4—61no—
100C2×C3×C4⋊C4—61no—
101C2×C3×C4⋊C4C2×D311yescharacteristic
102C2×C3×C4⋊C4C22×C311yescharacteristic
103C2×C3×C42—31no—
104#940—22no—
105#940C22×C314yes—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C22×C311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3C22×C311yescharacteristic
110C22×C3⋊C8C22×C311yescharacteristic
111#890—22no—
112#890C22×C314yescomplemented
113C3×#102—24no—
114C3×#102—34no—
115C3×#102—34no—
116C3×#102—34no—
117C3×#102C2×D314yescomplemented
118C3×#102C22×C314yescomplemented
119C3×#97—34no—
120#832C22×C314yescomplemented
121#821C22×C314yescomplemented
122#815C22×C314yescomplemented
123#809C22×C314yescomplemented
124C23×C32—22no—
125C23×C32—42no—
126C23×C32—42no—
127C23×C32—42no—
128C23×C32C2×D411yescharacteristic
129C22×C3×D3—22no—
130C22×C3×D3—42no—
131C22×C3×D3—42no—
132C22×C3×D3—42no—
133C32×D4—22no—
134C32×D4—22no—
135C32×D4—42no—
136C2×C32×C4—22no—
137C2×C32×C4—42no—
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4#4112yes—
140C2×C32×C4#4112yes—
141C2×C32×C4C2411yescharacteristic
142C3×#67—48no—
143C3×#67—48no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—42no—
148C2×C3×Dic3C2×D412yes—
149C2×C3×Dic3C2×D412yes—
150C3×D12—22no—
151C3×D12—22no—
152C3×D12—42no—
153C32×C8—22no—
154C3×C3⋊C8—22no—
155C2×#127—31no—
156C2×#125—31no—
157#490—38no—
158C2×#102—31no—
159C2×#102—31no—
160C2×#102—31no—
161C2×#102C3×D311yescharacteristic, complemented
162C2×#97—31no—
163C24×C3—21no—
164C24×C3—21no—
165C24×C3—41no—
166C24×C3—61no—
167C23×D3—21no—
168C2×C3×D4—24no—
169C2×C3×D4—22no—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—21no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—32no—
177C2×C3×D4—41no—
178C2×C3×D4—42no—
179C2×C3×D4—64no—
180C2×C3×D4—64no—
181C2×C3×D4—62no—
182C2×C3×D4—61no—
183C2×C3×D4—62no—
184C2×C3×D4—124no—
185C2×C3×D4—124no—
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—61no—
195C22×C3×C4C22×D311yescharacteristic
196C22×C3×C4C23×C311yescharacteristic
197C2×#67—24no—
198C2×#67—24no—
199C2×#67—44no—
200C2×#67—44no—
201C22×Dic3—21no—
202C22×Dic3C3×D411yescharacteristic
203C22×Dic3C3×D411yescharacteristic
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—21no—
207C2×D12C23×C312yes—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—32no—
212C2×C3×C8—42no—
213C2×C3×C8—62no—
214C2×C3×C8C22×D312yes—
215C2×C3×C8C23×C312yes—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—32no—
220C3×C4⋊C4—41no—
221C3×C4⋊C4—64no—
222C3×C4⋊C4—64no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4C22×D312yes—
225C3×C4⋊C4C23×C312yes—
226C3×C42—32no—
227C3×C42—32no—
228C4⋊Dic3—21no—
229C4⋊Dic3—24no—
230C4⋊Dic3C23×C312yes—
231Dic3⋊C4—24no—
232C4×Dic3C3×D412yes—
233C4×Dic3C23×C312yes—
234C2×C3⋊C8—22no—
235C2×C3⋊C8C23×C312yes—
236C22×C32—42no—
237C22×C32—42no—
238C22×C32—42no—
239C22×C32—84no—
240C22×C32#12412yes—
241C22×C32AGL(1,ℤ/8)12yes—
242C22×C32AGL(1,ℤ/8)12yes—
243C22×C32C22×D411yescharacteristic
244C2×C3×D3—42no—
245C2×C3×D3—42no—
246C2×C3×D3—42no—
247C2×C3×D3—84no—
248C32×C4—42no—
249C32×C4#12412yes—
250C32×C4C2×#4111yescharacteristic
251C32×C4C2×#4111yescharacteristic
252C3×Dic3—22no—
253C3×Dic3—22no—
254C3×Dic3—42no—
255C22×D4—31no—
256C22×D4—61no—
257C22×D4—61no—
258C22×D4C2×C3×D311yescharacteristic
259C22×C8—31no—
260C22×C8C2×C3×D311yescharacteristic
261#127—34no—
262#127—62no—
263#125—34no—
264#125—62no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4C2×C3×D311yescharacteristic
269C2×C42—31no—
270#102—34no—
271#102—34no—
272#102—34no—
273#102C2×C3×D314yescomplemented
274#97—34no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—21no—
278C23×C3—42no—
279C23×C3—44no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—62no—
287C23×C3—82no—
288C23×C3—82no—
289C23×C3—122no—
290C23×C3—122no—
291C23×C3—122no—
292C23×C3D3×D411yescharacteristic
293C23×C3C2×C3×D411yescharacteristic
294C22×D3—22no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—42no—
298C3×D4—24no—
299C3×D4—22no—
300C3×D4—22no—
301C3×D4—22no—
302C3×D4—22no—
303C3×D4—42no—
304C3×D4—44no—
305C3×D4—42no—
306C3×D4—42no—
307C3×D4—62no—
308C3×D4—62no—
309C3×D4—128no—
310C3×D4—128no—
311C3×D4—122no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—32no—
319C2×C3×C4—32no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—62no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—82no—
327C2×C3×C4—122no—
328C2×C3×C4#23512yes—
329C2×C3×C4D3×D411yescharacteristic
330C2×C3×C4#23712yes—
331C2×C3×C4C2×C3×D411yescharacteristic
332C2×C3×C4C3×#4112yes—
333C2×C3×C4C3×#4112yes—
334C2×C3×C4C23×D311yescharacteristic
335C2×C3×C4C24×C311yescharacteristic
336#67—48no—
337#67—48no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—42no—
342C2×Dic3C2×C3×D412yes—
343C2×Dic3C2×C3×D412yes—
344D12—22no—
345D12—22no—
346D12—42no—
347C3×C8—22no—
348C3×C8—22no—
349C3×C8—42no—
350C3×C8—62no—
351C3⋊C8—22no—
352C2×C32—82no—
353C2×C32#49014yescomplemented
354C2×C32C2×#12411yescharacteristic
355C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
356C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
357C3×D3—82no—
358C24—21no—
359C24—61no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—22no—
363C2×D4—32no—
364C2×D4—64no—
365C2×D4—64no—
366C2×D4—62no—
367C2×D4—61no—
368C2×D4—62no—
369C2×D4—124no—
370C2×D4—124no—
371C2×D4C22×C3×D312yes—
372C22×C4—21no—
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—61no—
376C22×C4C22×C3×D311yescharacteristic
377C2×C8—22no—
378C2×C8—32no—
379C2×C8—62no—
380C2×C8C22×C3×D312yes—
381C4⋊C4—21no—
382C4⋊C4—32no—
383C4⋊C4—64no—
384C4⋊C4—64no—
385C4⋊C4—61no—
386C4⋊C4C22×C3×D312yes—
387C42—32no—
388C42—32no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—21no—
393C22×C3—44no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—88no—
403C22×C3—84no—
404C22×C3—84no—
405C22×C3—122no—
406C22×C3—122no—
407C22×C3—122no—
408C22×C3—244no—
409C22×C3#88512yes—
410C22×C3#91112yes—
411C22×C3#91412yes—
412C22×C3C3×#12412yes—
413C22×C3C3×AGL(1,ℤ/8)12yes—
414C22×C3C3×AGL(1,ℤ/8)12yes—
415C22×C3C2×D3×D411yescharacteristic
416C22×C3C22×C3×D411yescharacteristic
417C2×D3—42no—
418C2×D3—42no—
419C2×D3—42no—
420C2×D3—84no—
421C3×C4—22no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—62no—
427C3×C4—62no—
428C3×C4—82no—
429C3×C4—122no—
430C3×C4#88512yes—
431C3×C4C3×#12412yes—
432C3×C4C2×#23511yescharacteristic
433C3×C4C2×#23711yescharacteristic
434C3×C4C2×C3×#4111yescharacteristic
435C3×C4C2×C3×#4111yescharacteristic
436Dic3—22no—
437Dic3—22no—
438Dic3—42no—
439C32C2×#49011yescharacteristic, complemented, Sylow
440C23—22no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—62no—
445C23—122no—
446C23—122no—
447C23—122no—
448C23C3×D3×D411yescharacteristic
449D4—22no—
450D4—22no—
451D4—42no—
452D4—62no—
453D4—62no—
454D4—128no—
455D4—128no—
456D4—122no—
457C2×C4—22no—
458C2×C4—32no—
459C2×C4—32no—
460C2×C4—42no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—122no—
465C2×C4C3×#23512yes—
466C2×C4C3×D3×D411yescharacteristic
467C2×C4C3×#23712yes—
468C2×C4C23×C3×D311yescharacteristic
469C8—22no—
470C8—62no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—24no—
475C2×C3—84no—
476C2×C3—82no—
477C2×C3—82no—
478C2×C3—242no—
479C2×C3#476414yesdirect product factor
480C2×C3C3×#49014yescomplemented
481C2×C3C2×#88511yescharacteristic
482C2×C3C2×#91111yescharacteristic
483C2×C3C2×#91411yescharacteristic
484C2×C3C2×C3×#12411yescharacteristic
485C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
486C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
487D3—82no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—84no—
492C22—122no—
493C22—122no—
494C22—122no—
495C22—244no—
496C22C3×#88512yes—
497C22C3×#91112yes—
498C22C3×#91412yes—
499C22C2×C3×D3×D411yescharacteristic
500C4—42no—
501C4—62no—
502C4—62no—
503C4—122no—
504C4C3×#88512yes—
505C4C2×C3×#23511yescharacteristic
506C4C2×C3×#23711yescharacteristic
507C3—21no—
508C3C2×#476411yescharacteristic, direct product factor
509C3C2×C3×#49011yescharacteristic, complemented
510C2—82no—
511C2—242no—
512C2C3×#476414yesdirect product factor
513C2C2×C3×#88511yescharacteristic
514C2C2×C3×#91111yescharacteristic
515C2C2×C3×#91411yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#102C2
socleC23×C32C2×D4
prime radicalC2×C3×#940C2
hypercenterC2×C3×#102D3
hypocenterC3C2×C3×#490
LCSG ▷ C2×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
2. C2×C3×#890C2semidirect
3. C2×C32×#102C2semidirect
4. C2×C3×#832C2semidirect
5. C2×C3×#821C2semidirect
6. C2×C3×#815C2semidirect
7. C2×C3×#809C2semidirect
8. C3×#4764C2direct (N × H)
10. C2×#4764C3direct (N × H)
25. C3×#890C22semidirect
26. C32×#102C22semidirect
27. C3×#832C22semidirect
28. C3×#821C22semidirect
29. C3×#815C22semidirect
30. C3×#809C22semidirect
34. C2×#890C2×C3semidirect
40. C2×C3×#102D3semidirect
41. C2×C3×#102C2×C3semidirect
43. C2×#832C2×C3semidirect
44. C2×#821C2×C3semidirect
45. C2×#815C2×C3semidirect
46. C2×#809C2×C3semidirect
47. #4764C2×C3direct (N × H)
112. #890C22×C3semidirect
117. C3×#102C2×D3semidirect
118. C3×#102C22×C3semidirect
120. #832C22×C3semidirect
121. #821C22×C3semidirect
122. #815C22×C3semidirect
123. #809C22×C3semidirect
161. C2×#102C3×D3semidirect
273. #102C2×C3×D3semidirect
353. C2×C32#490semidirect
439. C32C2×#490semidirect
479. C2×C3#4764direct (N × H)
480. C2×C3C3×#490semidirect
508. C3C2×#4764direct (N × H)
509. C3C2×C3×#490semidirect
512. C2C3×#4764direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9402
C2×C3×#8902
C2×C32×#1022
C2×C3×#8322
C2×C3×#8212
C2×C3×#8152
C2×C3×#8092
C3×#47642
C2×C3×#4903
C2×#47643