Finite Group Databasebeta

← C2×C3×#4753 #11864863

Subgroups — C2×C3×#4753 #11864863

Order 1152.

1529 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1007C211yesmaximal, characteristic, complemented
2C2×C3×#947C211yesmaximal, characteristic, complemented
3C2×C3×#908C212yesmaximal, complemented
4C2×C32×#101C211yesmaximal, characteristic, complemented
5C2×C3×#825C212yesmaximal, complemented
6C3×#4753C218yesmaximal, direct product factor
7C2×C3×#454—31nomaximal
8C2×#4753C311yesmaximal, characteristic, direct product factor
9C22×C32×Q8C2211yescharacteristic
10C3×#1007—22no—
11C3×#1007—24no—
12C3×#1007C2214yescomplemented
13C2×C3×#239—22no—
14C2×C3×D3×Q8—22no—
15C2×C3×#235—22no—
16C2×C3×#235C2212yes—
17C22×C3×D12C2211yescharacteristic
18C2×C32×M16C2212yes—
19C3×#947—22no—
20C3×#947C2214yescomplemented
21C2×C3×(C2×D3)⋊C4—22no—
22C2×C3×C4×Dic3C2211yescharacteristic
23C3×#908—24no—
24C3×#908C2218yescomplemented
25C2×C3×D24—22no—
26C2×C3×#204—22no—
27C32×#101C2214yescomplemented
28C3×#825C2218yescomplemented
29C2×C3×2−1+4—31no—
30C2×#1007C2×C311yescharacteristic, complemented
31C2×C3×AGL(1,ℤ/8)—32no—
32C2×C3×#124—31no—
33C2×#947C2×C311yescharacteristic, complemented
34C2×#908C2×C312yescomplemented
35C3×#454—38no—
36C2×C3×#104—32no—
37C2×C3×#101—21no—
38C2×C3×#101D311yescharacteristic, complemented
39C2×C3×#101C2×C311yescharacteristic, complemented
40C2×#825C2×C312yescomplemented
41#4753C2×C318yesdirect product factor
42C23×C3×D3—21no—
43C2×C32×Q8—22no—
44C2×C32×Q8—22no—
45C2×C32×Q8D412yes—
46C2×C32×Q8C2312yes—
47C22×C32×C4D412yes—
48C22×C32×C4C2311yescharacteristic
49C2×C3×#67—22no—
50C22×C3×Dic3—21no—
51C3×#239—28no—
52C3×#239—28no—
53C3×D3×Q8—28no—
54C3×D3×Q8—28no—
55C3×#235—28no—
56C3×#235—28no—
57C3×#235—22no—
58C3×#235—22no—
59C3×#235D414yes—
60C3×#235C2314yes—
61C2×C3×D12—22no—
62C2×C3×D12—22no—
63C2×C3×D12—22no—
64C2×C3×D12—21no—
65C2×C3×D12D412yes—
66C2×C3×D12C2312yes—
67C2×C3×C4×D3—24no—
68C2×C3×C4×D3—22no—
69C2×C3×Dic6—22no—
70C2×C3×Dic6D412yes—
71C32×M16—22no—
72C32×M16C2314yes—
73C2×C32×C8—22no—
74C3×(C2×D3)⋊C4—28no—
75C3×C4×Dic3—21no—
76C3×C4×Dic3C2312yes—
77C3×D24—28no—
78C3×#204—28no—
79C2×#454—31noSylow
80C3×2−1+4—34no—
81C3×2−1+4—64no—
82C3×2−1+4—62no—
83C2×C3×#41—32no—
84C2×C3×#41—62no—
85C2×C3×#41—62no—
86C22×C3×Q8—21no—
87C22×C3×Q8—62no—
88C22×C3×Q8C2×D311yescharacteristic
89C22×C3×Q8C22×C311yescharacteristic
90C22×C3×D4—31no—
91#1007—22no—
92#1007—24no—
93#1007C22×C314yescomplemented
94C2×#239—22no—
95C2×D3×Q8—22no—
96C2×#235—22no—
97C2×#235C22×C312yes—
98C22×D12C22×C311yescharacteristic
99C3×AGL(1,ℤ/8)—38no—
100C3×AGL(1,ℤ/8)—64no—
101C2×C3×SD16—62no—
102C2×C3×D8—62no—
103C2×C3×M16—22no—
104C2×C3×M16C2×D312yes—
105C2×C3×M16C22×C312yes—
106C3×#124—34no—
107C3×#124—62no—
108C2×C3×C22⋊C4—62no—
109C2×C3×C42—31no—
110#947—22no—
111#947C22×C314yescomplemented
112C2×(C2×D3)⋊C4—22no—
113C2×C4×Dic3C22×C311yescharacteristic
114#908—24no—
115#908C22×C318yescomplemented
116C2×D24—22no—
117C2×#204—22no—
118C3×#104—38no—
119C3×#101—24no—
120C3×#101C2×D314yescomplemented
121C3×#101C22×C314yescomplemented
122#825C22×C318yescomplemented
123C23×C32C2×D411yescharacteristic
124C22×C3×D3—22no—
125C22×C3×D3—22no—
126C22×C3×D3—44no—
127C22×C3×D3—41no—
128C22×C3×D3—41no—
129C32×Q8—24no—
130C32×Q8—24no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4C2×D414yes—
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D412yes—
136C2×C32×C4C2×D412yes—
137C2×C32×C4C2411yescharacteristic
138C3×#67—24no—
139C3×#67—24no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—42no—
143C3×D12—28no—
144C3×D12—22no—
145C3×D12—24no—
146C3×D12—42no—
147C3×D12C2×D414yes—
148C3×C4×D3—216no—
149C3×C4×D3—24no—
150C3×C4×D3—24no—
151C3×Dic6—28no—
152C3×Dic6—22no—
153C3×Dic6C2×D414yes—
154C32×C8—24no—
155C2×2−1+4—31no—
156C2×AGL(1,ℤ/8)—32no—
157C2×#124—31no—
158#454—38no—
159C2×#104—32no—
160C2×#101C3×D311yescharacteristic, complemented
161C24×C3—61no—
162C23×D3—21no—
163C3×#41—34no—
164C3×#41—34no—
165C3×#41—68no—
166C3×#41—68no—
167C3×#41—68no—
168C3×#41—62no—
169C3×#41—68no—
170C3×#41—62no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—32no—
178C2×C3×Q8—42no—
179C2×C3×Q8—42no—
180C2×C3×Q8—62no—
181C2×C3×Q8—68no—
182C2×C3×Q8—68no—
183C2×C3×Q8D1212yes—
184C2×C3×Q8C3×D412yes—
185C2×C3×Q8C22×D312yes—
186C2×C3×Q8C23×C312yes—
187C2×C3×D4—32no—
188C2×C3×D4—32no—
189C2×C3×D4—62no—
190C2×C3×D4—62no—
191C2×C3×D4—61no—
192C2×C3×D4—62no—
193C2×C3×D4—62no—
194C22×C3×C4—22no—
195C22×C3×C4—21no—
196C22×C3×C4—64no—
197C22×C3×C4—62no—
198C22×C3×C4—61no—
199C22×C3×C4D1212yes—
200C22×C3×C4C3×D412yes—
201C22×C3×C4C22×D311yescharacteristic
202C22×C3×C4C23×C311yescharacteristic
203C2×#67—22no—
204C22×Dic3—21no—
205#239—28no—
206#239—28no—
207D3×Q8—28no—
208D3×Q8—28no—
209#235—28no—
210#235—28no—
211#235—22no—
212#235—22no—
213#235C3×D414yes—
214#235C23×C314yes—
215C2×D12—22no—
216C2×D12—22no—
217C2×D12—22no—
218C2×D12—21no—
219C2×D12C3×D412yes—
220C2×D12C23×C312yes—
221C2×C4×D3—24no—
222C2×C4×D3—22no—
223C2×Dic6—22no—
224C2×Dic6C3×D412yes—
225C3×SD16—68no—
226C3×D8—68no—
227C3×M16—22no—
228C3×M16—22no—
229C3×M16—24no—
230C3×M16—42no—
231C3×M16C22×D314yes—
232C3×M16C23×C314yes—
233C2×C3×C8—22no—
234C2×C3×C8—22no—
235C2×C3×C8—42no—
236C3×C22⋊C4—68no—
237C3×C42—32no—
238C3×C42—61no—
239(C2×D3)⋊C4—28no—
240C4×Dic3—21no—
241C4×Dic3C23×C312yes—
242D24—28no—
243#204—28no—
244C22×C32—22no—
245C22×C32#12011yescharacteristic
246C22×C32#12011yescharacteristic
247C22×C32C22×D411yescharacteristic
248C2×C3×D3—24no—
249C2×C3×D3—44no—
250C2×C3×D3—44no—
251C2×C3×D3—42no—
252C2×C3×D3—82no—
253C2×C3×D3—82no—
254C32×C4—24no—
255C32×C4#12012yes—
256C32×C4C22×D412yes—
257C3×Dic3—24no—
258C3×Dic3—42no—
2592−1+4—34no—
2602−1+4—64no—
2612−1+4—62no—
262C2×#41—32no—
263C2×#41—62no—
264C2×#41—62no—
265C22×Q8—62no—
266C22×Q8C2×C3×D311yescharacteristic
267C22×D4—31no—
268AGL(1,ℤ/8)—38no—
269AGL(1,ℤ/8)—64no—
270C2×SD16—62no—
271C2×D8—62no—
272C2×M16C2×C3×D312yes—
273#124—34no—
274#124—62no—
275C2×C22⋊C4—62no—
276C2×C42—31no—
277#104—38no—
278#101C2×C3×D314yescomplemented
279C23×C3—21no—
280C23×C3—62no—
281C23×C3—62no—
282C23×C3—124no—
283C23×C3—121no—
284C23×C3—121no—
285C23×C3C2×D1211yescharacteristic
286C23×C3C2×C3×D411yescharacteristic
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—44no—
290C22×D3—41no—
291C22×D3—41no—
292C3×Q8—24no—
293C3×Q8—24no—
294C3×Q8—24no—
295C3×Q8—24no—
296C3×Q8—34no—
297C3×Q8—44no—
298C3×Q8—44no—
299C3×Q8—68no—
300C3×Q8—62no—
301C3×D4—34no—
302C3×D4—68no—
303C3×D4—64no—
304C3×D4—64no—
305C3×D4—62no—
306C3×D4—64no—
307C3×D4—122no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—24no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—616no—
320C2×C3×C4—62no—
321C2×C3×C4—64no—
322C2×C3×C4—64no—
323C2×C3×C4—62no—
324C2×C3×C4—122no—
325C2×C3×C4C2×D1214yes—
326C2×C3×C4C2×D1211yescharacteristic
327C2×C3×C4D3×D412yes—
328C2×C3×C4D3×D412yes—
329C2×C3×C4C2×C3×D414yes—
330C2×C3×C4C2×C3×D411yescharacteristic
331C2×C3×C4C2×C3×D412yes—
332C2×C3×C4C2×C3×D412yes—
333C2×C3×C4C23×D311yescharacteristic
334C2×C3×C4C24×C311yescharacteristic
335#67—24no—
336#67—24no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—42no—
340D12—28no—
341D12—22no—
342D12—24no—
343D12—42no—
344D12C2×C3×D414yes—
345C4×D3—216no—
346C4×D3—24no—
347C4×D3—24no—
348Dic6—28no—
349Dic6—22no—
350Dic6C2×C3×D414yes—
351C3×C8—24no—
352C3×C8—24no—
353C3×C8—44no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32#45412yescomplemented
357C2×C32C2×#12011yescharacteristic
358C3×D3—44no—
359C3×D3—82no—
360C24—61no—
361#41—34no—
362#41—34no—
363#41—68no—
364#41—68no—
365#41—68no—
366#41—62no—
367#41—68no—
368#41—62no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—32no—
372C2×Q8—62no—
373C2×Q8—68no—
374C2×Q8—68no—
375C2×Q8C3×D1212yes—
376C2×Q8C22×C3×D312yes—
377C2×D4—32no—
378C2×D4—32no—
379C2×D4—62no—
380C2×D4—62no—
381C2×D4—61no—
382C2×D4—62no—
383C2×D4—62no—
384C22×C4—64no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4C3×D1212yes—
388C22×C4C22×C3×D311yescharacteristic
389SD16—68no—
390D8—68no—
391M16—22no—
392M16C22×C3×D314yes—
393C2×C8—22no—
394C22⋊C4—68no—
395C42—32no—
396C42—61no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—42no—
403C22×C3—64no—
404C22×C3—124no—
405C22×C3—124no—
406C22×C3—122no—
407C22×C3—242no—
408C22×C3—242no—
409C22×C3#88211yescharacteristic
410C22×C3#88211yescharacteristic
411C22×C3C3×#12011yescharacteristic
412C22×C3C3×#12011yescharacteristic
413C22×C3C22×D1211yescharacteristic
414C22×C3C22×C3×D411yescharacteristic
415C2×D3—24no—
416C2×D3—44no—
417C2×D3—44no—
418C2×D3—42no—
419C2×D3—82no—
420C2×D3—82no—
421C3×C4—24no—
422C3×C4—24no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—44no—
426C3×C4—64no—
427C3×C4—122no—
428C3×C4#88212yes—
429C3×C4C3×#12012yes—
430C3×C4C2×D3×D412yes—
431C3×C4C22×C3×D412yes—
432Dic3—24no—
433Dic3—42no—
434C32C2×#45411yescharacteristic, complemented, Sylow
435C23—62no—
436C23—62no—
437C23—124no—
438C23—121no—
439C23—121no—
440C23C2×C3×D1211yescharacteristic
441Q8—24no—
442Q8—24no—
443Q8—34no—
444Q8—68no—
445Q8—62no—
446D4—34no—
447D4—68no—
448D4—64no—
449D4—64no—
450D4—62no—
451D4—64no—
452D4—122no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—616no—
456C2×C4—62no—
457C2×C4—64no—
458C2×C4—64no—
459C2×C4—62no—
460C2×C4—122no—
461C2×C4C2×C3×D1214yes—
462C2×C4C2×C3×D1211yescharacteristic
463C2×C4C3×D3×D412yes—
464C2×C4C3×D3×D412yes—
465C2×C4C23×C3×D311yescharacteristic
466C8—24no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—22no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—124no—
476C2×C3—242no—
477C2×C3#475312yesdirect product factor
478C2×C3C3×#45412yescomplemented
479C2×C3C2×#88211yescharacteristic
480C2×C3C2×C3×#12011yescharacteristic
481D3—44no—
482D3—82no—
483C22—22no—
484C22—64no—
485C22—124no—
486C22—124no—
487C22—122no—
488C22—242no—
489C22—242no—
490C22C3×#88211yescharacteristic
491C22C3×#88211yescharacteristic
492C22C22×C3×D1211yescharacteristic
493C4—24no—
494C4—64no—
495C4—122no—
496C4C3×#88212yes—
497C4C2×C3×D3×D412yes—
498C3—21no—
499C3C2×#475311yescharacteristic, direct product factor
500C3C2×C3×#45411yescharacteristic, complemented
501C2—21no—
502C2—21no—
503C2—124no—
504C2—242no—
505C2C3×#475312yesdirect product factor
506C2C2×C3×#88211yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#101C2
socleC22×C32#120
hypercenterC2×C3×#101D3
hypocenterC3C2×C3×#454
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#101

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1007C2semidirect
2. C2×C3×#947C2semidirect
3. C2×C3×#908C2semidirect
4. C2×C32×#101C2semidirect
5. C2×C3×#825C2semidirect
6. C3×#4753C2direct (N × H)
8. C2×#4753C3direct (N × H)
12. C3×#1007C22semidirect
20. C3×#947C22semidirect
24. C3×#908C22semidirect
27. C32×#101C22semidirect
28. C3×#825C22semidirect
30. C2×#1007C2×C3semidirect
33. C2×#947C2×C3semidirect
34. C2×#908C2×C3semidirect
38. C2×C3×#101D3semidirect
39. C2×C3×#101C2×C3semidirect
40. C2×#825C2×C3semidirect
41. #4753C2×C3direct (N × H)
93. #1007C22×C3semidirect
111. #947C22×C3semidirect
115. #908C22×C3semidirect
120. C3×#101C2×D3semidirect
121. C3×#101C22×C3semidirect
122. #825C22×C3semidirect
160. C2×#101C3×D3semidirect
278. #101C2×C3×D3semidirect
356. C2×C32#454semidirect
434. C32C2×#454semidirect
477. C2×C3#4753direct (N × H)
478. C2×C3C3×#454semidirect
499. C3C2×#4753direct (N × H)
500. C3C2×C3×#454semidirect
505. C2C3×#4753direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#10072
C2×C3×#9472
C2×C3×#9082
C2×C32×#1012
C2×C3×#8252
C3×#47532
C2×C3×#4543
C2×#47533