Finite Group Databasebeta

← C2×C3×#5003 #11869423

Subgroups — C2×C3×#5003 #11869423

Order 1152.

1101 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#117C211yesmaximal, characteristic, complemented
2C2×C3×#930C211yesmaximal, characteristic
3C2×C3×#830C211yesmaximal, characteristic, complemented
4C3×#5003C218yesmaximal, direct product factor
5C2×C3×#809C211yesmaximal, characteristic, complemented
6C2×C3×#809C211yesmaximal, characteristic, complemented
7C2×C3×#808C211yesmaximal, characteristic, complemented
8C2×C3×#807C211yesmaximal, characteristic, complemented
9C2×C3×#482—31nomaximal
10C2×#5003C311yesmaximal, characteristic, direct product factor
11C23×C32×C4C2211yescharacteristic
12C22×C3×#67—21no—
13C22×C3×D12C2211yescharacteristic
14C32×#117—22no—
15C32×#117C2214yescomplemented
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C22⋊C4—21no—
19C2×C32×C42—21no—
20C3×#930—22no—
21C3×#930C2214yes—
22C2×C3×(C2×D3)⋊C4—21no—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C22×C3×C3⋊C8C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C3×#830C2214yescomplemented
27C3×#809C2214yescomplemented
28C3×#809C2214yescomplemented
29C3×#808C2214yescomplemented
30C3×#807C2214yescomplemented
31C2×C3×#121—31no—
32C2×C3×#117—21no—
33C2×C3×#117D311yescharacteristic, complemented
34C2×C3×#117C2×C311yescharacteristic, complemented
35C2×#930C2×C311yescharacteristic
36C3×#482—38no—
37C2×C3×#107—31no—
38C2×C3×#106—31no—
39C2×C3×#102—31no—
40C2×C3×#102—31no—
41C2×C3×C22⋊C8—31no—
42C2×#830C2×C311yescharacteristic, complemented
43#5003C2×C318yesdirect product factor
44C2×#809C2×C311yescharacteristic, complemented
45C2×#809C2×C311yescharacteristic, complemented
46C2×#808C2×C311yescharacteristic, complemented
47C2×#807C2×C311yescharacteristic, complemented
48C24×C32D411yescharacteristic
49C23×C3×D3—21no—
50C22×C32×C4—22no—
51C22×C32×C4—22no—
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4D411yescharacteristic
55C22×C32×C4D412yes—
56C22×C32×C4C2312yes—
57C22×C32×C4C2311yescharacteristic
58C2×C3×#67—24no—
59C2×C3×#67—44no—
60C22×C3×Dic3—21no—
61C2×C3×D12—22no—
62C2×C3×D12—22no—
63C2×C3×D12—21no—
64C2×C3×D12C2312yes—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2312yes—
68C32×C4⋊C4C2312yes—
69C32×C22⋊C4—24no—
70C32×C42—24no—
71C3×(C2×D3)⋊C4—24no—
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3C2312yes—
74C2×C3×C3⋊C8—22no—
75C2×C3×C3⋊C8—22no—
76C2×C3×C3⋊C8C2312yes—
77C2×C3×C3⋊C8C2312yes—
78C2×#482—31noSylow
79C22×C3×D4—31no—
80C22×C3×D4—61no—
81C23×C3×C4—21no—
82C23×C3×C4C2×D311yescharacteristic
83C23×C3×C4C22×C311yescharacteristic
84C22×#67—21no—
85C22×D12C22×C311yescharacteristic
86C22×C3×C8—31no—
87C22×C3×C8—31no—
88C3×#121—34no—
89C3×#121—62no—
90C3×#117—24no—
91C3×#117—22no—
92C3×#117—22no—
93C3×#117—24no—
94C3×#117C2×D314yescomplemented
95C3×#117C22×C314yescomplemented
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—31no—
99C2×C3×C4⋊C4C2×D311yescharacteristic
100C2×C3×C4⋊C4C2×D311yescharacteristic
101C2×C3×C4⋊C4C22×C311yescharacteristic
102C2×C3×C4⋊C4C22×C311yescharacteristic
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4—61no—
107C2×C3×C42—22no—
108C2×C3×C42—21no—
109C2×C3×C42—21no—
110#930—22no—
111#930C22×C314yes—
112C2×(C2×D3)⋊C4—21no—
113C2×C4⋊Dic3C22×C311yescharacteristic
114C22×C3⋊C8C22×C311yescharacteristic
115C22×C3⋊C8C22×C311yescharacteristic
116C3×#107—34no—
117C3×#106—34no—
118C3×#102—34no—
119C3×#102—34no—
120C3×C22⋊C8—34no—
121#830C22×C314yescomplemented
122#809C22×C314yescomplemented
123#809C22×C314yescomplemented
124#808C22×C314yescomplemented
125#807C22×C314yescomplemented
126C23×C32—22no—
127C23×C32—22no—
128C23×C32—22no—
129C23×C32C2×D412yes—
130C23×C32C2×D411yescharacteristic
131C22×C3×D3—22no—
132C22×C3×D3—42no—
133C22×C3×D3—42no—
134C22×C3×D3—42no—
135C2×C32×C4—22no—
136C2×C32×C4—24no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—42no—
142C2×C32×C4—42no—
143C2×C32×C4C2×D412yes—
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4#4112yes—
146C2×C32×C4#4112yes—
147C2×C32×C4C2411yescharacteristic
148C3×#67—48no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—42no—
151C3×D12—22no—
152C3×D12—22no—
153C3×D12—42no—
154C3×C3⋊C8—22no—
155C3×C3⋊C8—22no—
156C2×#121—31no—
157C2×#117C3×D311yescharacteristic, complemented
158#482—38no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#102—31no—
162C2×#102—31no—
163C2×C22⋊C8—31no—
164C24×C3—21no—
165C24×C3—61no—
166C24×C3#6711yescharacteristic
167C24×C3C3×D411yescharacteristic
168C23×D3—21no—
169C2×C3×D4—32no—
170C2×C3×D4—64no—
171C2×C3×D4—62no—
172C2×C3×D4—62no—
173C2×C3×D4—61no—
174C2×C3×D4—124no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—61no—
192C22×C3×C4#6712yes—
193C22×C3×C4#6711yescharacteristic
194C22×C3×C4C3×D411yescharacteristic
195C22×C3×C4C3×D412yes—
196C22×C3×C4C22×D312yes—
197C22×C3×C4C22×D311yescharacteristic
198C22×C3×C4C23×C312yes—
199C22×C3×C4C23×C311yescharacteristic
200C2×#67—24no—
201C2×#67—44no—
202C22×Dic3—21no—
203C2×D12—22no—
204C2×D12—22no—
205C2×D12—21no—
206C2×D12C23×C312yes—
207C2×C3×C8—32no—
208C2×C3×C8—32no—
209C2×C3×C8—62no—
210C2×C3×C8—62no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—32no—
220C3×C4⋊C4—61no—
221C3×C4⋊C4C22×D312yes—
222C3×C4⋊C4C22×D312yes—
223C3×C4⋊C4C23×C312yes—
224C3×C4⋊C4C23×C312yes—
225C3×C22⋊C4—28no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—64no—
229C3×C42—28no—
230C3×C42—24no—
231C3×C42—24no—
232(C2×D3)⋊C4—24no—
233C4⋊Dic3—21no—
234C4⋊Dic3C23×C312yes—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8C23×C312yes—
238C2×C3⋊C8C23×C312yes—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—44no—
243C22×C32#12312yes—
244C22×C32#13512yes—
245C22×C32AGL(1,ℤ/8)12yes—
246C22×C32C22×D411yescharacteristic
247C2×C3×D3—42no—
248C2×C3×D3—42no—
249C2×C3×D3—42no—
250C2×C3×D3—84no—
251C32×C4—22no—
252C32×C4—42no—
253C32×C4—42no—
254C32×C4#12312yes—
255C32×C4C2×#4111yescharacteristic
256C32×C4C2×#4111yescharacteristic
257C3×Dic3—42no—
258C22×D4—31no—
259C22×D4—61no—
260C23×C4C2×C3×D311yescharacteristic
261C22×C8—31no—
262C22×C8—31no—
263#121—34no—
264#121—62no—
265#117—22no—
266#117C2×C3×D314yescomplemented
267C2×C4⋊C4—31no—
268C2×C4⋊C4C2×C3×D311yescharacteristic
269C2×C4⋊C4C2×C3×D311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—61no—
272C2×C42—21no—
273#107—34no—
274#106—34no—
275#102—34no—
276#102—34no—
277C22⋊C8—34no—
278C23×C3—24no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—21no—
286C23×C3—22no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—62no—
290C23×C3—122no—
291C23×C3—122no—
292C23×C3—122no—
293C23×C3C2×#6712yes—
294C23×C3C2×#6711yescharacteristic
295C23×C3C2×C3×D412yes—
296C23×C3C2×C3×D411yescharacteristic
297C22×D3—22no—
298C22×D3—42no—
299C22×D3—42no—
300C22×D3—42no—
301C3×D4—62no—
302C3×D4—62no—
303C3×D4—128no—
304C3×D4—122no—
305C2×C3×C4—24no—
306C2×C3×C4—28no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—44no—
329C2×C3×C4—44no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—62no—
335C2×C3×C4—122no—
336C2×C3×C4#23512yes—
337C2×C3×C4#23512yes—
338C2×C3×C4C2×#6712yes—
339C2×C3×C4C2×#6711yescharacteristic
340C2×C3×C4C2×C3×D412yes—
341C2×C3×C4C2×C3×D411yescharacteristic
342C2×C3×C4C3×#4112yes—
343C2×C3×C4C3×#4112yes—
344C2×C3×C4C23×D311yescharacteristic
345C2×C3×C4C24×C311yescharacteristic
346#67—48no—
347C2×Dic3—22no—
348C2×Dic3—42no—
349D12—22no—
350D12—22no—
351D12—42no—
352C3×C8—62no—
353C3×C8—62no—
354C3⋊C8—22no—
355C3⋊C8—22no—
356C2×C32—42no—
357C2×C32#48214yescomplemented
358C2×C32C2×#12311yescharacteristic
359C2×C32C2×#13511yescharacteristic
360C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
361C3×D3—82no—
362C24—61no—
363C24C3×#6711yescharacteristic
364C2×D4—32no—
365C2×D4—64no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—61no—
369C2×D4—124no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—61no—
375C22×C4C3×#6711yescharacteristic
376C22×C4C3×#6712yes—
377C22×C4C22×C3×D312yes—
378C22×C4C22×C3×D311yescharacteristic
379C2×C8—32no—
380C2×C8—32no—
381C2×C8—62no—
382C2×C8—62no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—32no—
386C4⋊C4—61no—
387C4⋊C4C22×C3×D312yes—
388C4⋊C4C22×C3×D312yes—
389C22⋊C4—24no—
390C22⋊C4—64no—
391C42—24no—
392C22×C3—24no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—21no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—48no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—122no—
409C22×C3—122no—
410C22×C3—122no—
411C22×C3—244no—
412C22×C3#92912yes—
413C22×C3#94912yes—
414C22×C3#95012yes—
415C22×C3C3×#12312yes—
416C22×C3C3×#13512yes—
417C22×C3C3×AGL(1,ℤ/8)12yes—
418C22×C3C22×#6711yescharacteristic
419C22×C3C22×C3×D411yescharacteristic
420C2×D3—42no—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—84no—
424C3×C4—24no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—44no—
431C3×C4—44no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—122no—
437C3×C4#92912yes—
438C3×C4C3×#12312yes—
439C3×C4C2×#23511yescharacteristic
440C3×C4C2×#23511yescharacteristic
441C3×C4C2×C3×#4111yescharacteristic
442C3×C4C2×C3×#4111yescharacteristic
443Dic3—42no—
444C32C2×#48211yescharacteristic, complemented, Sylow
445C23—22no—
446C23—22no—
447C23—22no—
448C23—62no—
449C23—122no—
450C23—122no—
451C23—122no—
452C23C2×C3×#6712yes—
453C23C2×C3×#6711yescharacteristic
454D4—62no—
455D4—62no—
456D4—128no—
457D4—122no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—62no—
467C2×C4—122no—
468C2×C4C3×#23512yes—
469C2×C4C3×#23512yes—
470C2×C4C2×C3×#6712yes—
471C2×C4C2×C3×#6711yescharacteristic
472C2×C4C23×C3×D311yescharacteristic
473C8—62no—
474C8—62no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—24no—
479C2×C3—44no—
480C2×C3—42no—
481C2×C3—42no—
482C2×C3—242no—
483C2×C3#500314yesdirect product factor
484C2×C3C3×#48214yescomplemented
485C2×C3C2×#92911yescharacteristic
486C2×C3C2×#94911yescharacteristic
487C2×C3C2×#95011yescharacteristic
488C2×C3C2×C3×#12311yescharacteristic
489C2×C3C2×C3×#13511yescharacteristic
490C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
491D3—82no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—44no—
496C22—122no—
497C22—122no—
498C22—122no—
499C22—244no—
500C22C3×#92912yes—
501C22C3×#94912yes—
502C22C3×#95012yes—
503C22C22×C3×#6711yescharacteristic
504C4—22no—
505C4—42no—
506C4—42no—
507C4—122no—
508C4C3×#92912yes—
509C4C2×C3×#23511yescharacteristic
510C4C2×C3×#23511yescharacteristic
511C3—21no—
512C3C2×#500311yescharacteristic, direct product factor
513C3C2×C3×#48211yescharacteristic, complemented
514C2—42no—
515C2—242no—
516C2C3×#500314yesdirect product factor
517C2C2×C3×#92911yescharacteristic
518C2C2×C3×#94911yescharacteristic
519C2C2×C3×#95011yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#117C2
socleC23×C32C2×D4
prime radicalC2×C3×#930C2
hypercenterC2×C3×#117D3
hypocenterC3C2×C3×#482
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×#117

Complements

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

NH ≅ G/Ntype
1. C2×C32×#117C2semidirect
3. C2×C3×#830C2semidirect
4. C3×#5003C2direct (N × H)
5. C2×C3×#809C2semidirect
6. C2×C3×#809C2semidirect
7. C2×C3×#808C2semidirect
8. C2×C3×#807C2semidirect
10. C2×#5003C3direct (N × H)
15. C32×#117C22semidirect
26. C3×#830C22semidirect
27. C3×#809C22semidirect
28. C3×#809C22semidirect
29. C3×#808C22semidirect
30. C3×#807C22semidirect
33. C2×C3×#117D3semidirect
34. C2×C3×#117C2×C3semidirect
42. C2×#830C2×C3semidirect
43. #5003C2×C3direct (N × H)
44. C2×#809C2×C3semidirect
45. C2×#809C2×C3semidirect
46. C2×#808C2×C3semidirect
47. C2×#807C2×C3semidirect
94. C3×#117C2×D3semidirect
95. C3×#117C22×C3semidirect
121. #830C22×C3semidirect
122. #809C22×C3semidirect
123. #809C22×C3semidirect
124. #808C22×C3semidirect
125. #807C22×C3semidirect
157. C2×#117C3×D3semidirect
266. #117C2×C3×D3semidirect
357. C2×C32#482semidirect
444. C32C2×#482semidirect
483. C2×C3#5003direct (N × H)
484. C2×C3C3×#482semidirect
512. C3C2×#5003direct (N × H)
513. C3C2×C3×#482semidirect
516. C2C3×#5003direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1172
C2×C3×#9302
C2×C3×#8302
C3×#50032
C2×C3×#8092
C2×C3×#8082
C2×C3×#8072
C2×C3×#4823
C2×#50033