Finite Group Databasebeta

← C2×C3×#5154 #11874336

Subgroups — C2×C3×#5154 #11874336

Order 1152.

1505 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×D8C211yesmaximal, characteristic, complemented
2C2×C3×#940C211yesmaximal, characteristic, complemented
3C2×C3×#936C211yesmaximal, characteristic, complemented
4C22×C3×#214C211yesmaximal, characteristic, complemented
5C22×C3×#213C211yesmaximal, characteristic, complemented
6C22×C3×#204C211yesmaximal, characteristic, complemented
7C3×#5154C218yesmaximal, direct product factor
8C2×C3×#815C211yesmaximal, characteristic, complemented
9C2×C3×#496—31nomaximal
10C2×#5154C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C22×C3×#67—22no—
14C22×C3×D12C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C2×C32×D8—24no—
17C2×C32×D8C2214yescomplemented
18C22×C32×C8C2211yescharacteristic
19C2×C3×#217—22no—
20C3×#940—22no—
21C3×#940C2214yescomplemented
22C3×#936—22no—
23C3×#936C2214yescomplemented
24C2×C3×#214—24no—
25C2×C3×#214C2214yescomplemented
26C2×C3×#213—24no—
27C2×C3×#213C2214yescomplemented
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C2×C3×#204—24no—
31C2×C3×#204C2214yescomplemented
32C3×#815C2214yescomplemented
33C22×C3×SD16—31no—
34C22×C3×SD16—31no—
35C22×C3×D8—21no—
36C22×C3×D8—31no—
37C22×C3×D8D311yescharacteristic, complemented
38C22×C3×D8C2×C311yescharacteristic, complemented
39C2×C3×#127—31no—
40C2×C3×#124—31no—
41C2×#940C2×C311yescharacteristic, complemented
42C2×#936C2×C311yescharacteristic, complemented
43C22×#214C2×C311yescharacteristic, complemented
44C22×#213C2×C311yescharacteristic, complemented
45C22×#204C2×C311yescharacteristic, complemented
46C3×#496—38no—
47C2×C3×#97—31no—
48#5154C2×C318yesdirect product factor
49C2×#815C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C24×C32—21no—
52C23×C3×D3—21no—
53C2×C32×D4—22no—
54C2×C32×D4—22no—
55C2×C32×D4—22no—
56C2×C32×D4—22no—
57C2×C32×D4—21no—
58C2×C32×D4—21no—
59C2×C32×D4C2312yes—
60C2×C32×D4C2312yes—
61C22×C32×C4C2311yescharacteristic
62C2×C3×#67—28no—
63C2×C3×#67—48no—
64C22×C3×Dic3—21no—
65C22×C3×Dic3D412yes—
66C2×C3×D12—22no—
67C2×C3×D12—22no—
68C2×C3×D12—21no—
69C2×C3×D12C2312yes—
70C2×C3×Dic6—22no—
71C2×C3×Dic6—22no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6C2312yes—
74C32×D8—28no—
75C2×C32×C8D414yescomplemented
76C2×C32×C8C2312yes—
77C3×#217—28no—
78C3×#214—28no—
79C3×#213—28no—
80C3×C4×Dic3D412yes—
81C3×C4×Dic3C2312yes—
82C2×C3×C3⋊C8D414yescomplemented
83C2×C3×C3⋊C8C2312yes—
84C3×#204—28no—
85C2×#496—31noSylow
86C22×C3×Q8—31no—
87C22×C3×D4—21no—
88C22×C3×D4—21no—
89C22×C3×D4—31no—
90C22×C3×D4—62no—
91C22×C3×D4C2×D311yescharacteristic
92C22×C3×D4C2×D311yescharacteristic
93C22×C3×D4C22×C311yescharacteristic
94C22×C3×D4C22×C311yescharacteristic
95C22×#67—22no—
96C22×D12C22×C311yescharacteristic
97C22×Dic6C22×C311yescharacteristic
98C2×C3×SD16—34no—
99C2×C3×SD16—34no—
100C2×C3×SD16—64no—
101C2×C3×SD16—64no—
102C2×C3×D8—28no—
103C2×C3×D8—24no—
104C2×C3×D8—24no—
105C2×C3×D8—24no—
106C2×C3×D8—34no—
107C2×C3×D8—64no—
108C2×C3×D8C2×D314yescomplemented
109C2×C3×D8C22×C314yescomplemented
110C22×C3×C8—21no—
111C22×C3×C8—31no—
112C22×C3×C8C2×D311yescharacteristic
113C22×C3×C8C22×C311yescharacteristic
114C3×#127—34no—
115C3×#127—62no—
116C3×#124—34no—
117C3×#124—62no—
118C2×C3×C22⋊C4—62no—
119C2×C3×C42—31no—
120C2×#217—22no—
121#940—22no—
122#940C22×C314yescomplemented
123#936—22no—
124#936C22×C314yescomplemented
125C2×#214—24no—
126C2×#214C22×C314yescomplemented
127C2×#213—24no—
128C2×#213C22×C314yescomplemented
129C2×C4×Dic3C22×C311yescharacteristic
130C22×C3⋊C8C22×C311yescharacteristic
131C2×#204—24no—
132C2×#204C22×C314yescomplemented
133C3×#97—34no—
134#815C22×C314yescomplemented
135C23×C32—22no—
136C23×C32—22no—
137C23×C32—42no—
138C23×C32—42no—
139C23×C32—44no—
140C23×C32—44no—
141C23×C32C2×D411yescharacteristic
142C22×C3×D3—22no—
143C22×C3×D3—42no—
144C22×C3×D3—44no—
145C32×D4—22no—
146C32×D4—22no—
147C32×D4—22no—
148C32×D4—22no—
149C32×D4—42no—
150C32×D4—42no—
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4C2×D412yes—
153C2×C32×C4C2×D412yes—
154C2×C32×C4C2411yescharacteristic
155C3×#67—416no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—42no—
159C2×C3×Dic3C2×D414yes—
160C3×D12—22no—
161C3×D12—22no—
162C3×D12—42no—
163C3×Dic6—22no—
164C3×Dic6—22no—
165C3×Dic6—42no—
166C32×C8C2×D414yescomplemented
167C3×C3⋊C8C2×D414yescomplemented
168C22×SD16—31no—
169C22×SD16—31no—
170C22×D8—31no—
171C22×D8C3×D311yescharacteristic, complemented
172C2×#127—31no—
173C2×#124—31no—
174#496—38no—
175C2×#97—31no—
176C24×C3—21no—
177C24×C3—21no—
178C24×C3—21no—
179C24×C3—21no—
180C24×C3—41no—
181C24×C3—41no—
182C24×C3—61no—
183C23×D3—21no—
184C2×C3×Q8—32no—
185C2×C3×Q8—61no—
186C2×C3×Q8—62no—
187C2×C3×Q8—62no—
188C2×C3×D4—24no—
189C2×C3×D4—24no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—22no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—22no—
203C2×C3×D4—22no—
204C2×C3×D4—32no—
205C2×C3×D4—42no—
206C2×C3×D4—42no—
207C2×C3×D4—41no—
208C2×C3×D4—41no—
209C2×C3×D4—68no—
210C2×C3×D4—62no—
211C2×C3×D4—61no—
212C2×C3×D4—62no—
213C2×C3×D4—128no—
214C2×C3×D4C22×D312yes—
215C2×C3×D4C22×D312yes—
216C2×C3×D4C23×C312yes—
217C2×C3×D4C23×C312yes—
218C22×C3×C4—21no—
219C22×C3×C4—32no—
220C22×C3×C4—61no—
221C22×C3×C4C22×D311yescharacteristic
222C22×C3×C4C23×C311yescharacteristic
223C2×#67—28no—
224C2×#67—48no—
225C22×Dic3—21no—
226C22×Dic3C3×D412yes—
227C2×D12—22no—
228C2×D12—22no—
229C2×D12—21no—
230C2×D12C23×C312yes—
231C2×Dic6—22no—
232C2×Dic6—22no—
233C2×Dic6—21no—
234C2×Dic6C23×C312yes—
235C3×SD16—68no—
236C3×SD16—68no—
237C3×D8—216no—
238C3×D8—28no—
239C3×D8—28no—
240C3×D8—68no—
241C2×C3×C8—24no—
242C2×C3×C8—22no—
243C2×C3×C8—34no—
244C2×C3×C8—32no—
245C2×C3×C8#6714yescomplemented
246C2×C3×C8C3×D414yescomplemented
247C2×C3×C8C22×D312yes—
248C2×C3×C8C23×C312yes—
249C3×C22⋊C4—68no—
250C3×C42—32no—
251C3×C42—32no—
252#217—28no—
253#214—28no—
254#213—28no—
255C4×Dic3C3×D412yes—
256C4×Dic3C23×C312yes—
257C2×C3⋊C8C3×D414yescomplemented
258C2×C3⋊C8C23×C312yes—
259#204—28no—
260C22×C32—42no—
261C22×C32—42no—
262C22×C32—44no—
263C22×C32—44no—
264C22×C32—84no—
265C22×C32—84no—
266C22×C32#12712yes—
267C22×C32AGL(1,ℤ/8)14yes—
268C22×C32C22×D411yescharacteristic
269C2×C3×D3—42no—
270C2×C3×D3—44no—
271C2×C3×D3—84no—
272C32×C4#12712yes—
273C32×C4C22×D411yescharacteristic
274C32×C4C22×D411yescharacteristic
275C3×Dic3—24no—
276C3×Dic3—42no—
277C22×Q8—31no—
278C22×D4—31no—
279C22×D4—62no—
280C22×D4C2×C3×D311yescharacteristic
281C22×D4C2×C3×D311yescharacteristic
282C2×SD16—34no—
283C2×SD16—34no—
284C2×SD16—64no—
285C2×SD16—64no—
286C2×D8—24no—
287C2×D8—34no—
288C2×D8—64no—
289C2×D8C2×C3×D314yescomplemented
290C22×C8—31no—
291C22×C8C2×C3×D311yescharacteristic
292#127—34no—
293#127—62no—
294#124—34no—
295#124—62no—
296C2×C22⋊C4—62no—
297C2×C42—31no—
298#97—34no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—21no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—44no—
307C23×C3—44no—
308C23×C3—44no—
309C23×C3—44no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—44no—
315C23×C3—44no—
316C23×C3—62no—
317C23×C3—84no—
318C23×C3—84no—
319C23×C3—124no—
320C23×C3—122no—
321C23×C3D3×D411yescharacteristic
322C23×C3C2×C3×D411yescharacteristic
323C22×D3—22no—
324C22×D3—42no—
325C22×D3—44no—
326C3×Q8—62no—
327C3×Q8—62no—
328C3×Q8—122no—
329C3×D4—24no—
330C3×D4—24no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—42no—
340C3×D4—42no—
341C3×D4—44no—
342C3×D4—44no—
343C3×D4—42no—
344C3×D4—42no—
345C3×D4—42no—
346C3×D4—42no—
347C3×D4—62no—
348C3×D4—62no—
349C3×D4—1216no—
350C3×D4—122no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—34no—
356C2×C3×C4—64no—
357C2×C3×C4—62no—
358C2×C3×C4—122no—
359C2×C3×C4D3×D411yescharacteristic
360C2×C3×C4D3×D412yes—
361C2×C3×C4C2×#6712yes—
362C2×C3×C4C2×C3×D411yescharacteristic
363C2×C3×C4C2×C3×D412yes—
364C2×C3×C4C2×C3×D412yes—
365C2×C3×C4C23×D311yescharacteristic
366C2×C3×C4C24×C311yescharacteristic
367#67—416no—
368C2×Dic3—24no—
369C2×Dic3—22no—
370C2×Dic3—42no—
371C2×Dic3C2×C3×D414yes—
372D12—22no—
373D12—22no—
374D12—42no—
375Dic6—22no—
376Dic6—22no—
377Dic6—42no—
378C3×C8—24no—
379C3×C8—34no—
380C3×C8C2×#6714yescomplemented
381C3×C8C2×C3×D414yescomplemented
382C3⋊C8C2×C3×D414yescomplemented
383C2×C32—82no—
384C2×C32—82no—
385C2×C32#49614yescomplemented
386C2×C32C2×#12711yescharacteristic
387C2×C32C2×AGL(1,ℤ/8)12yes—
388C3×D3—82no—
389C24—21no—
390C24—21no—
391C24—61no—
392C2×Q8—32no—
393C2×Q8—61no—
394C2×Q8—62no—
395C2×Q8—62no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—32no—
403C2×D4—68no—
404C2×D4—62no—
405C2×D4—61no—
406C2×D4—62no—
407C2×D4—128no—
408C2×D4C22×C3×D312yes—
409C2×D4C22×C3×D312yes—
410C22×C4—32no—
411C22×C4—61no—
412C22×C4C22×C3×D311yescharacteristic
413SD16—68no—
414SD16—68no—
415D8—28no—
416D8—68no—
417C2×C8—32no—
418C2×C8—34no—
419C2×C8C3×#6714yescomplemented
420C2×C8C22×C3×D312yes—
421C22⋊C4—68no—
422C42—32no—
423C42—32no—
424C22×C3—24no—
425C22×C3—22no—
426C22×C3—21no—
427C22×C3—44no—
428C22×C3—44no—
429C22×C3—44no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—84no—
438C22×C3—84no—
439C22×C3—88no—
440C22×C3—88no—
441C22×C3—84no—
442C22×C3—84no—
443C22×C3—84no—
444C22×C3—84no—
445C22×C3—124no—
446C22×C3—122no—
447C22×C3—244no—
448C22×C3#91114yes—
449C22×C3#94012yes—
450C22×C3C3×#12712yes—
451C22×C3C3×AGL(1,ℤ/8)14yes—
452C22×C3C2×D3×D411yescharacteristic
453C22×C3C22×C3×D411yescharacteristic
454C2×D3—44no—
455C2×D3—42no—
456C2×D3—84no—
457C3×C4—22no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—64no—
461C3×C4—122no—
462C3×C4#94012yes—
463C3×C4C3×#12712yes—
464C3×C4C2×D3×D411yescharacteristic
465C3×C4C22×#6711yescharacteristic
466C3×C4C22×C3×D411yescharacteristic
467C3×C4C22×C3×D411yescharacteristic
468Dic3—24no—
469Dic3—42no—
470C32C2×#49611yescharacteristic, complemented, Sylow
471C23—22no—
472C23—22no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—62no—
478C23—124no—
479C23—122no—
480C23C3×D3×D411yescharacteristic
481Q8—62no—
482Q8—62no—
483Q8—122no—
484D4—22no—
485D4—22no—
486D4—22no—
487D4—22no—
488D4—42no—
489D4—42no—
490D4—62no—
491D4—62no—
492D4—1216no—
493D4—122no—
494C2×C4—34no—
495C2×C4—64no—
496C2×C4—62no—
497C2×C4—122no—
498C2×C4C3×D3×D411yescharacteristic
499C2×C4C3×D3×D412yes—
500C2×C4C2×C3×#6712yes—
501C2×C4C23×C3×D311yescharacteristic
502C8—34no—
503C8C2×C3×#6714yescomplemented
504C2×C3—22no—
505C2×C3—21no—
506C2×C3—24no—
507C2×C3—84no—
508C2×C3—84no—
509C2×C3—82no—
510C2×C3—82no—
511C2×C3—82no—
512C2×C3—82no—
513C2×C3—242no—
514C2×C3#515414yesdirect product factor
515C2×C3C3×#49614yescomplemented
516C2×C3C2×#91112yes—
517C2×C3C2×#94011yescharacteristic
518C2×C3C2×C3×#12711yescharacteristic
519C2×C3C2×C3×AGL(1,ℤ/8)12yes—
520D3—82no—
521C22—44no—
522C22—44no—
523C22—42no—
524C22—42no—
525C22—84no—
526C22—84no—
527C22—124no—
528C22—122no—
529C22—244no—
530C22C3×#91114yes—
531C22C3×#94012yes—
532C22C2×C3×D3×D411yescharacteristic
533C4—64no—
534C4—122no—
535C4C3×#94012yes—
536C4C2×C3×D3×D411yescharacteristic
537C4C22×C3×#6711yescharacteristic
538C3—21no—
539C3C2×#515411yescharacteristic, direct product factor
540C3C2×C3×#49611yescharacteristic, complemented
541C2—82no—
542C2—82no—
543C2—242no—
544C2C3×#515414yesdirect product factor
545C2C2×C3×#91112yes—
546C2C2×C3×#94011yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×D8C2
socleC23×C32C2×D4
hypercenterC22×C3×D8D3
hypocenterC3C2×C3×#496
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×D8

Complements

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

NH ≅ G/Ntype
1. C22×C32×D8C2semidirect
2. C2×C3×#940C2semidirect
3. C2×C3×#936C2semidirect
4. C22×C3×#214C2semidirect
5. C22×C3×#213C2semidirect
6. C22×C3×#204C2semidirect
7. C3×#5154C2direct (N × H)
8. C2×C3×#815C2semidirect
10. C2×#5154C3direct (N × H)
17. C2×C32×D8C22semidirect
21. C3×#940C22semidirect
23. C3×#936C22semidirect
25. C2×C3×#214C22semidirect
27. C2×C3×#213C22semidirect
31. C2×C3×#204C22semidirect
32. C3×#815C22semidirect
37. C22×C3×D8D3semidirect
38. C22×C3×D8C2×C3semidirect
41. C2×#940C2×C3semidirect
42. C2×#936C2×C3semidirect
43. C22×#214C2×C3semidirect
44. C22×#213C2×C3semidirect
45. C22×#204C2×C3semidirect
48. #5154C2×C3direct (N × H)
49. C2×#815C2×C3semidirect
75. C2×C32×C8D4semidirect
82. C2×C3×C3⋊C8D4semidirect
108. C2×C3×D8C2×D3semidirect
109. C2×C3×D8C22×C3semidirect
122. #940C22×C3semidirect
124. #936C22×C3semidirect
126. C2×#214C22×C3semidirect
128. C2×#213C22×C3semidirect
132. C2×#204C22×C3semidirect
134. #815C22×C3semidirect
166. C32×C8C2×D4semidirect
167. C3×C3⋊C8C2×D4semidirect
171. C22×D8C3×D3semidirect
245. C2×C3×C8#67semidirect
246. C2×C3×C8C3×D4semidirect
257. C2×C3⋊C8C3×D4semidirect
289. C2×D8C2×C3×D3semidirect
380. C3×C8C2×#67semidirect
381. C3×C8C2×C3×D4semidirect
382. C3⋊C8C2×C3×D4semidirect
385. C2×C32#496semidirect
419. C2×C8C3×#67semidirect
470. C32C2×#496semidirect
503. C8C2×C3×#67semidirect
514. C2×C3#5154direct (N × H)
515. C2×C3C3×#496semidirect
539. C3C2×#5154direct (N × H)
540. C3C2×C3×#496semidirect
544. C2C3×#5154direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×D82
C2×C3×#9402
C2×C3×#9362
C22×C3×#2142
C22×C3×#2132
C22×C3×#2042
C3×#51542
C2×C3×#8152
C2×C3×#4963
C2×#51543