Finite Group Databasebeta

← C2×C3×#5165 #11865528

Subgroups — C2×C3×#5165 #11865528

Order 1152.

1277 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×SD16C211yesmaximal, characteristic, complemented
2C2×C3×#947C211yesmaximal, characteristic, complemented
3C2×C3×D4×Dic3C211yesmaximal, characteristic, complemented
4C22×C3×#213C211yesmaximal, characteristic
5C2×C3×#835C211yesmaximal, characteristic, complemented
6C3×#5165C218yesmaximal, direct product factor
7C2×C3×#821C211yesmaximal, characteristic, complemented
8C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
9C2×C3×#463—31nomaximal
10C2×#5165C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C23×C3×Dic3—21no—
14C22×C3×D12C2211yescharacteristic
15C2×C32×SD16—24no—
16C2×C32×SD16C2214yescomplemented
17C22×C32×C8C2211yescharacteristic
18C2×C3×#217—21no—
19C3×#947—22no—
20C3×#947C2214yescomplemented
21C3×D4×Dic3—22no—
22C3×D4×Dic3C2214yescomplemented
23C2×C3×#213—24no—
24C2×C3×#213C2214yes—
25C2×C3×(C2×D3)⋊C4—21no—
26C2×C3×(C2×D3)⋊C4—21no—
27C2×C3×C4⋊Dic3C2211yescharacteristic
28C2×C3×C4×Dic3C2211yescharacteristic
29C22×C3×C3⋊C8C2211yescharacteristic
30C3×#835C2214yescomplemented
31C3×#821C2214yescomplemented
32C3×Dic3⋊C8C2214yescomplemented
33C22×C3×SD16—21no—
34C22×C3×SD16D311yescharacteristic, complemented
35C22×C3×SD16C2×C311yescharacteristic, complemented
36C22×C3×D8—31no—
37C2×C3×#124—31no—
38C2×C3×C4×D4—31no—
39C2×#947C2×C311yescharacteristic, complemented
40C2×D4×Dic3C2×C311yescharacteristic, complemented
41C22×#213C2×C311yescharacteristic
42C3×#463—38no—
43C2×C3×C4⋊C8—31no—
44C2×C3×#103—31no—
45C2×C3×#102—31no—
46C2×#835C2×C311yescharacteristic, complemented
47#5165C2×C318yesdirect product factor
48C2×#821C2×C311yescharacteristic, complemented
49C2×Dic3⋊C8C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C23×C3×D3—21no—
52C2×C32×Q8—21no—
53C2×C32×Q8—22no—
54C2×C32×Q8—22no—
55C2×C32×Q8C2312yes—
56C2×C32×D4—21no—
57C2×C32×D4—22no—
58C2×C32×D4D414yes—
59C2×C32×D4C2312yes—
60C22×C32×C4—21no—
61C22×C32×C4C2311yescharacteristic
62C22×C3×Dic3—21no—
63C22×C3×Dic3—24no—
64C22×C3×Dic3—44no—
65C22×C3×Dic3D411yescharacteristic
66C22×C3×Dic3D411yescharacteristic
67C2×C3×D12—22no—
68C2×C3×D12—22no—
69C2×C3×D12—21no—
70C2×C3×D12C2312yes—
71C32×SD16—28no—
72C2×C32×C8—22no—
73C2×C32×C8C2312yes—
74C3×#217—24no—
75C3×#213—28no—
76C3×(C2×D3)⋊C4—24no—
77C3×(C2×D3)⋊C4—24no—
78C3×C4⋊Dic3—21no—
79C3×C4⋊Dic3C2312yes—
80C3×C4×Dic3D412yes—
81C3×C4×Dic3C2312yes—
82C2×C3×C3⋊C8—22no—
83C2×C3×C3⋊C8C2312yes—
84C2×#463—31noSylow
85C22×C3×Q8—21no—
86C22×C3×Q8C2×D311yescharacteristic
87C22×C3×Q8C22×C311yescharacteristic
88C22×C3×D4—21no—
89C22×C3×D4—31no—
90C22×C3×D4C2×D311yescharacteristic
91C22×C3×D4C22×C311yescharacteristic
92C23×C3×C4—61no—
93C23×Dic3—21no—
94C22×D12C22×C311yescharacteristic
95C2×C3×SD16—28no—
96C2×C3×SD16—24no—
97C2×C3×SD16—24no—
98C2×C3×SD16—24no—
99C2×C3×SD16C2×D314yescomplemented
100C2×C3×SD16C22×C314yescomplemented
101C2×C3×D8—34no—
102C2×C3×D8—64no—
103C22×C3×C8—21no—
104C22×C3×C8—31no—
105C22×C3×C8C2×D311yescharacteristic
106C22×C3×C8C22×C311yescharacteristic
107C3×#124—34no—
108C3×#124—62no—
109C3×C4×D4—34no—
110C3×C4×D4—62no—
111C2×C3×C4⋊C4—31no—
112C2×C3×C22⋊C4—61no—
113C2×C3×C22⋊C4—61no—
114C2×C3×C22⋊C4—61no—
115C2×C3×C42—31no—
116C2×#217—21no—
117#947—22no—
118#947C22×C314yescomplemented
119D4×Dic3—22no—
120D4×Dic3C22×C314yescomplemented
121C2×#213—24no—
122C2×#213C22×C314yes—
123C2×(C2×D3)⋊C4—21no—
124C2×(C2×D3)⋊C4—21no—
125C2×C4⋊Dic3C22×C311yescharacteristic
126C2×C4×Dic3C22×C311yescharacteristic
127C22×C3⋊C8C22×C311yescharacteristic
128C3×C4⋊C8—34no—
129C3×#103—34no—
130C3×#102—34no—
131#835C22×C314yescomplemented
132#821C22×C314yescomplemented
133Dic3⋊C8C22×C314yescomplemented
134C23×C32—24no—
135C23×C32—22no—
136C23×C32—42no—
137C23×C32—42no—
138C23×C32C2×D411yescharacteristic
139C22×C3×D3—22no—
140C22×C3×D3—42no—
141C22×C3×D3—42no—
142C22×C3×D3—42no—
143C32×Q8—22no—
144C32×Q8—22no—
145C32×Q8—42no—
146C32×D4—24no—
147C32×D4—22no—
148C32×D4C2×D414yes—
149C2×C32×C4—22no—
150C2×C32×C4—42no—
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4C2×D412yes—
153C2×C32×C4#4112yes—
154C2×C32×C4C2411yescharacteristic
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—48no—
159C2×C3×Dic3—42no—
160C2×C3×Dic3C2×D412yes—
161C2×C3×Dic3C2×D412yes—
162C3×D12—22no—
163C3×D12—22no—
164C3×D12—42no—
165C32×C8—22no—
166C3×C3⋊C8—22no—
167C22×SD16C3×D311yescharacteristic, complemented
168C22×D8—31no—
169C2×#124—31no—
170C2×C4×D4—31no—
171#463—38no—
172C2×C4⋊C8—31no—
173C2×#103—31no—
174C2×#102—31no—
175C24×C3—21no—
176C24×C3—21no—
177C24×C3—41no—
178C24×C3—61no—
179C23×D3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—24no—
183C2×C3×Q8—22no—
184C2×C3×Q8—22no—
185C2×C3×Q8—22no—
186C2×C3×Q8—22no—
187C2×C3×Q8—22no—
188C2×C3×Q8—41no—
189C2×C3×Q8—42no—
190C2×C3×Q8C22×D312yes—
191C2×C3×Q8C23×C312yes—
192C2×C3×D4—21no—
193C2×C3×D4—22no—
194C2×C3×D4—21no—
195C2×C3×D4—22no—
196C2×C3×D4—24no—
197C2×C3×D4—22no—
198C2×C3×D4—32no—
199C2×C3×D4—41no—
200C2×C3×D4—42no—
201C2×C3×D4—61no—
202C2×C3×D4—62no—
203C2×C3×D4—62no—
204C2×C3×D4#6714yes—
205C2×C3×D4C3×D414yes—
206C2×C3×D4C22×D312yes—
207C2×C3×D4C23×C312yes—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—31no—
212C22×C3×C4—31no—
213C22×C3×C4—41no—
214C22×C3×C4—64no—
215C22×C3×C4—61no—
216C22×C3×C4—124no—
217C22×C3×C4C22×D311yescharacteristic
218C22×C3×C4C23×C311yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—24no—
221C22×Dic3—44no—
222C22×Dic3C3×D411yescharacteristic
223C22×Dic3C3×D411yescharacteristic
224C2×D12—22no—
225C2×D12—22no—
226C2×D12—21no—
227C2×D12C23×C312yes—
228C3×SD16—216no—
229C3×SD16—28no—
230C3×SD16—28no—
231C3×D8—68no—
232C2×C3×C8—24no—
233C2×C3×C8—22no—
234C2×C3×C8—22no—
235C2×C3×C8—22no—
236C2×C3×C8—32no—
237C2×C3×C8—62no—
238C2×C3×C8C22×D312yes—
239C2×C3×C8C23×C312yes—
240C3×C4⋊C4—32no—
241C3×C4⋊C4—61no—
242C3×C22⋊C4—64no—
243C3×C22⋊C4—64no—
244C3×C22⋊C4—64no—
245C3×C42—32no—
246C3×C42—32no—
247#217—24no—
248#213—28no—
249(C2×D3)⋊C4—24no—
250(C2×D3)⋊C4—24no—
251C4⋊Dic3—21no—
252C4⋊Dic3C23×C312yes—
253C4×Dic3C3×D412yes—
254C4×Dic3C23×C312yes—
255C2×C3⋊C8—22no—
256C2×C3⋊C8C23×C312yes—
257C22×C32—24no—
258C22×C32—42no—
259C22×C32—42no—
260C22×C32—48no—
261C22×C32#12112yes—
262C22×C32#13512yes—
263C22×C32AGL(1,ℤ/8)12yes—
264C22×C32C22×D411yescharacteristic
265C2×C3×D3—42no—
266C2×C3×D3—42no—
267C2×C3×D3—42no—
268C2×C3×D3—84no—
269C32×C4—42no—
270C32×C4#12112yes—
271C32×C4C22×D411yescharacteristic
272C32×C4C2×#4111yescharacteristic
273C3×Dic3—22no—
274C3×Dic3—22no—
275C3×Dic3—42no—
276C22×Q8C2×C3×D311yescharacteristic
277C22×D4—31no—
278C22×D4C2×C3×D311yescharacteristic
279C23×C4—61no—
280C2×SD16—24no—
281C2×SD16C2×C3×D314yescomplemented
282C2×D8—34no—
283C2×D8—64no—
284C22×C8—31no—
285C22×C8C2×C3×D311yescharacteristic
286#124—34no—
287#124—62no—
288C4×D4—34no—
289C4×D4—62no—
290C2×C4⋊C4—31no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C42—31no—
295C4⋊C8—34no—
296#103—34no—
297#102—34no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—44no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—62no—
310C23×C3—82no—
311C23×C3—82no—
312C23×C3—122no—
313C23×C3—122no—
314C23×C3—122no—
315C23×C3D3×D411yescharacteristic
316C23×C3C2×C3×D411yescharacteristic
317C22×D3—22no—
318C22×D3—42no—
319C22×D3—42no—
320C22×D3—42no—
321C3×Q8—24no—
322C3×Q8—22no—
323C3×Q8—22no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—42no—
327C3×Q8—44no—
328C3×Q8—42no—
329C3×Q8—42no—
330C3×D4—24no—
331C3×D4—24no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—24no—
335C3×D4—44no—
336C3×D4—42no—
337C3×D4—62no—
338C3×D4—62no—
339C3×D4—122no—
340C3×D4C2×#6714yes—
341C3×D4C2×C3×D414yes—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—32no—
349C2×C3×C4—32no—
350C2×C3×C4—42no—
351C2×C3×C4—44no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—128no—
358C2×C3×C4—122no—
359C2×C3×C4D3×D411yescharacteristic
360C2×C3×C4#23712yes—
361C2×C3×C4C2×#6712yes—
362C2×C3×C4C2×C3×D411yescharacteristic
363C2×C3×C4C2×C3×D412yes—
364C2×C3×C4C3×#4112yes—
365C2×C3×C4C23×D311yescharacteristic
366C2×C3×C4C24×C311yescharacteristic
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—48no—
371C2×Dic3—42no—
372C2×Dic3C2×C3×D412yes—
373C2×Dic3C2×C3×D412yes—
374D12—22no—
375D12—22no—
376D12—42no—
377C3×C8—24no—
378C3×C8—22no—
379C3×C8—22no—
380C3×C8—62no—
381C3⋊C8—22no—
382C2×C32—44no—
383C2×C32#46314yescomplemented
384C2×C32C2×#12111yescharacteristic
385C2×C32C2×#13511yescharacteristic
386C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
387C3×D3—82no—
388C24—21no—
389C24—61no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8C22×C3×D312yes—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—32no—
397C2×D4—61no—
398C2×D4—62no—
399C2×D4—62no—
400C2×D4C3×#6714yes—
401C2×D4C22×C3×D312yes—
402C22×C4—21no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—64no—
406C22×C4—61no—
407C22×C4—124no—
408C22×C4C22×C3×D311yescharacteristic
409SD16—28no—
410D8—68no—
411C2×C8—22no—
412C2×C8—32no—
413C2×C8—62no—
414C2×C8C22×C3×D312yes—
415C4⋊C4—32no—
416C4⋊C4—61no—
417C22⋊C4—64no—
418C22⋊C4—64no—
419C22⋊C4—64no—
420C42—32no—
421C42—32no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—21no—
428C22×C3—44no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—48no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—48no—
435C22×C3—88no—
436C22×C3—82no—
437C22×C3—82no—
438C22×C3—122no—
439C22×C3—122no—
440C22×C3—122no—
441C22×C3—244no—
442C22×C3#91412yes—
443C22×C3#91612yes—
444C22×C3#93912yes—
445C22×C3C3×#12112yes—
446C22×C3C3×#13512yes—
447C22×C3C3×AGL(1,ℤ/8)12yes—
448C22×C3C2×D3×D411yescharacteristic
449C22×C3C22×C3×D411yescharacteristic
450C2×D3—42no—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—84no—
454C3×C4—22no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—44no—
458C3×C4—42no—
459C3×C4—42no—
460C3×C4—62no—
461C3×C4—62no—
462C3×C4—122no—
463C3×C4#93912yes—
464C3×C4C3×#12112yes—
465C3×C4C2×#23711yescharacteristic
466C3×C4C22×#6711yescharacteristic
467C3×C4C22×C3×D411yescharacteristic
468C3×C4C2×C3×#4111yescharacteristic
469Dic3—22no—
470Dic3—22no—
471Dic3—42no—
472C32C2×#46311yescharacteristic, complemented, Sylow
473C23—24no—
474C23—22no—
475C23—42no—
476C23—42no—
477C23—62no—
478C23—122no—
479C23—122no—
480C23—122no—
481C23C3×D3×D411yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—42no—
485D4—24no—
486D4—22no—
487D4—62no—
488D4—62no—
489D4—122no—
490D4C2×C3×#6714yes—
491C2×C4—22no—
492C2×C4—32no—
493C2×C4—32no—
494C2×C4—42no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—128no—
499C2×C4—122no—
500C2×C4C3×D3×D411yescharacteristic
501C2×C4C3×#23712yes—
502C2×C4C2×C3×#6712yes—
503C2×C4C23×C3×D311yescharacteristic
504C8—22no—
505C8—62no—
506C2×C3—24no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—44no—
511C2×C3—44no—
512C2×C3—84no—
513C2×C3—242no—
514C2×C3#516514yesdirect product factor
515C2×C3C3×#46314yescomplemented
516C2×C3C2×#91411yescharacteristic
517C2×C3C2×#91611yescharacteristic
518C2×C3C2×#93911yescharacteristic
519C2×C3C2×C3×#12111yescharacteristic
520C2×C3C2×C3×#13511yescharacteristic
521C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
522D3—82no—
523C22—24no—
524C22—48no—
525C22—42no—
526C22—42no—
527C22—122no—
528C22—122no—
529C22—122no—
530C22—244no—
531C22C3×#91412yes—
532C22C3×#91612yes—
533C22C3×#93912yes—
534C22C2×C3×D3×D411yescharacteristic
535C4—42no—
536C4—62no—
537C4—62no—
538C4—122no—
539C4C3×#93912yes—
540C4C2×C3×#23711yescharacteristic
541C4C22×C3×#6711yescharacteristic
542C3—21no—
543C3C2×#516511yescharacteristic, direct product factor
544C3C2×C3×#46311yescharacteristic, complemented
545C2—44no—
546C2—242no—
547C2C3×#516514yesdirect product factor
548C2C2×C3×#91411yescharacteristic
549C2C2×C3×#91611yescharacteristic
550C2C2×C3×#93911yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×SD16C2
socleC23×C32C2×D4
prime radicalC22×C3×#213C2
hypercenterC22×C3×SD16D3
hypocenterC3C2×C3×#463
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C22×C32×SD16C2semidirect
2. C2×C3×#947C2semidirect
3. C2×C3×D4×Dic3C2semidirect
5. C2×C3×#835C2semidirect
6. C3×#5165C2direct (N × H)
7. C2×C3×#821C2semidirect
8. C2×C3×Dic3⋊C8C2semidirect
10. C2×#5165C3direct (N × H)
16. C2×C32×SD16C22semidirect
20. C3×#947C22semidirect
22. C3×D4×Dic3C22semidirect
30. C3×#835C22semidirect
31. C3×#821C22semidirect
32. C3×Dic3⋊C8C22semidirect
34. C22×C3×SD16D3semidirect
35. C22×C3×SD16C2×C3semidirect
39. C2×#947C2×C3semidirect
40. C2×D4×Dic3C2×C3semidirect
46. C2×#835C2×C3semidirect
47. #5165C2×C3direct (N × H)
48. C2×#821C2×C3semidirect
49. C2×Dic3⋊C8C2×C3semidirect
99. C2×C3×SD16C2×D3semidirect
100. C2×C3×SD16C22×C3semidirect
118. #947C22×C3semidirect
120. D4×Dic3C22×C3semidirect
131. #835C22×C3semidirect
132. #821C22×C3semidirect
133. Dic3⋊C8C22×C3semidirect
167. C22×SD16C3×D3semidirect
281. C2×SD16C2×C3×D3semidirect
383. C2×C32#463semidirect
472. C32C2×#463semidirect
514. C2×C3#5165direct (N × H)
515. C2×C3C3×#463semidirect
543. C3C2×#5165direct (N × H)
544. C3C2×C3×#463semidirect
547. C2C3×#5165direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SD162
C2×C3×#9472
C2×C3×D4×Dic32
C22×C3×#2132
C2×C3×#8352
C3×#51652
C2×C3×#8212
C2×C3×Dic3⋊C82
C2×C3×#4633
C2×#51653