Finite Group Databasebeta

← C2×C3×#5016 #11865522

Subgroups — C2×C3×#5016 #11865522

Order 1152.

1341 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic, complemented
2C22×C3×#214C211yesmaximal, characteristic, complemented
3C22×C3×#213C211yesmaximal, characteristic
4C2×C3×#875C211yesmaximal, characteristic, complemented
5C3×#5016C218yesmaximal, direct product factor
6C2×C3×#810C211yesmaximal, characteristic, complemented
7C2×C3×#809C211yesmaximal, characteristic, complemented
8C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
9C2×C3×#463—31nomaximal
10C2×#5016C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C23×C32×C4—21no—
13C22×C3×D12C2211yescharacteristic
14C22×C3×Dic6C2211yescharacteristic
15C32×C4×D4—22no—
16C32×C4×D4C2214yescomplemented
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C22⋊C4—21no—
19C2×C32×C42C2211yescharacteristic
20C2×C3×#214—24no—
21C2×C3×#214C2214yescomplemented
22C2×C3×#213—24no—
23C2×C3×#213C2214yes—
24C2×C3×(C2×D3)⋊C4—21no—
25C2×C3×(C2×D3)⋊C4—21no—
26C22×C3×C3⋊C8C2211yescharacteristic
27C22×C3×C3⋊C8C2211yescharacteristic
28C3×#875—22no—
29C3×#875C2214yescomplemented
30C3×#810C2214yescomplemented
31C3×#809C2214yescomplemented
32C3×C4⋊(C3⋊C8)C2214yescomplemented
33C22×C3×SD16—31no—
34C22×C3×D8—31no—
35C2×C3×#124—31no—
36C2×C3×C4×D4—21no—
37C2×C3×C4×D4D311yescharacteristic, complemented
38C2×C3×C4×D4C2×C311yescharacteristic, complemented
39C22×#214C2×C311yescharacteristic, complemented
40C22×#213C2×C311yescharacteristic
41C2×#875C2×C311yescharacteristic, complemented
42C3×#463—38no—
43C2×C3×C4⋊C8—31no—
44C2×C3×#103—31no—
45C2×C3×#102—31no—
46#5016C2×C318yesdirect product factor
47C2×#810C2×C311yescharacteristic, complemented
48C2×#809C2×C311yescharacteristic, complemented
49C2×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
50C24×C32—21no—
51C23×C3×D3—21no—
52C2×C32×D4—21no—
53C2×C32×D4—22no—
54C2×C32×D4D414yes—
55C2×C32×D4C2312yes—
56C22×C32×C4—21no—
57C22×C32×C4—24no—
58C22×C32×C4—44no—
59C22×C32×C4D411yescharacteristic
60C22×C32×C4D411yescharacteristic
61C22×C32×C4C2311yescharacteristic
62C22×C3×Dic3—21no—
63C2×C3×D12—22no—
64C2×C3×D12—22no—
65C2×C3×D12—21no—
66C2×C3×D12C2312yes—
67C2×C3×Dic6—22no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2312yes—
71C32×C4⋊C4—21no—
72C32×C4⋊C4C2312yes—
73C32×C22⋊C4—24no—
74C32×C42D412yes—
75C32×C42C2312yes—
76C3×#214—28no—
77C3×#213—28no—
78C3×(C2×D3)⋊C4—24no—
79C3×(C2×D3)⋊C4—24no—
80C2×C3×C3⋊C8—22no—
81C2×C3×C3⋊C8—22no—
82C2×C3×C3⋊C8C2312yes—
83C2×C3×C3⋊C8C2312yes—
84C2×#463—31noSylow
85C22×C3×Q8—31no—
86C22×C3×D4—21no—
87C22×C3×D4—31no—
88C22×C3×D4C2×D311yescharacteristic
89C22×C3×D4C22×C311yescharacteristic
90C23×C3×C4—22no—
91C23×C3×C4—21no—
92C23×C3×C4—21no—
93C22×D12C22×C311yescharacteristic
94C22×Dic6C22×C311yescharacteristic
95C2×C3×SD16—34no—
96C2×C3×SD16—64no—
97C2×C3×D8—34no—
98C2×C3×D8—64no—
99C22×C3×C8—31no—
100C22×C3×C8—31no—
101C3×#124—34no—
102C3×#124—62no—
103C3×C4×D4—24no—
104C3×C4×D4—22no—
105C3×C4×D4—22no—
106C3×C4×D4—24no—
107C3×C4×D4C2×D314yescomplemented
108C3×C4×D4C22×C314yescomplemented
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4C2×D311yescharacteristic
111C2×C3×C4⋊C4C22×C311yescharacteristic
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—61no—
116C2×C3×C22⋊C4—61no—
117C2×C3×C42—21no—
118C2×C3×C42C2×D311yescharacteristic
119C2×C3×C42C22×C311yescharacteristic
120C2×#214—24no—
121C2×#214C22×C314yescomplemented
122C2×#213—24no—
123C2×#213C22×C314yes—
124C2×(C2×D3)⋊C4—21no—
125C2×(C2×D3)⋊C4—21no—
126C22×C3⋊C8C22×C311yescharacteristic
127C22×C3⋊C8C22×C311yescharacteristic
128#875—22no—
129#875C22×C314yescomplemented
130C3×C4⋊C8—34no—
131C3×#103—34no—
132C3×#102—34no—
133#810C22×C314yescomplemented
134#809C22×C314yescomplemented
135C4⋊(C3⋊C8)C22×C314yescomplemented
136C23×C32—24no—
137C23×C32—22no—
138C23×C32—42no—
139C23×C32—42no—
140C23×C32C2×D411yescharacteristic
141C22×C3×D3—22no—
142C22×C3×D3—42no—
143C22×C3×D3—42no—
144C22×C3×D3—42no—
145C32×D4—24no—
146C32×D4—22no—
147C32×D4C2×D414yes—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—48no—
152C2×C32×C4—42no—
153C2×C32×C4C2×D412yes—
154C2×C32×C4C2×D412yes—
155C2×C32×C4C2×D412yes—
156C2×C32×C4C2×D411yescharacteristic
157C2×C32×C4#4112yes—
158C2×C32×C4C2411yescharacteristic
159C2×C3×Dic3—22no—
160C2×C3×Dic3—42no—
161C3×D12—22no—
162C3×D12—22no—
163C3×D12—42no—
164C3×Dic6—22no—
165C3×Dic6—22no—
166C3×Dic6—42no—
167C3×C3⋊C8—22no—
168C3×C3⋊C8—22no—
169C22×SD16—31no—
170C22×D8—31no—
171C2×#124—31no—
172C2×C4×D4C3×D311yescharacteristic, complemented
173#463—38no—
174C2×C4⋊C8—31no—
175C2×#103—31no—
176C2×#102—31no—
177C24×C3—22no—
178C24×C3—21no—
179C24×C3—21no—
180C24×C3—61no—
181C23×D3—21no—
182C2×C3×Q8—32no—
183C2×C3×Q8—61no—
184C2×C3×Q8—62no—
185C2×C3×Q8—62no—
186C2×C3×D4—22no—
187C2×C3×D4—24no—
188C2×C3×D4—22no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—22no—
192C2×C3×D4—24no—
193C2×C3×D4—22no—
194C2×C3×D4—32no—
195C2×C3×D4—62no—
196C2×C3×D4—62no—
197C2×C3×D4—61no—
198C2×C3×D4D1214yes—
199C2×C3×D4C3×D414yes—
200C2×C3×D4C22×D312yes—
201C2×C3×D4C23×C312yes—
202C22×C3×C4—22no—
203C22×C3×C4—28no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—24no—
207C22×C3×C4—24no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—48no—
212C22×C3×C4—44no—
213C22×C3×C4—44no—
214C22×C3×C4—61no—
215C22×C3×C4#6711yescharacteristic
216C22×C3×C4#6711yescharacteristic
217C22×C3×C4C3×D411yescharacteristic
218C22×C3×C4C3×D411yescharacteristic
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C23×C311yescharacteristic
221C22×Dic3—21no—
222C2×D12—22no—
223C2×D12—22no—
224C2×D12—21no—
225C2×D12C23×C312yes—
226C2×Dic6—22no—
227C2×Dic6—22no—
228C2×Dic6—21no—
229C2×Dic6C23×C312yes—
230C3×SD16—68no—
231C3×D8—68no—
232C2×C3×C8—32no—
233C2×C3×C8—32no—
234C2×C3×C8—62no—
235C2×C3×C8—62no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4C22×D312yes—
241C3×C4⋊C4C23×C312yes—
242C3×C22⋊C4—28no—
243C3×C22⋊C4—24no—
244C3×C22⋊C4—24no—
245C3×C22⋊C4—64no—
246C3×C22⋊C4—64no—
247C3×C42—22no—
248C3×C42—22no—
249C3×C42#6712yes—
250C3×C42C3×D412yes—
251C3×C42C22×D312yes—
252C3×C42C23×C312yes—
253#214—28no—
254#213—28no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257C2×C3⋊C8—22no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8C23×C312yes—
260C2×C3⋊C8C23×C312yes—
261C22×C32—24no—
262C22×C32—42no—
263C22×C32—42no—
264C22×C32—48no—
265C22×C32#12112yes—
266C22×C32#13512yes—
267C22×C32AGL(1,ℤ/8)12yes—
268C22×C32C22×D411yescharacteristic
269C2×C3×D3—42no—
270C2×C3×D3—42no—
271C2×C3×D3—42no—
272C2×C3×D3—84no—
273C32×C4—22no—
274C32×C4—22no—
275C32×C4—42no—
276C32×C4#12112yes—
277C32×C4C22×D411yescharacteristic
278C32×C4C2×#4111yescharacteristic
279C3×Dic3—42no—
280C22×Q8—31no—
281C22×D4—31no—
282C22×D4C2×C3×D311yescharacteristic
283C23×C4—21no—
284C2×SD16—34no—
285C2×SD16—64no—
286C2×D8—34no—
287C2×D8—64no—
288C22×C8—31no—
289C22×C8—31no—
290#124—34no—
291#124—62no—
292C4×D4—22no—
293C4×D4C2×C3×D314yescomplemented
294C2×C4⋊C4C2×C3×D311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C42C2×C3×D311yescharacteristic
299C4⋊C8—34no—
300#103—34no—
301#102—34no—
302C23×C3—24no—
303C23×C3—28no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—21no—
309C23×C3—44no—
310C23×C3—44no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—62no—
316C23×C3—122no—
317C23×C3—122no—
318C23×C3—122no—
319C23×C3C2×#6711yescharacteristic
320C23×C3C2×C3×D411yescharacteristic
321C22×D3—22no—
322C22×D3—42no—
323C22×D3—42no—
324C22×D3—42no—
325C3×Q8—62no—
326C3×Q8—62no—
327C3×Q8—122no—
328C3×D4—28no—
329C3×D4—24no—
330C3×D4—24no—
331C3×D4—24no—
332C3×D4—22no—
333C3×D4—22no—
334C3×D4—24no—
335C3×D4—62no—
336C3×D4—62no—
337C3×D4—122no—
338C3×D4C2×D1214yes—
339C3×D4C2×C3×D414yes—
340C2×C3×C4—24no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—42no—
355C2×C3×C4—416no—
356C2×C3×C4—48no—
357C2×C3×C4—48no—
358C2×C3×C4—44no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—62no—
362C2×C3×C4—122no—
363C2×C3×C4C2×D1212yes—
364C2×C3×C4#23512yes—
365C2×C3×C4C2×#6712yes—
366C2×C3×C4C2×#6712yes—
367C2×C3×C4C2×#6711yescharacteristic
368C2×C3×C4C2×C3×D412yes—
369C2×C3×C4C2×C3×D412yes—
370C2×C3×C4C2×C3×D412yes—
371C2×C3×C4C2×C3×D411yescharacteristic
372C2×C3×C4C3×#4112yes—
373C2×C3×C4C23×D311yescharacteristic
374C2×C3×C4C24×C311yescharacteristic
375C2×Dic3—22no—
376C2×Dic3—42no—
377D12—22no—
378D12—22no—
379D12—42no—
380Dic6—22no—
381Dic6—22no—
382Dic6—42no—
383C3×C8—62no—
384C3×C8—62no—
385C3⋊C8—22no—
386C3⋊C8—22no—
387C2×C32—44no—
388C2×C32#46314yescomplemented
389C2×C32C2×#12111yescharacteristic
390C2×C32C2×#13511yescharacteristic
391C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
392C3×D3—82no—
393C24—21no—
394C24—61no—
395C2×Q8—32no—
396C2×Q8—61no—
397C2×Q8—62no—
398C2×Q8—62no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—32no—
402C2×D4—62no—
403C2×D4—62no—
404C2×D4—61no—
405C2×D4C3×D1214yes—
406C2×D4C22×C3×D312yes—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—44no—
410C22×C4—61no—
411C22×C4C3×#6711yescharacteristic
412C22×C4C3×#6711yescharacteristic
413C22×C4C22×C3×D311yescharacteristic
414SD16—68no—
415D8—68no—
416C2×C8—32no—
417C2×C8—32no—
418C2×C8—62no—
419C2×C8—62no—
420C4⋊C4—21no—
421C4⋊C4C22×C3×D312yes—
422C22⋊C4—24no—
423C22⋊C4—64no—
424C22⋊C4—64no—
425C42C3×#6712yes—
426C42C22×C3×D312yes—
427C22×C3—28no—
428C22×C3—24no—
429C22×C3—24no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—21no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—416no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—48no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—48no—
443C22×C3—122no—
444C22×C3—122no—
445C22×C3—122no—
446C22×C3—244no—
447C22×C3#93012yes—
448C22×C3#93212yes—
449C22×C3#95012yes—
450C22×C3C3×#12112yes—
451C22×C3C3×#13512yes—
452C22×C3C3×AGL(1,ℤ/8)12yes—
453C22×C3C22×#6711yescharacteristic
454C22×C3C22×C3×D411yescharacteristic
455C2×D3—42no—
456C2×D3—42no—
457C2×D3—42no—
458C2×D3—84no—
459C3×C4—24no—
460C3×C4—22no—
461C3×C4—22no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—22no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—42no—
468C3×C4—44no—
469C3×C4—42no—
470C3×C4—42no—
471C3×C4—122no—
472C3×C4#93012yes—
473C3×C4C3×#12112yes—
474C3×C4C22×D1211yescharacteristic
475C3×C4C2×#23511yescharacteristic
476C3×C4C22×C3×D411yescharacteristic
477C3×C4C2×C3×#4111yescharacteristic
478Dic3—42no—
479C32C2×#46311yescharacteristic, complemented, Sylow
480C23—24no—
481C23—22no—
482C23—42no—
483C23—42no—
484C23—62no—
485C23—122no—
486C23—122no—
487C23—122no—
488C23C2×C3×#6711yescharacteristic
489Q8—62no—
490Q8—62no—
491Q8—122no—
492D4—24no—
493D4—22no—
494D4—62no—
495D4—62no—
496D4—122no—
497D4C2×C3×D1214yes—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—62no—
504C2×C4—122no—
505C2×C4C2×C3×D1212yes—
506C2×C4C3×#23512yes—
507C2×C4C2×C3×#6712yes—
508C2×C4C2×C3×#6712yes—
509C2×C4C2×C3×#6711yescharacteristic
510C2×C4C23×C3×D311yescharacteristic
511C8—62no—
512C8—62no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—24no—
517C2×C3—48no—
518C2×C3—44no—
519C2×C3—44no—
520C2×C3—242no—
521C2×C3#501614yesdirect product factor
522C2×C3C3×#46314yescomplemented
523C2×C3C2×#93011yescharacteristic
524C2×C3C2×#93211yescharacteristic
525C2×C3C2×#95011yescharacteristic
526C2×C3C2×C3×#12111yescharacteristic
527C2×C3C2×C3×#13511yescharacteristic
528C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
529D3—82no—
530C22—24no—
531C22—42no—
532C22—42no—
533C22—48no—
534C22—122no—
535C22—122no—
536C22—122no—
537C22—244no—
538C22C3×#93012yes—
539C22C3×#93212yes—
540C22C3×#95012yes—
541C22C22×C3×#6711yescharacteristic
542C4—22no—
543C4—22no—
544C4—42no—
545C4—122no—
546C4C3×#93012yes—
547C4C22×C3×D1211yescharacteristic
548C4C2×C3×#23511yescharacteristic
549C3—21no—
550C3C2×#501611yescharacteristic, direct product factor
551C3C2×C3×#46311yescharacteristic, complemented
552C2—44no—
553C2—242no—
554C2C3×#501614yesdirect product factor
555C2C2×C3×#93011yescharacteristic
556C2C2×C3×#93211yescharacteristic
557C2C2×C3×#95011yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C4×D4C2
socleC23×C32C2×D4
prime radicalC22×C3×#213C2
hypercenterC2×C3×C4×D4D3
hypocenterC3C2×C3×#463
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×D4C2semidirect
2. C22×C3×#214C2semidirect
4. C2×C3×#875C2semidirect
5. C3×#5016C2direct (N × H)
6. C2×C3×#810C2semidirect
7. C2×C3×#809C2semidirect
8. C2×C3×C4⋊(C3⋊C8)C2semidirect
10. C2×#5016C3direct (N × H)
16. C32×C4×D4C22semidirect
21. C2×C3×#214C22semidirect
29. C3×#875C22semidirect
30. C3×#810C22semidirect
31. C3×#809C22semidirect
32. C3×C4⋊(C3⋊C8)C22semidirect
37. C2×C3×C4×D4D3semidirect
38. C2×C3×C4×D4C2×C3semidirect
39. C22×#214C2×C3semidirect
41. C2×#875C2×C3semidirect
46. #5016C2×C3direct (N × H)
47. C2×#810C2×C3semidirect
48. C2×#809C2×C3semidirect
49. C2×C4⋊(C3⋊C8)C2×C3semidirect
107. C3×C4×D4C2×D3semidirect
108. C3×C4×D4C22×C3semidirect
121. C2×#214C22×C3semidirect
129. #875C22×C3semidirect
133. #810C22×C3semidirect
134. #809C22×C3semidirect
135. C4⋊(C3⋊C8)C22×C3semidirect
172. C2×C4×D4C3×D3semidirect
293. C4×D4C2×C3×D3semidirect
388. C2×C32#463semidirect
479. C32C2×#463semidirect
521. C2×C3#5016direct (N × H)
522. C2×C3C3×#463semidirect
550. C3C2×#5016direct (N × H)
551. C3C2×C3×#463semidirect
554. C2C3×#5016direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C4×D42
C22×C3×#2142
C22×C3×#2132
C2×C3×#8752
C3×#50162
C2×C3×#8102
C2×C3×#8092
C2×C3×C4⋊(C3⋊C8)2
C2×C3×#4633
C2×#50163