Finite Group Databasebeta

← C2×C3×#5186 #11863642

Subgroups — C2×C3×#5186 #11863642

Order 1152.

1609 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D3×Q8C211yesmaximal, characteristic
2C22×C32×Q16C211yesmaximal, characteristic, complemented
3C2×C3×#946C211yesmaximal, characteristic
4C22×C3×#216C211yesmaximal, characteristic, complemented
5C2×C3×#835C211yesmaximal, characteristic, complemented
6C3×#5186C218yesmaximal, direct product factor
7C2×C3×(C2×D3)⋊C8C211yesmaximal, characteristic
8C2×C3×#816C211yesmaximal, characteristic, complemented
9C2×C3×#451—31nomaximal
10C2×#5186C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×Q8C2211yescharacteristic
13C2×C3×D3×Q8—24no—
14C2×C3×D3×Q8—22no—
15C2×C3×D3×Q8—24no—
16C2×C3×D3×Q8C2214yes—
17C22×C3×C4×D3—21no—
18C22×C3×C4×D3C2211yescharacteristic
19C22×C3×Dic6—21no—
20C22×C3×Dic6C2211yescharacteristic
21C2×C32×Q16—24no—
22C2×C32×Q16C2214yescomplemented
23C22×C32×C8C2211yescharacteristic
24C3×#946—22no—
25C3×#946C2214yes—
26C2×C3×#216—24no—
27C2×C3×#216C2214yescomplemented
28C2×C3×(C2×D3)⋊C4—21no—
29C2×C3×C4⋊Dic3C2211yescharacteristic
30C2×C3×Dic3⋊C4—21no—
31C22×C3×C3⋊C8C2211yescharacteristic
32C3×#835C2214yescomplemented
33C3×(C2×D3)⋊C8C2214yes—
34C3×#816C2214yescomplemented
35C23×C3×Q8—31no—
36C22×D3×Q8C2×C311yescharacteristic
37C22×C3×Q16—21no—
38C22×C3×Q16—31no—
39C22×C3×Q16D311yescharacteristic, complemented
40C22×C3×Q16C2×C311yescharacteristic, complemented
41C2×C3×C22⋊Q8—31no—
42C2×#946C2×C311yescharacteristic
43C22×#216C2×C311yescharacteristic, complemented
44C3×#451—38no—
45C2×C3×#103—31no—
46C2×C3×#103—31no—
47C2×C3×C22⋊C8—31no—
48C2×#835C2×C311yescharacteristic, complemented
49#5186C2×C318yesdirect product factor
50C2×(C2×D3)⋊C8C2×C311yescharacteristic
51C2×#816C2×C311yescharacteristic, complemented
52C23×C3×D3D411yescharacteristic
53C2×C32×Q8—22no—
54C2×C32×Q8—21no—
55C2×C32×Q8—21no—
56C2×C32×Q8—22no—
57C2×C32×Q8—22no—
58C2×C32×Q8D414yes—
59C2×C32×Q8C2312yes—
60C2×C32×Q8C2312yes—
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C22×C32×C4C2311yescharacteristic
64C22×C3×Dic3—21no—
65C22×C3×Dic3—21no—
66C22×C3×Dic3D411yescharacteristic
67C3×D3×Q8—216no—
68C3×D3×Q8—28no—
69C3×D3×Q8—28no—
70C2×C3×C4×D3—28no—
71C2×C3×C4×D3—24no—
72C2×C3×C4×D3—22no—
73C2×C3×C4×D3—22no—
74C2×C3×C4×D3D412yes—
75C2×C3×C4×D3C2312yes—
76C2×C3×Dic6—28no—
77C2×C3×Dic6—21no—
78C2×C3×Dic6—22no—
79C2×C3×Dic6—24no—
80C2×C3×Dic6D414yes—
81C2×C3×Dic6C2312yes—
82C32×Q16—28no—
83C2×C32×C8—22no—
84C2×C32×C8C2312yes—
85C3×#216—28no—
86C3×(C2×D3)⋊C4—24no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3C2312yes—
89C3×Dic3⋊C4—24no—
90C2×C3×C3⋊C8—22no—
91C2×C3×C3⋊C8C2312yes—
92C2×#451—31noSylow
93C22×C3×Q8—21no—
94C22×C3×Q8—21no—
95C22×C3×Q8—34no—
96C22×C3×Q8—31no—
97C22×C3×Q8—61no—
98C22×C3×Q8—62no—
99C22×C3×Q8—64no—
100C22×C3×Q8—64no—
101C22×C3×Q8C2×D311yescharacteristic
102C22×C3×Q8C2×D311yescharacteristic
103C22×C3×Q8C22×C311yescharacteristic
104C22×C3×Q8C22×C311yescharacteristic
105C23×C3×C4—31no—
106C23×C3×C4—61no—
107C2×D3×Q8—24no—
108C2×D3×Q8—22no—
109C2×D3×Q8—24no—
110C2×D3×Q8C22×C314yes—
111C22×C4×D3—21no—
112C22×C4×D3C22×C311yescharacteristic
113C22×Dic6—21no—
114C22×Dic6C22×C311yescharacteristic
115C2×C3×Q16—28no—
116C2×C3×Q16—24no—
117C2×C3×Q16—24no—
118C2×C3×Q16—24no—
119C2×C3×Q16—34no—
120C2×C3×Q16—64no—
121C2×C3×Q16C2×D314yescomplemented
122C2×C3×Q16C22×C314yescomplemented
123C22×C3×C8—21no—
124C22×C3×C8—31no—
125C22×C3×C8C2×D311yescharacteristic
126C22×C3×C8C22×C311yescharacteristic
127C3×C22⋊Q8—34no—
128C3×C22⋊Q8—62no—
129C2×C3×C4⋊C4—31no—
130C2×C3×C4⋊C4—61no—
131C2×C3×C22⋊C4—61no—
132#946—22no—
133#946C22×C314yes—
134C2×#216—24no—
135C2×#216C22×C314yescomplemented
136C2×(C2×D3)⋊C4—21no—
137C2×C4⋊Dic3C22×C311yescharacteristic
138C2×Dic3⋊C4—21no—
139C22×C3⋊C8C22×C311yescharacteristic
140C3×#103—34no—
141C3×#103—34no—
142C3×C22⋊C8—34no—
143#835C22×C314yescomplemented
144(C2×D3)⋊C8C22×C314yes—
145#816C22×C314yescomplemented
146C23×C32C2×D411yescharacteristic
147C22×C3×D3—22no—
148C22×C3×D3—22no—
149C22×C3×D3Q1614yescomplemented
150C22×C3×D3C2×D412yes—
151C32×Q8—22no—
152C32×Q8—24no—
153C32×Q8—22no—
154C32×Q8—22no—
155C32×Q8—42no—
156C32×Q8C2×D414yes—
157C2×C32×C4—24no—
158C2×C32×C4—22no—
159C2×C32×C4—22no—
160C2×C32×C4—42no—
161C2×C32×C4C2×D411yescharacteristic
162C2×C32×C4C2×D412yes—
163C2×C32×C4C2×D412yes—
164C2×C32×C4C2411yescharacteristic
165C2×C3×Dic3—22no—
166C2×C3×Dic3—24no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—42no—
170C2×C3×Dic3C2×D412yes—
171C3×C4×D3—216no—
172C3×C4×D3—24no—
173C3×C4×D3—22no—
174C3×C4×D3—22no—
175C3×Dic6—216no—
176C3×Dic6—24no—
177C3×Dic6—22no—
178C3×Dic6C2×D414yes—
179C32×C8—22no—
180C3×C3⋊C8—22no—
181C23×Q8—31no—
182C22×Q16—31no—
183C22×Q16C3×D311yescharacteristic, complemented
184C2×C22⋊Q8—31no—
185#451—38no—
186C2×#103—31no—
187C2×#103—31no—
188C2×C22⋊C8—31no—
189C24×C3—31no—
190C23×D3C3×D411yescharacteristic
191C2×C3×Q8—22no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—22no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—24no—
198C2×C3×Q8—22no—
199C2×C3×Q8—22no—
200C2×C3×Q8—22no—
201C2×C3×Q8—22no—
202C2×C3×Q8—24no—
203C2×C3×Q8—22no—
204C2×C3×Q8—22no—
205C2×C3×Q8—34no—
206C2×C3×Q8—32no—
207C2×C3×Q8—42no—
208C2×C3×Q8—41no—
209C2×C3×Q8—41no—
210C2×C3×Q8—42no—
211C2×C3×Q8—64no—
212C2×C3×Q8—68no—
213C2×C3×Q8—68no—
214C2×C3×Q8—616no—
215C2×C3×Q8—68no—
216C2×C3×Q8—61no—
217C2×C3×Q8—62no—
218C2×C3×Q8#6714yes—
219C2×C3×Q8C3×D414yes—
220C2×C3×Q8C22×D312yes—
221C2×C3×Q8C22×D312yes—
222C2×C3×Q8C23×C312yes—
223C2×C3×Q8C23×C312yes—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—31no—
230C22×C3×C4—32no—
231C22×C3×C4—32no—
232C22×C3×C4—41no—
233C22×C3×C4—41no—
234C22×C3×C4—64no—
235C22×C3×C4—68no—
236C22×C3×C4—61no—
237C22×C3×C4—62no—
238C22×C3×C4—62no—
239C22×C3×C4—61no—
240C22×C3×C4C22×D311yescharacteristic
241C22×C3×C4C23×C311yescharacteristic
242C22×Dic3—21no—
243C22×Dic3—21no—
244C22×Dic3C3×D411yescharacteristic
245D3×Q8—216no—
246D3×Q8—28no—
247D3×Q8—28no—
248C2×C4×D3—28no—
249C2×C4×D3—24no—
250C2×C4×D3—22no—
251C2×C4×D3—22no—
252C2×C4×D3C3×D412yes—
253C2×C4×D3C23×C312yes—
254C2×Dic6—28no—
255C2×Dic6—21no—
256C2×Dic6—22no—
257C2×Dic6—24no—
258C2×Dic6C3×D414yes—
259C2×Dic6C23×C312yes—
260C3×Q16—216no—
261C3×Q16—28no—
262C3×Q16—28no—
263C3×Q16—68no—
264C2×C3×C8—24no—
265C2×C3×C8—22no—
266C2×C3×C8—22no—
267C2×C3×C8—22no—
268C2×C3×C8—32no—
269C2×C3×C8—62no—
270C2×C3×C8C22×D312yes—
271C2×C3×C8C23×C312yes—
272C3×C4⋊C4—32no—
273C3×C4⋊C4—64no—
274C3×C4⋊C4—61no—
275C3×C22⋊C4—64no—
276#216—28no—
277(C2×D3)⋊C4—24no—
278C4⋊Dic3—21no—
279C4⋊Dic3C23×C312yes—
280Dic3⋊C4—24no—
281C2×C3⋊C8—22no—
282C2×C3⋊C8C23×C312yes—
283C22×C32#12012yes—
284C22×C32C2×Q1612yes—
285C22×C32#13712yes—
286C22×C32C22×D411yescharacteristic
287C2×C3×D3—22no—
288C2×C3×D3—22no—
289C2×C3×D3—28no—
290C2×C3×D3C2×Q1614yescomplemented
291C32×C4—24no—
292C32×C4—42no—
293C32×C4#12012yes—
294C32×C4C22×D411yescharacteristic
295C32×C4C22×D411yescharacteristic
296C3×Dic3—22no—
297C3×Dic3—24no—
298C3×Dic3—42no—
299C22×Q8—34no—
300C22×Q8—31no—
301C22×Q8—61no—
302C22×Q8—62no—
303C22×Q8—64no—
304C22×Q8—64no—
305C22×Q8C2×C3×D311yescharacteristic
306C22×Q8C2×C3×D311yescharacteristic
307C23×C4—31no—
308C23×C4—61no—
309C2×Q16—24no—
310C2×Q16—34no—
311C2×Q16—64no—
312C2×Q16C2×C3×D314yescomplemented
313C22×C8—31no—
314C22×C8C2×C3×D311yescharacteristic
315C22⋊Q8—34no—
316C22⋊Q8—62no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—61no—
319C2×C22⋊C4—61no—
320#103—34no—
321#103—34no—
322C22⋊C8—34no—
323C23×C3—21no—
324C23×C3—34no—
325C23×C3—32no—
326C23×C3—62no—
327C23×C3—62no—
328C23×C3D3×D411yescharacteristic
329C23×C3C2×C3×D411yescharacteristic
330C22×D3—22no—
331C22×D3—22no—
332C22×D3C3×Q1614yescomplemented
333C22×D3C2×C3×D412yes—
334C3×Q8—24no—
335C3×Q8—24no—
336C3×Q8—22no—
337C3×Q8—22no—
338C3×Q8—24no—
339C3×Q8—22no—
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—24no—
344C3×Q8—34no—
345C3×Q8—44no—
346C3×Q8—42no—
347C3×Q8—42no—
348C3×Q8—44no—
349C3×Q8—42no—
350C3×Q8—42no—
351C3×Q8—616no—
352C3×Q8—64no—
353C3×Q8—62no—
354C3×Q8C2×#6714yes—
355C3×Q8C2×C3×D414yes—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—32no—
367C2×C3×C4—44no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—44no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—616no—
374C2×C3×C4—62no—
375C2×C3×C4—64no—
376C2×C3×C4—62no—
377C2×C3×C4—64no—
378C2×C3×C4—62no—
379C2×C3×C4—62no—
380C2×C3×C4—62no—
381C2×C3×C4—122no—
382C2×C3×C4D3×D411yescharacteristic
383C2×C3×C4D3×D412yes—
384C2×C3×C4C2×#6712yes—
385C2×C3×C4C2×C3×D411yescharacteristic
386C2×C3×C4C2×C3×D412yes—
387C2×C3×C4C2×C3×D412yes—
388C2×C3×C4C23×D311yescharacteristic
389C2×C3×C4C24×C311yescharacteristic
390C2×Dic3—22no—
391C2×Dic3—24no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—42no—
395C2×Dic3C2×C3×D412yes—
396C4×D3—216no—
397C4×D3—24no—
398C4×D3—22no—
399C4×D3—22no—
400Dic6—216no—
401Dic6—24no—
402Dic6—22no—
403Dic6C2×C3×D414yes—
404C3×C8—24no—
405C3×C8—22no—
406C3×C8—22no—
407C3×C8—62no—
408C3⋊C8—22no—
409C2×C32#45114yescomplemented
410C2×C32C2×#12011yescharacteristic
411C2×C32C22×Q1611yescharacteristic
412C2×C32C2×#13711yescharacteristic
413C3×D3—24no—
414C24—31no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—32no—
421C2×Q8—34no—
422C2×Q8—64no—
423C2×Q8—68no—
424C2×Q8—68no—
425C2×Q8—616no—
426C2×Q8—68no—
427C2×Q8—61no—
428C2×Q8—62no—
429C2×Q8C3×#6714yes—
430C2×Q8C22×C3×D312yes—
431C2×Q8C22×C3×D312yes—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—31no—
435C22×C4—32no—
436C22×C4—32no—
437C22×C4—64no—
438C22×C4—68no—
439C22×C4—61no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—61no—
443C22×C4C22×C3×D311yescharacteristic
444Q16—28no—
445Q16—68no—
446C2×C8—22no—
447C2×C8—32no—
448C2×C8—62no—
449C2×C8C22×C3×D312yes—
450C4⋊C4—32no—
451C4⋊C4—64no—
452C4⋊C4—61no—
453C22⋊C4—64no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—21no—
458C22×C3—34no—
459C22×C3—68no—
460C22×C3—62no—
461C22×C3—62no—
462C22×C3D3×Q1612yes—
463C22×C3#91812yes—
464C22×C3#93712yes—
465C22×C3C3×#12012yes—
466C22×C3C2×C3×Q1612yes—
467C22×C3C3×#13712yes—
468C22×C3C2×D3×D411yescharacteristic
469C22×C3C22×C3×D411yescharacteristic
470C2×D3—22no—
471C2×D3—28no—
472C2×D3—22no—
473C2×D3C2×C3×Q1614yescomplemented
474C3×C4—24no—
475C3×C4—24no—
476C3×C4—22no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—44no—
480C3×C4—44no—
481C3×C4—42no—
482C3×C4—42no—
483C3×C4—62no—
484C3×C4—64no—
485C3×C4—122no—
486C3×C4#93712yes—
487C3×C4C3×#12012yes—
488C3×C4C2×D3×D411yescharacteristic
489C3×C4C22×#6711yescharacteristic
490C3×C4C22×C3×D411yescharacteristic
491C3×C4C22×C3×D411yescharacteristic
492Dic3—22no—
493Dic3—24no—
494Dic3—42no—
495C32C2×#45111yescharacteristic, complemented, Sylow
496C23—34no—
497C23—32no—
498C23—62no—
499C23—62no—
500C23C3×D3×D411yescharacteristic
501Q8—24no—
502Q8—22no—
503Q8—22no—
504Q8—22no—
505Q8—34no—
506Q8—42no—
507Q8—616no—
508Q8—64no—
509Q8—62no—
510Q8C2×C3×#6714yes—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—32no—
515C2×C4—42no—
516C2×C4—616no—
517C2×C4—62no—
518C2×C4—64no—
519C2×C4—62no—
520C2×C4—64no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—122no—
525C2×C4C3×D3×D411yescharacteristic
526C2×C4C3×D3×D412yes—
527C2×C4C2×C3×#6712yes—
528C2×C4C23×C3×D311yescharacteristic
529C8—22no—
530C8—62no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—24no—
534C2×C3—21no—
535C2×C3—64no—
536C2×C3#518614yesdirect product factor
537C2×C3C3×#45114yescomplemented
538C2×C3C2×D3×Q1611yescharacteristic
539C2×C3C2×#91811yescharacteristic
540C2×C3C2×#93711yescharacteristic
541C2×C3C2×C3×#12011yescharacteristic
542C2×C3C22×C3×Q1611yescharacteristic
543C2×C3C2×C3×#13711yescharacteristic
544D3—24no—
545C22—34no—
546C22—62no—
547C22—68no—
548C22—62no—
549C22C3×D3×Q1612yes—
550C22C3×#91812yes—
551C22C3×#93712yes—
552C22C2×C3×D3×D411yescharacteristic
553C4—24no—
554C4—42no—
555C4—62no—
556C4—64no—
557C4—122no—
558C4C3×#93712yes—
559C4C2×C3×D3×D411yescharacteristic
560C4C22×C3×#6711yescharacteristic
561C3—21no—
562C3C2×#518611yescharacteristic, direct product factor
563C3C2×C3×#45111yescharacteristic, complemented
564C2—64no—
565C2C3×#518614yesdirect product factor
566C2C2×C3×D3×Q1611yescharacteristic
567C2C2×C3×#91811yescharacteristic
568C2C2×C3×#93711yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC22×C32×Q16C2
socleC23×C32C2×D4
prime radicalC23×C3×D3D4
hypercenterC22×C3×Q16D3
hypocenterC3C2×C3×#451
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×Q16

Complements

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

NH ≅ G/Ntype
2. C22×C32×Q16C2semidirect
4. C22×C3×#216C2semidirect
5. C2×C3×#835C2semidirect
6. C3×#5186C2direct (N × H)
8. C2×C3×#816C2semidirect
10. C2×#5186C3direct (N × H)
22. C2×C32×Q16C22semidirect
27. C2×C3×#216C22semidirect
32. C3×#835C22semidirect
34. C3×#816C22semidirect
39. C22×C3×Q16D3semidirect
40. C22×C3×Q16C2×C3semidirect
43. C22×#216C2×C3semidirect
48. C2×#835C2×C3semidirect
49. #5186C2×C3direct (N × H)
51. C2×#816C2×C3semidirect
121. C2×C3×Q16C2×D3semidirect
122. C2×C3×Q16C22×C3semidirect
135. C2×#216C22×C3semidirect
143. #835C22×C3semidirect
145. #816C22×C3semidirect
149. C22×C3×D3Q16semidirect
183. C22×Q16C3×D3semidirect
290. C2×C3×D3C2×Q16semidirect
312. C2×Q16C2×C3×D3semidirect
332. C22×D3C3×Q16semidirect
409. C2×C32#451semidirect
473. C2×D3C2×C3×Q16semidirect
495. C32C2×#451semidirect
536. C2×C3#5186direct (N × H)
537. C2×C3C3×#451semidirect
562. C3C2×#5186direct (N × H)
563. C3C2×C3×#451semidirect
565. C2C3×#5186direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D3×Q82
C22×C32×Q162
C2×C3×#9462
C22×C3×#2162
C2×C3×#8352
C3×#51862
C2×C3×(C2×D3)⋊C82
C2×C3×#8162
C2×C3×#4513
C2×#51863