Finite Group Databasebeta

← C2×C3×#72985 #72276763

Subgroups — C2×C3×#72985 #72276763

Order 2304.

1409 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 1409 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#72985C214yesmaximal, direct product factor
2C2×C3×#4622C211yesmaximal, characteristic, complemented
3C2×C3×#4621C211yesmaximal, characteristic, complemented
4C22×C3×#95⋊C3C211yesmaximal, characteristic, complemented
5C2×#72985C311yesmaximal, characteristic, direct product factor
6C2×C3×#1504—31nomaximal
7C2×C3×GL(2,ℤ/4)—41nomaximal
8C2×C3×#95⋊C3C2211yescharacteristic
9C2×C3×#95⋊C3C2212yescomplemented
10C3×#4622C2212yescomplemented
11C3×#4621C2212yescomplemented
12C2×C3×#525—31no—
13C2×C3×#407—31no—
14C22×C3×#95D311yescharacteristic, complemented
15C22×#95⋊C3—21no—
16C22×#95⋊C3C2×C311yescharacteristic, complemented
17#72985C2×C314yesdirect product factor
18C2×#4622C2×C311yescharacteristic, complemented
19C2×#4621C2×C311yescharacteristic, complemented
20C3×#1504—34no—
21C3×#1504—34no—
22C2×C3×#328—31no—
23C2×C3×#328—31no—
24C2×C3×#327—31no—
25C2×C3×#327—31no—
26C23×C3×A4—41no—
27C22×C3×S4—41no—
28C3×GL(2,ℤ/4)—44no—
29C2×C3×SL(2,ℤ/4)—41no—
30C3×#95⋊C3C2311yescharacteristic
31C2×#1504—31noSylow
32C22×C3×#41—31no—
33C24×C3×C4—31no—
34C2×GL(2,ℤ/4)—41no—
35C22×C3×M16—61no—
36C3×#525—34no—
37C3×#525—34no—
38C3×#525—62no—
39C3×#525—62no—
40C2×C3×#123—61no—
41C2×C3×C22⋊Q8—31no—
42C2×C3×C22⋊Q8—31no—
43C2×C3×#121—31no—
44C2×C3×#121—31no—
45C2×C3×#120—61no—
46C2×C3×C4×D4—61no—
47C2×C3×C4×D4—61no—
48C2×C3×#117—31no—
49C3×#407—34no—
50C3×#407—62no—
51C2×C3×C22⋊C8—31no—
52C2×C3×C22⋊C8—31no—
53C2×C3×C22⋊C8—61no—
54C2×C3×#95—31no—
55C2×C3×#95—32no—
56C2×C3×#95C2×D311yescharacteristic
57C2×C3×#95C2×D312yescomplemented
58C2×#95⋊C3—21no—
59C2×#95⋊C3—22no—
60C2×#95⋊C3C22×C311yescharacteristic
61C2×#95⋊C3C22×C312yescomplemented
62#4622C22×C312yescomplemented
63#4621C22×C312yescomplemented
64C3×#328—32no—
65C3×#328—34no—
66C3×#328—32no—
67C3×#327—32no—
68C3×#327—34no—
69C3×#327—32no—
70C22×C3×A4—41no—
71C22×C3×A4—42no—
72C22×C3×A4—82no—
73C2×C3×S4—42no—
74C2×C3×S4—82no—
75C2×C3×#67—161no—
76C3×SL(2,ℤ/4)—42no—
77C2×#525—31no—
78C2×#407—31no—
79C22×#95C3×D311yescharacteristic, complemented
80#1504—34no—
81#1504—34no—
82C2×#328—31no—
83C2×#328—31no—
84C2×#327—31no—
85C2×#327—31no—
86C25×C3S411yescharacteristic
87C23×A4—41no—
88C23×A4—81no—
89C22×S4—41no—
90C2×C3×#41—34no—
91C2×C3×#41—62no—
92C2×C3×#41—124no—
93C22×C3×Q8—31no—
94C22×C3×D4—31no—
95C22×C3×D4—61no—
96C22×C3×D4—61no—
97C22×C3×D4—61no—
98C23×C3×C4—31no—
99C23×C3×C4—31no—
100C23×C3×C4—31no—
101C23×C3×C4—32no—
102C23×C3×C4—31no—
103C23×C3×C4—32no—
104C23×C3×C4—31no—
105C23×C3×C4—32no—
106C23×C3×C4—31no—
107C23×C3×C4—32no—
108C23×C3×C4—61no—
109C23×C3×C4—124no—
110GL(2,ℤ/4)—44no—
111C2×SL(2,ℤ/4)—41no—
112C2×C3×M16—64no—
113C2×C3×M16—124no—
114C22×C3×C8—61no—
115C22×C3×C8—61no—
116C3×#123—64no—
117C3×#123—64no—
118C3×C22⋊Q8—32no—
119C3×C22⋊Q8—32no—
120C3×C22⋊Q8—34no—
121C3×C22⋊Q8—61no—
122C3×C22⋊Q8—62no—
123C3×C22⋊Q8—61no—
124C3×#121—32no—
125C3×#121—32no—
126C3×#121—34no—
127C3×#121—61no—
128C3×#121—62no—
129C3×#121—61no—
130C3×#120—64no—
131C3×#120—64no—
132C3×C4×D4—64no—
133C3×C4×D4—64no—
134C3×C4×D4—64no—
135C3×C4×D4—64no—
136C3×#117—34no—
137C3×#117—62no—
138C2×C3×C4⋊C4—31no—
139C2×C3×C4⋊C4—31no—
140C2×C3×C4⋊C4—61no—
141C2×C3×C4⋊C4—61no—
142C2×C3×C22⋊C4—61no—
143C2×C3×C22⋊C4—61no—
144C2×C3×C22⋊C4—61no—
145C2×C3×C22⋊C4—61no—
146C2×C3×C22⋊C4—61no—
147C2×C3×C42—61no—
148C3×C22⋊C8—32no—
149C3×C22⋊C8—32no—
150C3×C22⋊C8—61no—
151C3×C22⋊C8—61no—
152C3×C22⋊C8—64no—
153C3×#95—31no—
154C3×#95—32no—
155C3×#95—61no—
156C3×#95C22×D311yescharacteristic
157#95⋊C3—21no—
158#95⋊C3C23×C311yescharacteristic
159C23×C32—161no—
160C22×C3×D3—161no—
161C2×C3×A4—42no—
162C2×C3×A4—41no—
163C2×C3×A4—82no—
164C3×S4—82no—
165C3×#67—164no—
166C2×C3×Dic3—161no—
167C22×#41—31no—
168C24×C4—31no—
169C22×M16—61no—
170#525—34no—
171#525—34no—
172#525—62no—
173#525—62no—
174C2×#123—61no—
175C2×C22⋊Q8—31no—
176C2×C22⋊Q8—31no—
177C2×#121—31no—
178C2×#121—31no—
179C2×#120—61no—
180C2×C4×D4—61no—
181C2×C4×D4—61no—
182C2×#117—31no—
183#407—34no—
184#407—62no—
185C2×C22⋊C8—31no—
186C2×C22⋊C8—31no—
187C2×C22⋊C8—61no—
188C2×#95—31no—
189C2×#95—32no—
190C2×#95C2×C3×D311yescharacteristic
191C2×#95C2×C3×D312yescomplemented
192#328—32no—
193#328—34no—
194#328—32no—
195#327—32no—
196#327—34no—
197#327—32no—
198C24×C3—22no—
199C24×C3—31no—
200C24×C3—31no—
201C24×C3—32no—
202C24×C3—62no—
203C24×C3—61no—
204C24×C3C2×S412yes—
205C24×C3C2×S411yescharacteristic
206C22×A4—41no—
207C22×A4—42no—
208C22×A4—82no—
209C22×A4—84no—
210C22×A4—82no—
211C22×A4—81no—
212C2×S4—42no—
213C2×S4—82no—
214C3×#41—128no—
215C3×#41—128no—
216C2×C3×Q8—32no—
217C2×C3×Q8—61no—
218C2×C3×Q8—122no—
219C2×C3×D4—32no—
220C2×C3×D4—61no—
221C2×C3×D4—62no—
222C2×C3×D4—64no—
223C2×C3×D4—64no—
224C2×C3×D4—62no—
225C2×C3×D4—122no—
226C2×C3×D4—122no—
227C2×C3×D4—122no—
228C2×C3×D4—121no—
229C2×C3×D4—124no—
230C2×C3×D4—121no—
231C2×C3×D4—122no—
232C22×C3×C4—31no—
233C22×C3×C4—31no—
234C22×C3×C4—31no—
235C22×C3×C4—31no—
236C22×C3×C4—32no—
237C22×C3×C4—32no—
238C22×C3×C4—32no—
239C22×C3×C4—32no—
240C22×C3×C4—32no—
241C22×C3×C4—31no—
242C22×C3×C4—32no—
243C22×C3×C4—32no—
244C22×C3×C4—31no—
245C22×C3×C4—32no—
246C22×C3×C4—32no—
247C22×C3×C4—31no—
248C22×C3×C4—32no—
249C22×C3×C4—31no—
250C22×C3×C4—64no—
251C22×C3×C4—64no—
252C22×C3×C4—64no—
253C22×C3×C4—61no—
254C22×C3×C4—61no—
255C22×C3×C4—61no—
256C22×C3×C4—122no—
257C22×C3×C4—122no—
258C22×C3×C4—122no—
259C22×C3×C4—124no—
260C22×C3×C4—124no—
261C22×C3×C4—122no—
262C22×C3×C4—124no—
263C22×C3×C4—124no—
264C22×C3×C4—124no—
265C2×#67—161no—
266SL(2,ℤ/4)—42no—
267C3×M16—128no—
268C2×C3×C8—62no—
269C2×C3×C8—62no—
270C2×C3×C8—122no—
271C2×C3×C8—122no—
272C3×C4⋊C4—32no—
273C3×C4⋊C4—32no—
274C3×C4⋊C4—61no—
275C3×C4⋊C4—61no—
276C3×C4⋊C4—62no—
277C3×C4⋊C4—62no—
278C3×C4⋊C4—64no—
279C3×C22⋊C4—62no—
280C3×C22⋊C4—62no—
281C3×C22⋊C4—64no—
282C3×C22⋊C4—64no—
283C3×C22⋊C4—62no—
284C3×C22⋊C4—64no—
285C3×C22⋊C4—62no—
286C3×C42—64no—
287C22×C32—162no—
288C22×C32—161no—
289C22×C32—322no—
290C2×C3×D3—162no—
291C2×C3×D3—322no—
292C3×A4—41no—
293C3×Dic3—162no—
294C25C3×S411yescharacteristic
295C2×#41—34no—
296C2×#41—62no—
297C2×#41—124no—
298C22×Q8—31no—
299C22×D4—31no—
300C22×D4—61no—
301C22×D4—61no—
302C22×D4—61no—
303C23×C4—31no—
304C23×C4—31no—
305C23×C4—31no—
306C23×C4—31no—
307C23×C4—32no—
308C23×C4—32no—
309C23×C4—31no—
310C23×C4—31no—
311C23×C4—32no—
312C23×C4—32no—
313C23×C4—61no—
314C23×C4—124no—
315C2×M16—64no—
316C2×M16—124no—
317C22×C8—61no—
318C22×C8—61no—
319#123—64no—
320#123—64no—
321C22⋊Q8—32no—
322C22⋊Q8—32no—
323C22⋊Q8—34no—
324C22⋊Q8—61no—
325C22⋊Q8—62no—
326C22⋊Q8—61no—
327#121—32no—
328#121—32no—
329#121—34no—
330#121—61no—
331#121—62no—
332#121—61no—
333#120—64no—
334#120—64no—
335C4×D4—64no—
336C4×D4—64no—
337C4×D4—64no—
338C4×D4—64no—
339#117—34no—
340#117—62no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—31no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C22⋊C4—61no—
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C42—61no—
351C22⋊C8—32no—
352C22⋊C8—32no—
353C22⋊C8—61no—
354C22⋊C8—61no—
355C22⋊C8—64no—
356#95—31no—
357#95—32no—
358#95—61no—
359#95C22×C3×D311yescharacteristic
360C23×C3—22no—
361C23×C3—31no—
362C23×C3—31no—
363C23×C3—32no—
364C23×C3—32no—
365C23×C3—34no—
366C23×C3—32no—
367C23×C3—32no—
368C23×C3—31no—
369C23×C3—62no—
370C23×C3—62no—
371C23×C3—62no—
372C23×C3—61no—
373C23×C3—64no—
374C23×C3—62no—
375C23×C3—64no—
376C23×C3—62no—
377C23×C3—122no—
378C23×C3—122no—
379C23×C3—121no—
380C23×C3—121no—
381C23×C3—161no—
382C23×C3—321no—
383C23×C3#85711yescharacteristic
384C23×C3#98312yes—
385C23×C3C22×S411yescharacteristic
386C22×D3—161no—
387C2×A4—41no—
388C2×A4—42no—
389C2×A4—82no—
390C2×A4—84no—
391C2×A4—81no—
392C2×A4—82no—
393S4—82no—
394C3×Q8—122no—
395C3×Q8—122no—
396C3×D4—122no—
397C3×D4—122no—
398C3×D4—122no—
399C3×D4—124no—
400C3×D4—124no—
401C3×D4—128no—
402C3×D4—124no—
403C3×D4—122no—
404C2×C3×C4—32no—
405C2×C3×C4—31no—
406C2×C3×C4—31no—
407C2×C3×C4—32no—
408C2×C3×C4—31no—
409C2×C3×C4—31no—
410C2×C3×C4—62no—
411C2×C3×C4—64no—
412C2×C3×C4—62no—
413C2×C3×C4—64no—
414C2×C3×C4—62no—
415C2×C3×C4—62no—
416C2×C3×C4—64no—
417C2×C3×C4—64no—
418C2×C3×C4—62no—
419C2×C3×C4—62no—
420C2×C3×C4—62no—
421C2×C3×C4—121no—
422C2×C3×C4—122no—
423C2×C3×C4—121no—
424C2×C3×C4—122no—
425C2×C3×C4—121no—
426C2×C3×C4—122no—
427C2×C3×C4—122no—
428C2×C3×C4—122no—
429C2×C3×C4—122no—
430C2×C3×C4—121no—
431C2×C3×C4—128no—
432C2×C3×C4—122no—
433C2×C3×C4—122no—
434C2×C3×C4—121no—
435C2×C3×C4—121no—
436#67—164no—
437C2×Dic3—161no—
438C3×C8—122no—
439C3×C8—122no—
440C2×C32—162no—
441C2×C32—161no—
442C2×C32—322no—
443C3×D3—322no—
444C24—22no—
445C24—31no—
446C24—31no—
447C24—32no—
448C24—62no—
449C24—61no—
450C24C2×C3×S412yes—
451C24C2×C3×S411yescharacteristic
452#41—128no—
453#41—128no—
454C2×Q8—32no—
455C2×Q8—61no—
456C2×Q8—122no—
457C2×D4—32no—
458C2×D4—61no—
459C2×D4—62no—
460C2×D4—64no—
461C2×D4—64no—
462C2×D4—62no—
463C2×D4—122no—
464C2×D4—124no—
465C2×D4—122no—
466C2×D4—122no—
467C2×D4—121no—
468C2×D4—121no—
469C2×D4—122no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—32no—
475C22×C4—32no—
476C22×C4—32no—
477C22×C4—32no—
478C22×C4—32no—
479C22×C4—31no—
480C22×C4—32no—
481C22×C4—32no—
482C22×C4—31no—
483C22×C4—32no—
484C22×C4—32no—
485C22×C4—31no—
486C22×C4—32no—
487C22×C4—31no—
488C22×C4—64no—
489C22×C4—64no—
490C22×C4—64no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—61no—
494C22×C4—122no—
495C22×C4—122no—
496C22×C4—122no—
497C22×C4—124no—
498C22×C4—124no—
499C22×C4—122no—
500C22×C4—124no—
501C22×C4—124no—
502C22×C4—124no—
503M16—128no—
504C2×C8—62no—
505C2×C8—62no—
506C2×C8—122no—
507C2×C8—122no—
508C4⋊C4—32no—
509C4⋊C4—32no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—62no—
513C4⋊C4—62no—
514C4⋊C4—64no—
515C22⋊C4—62no—
516C22⋊C4—62no—
517C22⋊C4—64no—
518C22⋊C4—64no—
519C22⋊C4—62no—
520C22⋊C4—64no—
521C22⋊C4—62no—
522C42—64no—
523C22×C3—22no—
524C22×C3—31no—
525C22×C3—32no—
526C22×C3—32no—
527C22×C3—32no—
528C22×C3—31no—
529C22×C3—31no—
530C22×C3—34no—
531C22×C3—32no—
532C22×C3—62no—
533C22×C3—62no—
534C22×C3—64no—
535C22×C3—61no—
536C22×C3—62no—
537C22×C3—64no—
538C22×C3—62no—
539C22×C3—122no—
540C22×C3—122no—
541C22×C3—122no—
542C22×C3—161no—
543C22×C3—162no—
544C22×C3—242no—
545C22×C3—242no—
546C22×C3—322no—
547C22×C3—324no—
548C22×C3—322no—
549C22×C3—321no—
550C22×C3C2×#85712yes—
551C22×C3C2×#85711yescharacteristic
552C22×C3C2×#98311yescharacteristic
553C2×D3—162no—
554C2×D3—322no—
555A4—41no—
556A4—81no—
557C3×C4—61no—
558C3×C4—61no—
559C3×C4—62no—
560C3×C4—62no—
561C3×C4—61no—
562C3×C4—61no—
563C3×C4—122no—
564C3×C4—122no—
565C3×C4—122no—
566Dic3—162no—
567C32—161noSylow
568C23—22no—
569C23—31no—
570C23—31no—
571C23—32no—
572C23—32no—
573C23—34no—
574C23—32no—
575C23—32no—
576C23—31no—
577C23—62no—
578C23—62no—
579C23—62no—
580C23—61no—
581C23—64no—
582C23—62no—
583C23—64no—
584C23—62no—
585C23—122no—
586C23—122no—
587C23—121no—
588C23—121no—
589C23C3×#85711yescharacteristic
590C23C3×#98312yes—
591C23C22×C3×S411yescharacteristic
592Q8—122no—
593Q8—122no—
594D4—122no—
595D4—122no—
596D4—128no—
597D4—122no—
598D4—124no—
599D4—124no—
600D4—124no—
601D4—122no—
602C2×C4—32no—
603C2×C4—31no—
604C2×C4—31no—
605C2×C4—32no—
606C2×C4—31no—
607C2×C4—31no—
608C2×C4—62no—
609C2×C4—64no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—64no—
613C2×C4—64no—
614C2×C4—62no—
615C2×C4—64no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—121no—
620C2×C4—122no—
621C2×C4—121no—
622C2×C4—122no—
623C2×C4—121no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4—122no—
628C2×C4—121no—
629C2×C4—128no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—121no—
633C2×C4—121no—
634C8—122no—
635C8—122no—
636C2×C3—22no—
637C2×C3—31no—
638C2×C3—31no—
639C2×C3—32no—
640C2×C3—62no—
641C2×C3—162no—
642C2×C3—161no—
643C2×C3—242no—
644C2×C3—322no—
645C2×C3—324no—
646C2×C3—322no—
647C2×C3—321no—
648C2×C3#7298512yesdirect product factor
649C2×C3C22×#85711yescharacteristic
650D3—322no—
651C22—22no—
652C22—31no—
653C22—32no—
654C22—32no—
655C22—32no—
656C22—31no—
657C22—31no—
658C22—34no—
659C22—32no—
660C22—62no—
661C22—62no—
662C22—64no—
663C22—61no—
664C22—62no—
665C22—64no—
666C22—62no—
667C22—122no—
668C22—122no—
669C22—122no—
670C22—242no—
671C22—242no—
672C22C2×C3×#85712yes—
673C22C2×C3×#85711yescharacteristic
674C22C2×C3×#98311yescharacteristic
675C4—61no—
676C4—62no—
677C4—61no—
678C4—61no—
679C4—61no—
680C4—62no—
681C4—122no—
682C4—122no—
683C4—122no—
684C3—161no—
685C3—321no—
686C3C2×#7298511yescharacteristic, direct product factor
687C2—22no—
688C2—31no—
689C2—31no—
690C2—32no—
691C2—62no—
692C2—242no—
693C2C3×#7298512yesdirect product factor
694C2C22×C3×#85711yescharacteristic
subgroupG/H
centerC22×C3C2×#857
commutator, cosocle, hypocenter#95⋊C3C23×C3
FrattiniC23C22×C3×S4
FittingC22×C3×#95D3
socleC24×C3C2×S4
hypercenterC23×C3#857
LCSG ▷ #95⋊C3
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C3×#72985C2direct (N × H)
2. C2×C3×#4622C2semidirect
3. C2×C3×#4621C2semidirect
4. C22×C3×#95⋊C3C2semidirect
5. C2×#72985C3direct (N × H)
9. C2×C3×#95⋊C3C22semidirect
10. C3×#4622C22semidirect
11. C3×#4621C22semidirect
14. C22×C3×#95D3semidirect
16. C22×#95⋊C3C2×C3semidirect
17. #72985C2×C3direct (N × H)
18. C2×#4622C2×C3semidirect
19. C2×#4621C2×C3semidirect
57. C2×C3×#95C2×D3semidirect
61. C2×#95⋊C3C22×C3semidirect
62. #4622C22×C3semidirect
63. #4621C22×C3semidirect
79. C22×#95C3×D3semidirect
191. C2×#95C2×C3×D3semidirect
648. C2×C3#72985direct (N × H)
686. C3C2×#72985direct (N × H)
693. C2C3×#72985direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#729852
C2×C3×#46222
C2×C3×#46212
C22×C3×#95⋊C32
C2×#729853
C2×C3×#15043
C2×C3×GL(2,ℤ/4)4