Finite Group Databasebeta

← #67×D24 #11885756

Subgroups — #67×D24 #11885756

Order 1152.

1307 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D24C211yesmaximal, characteristic, complemented
2D12×#67C212yesmaximal, complemented
3C2×D3×D24C211yesmaximal, characteristic, complemented
4#10592814C211yesmaximal, characteristic, complemented
5C3×C8×#67C211yesmaximal, characteristic, complemented
6#10591162C212yesmaximal, complemented
7#10590944C212yesmaximal, complemented
8#10590880C211yesmaximal, characteristic, complemented
9#10590831C211yesmaximal, characteristic, complemented
10Dic3×D24C211yesmaximal, characteristic, complemented
11#10590753C212yesmaximal, complemented
12D8×#67—31nomaximal
13D4×D24—31nomaximal
14C22×C3×D12C2212yescomplemented
15C2×D3×#67—22no—
16C2×D3×D12C2212yescomplemented
17C22×C32×C8C2211yescharacteristic, complemented
18#64139C2212yescomplemented
19C2×#3745C2211yescharacteristic, complemented
20C3×C4×#67C2211yescharacteristic
21C2×C3×D24—24no—
22C2×C3×D24C2211yescharacteristic
23C2×C3×D24C2212yescomplemented
24C2×C3×D24C2211yescharacteristic, complemented
25C2×C3×D3×C8C2211yescharacteristic, complemented
26#63977—22no—
27#63922—22no—
28#63911C2212yescomplemented
29#63908C2212yescomplemented
30Dic3×D12C2212yescomplemented
31#63886—22no—
32C2×#3714C2212yescomplemented
33C2×#3714C2212yescomplemented
34D3×D24—24no—
35#63648C2212yescomplemented
36#63645C2211yescharacteristic, complemented
37C3×#830C2211yescharacteristic, complemented
38C3×(C2×D3)⋊C8C2211yescharacteristic, complemented
39C3×Dic3⋊C8C2211yescharacteristic, complemented
40C3×C8×Dic3C2211yescharacteristic, complemented
41#63563C2212yescomplemented
42#63563C2212yescomplemented
43C22×C3×D8—31no—
44D4×#67—32no—
45C2×D3×D8—31no—
46C22×D24—31no—
47C22×D24D311yescharacteristic, complemented
48D4×D12—32no—
49C3×C8×D4—31no—
50#5217—32no—
51#5157—31no—
52#5156—32no—
53#5151—31no—
54#5150—32no—
55Dic3×D8—31no—
56#5115—31no—
57#5115—31no—
58#5115—31no—
59#5115—61no—
60C8×#67D311yescharacteristic, complemented
61#4843—32no—
62#4732—32no—
63#4732—32no—
64#4695—31no—
65C4×D24—31no—
66C23×C3×D3—22no—
67C22×D32—22no—
68C22×C32×C4C2311yescharacteristic
69C2×#638—22no—
70C2×C4⋊(C3⋊D3)C2312yes—
71C2×C3×#67D411yescharacteristic
72C2×C3×D12—24no—
73C2×C3×D12—22no—
74C2×C3×D12—42no—
75C2×C3×D12C2312yes—
76C2×C3×D12C2312yes—
77C2×C3×C4×D3C2311yescharacteristic
78D3×#67—48no—
79C2×#626—22no—
80C2×#626—22no—
81C2×#626—22no—
82C2×#625—22no—
83C2×D3×Dic3—22no—
84D3×D12—24no—
85D3×D12—42no—
86C2×C32×C8—22no—
87C2×C32×C8C2311yescharacteristic
88C2×C32×C8C2311yescharacteristic
89#3752—22no—
90C4⋊(C3⋊Dic3)C2312yes—
91#3745—22no—
92C3×#217D411yescharacteristic
93C3×(C2×D3)⋊C4D411yescharacteristic
94C3×Dic3⋊C4D411yescharacteristic
95C3×C4×Dic3C2311yescharacteristic
96C2×C3×C3⋊C8C2311yescharacteristic
97C2×C3×C3⋊C8C2311yescharacteristic
98C3×D24—24no—
99C3×D24—22no—
100C3×D24D414yescomplemented
101C3×D3×C8—22no—
102(C2×D3)⋊Dic3—22no—
103(C2×D3)⋊Dic3—22no—
104#3714—24no—
105#3714—42no—
106D4×D8—91noSylow
107C22×C3×D4—32no—
108C22×#67—62no—
109C2×D3×D4—32no—
110C2×D3×D4—62no—
111C22×D12—32no—
112C22×D12C2×D312yescomplemented
113C2×C3×D8—31no—
114C2×C3×D8—32no—
115C2×C3×D8—31no—
116C2×C3×D8—64no—
117C22×C3×C8—21no—
118C22×C3×C8—31no—
119C22×C3×C8C2×D311yescharacteristic, complemented
120C22×C3×C8C2×D311yescharacteristic, complemented
121C3×C4×D4—31no—
122#953—62no—
123#940—32no—
124#939—62no—
125#938—32no—
126#937—62no—
127D4×Dic3—32no—
128C2×#213—32no—
129C2×#213—32no—
130#930—32no—
131#930—32no—
132#930—32no—
133#930—62no—
134C4×#67C2×D311yescharacteristic
135D3×D8—64no—
136C2×D24—24no—
137C2×D24—32no—
138C2×D24—31no—
139C2×D24—31no—
140C2×D24—32no—
141C2×D24—61no—
142C2×D24—64no—
143C2×D24C2×D311yescharacteristic
144C2×D24C2×D311yescharacteristic, complemented
145C2×D24C2×D312yescomplemented
146C2×D3×C8C2×D311yescharacteristic, complemented
147#895—62no—
148#882—62no—
149#882—62no—
150#874—32no—
151C4×D12—32no—
152C3×C4⋊C8—31no—
153C3×C22⋊C8—31no—
154C3×C22⋊C8—31no—
155C3×C4×C8—31no—
156#832—32no—
157#832—32no—
158#830C2×D311yescharacteristic, complemented
159#821—32no—
160#821—32no—
161#821—32no—
162#821—32no—
163#821—62no—
164(C2×D3)⋊C8C2×D311yescharacteristic, complemented
165#818—31no—
166#818—31no—
167#818—61no—
168Dic3⋊C8C2×D311yescharacteristic, complemented
169C8×Dic3C2×D311yescharacteristic, complemented
170C23×C32C2×D411yescharacteristic
171C22×C3⋊D3—22no—
172C22×C3×D3—22no—
173C22×C3×D3—22no—
174C22×C3×D3—44no—
175C22×C3×D3—44no—
176C22×C3×D3—82no—
177C22×C3×D3C2×D411yescharacteristic
178C2×D32—44no—
179C2×D32—44no—
180C2×D32—82no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4C2×D411yescharacteristic
184C2×C32×C4C2411yescharacteristic
185#638—44no—
186C2×C3⋊Dic3—22no—
187C4⋊(C3⋊D3)—22no—
188C4⋊(C3⋊D3)—22no—
189C3×#67D814yescomplemented
190C2×C3×Dic3C2×D411yescharacteristic
191C2×C3×Dic3C2×D411yescharacteristic
192C2×C3×Dic3C2×D411yescharacteristic
193C3×D12—22no—
194C3×D12—22no—
195C3×D12—22no—
196C3×D12—44no—
197C3×D12C2×D414yescomplemented
198C3×C4×D3—21no—
199C3×C4×D3—21no—
200#626—44no—
201#626—44no—
202#626—82no—
203#625—44no—
204D3×Dic3—44no—
205C32×C8—21no—
206C32×C8C2×D412yescomplemented
207C3×C3⋊C8—21no—
208C3×C3⋊C8—21no—
209C22×D8—31no—
210C22×D8—91no—
211D42—92no—
212#493—91no—
213#466—91no—
214#466—91no—
215#459—92no—
216#447—92no—
217#447—92no—
218C4×D8—91no—
219C8×D4—31no—
220C24×C3—62no—
221C23×D3—22no—
222C23×D3—62no—
223C23×D3—62no—
224C2×C3×D4—31no—
225C2×C3×D4—32no—
226C2×C3×D4—32no—
227C2×C3×D4—64no—
228C2×C3×D4—62no—
229C2×C3×D4—122no—
230C22×C3×C4—21no—
231C22×C3×C4—31no—
232C22×C3×C4C22×D311yescharacteristic
233C22×C3×C4C22×D311yescharacteristic
234C2×#67—62no—
235C2×#67—62no—
236C2×#67—62no—
237C2×#67—62no—
238C2×#67—62no—
239C2×#67—62no—
240C2×#67—62no—
241C2×#67—122no—
242C2×#67—128no—
243C2×#67D1211yescharacteristic
244C22×Dic3—62no—
245D3×D4—64no—
246D3×D4—122no—
247D3×D4—128no—
248C2×D12—24no—
249C2×D12—22no—
250C2×D12—32no—
251C2×D12—32no—
252C2×D12—42no—
253C2×D12—62no—
254C2×D12—64no—
255C2×D12—62no—
256C2×D12—62no—
257C2×D12—62no—
258C2×D12—122no—
259C2×D12C22×D312yes—
260C2×D12C22×D312yes—
261C2×C4×D3—62no—
262C2×C4×D3C22×D311yescharacteristic
263C3×D8—34no—
264C3×D8—64no—
265C3×D8—62no—
266C2×C3×C8—22no—
267C2×C3×C8—22no—
268C2×C3×C8—21no—
269C2×C3×C8—21no—
270C2×C3×C8—31no—
271C2×C3×C8—31no—
272C2×C3×C8—42no—
273C2×C3×C8—62no—
274C2×C3×C8C22×D311yescharacteristic
275C2×C3×C8C22×D311yescharacteristic
276C2×C3×C8C22×D311yescharacteristic
277C2×C3×C8C22×D311yescharacteristic
278C3×C4⋊C4—31no—
279C3×C22⋊C4—31no—
280C3×C22⋊C4—31no—
281C3×C42—31no—
282#217—62no—
283#217—62no—
284#217D1211yescharacteristic
285#213—64no—
286#213—122no—
287(C2×D3)⋊C4—62no—
288(C2×D3)⋊C4—62no—
289(C2×D3)⋊C4—62no—
290(C2×D3)⋊C4—62no—
291(C2×D3)⋊C4—122no—
292(C2×D3)⋊C4D1211yescharacteristic
293C4⋊Dic3—32no—
294C4⋊Dic3—32no—
295C4⋊Dic3—62no—
296Dic3⋊C4D1211yescharacteristic
297C4×Dic3C22×D311yescharacteristic
298C2×C3⋊C8C22×D311yescharacteristic
299C2×C3⋊C8C22×D311yescharacteristic
300D24—24no—
301D24—22no—
302D24—64no—
303D24—62no—
304D24—62no—
305D24—122no—
306D24—122no—
307D24#6714yesdirect product factor
308D3×C8—22no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32C2×D812yescomplemented
312C22×C32C22×D411yescharacteristic
313C2×C3⋊D3—42no—
314C2×C3⋊D3—42no—
315C2×C3⋊D3—42no—
316C2×C3×D3—21no—
317C2×C3×D3—21no—
318C2×C3×D3—24no—
319C2×C3×D3—42no—
320C2×C3×D3—42no—
321C2×C3×D3—42no—
322C2×C3×D3—42no—
323C2×C3×D3—42no—
324C2×C3×D3—88no—
325C2×C3×D3C2×D812yescomplemented
326D32—88no—
327C32×C4—21no—
328C32×C4C22×D411yescharacteristic
329C32×C4H(5,𝔽2)11yescharacteristic
330C3⋊Dic3—42no—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3C2×D812yescomplemented
334C22×D4—32no—
335C22×D4—92no—
336C22×D4—182no—
337C2×D8—31no—
338C2×D8—32no—
339C2×D8—31no—
340C2×D8—64no—
341C2×D8—92no—
342C2×D8—92no—
343C2×D8—91no—
344C2×D8—184no—
345C22×C8—31no—
346C22×C8D3211yescharacteristic, complemented
347#127—92no—
348#121—92no—
349#121—92no—
350#121—182no—
351#120—182no—
352#120—182no—
353C4×D4—31no—
354C4×D4—92no—
355#107—91no—
356C4⋊C8—31no—
357#102—92no—
358#102—92no—
359#102—92no—
360C22⋊C8—31no—
361C22⋊C8—31no—
362C4×C8—31no—
363C23×C3—21no—
364C23×C3—31no—
365C23×C3—62no—
366C23×C3—62no—
367C23×C3—124no—
368C23×C3—124no—
369C23×C3—242no—
370C23×C3C2×D1211yescharacteristic
371C23×C3D3×D411yescharacteristic
372C22×D3—22no—
373C22×D3—22no—
374C22×D3—44no—
375C22×D3—44no—
376C22×D3—62no—
377C22×D3—62no—
378C22×D3—82no—
379C22×D3—122no—
380C22×D3—124no—
381C22×D3—124no—
382C22×D3—124no—
383C22×D3—124no—
384C22×D3—242no—
385C22×D3—242no—
386C22×D3C2×D1211yescharacteristic
387C3×D4—34no—
388C3×D4—34no—
389C3×D4—62no—
390C3×D4—62no—
391C3×D4—62no—
392C3×D4—124no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—31no—
400C2×C3×C4—31no—
401C2×C3×C4—31no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—61no—
405C2×C3×C4—61no—
406C2×C3×C4C2×D1211yescharacteristic
407C2×C3×C4D3×D411yescharacteristic
408C2×C3×C4C23×D311yescharacteristic
409C2×C3×C4C23×D311yescharacteristic
410#67—124no—
411#67—124no—
412#67—124no—
413#67—124no—
414#67—124no—
415#67—124no—
416#67—244no—
417#67—242no—
418#67D2414yesdirect product factor
419C2×Dic3—62no—
420C2×Dic3—62no—
421C2×Dic3—122no—
422C2×Dic3—124no—
423C2×Dic3C2×D1211yescharacteristic
424C2×Dic3C2×D1211yescharacteristic
425C2×Dic3C2×D1211yescharacteristic
426D12—22no—
427D12—22no—
428D12—22no—
429D12—44no—
430D12—62no—
431D12—62no—
432D12—62no—
433D12—62no—
434D12—122no—
435D12—122no—
436D12—124no—
437D12—124no—
438D12—242no—
439D12C2×#6714yescomplemented
440C4×D3—21no—
441C4×D3—21no—
442C4×D3—124no—
443C3×C8—21no—
444C3×C8—21no—
445C3×C8—22no—
446C3×C8—41no—
447C3×C8—61no—
448C3×C8—61no—
449C3×C8D3×D412yescomplemented
450C3×C8C2×#6712yescomplemented
451C3⋊C8—21no—
452C3⋊C8—21no—
453C2×C32—22no—
454C2×C32D4211yescharacteristic
455C2×C32C22×D811yescharacteristic
456C2×C32#57611yescharacteristic
457C3⋊D3—82no—
458C3×D3—22no—
459C3×D3—44no—
460C3×D3—82no—
461C24—62no—
462C24—182no—
463C2×D4—31no—
464C2×D4—32no—
465C2×D4—32no—
466C2×D4—64no—
467C2×D4—62no—
468C2×D4—92no—
469C2×D4—122no—
470C2×D4—182no—
471C2×D4—182no—
472C2×D4—182no—
473C2×D4—184no—
474C2×D4—182no—
475C2×D4—182no—
476C2×D4—362no—
477C2×D4—368no—
478C22×C4—31no—
479C22×C4—182no—
480C22×C4C2×D3211yescharacteristic
481D8—34no—
482D8—64no—
483D8—62no—
484D8—184no—
485D8—182no—
486D8—362no—
487C2×C8—22no—
488C2×C8—31no—
489C2×C8—31no—
490C2×C8—62no—
491C2×C8C2×D3211yescharacteristic
492C2×C8C2×D3211yescharacteristic
493C4⋊C4—31no—
494C4⋊C4—92no—
495C22⋊C4—31no—
496C22⋊C4—31no—
497C22⋊C4—182no—
498C22⋊C4—182no—
499C22⋊C4—182no—
500C42—31no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—22no—
507C22×C3—32no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—61no—
511C22×C3—61no—
512C22×C3—64no—
513C22×C3—122no—
514C22×C3—122no—
515C22×C3—122no—
516C22×C3—122no—
517C22×C3—122no—
518C22×C3—248no—
519C22×C3C2×D2412yescomplemented
520C22×C3D3×D812yescomplemented
521C22×C3C22×D1211yescharacteristic
522C22×C3C2×D3×D411yescharacteristic
523C2×D3—21no—
524C2×D3—21no—
525C2×D3—24no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C2×D3—42no—
531C2×D3—88no—
532C2×D3—122no—
533C2×D3—122no—
534C2×D3—122no—
535C2×D3—122no—
536C2×D3—122no—
537C2×D3—122no—
538C2×D3—242no—
539C2×D3—242no—
540C2×D3—242no—
541C2×D3—248no—
542C2×D3—248no—
543C2×D3C2×D2412yescomplemented
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—32no—
549C3×C4—41no—
550C3×C4—61no—
551C3×C4—61no—
552C3×C4C2×D3×D411yescharacteristic
553C3×C4#100411yescharacteristic
554C3×C4#100911yescharacteristic
555C3×C4C22×#6711yescharacteristic
556Dic3—21no—
557Dic3—21no—
558Dic3—122no—
559Dic3—122no—
560Dic3—242no—
561Dic3C2×D2412yescomplemented
562C32D4×D811yescharacteristic, complemented, Sylow
563C23—31no—
564C23—62no—
565C23—62no—
566C23—124no—
567C23—124no—
568C23—182no—
569C23—242no—
570C23—364no—
571C23—364no—
572C23—722no—
573C23D3×D1211yescharacteristic
574D4—34no—
575D4—34no—
576D4—62no—
577D4—62no—
578D4—62no—
579D4—124no—
580D4—182no—
581D4—182no—
582D4—364no—
583D4—364no—
584D4—364no—
585D4—364no—
586D4—722no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—31no—
590C2×C4—31no—
591C2×C4—31no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—182no—
595C2×C4—364no—
596C2×C4D3×D1211yescharacteristic
597C2×C4C22×D3211yescharacteristic
598C8—21no—
599C8—61no—
600C8—61no—
601C8D3×#6712yescomplemented
602C2×C3—22no—
603C2×C3—22no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—42no—
608C2×C3—62no—
609C2×C3—124no—
610C2×C3—242no—
611C2×C3D4×D1211yescharacteristic
612C2×C3C22×D2411yescharacteristic
613C2×C3#575211yescharacteristic
614C2×C3C2×D3×D811yescharacteristic
615C2×C3#575711yescharacteristic
616C2×C3D4×#6711yescharacteristic
617D3—22no—
618D3—44no—
619D3—82no—
620D3—242no—
621D3—242no—
622D3—482no—
623C22—21no—
624C22—21no—
625C22—32no—
626C22—61no—
627C22—61no—
628C22—64no—
629C22—122no—
630C22—122no—
631C22—122no—
632C22—122no—
633C22—122no—
634C22—248no—
635C22—362no—
636C22—362no—
637C22—362no—
638C22—728no—
639C22D3×D2412yescomplemented
640C22C2×D3×D1211yescharacteristic
641C4—21no—
642C4—32no—
643C4—61no—
644C4—61no—
645C4—362no—
646C4#6430711yescharacteristic
647C4C2×D3×#6711yescharacteristic
648C3—21no—
649C3D4×D2411yescharacteristic, complemented
650C3D8×#6711yescharacteristic, complemented
651C2—22no—
652C2—62no—
653C2—124no—
654C2—242no—
655C2—722no—
656C2C2×D3×D2411yescharacteristic
657C2#1059467811yescharacteristic
658C2D12×#6711yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC22×C32×C8C22
socleC22×C32C22×D4
hypercenterC22×C8D32
hypocenterC32D4×D8
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C22×C3×D24C2semidirect
2. D12×#67C2semidirect
3. C2×D3×D24C2semidirect
4. #10592814C2semidirect
5. C3×C8×#67C2semidirect
6. #10591162C2semidirect
7. #10590944C2semidirect
8. #10590880C2semidirect
9. #10590831C2semidirect
10. Dic3×D24C2semidirect
11. #10590753C2semidirect
14. C22×C3×D12C22semidirect
16. C2×D3×D12C22semidirect
17. C22×C32×C8C22semidirect
18. #64139C22semidirect
19. C2×#3745C22semidirect
23. C2×C3×D24C22semidirect
24. C2×C3×D24C22semidirect
25. C2×C3×D3×C8C22semidirect
28. #63911C22semidirect
29. #63908C22semidirect
30. Dic3×D12C22semidirect
32. C2×#3714C22semidirect
33. C2×#3714C22semidirect
35. #63648C22semidirect
36. #63645C22semidirect
37. C3×#830C22semidirect
38. C3×(C2×D3)⋊C8C22semidirect
39. C3×Dic3⋊C8C22semidirect
40. C3×C8×Dic3C22semidirect
41. #63563C22semidirect
42. #63563C22semidirect
47. C22×D24D3semidirect
60. C8×#67D3semidirect
100. C3×D24D4semidirect
112. C22×D12C2×D3semidirect
119. C22×C3×C8C2×D3semidirect
120. C22×C3×C8C2×D3semidirect
144. C2×D24C2×D3semidirect
145. C2×D24C2×D3semidirect
146. C2×D3×C8C2×D3semidirect
158. #830C2×D3semidirect
164. (C2×D3)⋊C8C2×D3semidirect
168. Dic3⋊C8C2×D3semidirect
169. C8×Dic3C2×D3semidirect
189. C3×#67D8semidirect
197. C3×D12C2×D4semidirect
206. C32×C8C2×D4semidirect
307. D24#67direct (N × H)
311. C22×C32C2×D8semidirect
325. C2×C3×D3C2×D8semidirect
333. C3×Dic3C2×D8semidirect
346. C22×C8D32semidirect
418. #67D24direct (N × H)
439. D12C2×#67semidirect
449. C3×C8D3×D4semidirect
450. C3×C8C2×#67semidirect
519. C22×C3C2×D24semidirect
520. C22×C3D3×D8semidirect
543. C2×D3C2×D24semidirect
561. Dic3C2×D24semidirect
562. C32D4×D8semidirect
601. C8D3×#67semidirect
639. C22D3×D24semidirect
649. C3D4×D24semidirect
650. C3D8×#67semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D242
D12×#672
C2×D3×D242
#105928142
C3×C8×#672
#105911622
#105909442
#105908802
#105908312
Dic3×D242
#105907532
D8×#673
D4×D243