Finite Group Databasebeta

← #11863953

Subgroups — #11863953

Order 1152.

2137 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D3×D4C211yesmaximal, characteristic, complemented
2C2×#64141C211yesmaximal, characteristic, complemented
3#10595023C211yesmaximal, characteristic, complemented
4C2×#63830C211yesmaximal, characteristic, complemented
5C3×#5215C211yesmaximal, characteristic, complemented
6#10591459C214yesmaximal, complemented
7#10591448C214yesmaximal, complemented
8#10591085C211yesmaximal, characteristic, complemented
9#10591069C211yesmaximal, characteristic, complemented
10#84038—31nomaximal
11#78926—31nomaximal
12C24×C3×D3—21no—
13C22×C32×D4C2211yescharacteristic
14C2×#3828C2211yescharacteristic, complemented
15C22×C3×#67—21no—
16C2×C3×D3×D4—24no—
17C2×C3×D3×D4—24no—
18C2×C3×D3×D4—22no—
19C2×C3×D3×D4C2214yes—
20C22×C3×D12C2211yescharacteristic
21C22×C3×C4×D3C2211yescharacteristic
22C2×#3754C2212yescomplemented
23#64141—24no—
24C2×#3753C2212yescomplemented
25C2×#3748C2211yescharacteristic, complemented
26C3×#926C2211yescharacteristic, complemented
27C2×C3×#208C2211yescharacteristic, complemented
28#63977—21no—
29#63966—21no—
30#63909C2212yescomplemented
31C4×#626—22no—
32#63857C2212yescomplemented
33#63830—24no—
34C2×#3716C2212yescomplemented
35C2×#3714C2212yescomplemented
36C3×#832C2214yescomplemented
37#63562C2214yescomplemented
38(C2×D3)⋊(C3⋊C8)C2214yescomplemented
39C23×C3×D4—31no—
40C22×D3×D4D311yescharacteristic, complemented
41C2×#932—31no—
42C2×#932—31no—
43C2×#932—31no—
44C2×#932—61no—
45#5791—31no—
46C2×#908—31no—
47#5524—31no—
48C3×#418—31no—
49#5218—34no—
50#5217—34no—
51#5215—31no—
52#5215D311yescharacteristic, complemented
53#5207—31no—
54#5126—31no—
55#5001—31no—
56#4776—34no—
57#4773—34no—
58C24×C32—21no—
59C23×C3×D3—28no—
60C23×C3×D3—21no—
61C23×C3×D3—24no—
62C23×C3×D3—48no—
63C23×C3×D3D411yescharacteristic
64C2×C32×D4—21no—
65C2×C32×D4—22no—
66C2×C32×D4D414yescomplemented
67C2×C32×D4C2312yes—
68C22×C32×C4C2311yescharacteristic
69C2×#638—21no—
70#3828—22no—
71#3828—22no—
72C2×C4⋊(C3⋊D3)C2311yescharacteristic, complemented
73C2×C4×C3⋊D3—21no—
74C2×#634C2311yescharacteristic, complemented
75C2×C3×#67—28no—
76C2×C3×#67—24no—
77C22×C3×Dic3D411yescharacteristic
78C3×D3×D4—216no—
79C3×D3×D4—28no—
80C3×D3×D4—28no—
81C2×C3×D12—21no—
82C2×C3×D12—22no—
83C2×C3×D12D414yescomplemented
84C2×C3×D12C2312yes—
85C2×C3×C4×D3—22no—
86C2×C3×C4×D3—22no—
87C2×C3×C4×D3D412yes—
88C2×C3×C4×D3C2312yes—
89C2×#626—22no—
90#3754—24no—
91#3753—24no—
92#3748—22no—
93C2×C3⋊(C3⋊C8)C2312yescomplemented
94C3×#217—21no—
95C3×C4⋊Dic3C2312yescomplemented
96C3×C4×Dic3—21no—
97C3×#208—22no—
98C2×C3×C3⋊C8C2312yescomplemented
99Dic3⋊Dic3—22no—
100#3722—22no—
101(C2×D3)⋊Dic3—22no—
102#3716—24no—
103#3714—24no—
104#3003—91noSylow
105C25×C3—61no—
106C24×D3—21no—
107C22×C3×D4—21no—
108C22×C3×D4—34no—
109C22×C3×D4—31no—
110C22×C3×D4—62no—
111C22×C3×D4—61no—
112C22×C3×D4—64no—
113C22×C3×D4—64no—
114C22×C3×D4C2×D311yescharacteristic
115C22×C3×D4C2×D311yescharacteristic
116C23×C3×C4—31no—
117C22×#67—21no—
118C2×D3×D4—24no—
119C2×D3×D4—24no—
120C2×D3×D4—22no—
121C2×D3×D4C2×D314yes—
122C2×#235—31no—
123C2×#235—31no—
124C2×#235—61no—
125C22×D12C2×D311yescharacteristic
126C22×C4×D3C2×D311yescharacteristic
127C2×C3×M16—31no—
128C3×#117—31no—
129#953—61no—
130C2×#214—32no—
131C2×#214—32no—
132C2×#214—32no—
133C2×#214—62no—
134#932—64no—
135#932—68no—
136#932—64no—
137#932—64no—
138C2×#213—32no—
139C2×#213—32no—
140C2×#213—32no—
141C2×#213—62no—
142#930—32no—
143#929—61no—
144C4×#67—62no—
145#926C2×D311yescharacteristic, complemented
146#924—32no—
147C2×#208—31no—
148C2×#208—31no—
149C2×#208—61no—
150C2×#208C2×D311yescharacteristic, complemented
151#908—64no—
152C2×D24—32no—
153C2×#204—32no—
154#897—32no—
155#895—32no—
156#886—61no—
157#882—61no—
158C4×D12—62no—
159C3×#102—34no—
160#832—34no—
161#832C2×D314yescomplemented
162#830—34no—
163(C2×D3)⋊C8—34no—
164#809—34no—
165C23×C32—24no—
166C23×C32—22no—
167C23×C32—44no—
168C23×C32C2×D411yescharacteristic
169C22×C3⋊D3—21no—
170C22×C3×D3—22no—
171C22×C3×D3—24no—
172C22×C3×D3—216no—
173C22×C3×D3—24no—
174C22×C3×D3—22no—
175C22×C3×D3—432no—
176C22×C3×D3—48no—
177C22×C3×D3—44no—
178C22×C3×D3—48no—
179C22×C3×D3C2×D412yes—
180C32×D4—24no—
181C32×D4—22no—
182C32×D4C2×D414yes—
183C2×C32×C4C2×D412yes—
184C2×C32×C4C2×D411yescharacteristic
185C2×C32×C4C2×D412yes—
186C2×C32×C4C2411yescharacteristic
187#638—42no—
188C2×C3⋊Dic3—21no—
189C4⋊(C3⋊D3)—21no—
190C4⋊(C3⋊D3)—21no—
191C4×C3⋊D3—42no—
192#634—21no—
193#634—21no—
194C3×#67—216no—
195C2×C3×Dic3—22no—
196C2×C3×Dic3—22no—
197C2×C3×Dic3C2×D412yes—
198C3×D12—24no—
199C3×D12—22no—
200C3×D12C2×D414yes—
201C3×C4×D3—24no—
202C3×C4×D3—22no—
203C3×C4×D3—22no—
204#626—44no—
205C3⋊(C3⋊C8)—22no—
206C3×C3⋊C8—22no—
207C23×D4—31no—
208C2×AGL(1,ℤ/8)—91no—
209C2×AGL(1,ℤ/8)—91no—
210#525—91no—
211#450—94no—
212#447—94no—
213#418—31no—
214#418—91no—
215#407—91no—
216C24×C3—21no—
217C24×C3—21no—
218C24×C3—31no—
219C24×C3—41no—
220C24×C3—64no—
221C24×C3—68no—
222C24×C3—61no—
223C24×C3—128no—
224C23×D3—28no—
225C23×D3—21no—
226C23×D3—24no—
227C23×D3—48no—
228C23×D3#6711yescharacteristic
229C2×C3×D4—21no—
230C2×C3×D4—22no—
231C2×C3×D4—21no—
232C2×C3×D4—22no—
233C2×C3×D4—24no—
234C2×C3×D4—22no—
235C2×C3×D4—34no—
236C2×C3×D4—32no—
237C2×C3×D4—41no—
238C2×C3×D4—42no—
239C2×C3×D4—64no—
240C2×C3×D4—68no—
241C2×C3×D4—616no—
242C2×C3×D4—68no—
243C2×C3×D4—68no—
244C2×C3×D4—62no—
245C2×C3×D4—61no—
246C2×C3×D4D1214yescomplemented
247C2×C3×D4#6714yescomplemented
248C2×C3×D4C22×D312yes—
249C2×C3×D4C22×D312yes—
250C22×C3×C4—21no—
251C22×C3×C4—31no—
252C22×C3×C4—32no—
253C22×C3×C4—32no—
254C22×C3×C4—62no—
255C22×C3×C4—62no—
256C22×C3×C4C22×D311yescharacteristic
257C22×C3×C4C22×D311yescharacteristic
258C2×#67—28no—
259C2×#67—24no—
260C2×#67—62no—
261C2×#67—61no—
262C2×#67—61no—
263C2×#67—121no—
264C22×Dic3#6711yescharacteristic
265D3×D4—216no—
266D3×D4—28no—
267D3×D4—28no—
268#235—64no—
269#235—62no—
270#235—62no—
271#235—62no—
272#235—62no—
273#235—122no—
274C2×D12—22no—
275C2×D12—21no—
276C2×D12—31no—
277C2×D12—31no—
278C2×D12—61no—
279C2×D12—62no—
280C2×D12#6714yescomplemented
281C2×D12C22×D312yes—
282C2×C4×D3—22no—
283C2×C4×D3—22no—
284C2×C4×D3—61no—
285C2×C4×D3—61no—
286C2×C4×D3—121no—
287C2×C4×D3#6712yes—
288C2×C4×D3C22×D312yes—
289C2×Dic6—31no—
290C2×Dic6—31no—
291C2×Dic6—61no—
292C3×M16—62no—
293C2×C3×C8—32no—
294C3×C4⋊C4—32no—
295C3×C22⋊C4—61no—
296C3×C42—61no—
297#217—21no—
298#217—62no—
299#214—64no—
300#214—68no—
301#214—64no—
302#214—64no—
303#213—64no—
304#213—68no—
305#213—64no—
306#213—64no—
307(C2×D3)⋊C4—62no—
308(C2×D3)⋊C4—62no—
309(C2×D3)⋊C4—62no—
310C4⋊Dic3—62no—
311C4⋊Dic3C22×D312yescomplemented
312Dic3⋊C4—62no—
313C4×Dic3—21no—
314#208—22no—
315#208—64no—
316#208—62no—
317#208—62no—
318C2×C3⋊C8—32no—
319C2×C3⋊C8—32no—
320C2×C3⋊C8—62no—
321C2×C3⋊C8C22×D312yescomplemented
322D24—64no—
323#204—64no—
324C22×C32—22no—
325C22×C32—24no—
326C22×C32—48no—
327C22×C32—44no—
328C22×C32#12012yes—
329C22×C32C22×D411yescharacteristic
330C2×C3⋊D3—41no—
331C2×C3⋊D3—42no—
332C2×C3×D3—28no—
333C2×C3×D3—22no—
334C2×C3×D3—22no—
335C2×C3×D3—24no—
336C2×C3×D3—432no—
337C2×C3×D3—48no—
338C2×C3×D3—44no—
339C2×C3×D3AGL(1,ℤ/8)14yescomplemented
340C32×C4#12012yes—
341C32×C4C22×D411yescharacteristic
342C32×C4C22×D411yescharacteristic
343C3⋊Dic3—41no—
344C3×Dic3—22no—
345C3×Dic3—42no—
346C25—61no—
347C2×#41—91no—
348C22×D4—34no—
349C22×D4—31no—
350C22×D4—62no—
351C22×D4—61no—
352C22×D4—64no—
353C22×D4—64no—
354C22×D4D3211yescharacteristic
355C23×C4—31no—
356AGL(1,ℤ/8)—184no—
357AGL(1,ℤ/8)—184no—
358C2×SD16—92no—
359C2×SD16—92no—
360C2×D8—92no—
361C2×D8—92no—
362C2×M16—31no—
363C2×M16—91no—
364#123—181no—
365C22⋊Q8—92no—
366#121—92no—
367#120—181no—
368C4×D4—182no—
369#117—31no—
370#102—34no—
371#102—94no—
372C22⋊C8—94no—
373C23×C3—24no—
374C23×C3—24no—
375C23×C3—22no—
376C23×C3—22no—
377C23×C3—21no—
378C23×C3—32no—
379C23×C3—42no—
380C23×C3—44no—
381C23×C3—44no—
382C23×C3—44no—
383C23×C3—64no—
384C23×C3—616no—
385C23×C3—62no—
386C23×C3—64no—
387C23×C3—62no—
388C23×C3—84no—
389C23×C3—124no—
390C23×C3—128no—
391C23×C3—1232no—
392C23×C3—128no—
393C23×C3D3×D411yescharacteristic
394C23×C3C2×#6711yescharacteristic
395C22×D3—22no—
396C22×D3—24no—
397C22×D3—216no—
398C22×D3—24no—
399C22×D3—22no—
400C22×D3—432no—
401C22×D3—48no—
402C22×D3—44no—
403C22×D3—48no—
404C22×D3—61no—
405C22×D3—61no—
406C22×D3—121no—
407C22×D3C2×#6712yes—
408C3×D4—24no—
409C3×D4—24no—
410C3×D4—22no—
411C3×D4—22no—
412C3×D4—24no—
413C3×D4—34no—
414C3×D4—42no—
415C3×D4—44no—
416C3×D4—616no—
417C3×D4—64no—
418C3×D4—62no—
419C3×D4C2×D1214yes—
420C3×D4C2×#6714yes—
421C2×C3×C4—21no—
422C2×C3×C4—22no—
423C2×C3×C4—21no—
424C2×C3×C4—22no—
425C2×C3×C4—32no—
426C2×C3×C4—62no—
427C2×C3×C4—64no—
428C2×C3×C4—62no—
429C2×C3×C4—62no—
430C2×C3×C4—62no—
431C2×C3×C4C2×D1212yes—
432C2×C3×C4D3×D411yescharacteristic
433C2×C3×C4D3×D412yes—
434C2×C3×C4C2×#6712yes—
435C2×C3×C4C2×#6711yescharacteristic
436C2×C3×C4C2×#6712yes—
437C2×C3×C4C23×D311yescharacteristic
438C2×C3×C4C23×D311yescharacteristic
439#67—216no—
440#67—124no—
441#67—124no—
442#67—122no—
443#67—122no—
444C2×Dic3—22no—
445C2×Dic3—22no—
446C2×Dic3—61no—
447C2×Dic3—61no—
448C2×Dic3—121no—
449C2×Dic3C2×#6712yes—
450D12—24no—
451D12—22no—
452D12—62no—
453D12—61no—
454D12—61no—
455D12—61no—
456D12—61no—
457D12—121no—
458D12—124no—
459D12C2×#6714yes—
460C4×D3—24no—
461C4×D3—22no—
462C4×D3—22no—
463C4×D3—124no—
464C4×D3—122no—
465C4×D3—122no—
466Dic6—62no—
467Dic6—61no—
468Dic6—61no—
469Dic6—61no—
470Dic6—61no—
471Dic6—121no—
472C3×C8—62no—
473C3⋊C8—22no—
474C3⋊C8—64no—
475C3⋊C8—62no—
476C3⋊C8—62no—
477C2×C32—22no—
478C2×C32—44no—
479C2×C32C2×#12011yescharacteristic
480C2×C32C2×AGL(1,ℤ/8)12yes—
481C3⋊D3—81no—
482C3×D3—24no—
483C3×D3—44no—
484C24—21no—
485C24—31no—
486C24—64no—
487C24—68no—
488C24—61no—
489C24—128no—
490#41—182no—
491#41—182no—
492C2×Q8—91no—
493C2×D4—22no—
494C2×D4—21no—
495C2×D4—32no—
496C2×D4—34no—
497C2×D4—64no—
498C2×D4—68no—
499C2×D4—616no—
500C2×D4—68no—
501C2×D4—68no—
502C2×D4—62no—
503C2×D4—61no—
504C2×D4—91no—
505C2×D4—182no—
506C2×D4—181no—
507C2×D4#62614yescomplemented
508C2×D4C2×D3212yes—
509C22×C4—31no—
510C22×C4—32no—
511C22×C4—32no—
512C22×C4—62no—
513C22×C4—62no—
514C22×C4—181no—
515C22×C4C2×D3211yescharacteristic
516SD16—184no—
517SD16—184no—
518D8—184no—
519D8—184no—
520M16—62no—
521M16—182no—
522C2×C8—32no—
523C2×C8—92no—
524C4⋊C4—32no—
525C4⋊C4—182no—
526C22⋊C4—61no—
527C22⋊C4—182no—
528C22⋊C4—182no—
529C42—61no—
530C22×C3—22no—
531C22×C3—22no—
532C22×C3—24no—
533C22×C3—24no—
534C22×C3—21no—
535C22×C3—22no—
536C22×C3—34no—
537C22×C3—42no—
538C22×C3—44no—
539C22×C3—48no—
540C22×C3—48no—
541C22×C3—44no—
542C22×C3—44no—
543C22×C3—68no—
544C22×C3—62no—
545C22×C3—62no—
546C22×C3—64no—
547C22×C3—84no—
548C22×C3—88no—
549C22×C3—128no—
550C22×C3—1232no—
551C22×C3—124no—
552C22×C3#88212yes—
553C22×C3#95312yes—
554C22×C3C2×D3×D411yescharacteristic
555C22×C3C22×#6711yescharacteristic
556C2×D3—28no—
557C2×D3—22no—
558C2×D3—22no—
559C2×D3—24no—
560C2×D3—432no—
561C2×D3—48no—
562C2×D3—44no—
563C2×D3—122no—
564C2×D3—121no—
565C2×D3—122no—
566C2×D3—122no—
567C2×D3—121no—
568C2×D3—242no—
569C2×D3#93214yescomplemented
570C3×C4—21no—
571C3×C4—21no—
572C3×C4—22no—
573C3×C4—62no—
574C3×C4—122no—
575C3×C4#88212yes—
576C3×C4#95312yes—
577C3×C4C22×D1211yescharacteristic
578C3×C4C2×D3×D411yescharacteristic
579C3×C4C22×#6711yescharacteristic
580C3×C4C22×#6711yescharacteristic
581Dic3—22no—
582Dic3—42no—
583Dic3—122no—
584Dic3—121no—
585Dic3—121no—
586C32#300311yescharacteristic, complemented, Sylow
587C23—22no—
588C23—24no—
589C23—32no—
590C23—44no—
591C23—64no—
592C23—616no—
593C23—62no—
594C23—62no—
595C23—64no—
596C23—124no—
597C23—128no—
598C23—1232no—
599C23—128no—
600C23—181no—
601C23D3×#6711yescharacteristic
602Q8—181no—
603Q8—181no—
604D4—22no—
605D4—24no—
606D4—34no—
607D4—616no—
608D4—62no—
609D4—64no—
610D4—181no—
611D4—181no—
612D4—364no—
613D4—362no—
614D4C2×#62614yes—
615C2×C4—32no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—64no—
620C2×C4—62no—
621C2×C4—181no—
622C2×C4—362no—
623C2×C4C2×#62612yes—
624C2×C4D3×#6712yes—
625C2×C4D3×#6711yescharacteristic
626C2×C4C22×D3211yescharacteristic
627C8—62no—
628C8—182no—
629C2×C3—22no—
630C2×C3—22no—
631C2×C3—22no—
632C2×C3—21no—
633C2×C3—42no—
634C2×C3—44no—
635C2×C3—44no—
636C2×C3—64no—
637C2×C3—84no—
638C2×C3—124no—
639C2×C3C2×#88211yescharacteristic
640C2×C3D3×AGL(1,ℤ/8)12yes—
641C2×C3C2×#93212yes—
642C2×C3C2×#95311yescharacteristic
643D3—24no—
644D3—44no—
645D3—242no—
646D3—241no—
647D3—241no—
648C22—22no—
649C22—24no—
650C22—34no—
651C22—48no—
652C22—44no—
653C22—68no—
654C22—62no—
655C22—62no—
656C22—64no—
657C22—128no—
658C22—1232no—
659C22—124no—
660C22—362no—
661C22—361no—
662C22#6397712yes—
663C22C2×D3×#6711yescharacteristic
664C4—62no—
665C4—122no—
666C4—361no—
667C4#6397712yes—
668C4C22×#62611yescharacteristic
669C4C2×D3×#6711yescharacteristic
670C3—21no—
671C3#7892611yescharacteristic, complemented
672C3#8403811yescharacteristic, complemented
673C2—22no—
674C2—44no—
675C2—64no—
676C2—124no—
677C2—721no—
678C2D3×#93212yes—
679C2C2×#6397711yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC22×C32×D4C22
socleC22×C32C22×D4
hypercenterC22×D4D32
hypocenterC32#3003
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×D3×D4C2semidirect
2. C2×#64141C2semidirect
3. #10595023C2semidirect
4. C2×#63830C2semidirect
5. C3×#5215C2semidirect
6. #10591459C2semidirect
7. #10591448C2semidirect
8. #10591085C2semidirect
9. #10591069C2semidirect
14. C2×#3828C22semidirect
22. C2×#3754C22semidirect
24. C2×#3753C22semidirect
25. C2×#3748C22semidirect
26. C3×#926C22semidirect
27. C2×C3×#208C22semidirect
30. #63909C22semidirect
32. #63857C22semidirect
34. C2×#3716C22semidirect
35. C2×#3714C22semidirect
36. C3×#832C22semidirect
37. #63562C22semidirect
38. (C2×D3)⋊(C3⋊C8)C22semidirect
40. C22×D3×D4D3semidirect
52. #5215D3semidirect
66. C2×C32×D4D4semidirect
72. C2×C4⋊(C3⋊D3)C23semidirect
74. C2×#634C23semidirect
83. C2×C3×D12D4semidirect
93. C2×C3⋊(C3⋊C8)C23semidirect
95. C3×C4⋊Dic3C23semidirect
98. C2×C3×C3⋊C8C23semidirect
145. #926C2×D3semidirect
150. C2×#208C2×D3semidirect
161. #832C2×D3semidirect
246. C2×C3×D4D12semidirect
247. C2×C3×D4#67semidirect
280. C2×D12#67semidirect
311. C4⋊Dic3C22×D3semidirect
321. C2×C3⋊C8C22×D3semidirect
339. C2×C3×D3AGL(1,ℤ/8)semidirect
507. C2×D4#626semidirect
569. C2×D3#932semidirect
586. C32#3003semidirect
671. C3#78926semidirect
672. C3#84038semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D3×D42
C2×#641412
#105950232
C2×#638302
C3×#52152
#105914592
#105914482
#105910852
#105910692
#840383
#789263