Finite Group Databasebeta

← #6376023

Subgroups — #6376023

Order 512.

2044 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2#34226C211yesmaximal, characteristic, complemented
3#34058C211yesmaximal, characteristic, complemented
4#22151C212yesmaximal, complemented
5#22145C212yesmaximal, complemented
6#22116C212yesmaximal, complemented
7#22110C212yesmaximal, complemented
8#22093C214yesmaximal, complemented
9#22056C214yesmaximal, complemented
10#21393C211yesmaximal, characteristic, complemented
11#21362C211yesmaximal, characteristic, complemented
12#21284C212yesmaximal, complemented
13#21282C212yesmaximal, complemented
14#21134C212yesmaximal, complemented
15#21129C212yesmaximal, complemented
16#20738C211yesmaximal, characteristic, complemented
17#20470C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3100C4116yescomplemented
21C2×#480C2212yes—
22C2×#474C2211yescharacteristic
23C2×#474C2211yescharacteristic
24C2×#459C2211yescharacteristic
25C2×#459C2211yescharacteristic
26C2×#447C2212yes—
27C2×C4×D8C2212yes—
28C2×C4×D8C2212yes—
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2367—22no—
32#2363—22no—
33#2357—21no—
34#2349—21no—
35#2341—21no—
36#2339—21no—
37#2324C2214yes—
38#2319—21no—
39#2319—21no—
40#2319C2211yescharacteristic
41#2319C2211yescharacteristic
42#2319C2211yescharacteristic
43#2319C2211yescharacteristic
44#2319C2212yescomplemented
45#2316C2212yes—
46#2312—22no—
47#2309C2212yes—
48C4×C4⋊Q8C2212yescomplemented
49C4×#127C2212yescomplemented
50#2293—22no—
51#2293C2212yes—
52#2293C2212yes—
53#2285C2211yescharacteristic
54#2280—21no—
55#2280—21no—
56#2275C2212yescomplemented
57C4×#117C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#1988C2212yescomplemented
60#1982C2212yescomplemented
61#1972C2214yes—
62#1970C2214yes—
63#1948C2212yescomplemented
64#1947C2212yescomplemented
65#1943C2214yescomplemented
66#1942C2214yescomplemented
67#1936C2214yes—
68#1936C2214yescomplemented
69#1934C2214yes—
70#1934C2214yescomplemented
71#1926C2214yes—
72#1925C2214yes—
73#1921C2214yescomplemented
74#1918C2214yescomplemented
75#1879C2214yes—
76#1879C2214yes—
77#1875C2214yes—
78#1864C2214yescomplemented
79#1862C2214yescomplemented
80#1854C2214yes—
81#1824C2212yes—
82#1802C2212yes—
83#1802C2212yes—
84#1786C2212yes—
85C4×#107C2212yescomplemented
86C4×#106C2212yescomplemented
87#1767C2212yescomplemented
88C4×C4⋊C8C2212yescomplemented
89#1762C2212yes—
90#1762C2212yes—
91#1762C2212yescomplemented
92#1762C2212yescomplemented
93#1750C2214yes—
94#1746C2212yescomplemented
95#1726C2212yescomplemented
96C22×#41—21no—
97C23×D4C2311yescharacteristic
98C24×C4—21no—
99C22×D8C2312yes—
100C22×D8C2312yes—
101#532—24no—
102#532C2×C418yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#124—21no—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×C4×Q8—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—24no—
115C2×C4×D4C2×C418yescomplemented
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2314yes—
122C2×#117—21no—
123C2×#117—22no—
124C2×#117—22no—
125C2×#117C2312yes—
126C2×#117C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C42C2311yescharacteristic
130#480C2×C4116yescomplemented
131#474C2×C418yescomplemented
132#474C2×C418yescomplemented
133#459C2×C418yes—
134#459C2×C418yes—
135#447C2×C4116yescomplemented
136C4×D8C2×C4116yescomplemented
137C4×D8C2×C4116yescomplemented
138#432—24no—
139#431C2×C418yescomplemented
140C2×#107C2314yes—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×#97C2312yes—
158C2×#97C2314yes—
159C2×C4×C8C2314yes—
160C2×C4×C8C2312yes—
161#391—22no—
162#390—22no—
163#390—22no—
164#390—24no—
165#390—22no—
166#390—22no—
167#390C2314yes—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2314yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#388—24no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—24no—
190#386—24no—
191#385—24no—
192#384C2312yes—
193#382—21no—
194#381—21no—
195#381—21no—
196#381—21no—
197#381—21no—
198#381—22no—
199#380—21no—
200#380—22no—
201#380—22no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—22no—
207#380C2312yes—
208#379—22no—
209#379—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—24no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D411yescharacteristic
228C4×C22⋊C4D411yescharacteristic
229#376—21no—
230#376C2311yescharacteristic
231#376C2311yescharacteristic
232C2×#95—22no—
233C43—21no—
234C43C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C25—21no—
244C2×#41—28no—
245C2×#41—84no—
246C22×Q8—21no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—42no—
253C22×D4—42no—
254C22×D4C22×C414yes—
255C22×D4C2412yes—
256C22×D4C2411yescharacteristic
257C22×D4C2411yescharacteristic
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—28no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—22no—
265C23×C4—24no—
266C23×C4—84no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×D8—44no—
271C2×D8—44no—
272C2×D8C22×C418yes—
273C2×D8C22×C418yes—
274C22×C8C2414yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C4⋊Q8—22no—
278C4⋊Q8C22×C414yes—
279#127—22no—
280#127C22×C414yes—
281#124—28no—
282#121—28no—
283#121—24no—
284#121C22×C418yes—
285C4×Q8—84no—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—24no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—84no—
297C4×D4—84no—
298C4×D4C22×C414yes—
299C4×D4C22×C414yes—
300C4×D4C22×C414yes—
301C4×D4C22×C414yes—
302C4×D4C22×C418yes—
303#117—42no—
304#117—42no—
305#117—42no—
306#117—42no—
307#117—84no—
308#117—84no—
309#117—82no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4C22×C414yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42C22×C414yes—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42#4114yes—
365C2×C42C2412yes—
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#107—44no—
370#107C22×C418yes—
371#107C22×C418yes—
372#106—44no—
373C4⋊C8—24no—
374C4⋊C8C22×C414yes—
375C4⋊C8C22×C414yes—
376#102—44no—
377#102—44no—
378#102—44no—
379#102—44no—
380#102C22×C418yes—
381#102C22×C418yes—
382#102C22×C418yes—
383#102C22×C418yes—
384C22⋊C8—24no—
385C22⋊C8C22×C418yes—
386#97—24no—
387#97—44no—
388C4×C8—44no—
389C4×C8C22×C418yes—
390#95—22no—
391#95—24no—
392#95—21no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—24no—
401#95C2×D411yescharacteristic
402#95C2×D411yescharacteristic
403#95C2×D412yes—
404#95C2×D411yescharacteristic
405#95C2×D411yescharacteristic
406#95C2×D412yes—
407C24—21no—
408C24—21no—
409C24—22no—
410C24—24no—
411C24—44no—
412C24—82no—
413C24—81no—
414C24—81no—
415C24C22×D411yescharacteristic
416#41—816no—
417C2×Q8—24no—
418C2×Q8—44no—
419C2×D4—28no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—48no—
438C2×D4—44no—
439C2×D4—84no—
440C2×D4C23×C412yes—
441C2×D4C23×C412yes—
442C2×D4C23×C414yes—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—28no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—816no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—82no—
470C22×C4C4×D414yes—
471C22×C4C4×D414yes—
472C22×C4C23×C412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C2×#4112yes—
483C22×C4C2511yescharacteristic
484D8—48no—
485D8—48no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8C23×C414yes—
492C2×C8C23×C414yes—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—84no—
513C4⋊C4C23×C412yes—
514C4⋊C4C23×C412yes—
515C4⋊C4C23×C414yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—28no—
521C22⋊C4—28no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4—82no—
537C42—24no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42—44no—
547C42—81no—
548C42—81no—
549C42—82no—
550C42—84no—
551C42C4×D414yes—
552C42C4×D414yes—
553C42C23×C412yes—
554C42C23×C412yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—48no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—44no—
577C23—84no—
578C23—84no—
579C23—88no—
580C23—88no—
581C23C2×C4×D412yes—
582C23C23×D411yescharacteristic
583Q8—48no—
584D4—22no—
585D4—22no—
586D4—24no—
587D4—24no—
588D4—22no—
589D4—22no—
590D4—48no—
591D4—48no—
592D4—44no—
593D4—42no—
594D4—42no—
595D4—48no—
596D4—88no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—88no—
640C2×C4—84no—
641C2×C4—168no—
642C2×C4C2×C4×D412yes—
643C2×C4C2×C4×D412yes—
644C2×C4C2×C4×D412yes—
645C2×C4C2×C4×D412yes—
646C2×C4C2×C4×D412yes—
647C2×C4#57614yes—
648C2×C4C24×C411yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4#58512yes—
651C2×C4#58512yes—
652C8—24no—
653C8—24no—
654C8—44no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—44no—
669C22—816no—
670C22—84no—
671C22—82no—
672C22—82no—
673C22—84no—
674C22—82no—
675C22—82no—
676C22C22×C4×D411yescharacteristic
677C22#344912yes—
678C22C2×#57612yes—
679C22#358511yescharacteristic
680C22#358611yescharacteristic
681C4—22no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—82no—
693C4—82no—
694C4—84no—
695C4#294814yes—
696C4#342811yescharacteristic
697C4#342811yescharacteristic
698C4#345512yes—
699C2—42no—
700C2—84no—
701C2—82no—
702C2—82no—
703C2#3406511yescharacteristic
704C2C2×#294811yescharacteristic
705C2#6017011yescharacteristic
706C2#6098812yes—
707C2#6099012yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3100C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34226C2semidirect
3. #34058C2semidirect
4. #22151C2semidirect
5. #22145C2semidirect
6. #22116C2semidirect
7. #22110C2semidirect
8. #22093C2semidirect
9. #22056C2semidirect
10. #21393C2semidirect
11. #21362C2semidirect
12. #21284C2semidirect
13. #21282C2semidirect
14. #21134C2semidirect
15. #21129C2semidirect
16. #20738C2semidirect
17. #20470C2semidirect
20. #3100C4semidirect
44. #2319C22semidirect
48. C4×C4⋊Q8C22semidirect
49. C4×#127C22semidirect
56. #2275C22semidirect
59. #1988C22semidirect
60. #1982C22semidirect
63. #1948C22semidirect
64. #1947C22semidirect
65. #1943C22semidirect
66. #1942C22semidirect
68. #1936C22semidirect
70. #1934C22semidirect
73. #1921C22semidirect
74. #1918C22semidirect
78. #1864C22semidirect
79. #1862C22semidirect
85. C4×#107C22semidirect
86. C4×#106C22semidirect
87. #1767C22semidirect
88. C4×C4⋊C8C22semidirect
91. #1762C22semidirect
92. #1762C22semidirect
94. #1746C22semidirect
95. #1726C22semidirect
115. C2×C4×D4C2×C4semidirect
130. #480C2×C4semidirect
131. #474C2×C4semidirect
132. #474C2×C4semidirect
135. #447C2×C4semidirect
136. C4×D8C2×C4semidirect
137. C4×D8C2×C4semidirect
139. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
#342262
#340582
#221512
#221452
#221162
#221102
#220932
#220562
#213932
#213622
#212842
#212822
#211342
#211292
#207382
#204702