Finite Group Databasebeta

← #6519211

Subgroups — #6519211

Order 512.

1658 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34946C211yesmaximal, characteristic
3#29140C212yesmaximal, complemented
4#29122C212yesmaximal, complemented
5#29122C212yesmaximal, complemented
6#28721C212yesmaximal, complemented
7#27814C212yesmaximal, complemented
8#27768C212yesmaximal, complemented
9#27748C212yesmaximal, complemented
10#27728C212yesmaximal, complemented
11#27699C214yesmaximal, complemented
12#27256C214yesmaximal
13#21885C211yesmaximal, characteristic, complemented
14#21860C211yesmaximal, characteristic, complemented
15#20732C211yesmaximal, characteristic
16C2×#1802C211yesmaximal, characteristic
17#20543C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#488C2214yes—
21C2×#431C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#102C2212yes—
24#2859C2212yescomplemented
25#2859C2212yescomplemented
26#2647—22no—
27#2571—21no—
28#2571C2212yescomplemented
29#2563C2211yescharacteristic
30#2563C2211yescharacteristic
31#2562—21no—
32#2562—21no—
33#2561—22no—
34#2552—21no—
35#2552C2212yescomplemented
36#2547C2211yescharacteristic
37#2547C2211yescharacteristic
38#2503C2212yes—
39#2442—22no—
40#2442C2214yes—
41#2435C2212yes—
42#2398—22no—
43#2317—21no—
44#2316—21no—
45C4×#124C2212yes—
46C4×#124C2212yes—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49#2280—21no—
50#2280—21no—
51C4×#117C2211yescharacteristic
52#2101C2214yescomplemented
53#2101C2214yescomplemented
54#2074C2214yescomplemented
55#2073C2214yes—
56#2042C2214yes—
57#2042C2214yes—
58#2026C2214yescomplemented
59#2025C2214yes—
60#1991C2214yescomplemented
61#1990C2212yescomplemented
62#1987C2212yescomplemented
63#1981C2212yescomplemented
64#1958C2214yes—
65#1956C2212yescomplemented
66#1934C2214yes—
67#1934C2214yes—
68#1933C2214yescomplemented
69#1933C2214yescomplemented
70#1892C2214yes—
71#1872C2214yes—
72#1835C2214yescomplemented
73#1823C2212yes—
74#1803C2212yescomplemented
75#1803C2212yescomplemented
76#1803C2212yescomplemented
77#1803C2212yescomplemented
78#1802C2211yescharacteristic
79#1802C2212yes—
80#1802C2212yes—
81#1802C2212yes—
82#1802C2212yes—
83#1802C2211yescharacteristic
84#1802C2211yescharacteristic
85#1802C2211yescharacteristic
86#1786C2212yes—
87C8×C4⋊C4C2212yescomplemented
88C8×C4⋊C4C2212yescomplemented
89C4×C4⋊C8C2212yescomplemented
90C4×C4⋊C8C2212yescomplemented
91C4×#102C2212yes—
92C4×#102C2212yes—
93C8×C22⋊C4C2214yes—
94C2×#340C2212yes—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C23×C8C2312yes—
98#533—24no—
99#533—24no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—21no—
103C2×#125—22no—
104C2×#125C2314yes—
105C2×#124C2312yes—
106C2×#124C2312yes—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8C2312yes—
109C2×C4×Q8—21no—
110C2×C4×Q8C2312yes—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×#117—21no—
120C2×#117—21no—
121C2×#117D411yescharacteristic
122C2×#117D411yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C22⋊C4—21no—
128C22×C42C2311yescharacteristic
129#488—48no—
130#431—24no—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102—44no—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150C2×C4×C8C2314yes—
151#400—22no—
152#400—22no—
153#400—24no—
154#400—24no—
155#398—24no—
156#398—24no—
157#397—22no—
158#397—22no—
159#394—22no—
160#394—22no—
161#393—22no—
162#393—22no—
163#391C2312yes—
164#391C2312yes—
165#390C2312yes—
166#390C2312yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2314yes—
176#389C2314yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#385—24no—
182#384—22no—
183#384—21no—
184#384—21no—
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#382—22no—
190#382—22no—
191#382—21no—
192#382C2312yes—
193#382C2312yes—
194#382C2314yes—
195#381—22no—
196#381—22no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—24no—
207#380C2312yes—
208#379—22no—
209#379—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222#376D412yes—
223C43C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236C25—21no—
237C22×Q8—21no—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—41no—
245C22×D4—41no—
246C22×D4C2411yescharacteristic
247C22×D4C2411yescharacteristic
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C22×C8—24no—
261C22×C8—42no—
262C22×C8—42no—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270#125—44no—
271#125—44no—
272#125—44no—
273#125—84no—
274#124—24no—
275#124—24no—
276#124—42no—
277#124—42no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—44no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—44no—
293C4×D4—44no—
294#117—24no—
295#117—24no—
296#117—44no—
297#117—44no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42#4114yes—
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—22no—
367C4⋊C8—22no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103—44no—
372#102—28no—
373#102—24no—
374#102—24no—
375#102—44no—
376#102—42no—
377#102—42no—
378C22⋊C8—24no—
379C4×C8—24no—
380C4×C8—44no—
381C4×C8—44no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—21no—
401C24—21no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—81no—
408C24—81no—
409C24C22×D411yescharacteristic
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—42no—
417C2×Q8—81no—
418C2×Q8—81no—
419C2×D4—28no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4H(5,𝔽2)12yes—
437C2×D42−1+412yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—816no—
470C22×C4#12414yes—
471C22×C4#12414yes—
472C22×C4#13514yes—
473C22×C4#13514yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2511yescharacteristic
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—84no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—42no—
554C42—42no—
555C42—44no—
556C42—44no—
557C42#12414yes—
558C42#12414yes—
559C42C2×#4112yes—
560C42C2×#4112yes—
561C42C2×#4112yes—
562C42C2×#4112yes—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—22no—
569C23—48no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23C2×#12412yes—
589C23C2×#13512yes—
590C23C2×#13512yes—
591C23C23×D411yescharacteristic
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—84no—
597D4—22no—
598D4—22no—
599D4—22no—
600D4—22no—
601D4—48no—
602D4—41no—
603D4—41no—
604D4—41no—
605D4—41no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
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—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—164no—
656C2×C4#488116yescomplemented
657C2×C4C2×#12412yes—
658C2×C4C2×#12412yes—
659C2×C4C2×#12412yes—
660C2×C4C2×#12412yes—
661C2×C4C2×#12412yes—
662C2×C4C2×#13512yes—
663C2×C4C2×#13512yes—
664C2×C4C2×#13512yes—
665C2×C4C2×#13512yes—
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×2−1+411yescharacteristic
670C2×C4#58511yescharacteristic
671C2×C4#58511yescharacteristic
672C8—42no—
673C8—42no—
674C8—44no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—816no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22C2×#48814yes—
697C22C22×#12411yescharacteristic
698C22#348811yescharacteristic
699C22#348911yescharacteristic
700C22C22×#13511yescharacteristic
701C22C22×#13511yescharacteristic
702C22#358511yescharacteristic
703C22#358611yescharacteristic
704C4—22no—
705C4—22no—
706C4—22no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—42no—
711C4—82no—
712C4—81no—
713C4—81no—
714C4—82no—
715C4—82no—
716C4—84no—
717C4—82no—
718C4C2×#48814yes—
719C4C2×#48814yes—
720C4#349211yescharacteristic
721C4#349311yescharacteristic
722C4#349411yescharacteristic
723C4#349511yescharacteristic
724C2—24no—
725C2—82no—
726C2—82no—
727C2#3554211yescharacteristic
728C2C22×#48811yescharacteristic
729C2#6062111yescharacteristic
730C2#6194911yescharacteristic
731C2#6195011yescharacteristic
732C2#6209711yescharacteristic
733C2#6209811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×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
1. #35286C2semidirect
3. #29140C2semidirect
4. #29122C2semidirect
5. #29122C2semidirect
6. #28721C2semidirect
7. #27814C2semidirect
8. #27768C2semidirect
9. #27748C2semidirect
10. #27728C2semidirect
11. #27699C2semidirect
13. #21885C2semidirect
14. #21860C2semidirect
17. #20543C2semidirect
24. #2859C22semidirect
25. #2859C22semidirect
28. #2571C22semidirect
35. #2552C22semidirect
52. #2101C22semidirect
53. #2101C22semidirect
54. #2074C22semidirect
58. #2026C22semidirect
60. #1991C22semidirect
61. #1990C22semidirect
62. #1987C22semidirect
63. #1981C22semidirect
65. #1956C22semidirect
68. #1933C22semidirect
69. #1933C22semidirect
72. #1835C22semidirect
74. #1803C22semidirect
75. #1803C22semidirect
76. #1803C22semidirect
77. #1803C22semidirect
87. C8×C4⋊C4C22semidirect
88. C8×C4⋊C4C22semidirect
89. C4×C4⋊C8C22semidirect
90. C4×C4⋊C8C22semidirect
656. C2×C4#488semidirect

Maximal subgroups

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

subgroupindex
#352862
#349462
#291402
#291222
#287212
#278142
#277682
#277482
#277282
#276992
#272562
#218852
#218602
#207322
C2×#18022
#205432