Finite Group Databasebeta

← #6605587

Subgroups — #6605587

Order 512.

1998 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36160C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#32981C212yesmaximal, complemented
5#27948C211yesmaximal, characteristic, complemented
6#27677C211yesmaximal, characteristic, complemented
7#27604C212yesmaximal, complemented
8#26299C212yesmaximal, complemented
9#26137C214yesmaximal, complemented
10#25013C212yesmaximal, complemented
11#24944C214yesmaximal, complemented
12#24553C212yesmaximal, complemented
13#22657C212yesmaximal, complemented
14#22489C212yesmaximal, complemented
15D4×C22⋊C8C211yesmaximal, characteristic, complemented
16C4×#481C211yesmaximal, characteristic, complemented
17#20843C212yesmaximal, complemented
18C2×#555C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21C2×#481C2211yescharacteristic
22C2×#482C2214yescomplemented
23C2×C8×D4C2212yes—
24C2×#427C2214yescomplemented
25C22×C22⋊C8C2212yes—
26#2857C2211yescharacteristic
27#2847C2212yes—
28#2563C2211yescharacteristic
29#2537C2211yescharacteristic
30#2485C2212yes—
31#2472—22no—
32#2459—22no—
33#2420—21no—
34#2414C2212yes—
35#2408—21no—
36#2363—21no—
37#2354—21no—
38#2354—21no—
39#2354—22no—
40#2354C2214yes—
41#2354C2211yescharacteristic
42#2354C2212yes—
43#2354C2212yes—
44#2352—21no—
45C4×#121C2211yescharacteristic
46C2×#390—21no—
47C2×#388—22no—
48C2×#384C2211yescharacteristic
49#2280C2212yes—
50C2×#380C2212yes—
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2099C2212yescomplemented
54#2093C2212yescomplemented
55#2084C2214yescomplemented
56#2077C2214yescomplemented
57#2077C2214yescomplemented
58#2075C2214yescomplemented
59#2034C2214yescomplemented
60#1990C2212yescomplemented
61#1987C2212yescomplemented
62#1954C2212yes—
63#1947C2212yescomplemented
64#1947C2212yescomplemented
65#1926C2212yescomplemented
66#1921C2214yescomplemented
67#1919C2212yescomplemented
68#1893C2214yescomplemented
69#1875C2214yescomplemented
70#1874C2212yes—
71#1874C2211yescharacteristic
72#1869C2214yes—
73#1854C2214yescomplemented
74#1854C2214yescomplemented
75#1854C2214yescomplemented
76#1854C2214yescomplemented
77#1853C2212yes—
78#1853C2212yes—
79#1853C2211yescharacteristic
80#1852C2212yes—
81#1852C2212yes—
82#1844C2212yescomplemented
83#1825C2214yescomplemented
84#1825C2214yescomplemented
85#1824C2211yescharacteristic
86#1824C2212yes—
87#1807C2214yescomplemented
88#1802C2212yescomplemented
89#1797C2211yescharacteristic
90#1779C2212yes—
91C4×#106C2212yescomplemented
92C4×#102C2212yescomplemented
93C4×C22⋊C8C2211yescharacteristic
94#1735C2214yescomplemented
95C23×D4D411yescharacteristic
96C24×C4C2312yes—
97C23×C8C2314yes—
98#555—28no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#124—22no—
101C2×C22⋊Q8—21no—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2311yescharacteristic
105C2×#121C2312yes—
106C2×#121C2312yes—
107C2×#120—22no—
108C2×C4×D4—28no—
109C2×C4×D4—21no—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D412yes—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×#117—22no—
118C2×#117—22no—
119C2×#117—21no—
120C2×#117—48no—
121C2×#117C2312yes—
122C2×#117C2314yes—
123C2×#117C2314yes—
124C2×#117C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4D412yes—
134C22×C42—22no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#482—48no—
138#481—24no—
139C8×D4—28no—
140#427—48no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2312yes—
144C2×#106C2312yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×C22⋊C8—28no—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C4×C8C2312yes—
159#397—22no—
160#394—22no—
161#394—22no—
162#394—22no—
163#393—22no—
164#392—22no—
165#391—21no—
166#390—21no—
167#390—22no—
168#390—24no—
169#390—22no—
170#390—21no—
171#390—22no—
172#390C2311yescharacteristic
173#389—21no—
174#389C2312yes—
175#389C2311yescharacteristic
176#388—22no—
177#388—24no—
178#388—24no—
179#388—24no—
180#388—24no—
181#388—22no—
182#388—22no—
183#388—21no—
184#387—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—21no—
189#384—21no—
190#384—21no—
191#384C2312yes—
192#384C2312yes—
193#384C2311yescharacteristic
194#382—21no—
195#381—22no—
196#380—21no—
197#380—22no—
198#380—22no—
199#380—21no—
200#380C2312yes—
201#380C2312yes—
202#380C2312yes—
203#380C2314yes—
204#380C2314yes—
205#380C2312yes—
206#380C2312yes—
207#380C2311yescharacteristic
208#379—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—24no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C2×#95—22no—
222C2×#95—21no—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#336C2312yes—
235#336C2312yes—
236C25C2×D412yes—
237C22×Q8—21no—
238C22×D4—24no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—24no—
243C22×D4—42no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2×D412yes—
246C22×D4C2×D411yescharacteristic
247C22×D4C2411yescharacteristic
248C23×C4—21no—
249C23×C4—28no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—44no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2412yes—
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C22×C8—28no—
265C22×C8—28no—
266C22×C8—44no—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271#124—48no—
272C22⋊Q8—82no—
273#121—28no—
274#121—22no—
275#121—22no—
276#121—44no—
277#121—44no—
278#121—44no—
279#120—84no—
280C4×D4—24no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—44no—
287C4×D4#4118yes—
288#117—28no—
289#117—28no—
290#117—28no—
291#117—44no—
292#117—44no—
293#117—48no—
294#117—48no—
295#117—44no—
296#117—44no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
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—22no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—28no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4—84no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—48no—
361C2×C42—44no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—28no—
367#107—44no—
368#106—24no—
369#106—24no—
370#106—44no—
371#106—44no—
372C4⋊C8—24no—
373#102—24no—
374#102—24no—
375#102—44no—
376C22⋊C8—24no—
377C22⋊C8—28no—
378C22⋊C8—22no—
379C22⋊C8—28no—
380C22⋊C8—44no—
381C22⋊C8#4114yes—
382C4×C8—24no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—21no—
394#95—22no—
395#95—22no—
396C24—24no—
397C24—24no—
398C24—24no—
399C24—24no—
400C24—21no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24C22×D412yes—
405C24C22×D412yes—
406C24C22×D411yescharacteristic
407C2×Q8—42no—
408C2×Q8—81no—
409C2×Q8—81no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—416no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4#12314yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—42no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4#13514yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C2×#4118yes—
487C22×C4C2×#4112yes—
488C22×C42−1+412yes—
489C22×C4C2511yescharacteristic
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—416no—
499C2×C8C2×#4114yes—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—81no—
524C4⋊C4—82no—
525C4⋊C4#12314yes—
526C4⋊C4H(5,𝔽2)14yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—42no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—81no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4#12318yes—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—48no—
553C42—48no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—44no—
558C42—82no—
559C42C2×#4114yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—48no—
578C23—48no—
579C23—48no—
580C23—48no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23C2×#12314yes—
592C23#54712yes—
593C23C2×#13512yes—
594C23#57712yes—
595C23C23×D411yescharacteristic
596Q8—162no—
597D4—28no—
598D4—48no—
599D4—44no—
600D4—42no—
601D4—42no—
602D4—84no—
603D4—164no—
604D4—164no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—28no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—84no—
656C2×C4—88no—
657C2×C4—84no—
658C2×C4C2×#12312yes—
659C2×C4C2×#12314yes—
660C2×C4C2×#12312yes—
661C2×C4#54712yes—
662C2×C4C2×#13512yes—
663C2×C4C2×#13512yes—
664C2×C4#57512yes—
665C2×C4C22×#4111yescharacteristic
666C2×C4C22×#4112yes—
667C2×C4C2×H(5,𝔽2)12yes—
668C2×C4C2×2−1+411yescharacteristic
669C8—24no—
670C8—44no—
671C8—42no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—816no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22—81no—
700C22#334014yes—
701C22C22×#12311yescharacteristic
702C22C2×#54711yescharacteristic
703C22#348411yescharacteristic
704C22C22×#13511yescharacteristic
705C22C2×#57511yescharacteristic
706C22C2×#57711yescharacteristic
707C22#358511yescharacteristic
708C4—22no—
709C4—22no—
710C4—22no—
711C4—24no—
712C4—41no—
713C4—42no—
714C4—44no—
715C4—81no—
716C4—82no—
717C4—81no—
718C4—82no—
719C4—81no—
720C4—84no—
721C4—82no—
722C4#309114yes—
723C4#348112yes—
724C4#352012yes—
725C2—24no—
726C2—44no—
727C2—82no—
728C2—161no—
729C2#3623711yescharacteristic
730C2C2×#309111yescharacteristic
731C2#6052511yescharacteristic
732C2C2×#334011yescharacteristic
733C2#6125611yescharacteristic
734C2#6156611yescharacteristic
735C2#6160411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36160C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2354C2semidirect
3. #33149C2semidirect
4. #32981C2semidirect
5. #27948C2semidirect
6. #27677C2semidirect
7. #27604C2semidirect
8. #26299C2semidirect
9. #26137C2semidirect
10. #25013C2semidirect
11. #24944C2semidirect
12. #24553C2semidirect
13. #22657C2semidirect
14. #22489C2semidirect
15. D4×C22⋊C8C2semidirect
16. C4×#481C2semidirect
17. #20843C2semidirect
22. C2×#482C22semidirect
24. C2×#427C22semidirect
53. #2099C22semidirect
54. #2093C22semidirect
55. #2084C22semidirect
56. #2077C22semidirect
57. #2077C22semidirect
58. #2075C22semidirect
59. #2034C22semidirect
60. #1990C22semidirect
61. #1987C22semidirect
63. #1947C22semidirect
64. #1947C22semidirect
65. #1926C22semidirect
66. #1921C22semidirect
67. #1919C22semidirect
68. #1893C22semidirect
69. #1875C22semidirect
73. #1854C22semidirect
74. #1854C22semidirect
75. #1854C22semidirect
76. #1854C22semidirect
82. #1844C22semidirect
83. #1825C22semidirect
84. #1825C22semidirect
87. #1807C22semidirect
88. #1802C22semidirect
91. C4×#106C22semidirect
92. C4×#102C22semidirect
94. #1735C22semidirect

Maximal subgroups

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

subgroupindex
#361602
C2×#23542
#331492
#329812
#279482
#276772
#276042
#262992
#261372
#250132
#249442
#245532
#226572
#224892
D4×C22⋊C82
C4×#4812
#208432