Finite Group Databasebeta

← #6556195

Subgroups — #6556195

Order 512.

1884 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36289C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic, complemented
3#30203C211yesmaximal, characteristic, complemented
4#30188C212yesmaximal, complemented
5#29354C211yesmaximal, characteristic, complemented
6#29109C212yesmaximal, complemented
7#28726C211yesmaximal, characteristic, complemented
8#27978C212yesmaximal, complemented
9#27758C212yesmaximal, complemented
10#27735C212yesmaximal, complemented
11#27711C214yesmaximal, complemented
12#27262C214yesmaximal, complemented
13#23817C211yesmaximal, characteristic, complemented
14C2×#2071C212yesmaximal, complemented
15#22487C212yesmaximal, complemented
16D4×#102C212yesmaximal
17#20558C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×#554C2211yescharacteristic
20C2×D42C2211yescharacteristic
21C2×#489C2212yes—
22C2×C8×D4C2212yes—
23C22×#102C2214yes—
24#2868C2211yescharacteristic, complemented
25#2858C2211yescharacteristic
26#2857C2212yescomplemented
27#2850C2212yes—
28#2567—21no—
29#2563C2211yescharacteristic
30#2552—21no—
31#2552C2212yescomplemented
32#2548C2211yescharacteristic
33#2504C2212yes—
34#2502—22no—
35#2490—21no—
36#2490—21no—
37#2472—22no—
38#2472—22no—
39#2440—22no—
40#2440C2214yescomplemented
41#2435C2212yes—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44D4×C22⋊C4—21no—
45D4×C22⋊C4—21no—
46C4×C4⋊Q8C2211yescharacteristic, complemented
47C4×#124C2211yescharacteristic
48C2×#390C2212yes—
49C2×#384C2212yescomplemented
50C42×D4C2211yescharacteristic
51#2101C2212yescomplemented
52#2101C2214yescomplemented
53#2072C2211yescharacteristic, complemented
54#2072C2212yescomplemented
55#2071C2211yescharacteristic
56#2071C2212yes—
57#2071C2214yes—
58#2071C2214yescomplemented
59#2071C2212yescomplemented
60#2071C2212yescomplemented
61#2071C2212yescomplemented
62#2042C2212yes—
63#2042C2214yescomplemented
64#2024C2211yescharacteristic, complemented
65#2023C2211yescharacteristic
66#2023C2214yescomplemented
67#2023C2212yescomplemented
68#1993C2212yescomplemented
69#1990C2212yescomplemented
70#1987C2212yescomplemented
71#1982C2212yes—
72#1959C2214yescomplemented
73#1957C2212yes—
74#1936C2214yes—
75#1936C2214yescomplemented
76#1936C2214yescomplemented
77#1893C2214yes—
78#1892C2214yescomplemented
79#1874C2214yes—
80#1872C2214yescomplemented
81#1836C2214yescomplemented
82#1824C2214yescomplemented
83#1803C2212yescomplemented
84#1802C2212yes—
85#1786C2214yes—
86#1770C2212yescomplemented
87#1767C2211yescharacteristic, complemented
88C4×#103C2212yescomplemented
89C4×#102C2212yes—
90#1752C2214yescomplemented
91#1749C2212yescomplemented
92#1744C2211yescharacteristic, complemented
93C2×#340C2214yescomplemented
94C2×#336C2212yescomplemented
95C23×D4—21no—
96C23×D4C2312yes—
97C23×C8C2314yes—
98#554—28no—
99#554—48no—
100D42—28no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#125—22no—
108C2×#125—21no—
109C2×#124—21no—
110C2×#124—21no—
111C2×#124C2311yescharacteristic
112C2×#124C2312yes—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2312yes—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×#120—22no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—22no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C42C2312yes—
135#489—44no—
136C8×D4—28no—
137C2×C4⋊C8C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#102—48no—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×C22⋊C8C2314yes—
150C2×#97C2314yes—
151C2×#97C2312yes—
152C2×C4×C8C2312yes—
153#398—24no—
154#398—24no—
155#397—22no—
156#397—22no—
157#395—24no—
158#394—22no—
159#394—22no—
160#394—22no—
161#393—22no—
162#391—21no—
163#391—22no—
164#391C2311yescharacteristic
165#390—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—22no—
170#390—22no—
171#390C2312yes—
172#390C2311yescharacteristic
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—22no—
180#389C2312yes—
181#389C2314yes—
182#388—22no—
183#388—22no—
184#388—21no—
185#388—21no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—24no—
191#385—24no—
192#384—22no—
193#384—21no—
194#384—21no—
195#384—22no—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#384C2314yes—
200#382C2312yes—
201#382C2311yescharacteristic
202#380—22no—
203#380—21no—
204#380—21no—
205#380C2314yes—
206#379—22no—
207#379—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2312yes—
234C25—22no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2411yescharacteristic
239C22×D4—24no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—28no—
245C22×D4—44no—
246C22×D4—42no—
247C22×D4—44no—
248C22×D4—44no—
249C22×D4—44no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2412yes—
252C22×D4C2411yescharacteristic
253C23×C4—22no—
254C23×C4—24no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C22×C8—28no—
260C22×C8—44no—
261C22×C8—44no—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8—44no—
270C4⋊Q8—82no—
271#127—24no—
272#125—48no—
273#125—44no—
274#124—22no—
275#124—41no—
276#124—44no—
277#124—82no—
278#124—82no—
279C22⋊Q8—28no—
280C22⋊Q8—416no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283#121—28no—
284#121—44no—
285#120—48no—
286C4×Q8—24no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—44no—
292C4×D4—48no—
293C4×D4#4118yes—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—84no—
344C2×C22⋊C4—84no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—44no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C4⋊C8—24no—
362#103—42no—
363#103—42no—
364#103—44no—
365#102—24no—
366#102—28no—
367#102—44no—
368#102—44no—
369#102—44no—
370#102—42no—
371#102—44no—
372C22⋊C8—28no—
373#97—44no—
374#97—42no—
375C4×C8—24no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C24—22no—
391C24—44no—
392C24—44no—
393C24—44no—
394C24—44no—
395C24—82no—
396C24—82no—
397C24—82no—
398C24C22×D412yes—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—81no—
409C2×Q8—81no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—416no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—88no—
430C2×D4—82no—
431C2×D4—81no—
432C2×D4—81no—
433C2×D4—816no—
434C2×D4#12414yes—
435C2×D4AGL(1,ℤ/8)14yes—
436C2×D4#13714yes—
437C2×D4H(5,𝔽2)12yes—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4118yes—
473C22×C4C2511yescharacteristic
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—88no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4#12414yes—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4#12418yes—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—24no—
540C42—22no—
541C42—21no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42C2×#4114yes—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—88no—
566C23—88no—
567C23—88no—
568C23—88no—
569C23—164no—
570C23—164no—
571C23—164no—
572C23C2×#12414yes—
573C23C2×AGL(1,ℤ/8)14yes—
574C23C2×#13714yes—
575C23C23×D411yescharacteristic
576Q8—42no—
577Q8—42no—
578Q8—84no—
579Q8—162no—
580Q8—164no—
581D4—24no—
582D4—44no—
583D4—42no—
584D4—44no—
585D4—88no—
586D4—88no—
587D4—84no—
588D4—162no—
589D4#489116yescomplemented
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—84no—
636C2×C4—168no—
637C2×C4—164no—
638C2×C4C2×#12412yes—
639C2×C4C2×#12414yes—
640C2×C4C2×#12412yes—
641C2×C4C2×AGL(1,ℤ/8)12yes—
642C2×C4C2×#13712yes—
643C2×C4#57612yes—
644C2×C4#57712yes—
645C2×C4C22×#4112yes—
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4#58512yes—
648C2×C4#58511yescharacteristic
649C8—44no—
650C8—42no—
651C8—42no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—1616no—
673C22C2×#48918yes—
674C22C22×#12411yescharacteristic
675C22#348511yescharacteristic
676C22#348811yescharacteristic
677C22C22×AGL(1,ℤ/8)11yescharacteristic
678C22C22×#13711yescharacteristic
679C22C2×#57611yescharacteristic
680C22C2×#57711yescharacteristic
681C4—24no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—41no—
686C4—41no—
687C4—42no—
688C4—42no—
689C4—81no—
690C4—82no—
691C4—84no—
692C4—82no—
693C4—81no—
694C4—81no—
695C4—82no—
696C4C2×#48914yes—
697C4#351312yes—
698C4#352112yes—
699C2—28no—
700C2—82no—
701C2—162no—
702C2#3584711yescharacteristic
703C2C22×#48911yescharacteristic
704C2#6062211yescharacteristic
705C2#6172611yescharacteristic
706C2#6172811yescharacteristic
707C2#6197911yescharacteristic
708C2#6198011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36180C2semidirect
3. #30203C2semidirect
4. #30188C2semidirect
5. #29354C2semidirect
6. #29109C2semidirect
7. #28726C2semidirect
8. #27978C2semidirect
9. #27758C2semidirect
10. #27735C2semidirect
11. #27711C2semidirect
12. #27262C2semidirect
13. #23817C2semidirect
14. C2×#2071C2semidirect
15. #22487C2semidirect
17. #20558C2semidirect
24. #2868C22semidirect
26. #2857C22semidirect
31. #2552C22semidirect
40. #2440C22semidirect
46. C4×C4⋊Q8C22semidirect
49. C2×#384C22semidirect
51. #2101C22semidirect
52. #2101C22semidirect
53. #2072C22semidirect
54. #2072C22semidirect
58. #2071C22semidirect
59. #2071C22semidirect
60. #2071C22semidirect
61. #2071C22semidirect
63. #2042C22semidirect
64. #2024C22semidirect
66. #2023C22semidirect
67. #2023C22semidirect
68. #1993C22semidirect
69. #1990C22semidirect
70. #1987C22semidirect
72. #1959C22semidirect
75. #1936C22semidirect
76. #1936C22semidirect
78. #1892C22semidirect
80. #1872C22semidirect
81. #1836C22semidirect
82. #1824C22semidirect
83. #1803C22semidirect
86. #1770C22semidirect
87. #1767C22semidirect
88. C4×#103C22semidirect
90. #1752C22semidirect
91. #1749C22semidirect
92. #1744C22semidirect
93. C2×#340C22semidirect
94. C2×#336C22semidirect
589. D4#489semidirect

Maximal subgroups

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

subgroupindex
#362892
#361802
#302032
#301882
#293542
#291092
#287262
#279782
#277582
#277352
#277112
#272622
#238172
C2×#20712
#224872
D4×#1022
#205582