Finite Group Databasebeta

← #6616247

Subgroups — #6616247

Order 512.

1947 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36126C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#33106C211yesmaximal, characteristic, complemented
4#33096C212yesmaximal, complemented
5#27019C214yesmaximal, complemented
6#26709C214yesmaximal, complemented
7#26446C212yesmaximal, complemented
8#25903C211yesmaximal, characteristic, complemented
9#25274C212yesmaximal, complemented
10#24608C212yesmaximal, complemented
11#24299C211yesmaximal, characteristic, complemented
12#24060C212yesmaximal, complemented
13#23223C212yesmaximal, complemented
14#22969C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16C4×#480C211yesmaximal, characteristic, complemented
17#21395C212yesmaximal, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C2×#486C2212yes—
22C2×#480C2211yescharacteristic
23C2×#480C2212yes—
24C2×#476C2212yes—
25C2×#432C2212yes—
26#2856C2211yescharacteristic
27#2854C2212yes—
28#2551C2211yescharacteristic
29#2541C2211yescharacteristic
30#2491—21no—
31#2461—22no—
32#2452C2212yes—
33#2445C2212yes—
34#2430—22no—
35#2416—21no—
36#2369—21no—
37#2369—22no—
38#2369—21no—
39#2369C2212yes—
40#2369C2214yes—
41#2369C2212yes—
42#2366C2211yescharacteristic
43#2361—21no—
44D4×C4⋊C4C2211yescharacteristic
45C4×#121C2211yescharacteristic
46C2×#391—22no—
47C2×#389—21no—
48C2×#389C2212yes—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2087C2214yes—
54#2065C2214yes—
55#2058C2212yes—
56#2057C2214yes—
57#2041C2214yes—
58#2023C2214yes—
59#2023C2212yes—
60#1990C2212yes—
61#1987C2214yes—
62#1987C2212yes—
63#1982C2212yes—
64#1982C2214yes—
65#1947C2212yes—
66#1944C2214yes—
67#1939C2212yes—
68#1936C2214yes—
69#1926C2214yes—
70#1925C2212yes—
71#1925C2214yes—
72#1925C2214yes—
73#1921C2214yes—
74#1920C2214yes—
75#1918C2212yes—
76#1917C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2211yescharacteristic
80#1874C2212yes—
81#1874C2212yes—
82#1862C2214yes—
83#1862C2214yes—
84#1853C2212yes—
85#1852C2211yescharacteristic
86#1823C2211yescharacteristic
87#1817C2212yes—
88#1797C2211yescharacteristic
89C4×#107C2212yes—
90#1770C2212yes—
91#1762C2214yes—
92#1762C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41—22no—
96C23×Q8—21no—
97#547—48no—
98#532—28no—
99#532—44no—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#127C2311yescharacteristic
108C2×#127C2312yes—
109C2×#124—22no—
110C2×#123—22no—
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×#121—21no—
114C2×#121—22no—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×#121C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C42—21no—
135C22×C42—22no—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#486—28no—
139#480—24no—
140#480—44no—
141#476—28no—
142#432—44no—
143C2×#107C2312yes—
144C2×#107C2312yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2311yescharacteristic
159C2×#97C2314yes—
160C2×C4×C8C2312yes—
161#396—22no—
162#395—22no—
163#394—22no—
164#394—22no—
165#393—22no—
166#392—22no—
167#391—24no—
168#391—24no—
169#391—24no—
170#391—24no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#390C2311yescharacteristic
178#389—21no—
179#389—22no—
180#389—22no—
181#389—24no—
182#389—22no—
183#389—22no—
184#389C2314yes—
185#389C2311yescharacteristic
186#389C2312yes—
187#389C2312yes—
188#389C2314yes—
189#388—22no—
190#388—22no—
191#388—21no—
192#386—22no—
193#386—21no—
194#386—22no—
195#386—22no—
196#385—22no—
197#385—22no—
198#384C2311yescharacteristic
199#380—21no—
200#380C2311yescharacteristic
201C4×C4⋊C4—22no—
202C4×C4⋊C4—24no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—21no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C2×#41—48no—
238C2×#41—48no—
239C2×#41—84no—
240C22×Q8—28no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—82no—
245C22×Q8—82no—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—21no—
252C22×D4C2412yes—
253C22×D4C2411yescharacteristic
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C22×C8C2412yes—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C4⋊Q8—28no—
269C4⋊Q8—28no—
270C4⋊Q8—28no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—42no—
274C4⋊Q8—44no—
275C4⋊Q8#4118yescomplemented
276#127—24no—
277#127—24no—
278#127—42no—
279#124—48no—
280#123—48no—
281#123—48no—
282C22⋊Q8—48no—
283#121—22no—
284#121—24no—
285#121—24no—
286#121—44no—
287#121—48no—
288#121—44no—
289#121—44no—
290#121—82no—
291#121#4114yes—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—48no—
296C4×D4—42no—
297C4×D4—42no—
298C4×D4—44no—
299C4×D4—44no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—44no—
358C2×C42—84no—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2412yes—
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—24no—
368#107—24no—
369#107—44no—
370#106—28no—
371C4⋊C8—24no—
372C4⋊C8—44no—
373#102—24no—
374#102—28no—
375#102—28no—
376#102—28no—
377#102—24no—
378#102—28no—
379#102—44no—
380C22⋊C8—22no—
381C22⋊C8—22no—
382C22⋊C8—24no—
383#97—44no—
384C4×C8—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—21no—
401C24—22no—
402C24C22×D411yescharacteristic
403#41—832no—
404C2×Q8—28no—
405C2×Q8—28no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×Q8—82no—
413C2×Q8—816no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—81no—
438C2×D4—81no—
439C2×D4C2×#4112yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—28no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—42no—
461C22×C4—41no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4AGL(1,ℤ/8)14yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4C2511yescharacteristic
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C4⋊C4—24no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—81no—
527C4⋊C4#13714yes—
528C4⋊C4C2×#4118yes—
529C4⋊C4C2×#4112yes—
530C4⋊C42−1+414yes—
531C22⋊C4—24no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—82no—
543C22⋊C4—81no—
544C22⋊C4—82no—
545C42—22no—
546C42—22no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—84no—
558C42—84no—
559C42C2×#4114yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23#54612yes—
579C23C2×AGL(1,ℤ/8)12yes—
580C23#57612yes—
581C23C23×D411yescharacteristic
582Q8—88no—
583Q8—88no—
584Q8—88no—
585D4—24no—
586D4—48no—
587D4—44no—
588D4—42no—
589D4—44no—
590D4—88no—
591D4—88no—
592D4—88no—
593D4—84no—
594D4—162no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—28no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—48no—
624C2×C4—416no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—48no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4#52512yes—
649C2×C4#52512yes—
650C2×C4#54612yes—
651C2×C4C2×AGL(1,ℤ/8)12yes—
652C2×C4C2×AGL(1,ℤ/8)12yes—
653C2×C4C2×#13714yes—
654C2×C4C2×#13712yes—
655C2×C4C22×#4111yescharacteristic
656C2×C4C22×#4112yes—
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4C2×2−1+412yes—
659C8—42no—
660C8—42no—
661C8—44no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22#332514yes—
683C22C2×#52511yescharacteristic
684C22C2×#54611yescharacteristic
685C22#348611yescharacteristic
686C22C22×AGL(1,ℤ/8)11yescharacteristic
687C22C22×#13711yescharacteristic
688C22C2×#57611yescharacteristic
689C22#358511yescharacteristic
690C4—22no—
691C4—22no—
692C4—22no—
693C4—24no—
694C4—44no—
695C4—41no—
696C4—48no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—81no—
702C4—81no—
703C4—82no—
704C4—82no—
705C4#311214yes—
706C4#348212yes—
707C4#352212yes—
708C2—24no—
709C2—82no—
710C2—162no—
711C2#3642211yescharacteristic
712C2C2×#311211yescharacteristic
713C2#6057811yescharacteristic
714C2C2×#332511yescharacteristic
715C2#6116211yescharacteristic
716C2#6182511yescharacteristic
717C2#6184411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36126C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2369C2semidirect
3. #33106C2semidirect
4. #33096C2semidirect
5. #27019C2semidirect
6. #26709C2semidirect
7. #26446C2semidirect
8. #25903C2semidirect
9. #25274C2semidirect
10. #24608C2semidirect
11. #24299C2semidirect
12. #24060C2semidirect
13. #23223C2semidirect
14. #22969C2semidirect
15. #22603C2semidirect
16. C4×#480C2semidirect
17. #21395C2semidirect
275. C4⋊Q8#41semidirect

Maximal subgroups

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

subgroupindex
#361262
C2×#23692
#331062
#330962
#270192
#267092
#264462
#259032
#252742
#246082
#242992
#240602
#232232
#229692
#226032
C4×#4802
#213952