Finite Group Databasebeta

← #6800730

Subgroups — #6800730

Order 512.

1849 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#127C211yesmaximal, characteristic, complemented
2#34954C211yesmaximal, characteristic, complemented
3#33196C211yesmaximal, characteristic, complemented
4#32446C214yesmaximal, complemented
5#31656C214yesmaximal, complemented
6#29981C212yesmaximal, complemented
7#29148C212yesmaximal, complemented
8#28716C211yesmaximal, characteristic, complemented
9#28415C212yesmaximal, complemented
10#28251C211yesmaximal, characteristic
11#28158C212yesmaximal, complemented
12#27736C212yesmaximal, complemented
13#25933C212yesmaximal, complemented
14#24330C212yesmaximal, complemented
15#22825C211yesmaximal, characteristic, complemented
16#22151C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C2×D4×Q8—21no—
20C2×D4×Q8C2212yes—
21C2×#486C2212yes—
22C2×#475C2212yes—
23C2×#460C2211yescharacteristic
24C2×#459C2212yes—
25C2×#459C2212yes—
26C2×C4×SD16C2212yes—
27#2868C2212yes—
28#2868C2211yescharacteristic
29#2593—22no—
30#2582C2211yescharacteristic
31#2571C2211yescharacteristic
32#2548C2211yescharacteristic
33#2548C2211yescharacteristic
34#2542—22no—
35#2542—22no—
36#2542—21no—
37#2542—21no—
38#2542C2212yes—
39#2542C2214yes—
40#2513—21no—
41#2513C2212yes—
42#2448—21no—
43#2448C2212yes—
44#2395—22no—
45C4×#127C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#127C2212yes—
48C4×#124—21no—
49C4×#124C2212yes—
50#2293—21no—
51#2293C2212yes—
52C42×Q8C2211yescharacteristic
53C42×Q8C2211yescharacteristic
54#2100C2214yes—
55#2094C2214yes—
56#2073C2214yes—
57#2071C2214yes—
58#2058C2214yes—
59#2054C2214yes—
60#2024C2214yes—
61#2023C2214yes—
62#2023C2214yes—
63#2012C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1972C2214yes—
69#1959C2214yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1948C2212yes—
73#1936C2214yes—
74#1926C2212yes—
75#1917C2214yes—
76#1880C2212yes—
77#1879C2214yes—
78#1879C2214yes—
79#1877C2214yes—
80#1864C2214yes—
81#1842C2211yescharacteristic
82#1836C2214yes—
83#1831C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C23×Q8—22no—
96C22×SD16C2312yes—
97C22×D8C2314yes—
98D4×Q8—44no—
99D4×Q8—44no—
100D4×Q8—48no—
101D4×Q8—48no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×#127C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#124—21no—
111C2×#124C2312yes—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×C4×D4C2314yes—
133C2×#117—22no—
134#486—44no—
135#475—44no—
136#460—24no—
137#459—28no—
138#459—44no—
139C4×SD16—44no—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158#400—22no—
159#399—22no—
160#399—24no—
161#399—24no—
162#399—24no—
163#398—22no—
164#397—22no—
165#397—22no—
166#397—22no—
167#394—22no—
168#394—22no—
169#391—21no—
170#391C2312yes—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—22no—
177#390—21no—
178#390—21no—
179#390—22no—
180#390C2312yes—
181#390C2312yes—
182#390C2314yes—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#389—21no—
187#389—21no—
188#389C2312yes—
189#389C2312yes—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#384—22no—
199#384—21no—
200#384—21no—
201#384—22no—
202#384C2311yescharacteristic
203#384C2312yes—
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2314yes—
207#382—21no—
208#382C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C43—21no—
225C43C2311yescharacteristic
226C43C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C22×Q8—24no—
239C22×Q8—22no—
240C22×Q8—48no—
241C22×Q8—48no—
242C22×Q8—48no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2411yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2412yes—
250C22×D4C2412yes—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4—22no—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C2×D8—28no—
258C2×D8—44no—
259C2×D8—44no—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—48no—
268C4⋊Q8—44no—
269C4⋊Q8—48no—
270C4⋊Q8—44no—
271#127—22no—
272#127—22no—
273#127—22no—
274#127#4114yes—
275#124—24no—
276#124—42no—
277#124—44no—
278C22⋊Q8—48no—
279C22⋊Q8—416no—
280C22⋊Q8—48no—
281C22⋊Q8—48no—
282C22⋊Q8—416no—
283C22⋊Q8—416no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—41no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—48no—
301C4×D4—44no—
302C4×D4—48no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—82no—
306#117—84no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D412yes—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#106—42no—
373#106—44no—
374C4⋊C8—24no—
375C4⋊C8—22no—
376C4⋊C8—24no—
377C4⋊C8—22no—
378C4⋊C8—42no—
379#103—42no—
380#103—44no—
381#102—28no—
382#102—24no—
383#102—42no—
384#102—42no—
385#102—44no—
386#102—44no—
387#102—44no—
388#97—44no—
389C4×C8—24no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24—22no—
403C24—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—44no—
414C2×Q8—48no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—832no—
418C2×Q8AGL(1,ℤ/8)14yes—
419C2×Q82−1+412yes—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—81no—
442C2×D4—81no—
443C2×D4C2×#4114yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2511yescharacteristic
471SD16—48no—
472D8—88no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—81no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C42—24no—
535C42—22no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—21no—
542C42—22no—
543C42—22no—
544C42—21no—
545C42—41no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—81no—
553C42AGL(1,ℤ/8)14yes—
554C42C2×#4112yes—
555C42H(5,𝔽2)12yes—
556C42H(5,𝔽2)12yes—
557C422−1+412yes—
558C23—24no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23C23×D411yescharacteristic
573Q8—28no—
574Q8—22no—
575Q8—22no—
576Q8—41no—
577Q8—41no—
578Q8—816no—
579Q8—88no—
580D4—24no—
581D4—24no—
582D4—48no—
583D4—48no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—84no—
591D4—84no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—816no—
634C2×C4—88no—
635C2×C4—88no—
636C2×C4#53512yes—
637C2×C4#53512yes—
638C2×C4#54812yes—
639C2×C4#54812yes—
640C2×C4C2×AGL(1,ℤ/8)14yes—
641C2×C4C2×AGL(1,ℤ/8)12yes—
642C2×C4C2×AGL(1,ℤ/8)12yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4#57712yes—
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×2−1+411yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C2×C4#58511yescharacteristic
652C8—44no—
653C8—42no—
654C8—42no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22C2×#53511yescharacteristic
670C22C2×#54811yescharacteristic
671C22#348911yescharacteristic
672C22C22×AGL(1,ℤ/8)11yescharacteristic
673C22C22×AGL(1,ℤ/8)11yescharacteristic
674C22C2×#57711yescharacteristic
675C22#358611yescharacteristic
676C4—24no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—41no—
684C4—42no—
685C4—84no—
686C4—82no—
687C4—82no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4#321714yes—
692C4#337314yes—
693C4#351411yescharacteristic
694C4#352011yescharacteristic
695C4#354711yescharacteristic
696C4#355211yescharacteristic
697C2—84no—
698C2—162no—
699C2#3912411yescharacteristic
700C2C2×#321711yescharacteristic
701C2#6083611yescharacteristic
702C2C2×#337311yescharacteristic
703C2#6131711yescharacteristic
704C2#6211111yescharacteristic
705C2#6215011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28251C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. Q8×#127C2semidirect
2. #34954C2semidirect
3. #33196C2semidirect
4. #32446C2semidirect
5. #31656C2semidirect
6. #29981C2semidirect
7. #29148C2semidirect
8. #28716C2semidirect
9. #28415C2semidirect
11. #28158C2semidirect
12. #27736C2semidirect
13. #25933C2semidirect
14. #24330C2semidirect
15. #22825C2semidirect
16. #22151C2semidirect
17. C4×#460C2semidirect

Maximal subgroups

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

subgroupindex
Q8×#1272
#349542
#331962
#324462
#316562
#299812
#291482
#287162
#284152
#282512
#281582
#277362
#259332
#243302
#228252
#221512
C4×#4602