Finite Group Databasebeta

← #6852595

Subgroups — #6852595

Order 512.

1571 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic, complemented
2#34941C211yesmaximal, characteristic, complemented
3#33197C211yesmaximal, characteristic, complemented
4#32354C214yesmaximal, complemented
5#31530C214yesmaximal, complemented
6#29986C212yesmaximal, complemented
7#29256C212yesmaximal, complemented
8#29210C212yesmaximal, complemented
9#28689C211yesmaximal, characteristic, complemented
10#28528C212yesmaximal, complemented
11#28280C211yesmaximal, characteristic, complemented
12#27814C212yesmaximal, complemented
13#25839C212yesmaximal, complemented
14#24327C212yesmaximal, complemented
15#22821C211yesmaximal, characteristic, complemented
16#22145C212yesmaximal
17C4×#459C211yesmaximal, characteristic
18C2×#554C2212yes—
19C2×#500C2212yes—
20C2×#476C2212yes—
21C2×#476C2212yes—
22C2×#459C2211yescharacteristic
23C2×C4×D8C2212yes—
24#2868C2211yescharacteristic
25#2859C2212yes—
26#2574—22no—
27#2567C2211yescharacteristic
28#2566C2214yes—
29#2563C2211yescharacteristic
30#2563C2212yes—
31#2552C2211yescharacteristic
32#2538—21no—
33#2538—21no—
34#2538C2211yescharacteristic
35#2508C2212yes—
36#2497—21no—
37#2454C2212yes—
38#2452—22no—
39#2452—22no—
40#2423—21no—
41#2396—22no—
42D4×C4⋊C4—21no—
43D4×C4⋊C4—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2212yes—
46C4×#127C2211yescharacteristic
47C4×#125C2212yes—
48C4×C22⋊Q8—21no—
49#2293C2212yes—
50#2285—21no—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2101C2214yes—
54#2092C2214yes—
55#2092C2214yes—
56#2083C2214yes—
57#2065C2214yes—
58#2059C2214yes—
59#2059C2214yes—
60#2059C2214yes—
61#2025C2214yes—
62#2025C2214yes—
63#1991C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1982C2212yes—
69#1970C2214yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1947C2211yescharacteristic
76#1934C2214yes—
77#1925C2212yes—
78#1916C2214yes—
79#1879C2214yes—
80#1879C2212yes—
81#1864C2214yes—
82#1862C2214yes—
83#1862C2214yes—
84#1846C2214yes—
85#1842C2211yescharacteristic
86#1831C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#102C2212yes—
95C22×D8C2312yes—
96#554—44no—
97#554—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#125C2312yes—
104C2×#125C2314yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—24no—
108C2×C4×Q8—21no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2314yes—
121C2×#117—22no—
122C22×C4⋊C4—22no—
123C22×C42—21no—
124#500—44no—
125#476—44no—
126#476—44no—
127#459Q818yescomplemented
128C4×D8—44no—
129C2×#107C2312yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2314yes—
148#400—22no—
149#399—22no—
150#398—22no—
151#398—22no—
152#396—24no—
153#396—24no—
154#396—22no—
155#396—22no—
156#395—22no—
157#393—22no—
158#393—24no—
159#391—21no—
160#391—21no—
161#390—21no—
162#390—21no—
163#390—22no—
164#390—22no—
165#390C2314yes—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—24no—
187#385—24no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2312yes—
196#382—22no—
197#382C2312yes—
198#382C2314yes—
199#380—22no—
200#380—22no—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C43C2312yes—
223C43C2311yescharacteristic
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—24no—
247C2×D8—42no—
248C2×D8—42no—
249C2×D8C2×Q818yes—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—82no—
258C4⋊Q8—82no—
259#127—22no—
260#127C2×Q814yes—
261#125—24no—
262#125—42no—
263#125—44no—
264#125—44no—
265C22⋊Q8—416no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—42no—
272C4×Q8—82no—
273C4×D4—22no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—41no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—84no—
285C4×D4—84no—
286C4×D4C2×Q814yes—
287#117—84no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—44no—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—42no—
353#107—44no—
354#107—44no—
355#106—44no—
356#106—44no—
357C4⋊C8—24no—
358C4⋊C8—22no—
359C4⋊C8—24no—
360C4⋊C8—42no—
361C4⋊C8C2×Q814yes—
362#102—24no—
363#102—44no—
364#102—42no—
365#102—42no—
366#102—44no—
367#102—44no—
368#102C2×Q818yes—
369C4×C8—24no—
370C4×C8—44no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—21no—
388C24—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—81no—
395C2×Q8—81no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C22×Q814yes—
415C2×D4C22×Q812yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—42no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—81no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449D8—48no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8C22×Q814yes—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4C22×Q812yes—
497C4⋊C42−1+412yes—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C42—24no—
512C42—22no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—22no—
519C42—22no—
520C42—21no—
521C42—42no—
522C42—42no—
523C42—41no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—44no—
528C42#13514yes—
529C42C22×Q812yes—
530C42H(5,𝔽2)12yes—
531C42H(5,𝔽2)12yes—
532C422−1+412yes—
533C23—22no—
534C23—24no—
535C23—42no—
536C23—41no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23C23×D411yescharacteristic
548Q8—44no—
549Q8—84no—
550Q8—164no—
551D4—22no—
552D4—24no—
553D4—22no—
554D4—24no—
555D4—48no—
556D4—48no—
557D4—41no—
558D4—42no—
559D4—41no—
560D4—42no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—84no—
605C2×C4—88no—
606C2×C4—84no—
607C2×C4—168no—
608C2×C4#53512yes—
609C2×C4#53512yes—
610C2×C4D4×Q812yes—
611C2×C4D4×Q812yes—
612C2×C4C2×#13512yes—
613C2×C4C2×#13512yes—
614C2×C4C2×#13512yes—
615C2×C4#57512yes—
616C2×C4#57612yes—
617C2×C4C23×Q811yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4C2×2−1+411yescharacteristic
621C2×C4C2×2−1+411yescharacteristic
622C2×C4#58511yescharacteristic
623C8—24no—
624C8—42no—
625C8—44no—
626C22—42no—
627C22—44no—
628C22—42no—
629C22—44no—
630C22—44no—
631C22—42no—
632C22—82no—
633C22—81no—
634C22—81no—
635C22—82no—
636C22—82no—
637C22—81no—
638C22—81no—
639C22—82no—
640C22C2×#53511yescharacteristic
641C22C2×D4×Q811yescharacteristic
642C22#348911yescharacteristic
643C22C22×#13511yescharacteristic
644C22C2×#57511yescharacteristic
645C22C2×#57611yescharacteristic
646C22#358511yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—82no—
658C4—81no—
659C4—84no—
660C4—82no—
661C4—82no—
662C4#321214yes—
663C4#338014yes—
664C4#350111yescharacteristic
665C4#350311yescharacteristic
666C4#355011yescharacteristic
667C4#355411yescharacteristic
668C2—82no—
669C2—84no—
670C2#3959611yescharacteristic
671C2C2×#321211yescharacteristic
672C2#6083211yescharacteristic
673C2C2×#338011yescharacteristic
674C2#6136711yescharacteristic
675C2#6209811yescharacteristic
676C2#6213611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
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. #36559C2semidirect
2. #34941C2semidirect
3. #33197C2semidirect
4. #32354C2semidirect
5. #31530C2semidirect
6. #29986C2semidirect
7. #29256C2semidirect
8. #29210C2semidirect
9. #28689C2semidirect
10. #28528C2semidirect
11. #28280C2semidirect
12. #27814C2semidirect
13. #25839C2semidirect
14. #24327C2semidirect
15. #22821C2semidirect
127. #459Q8semidirect

Maximal subgroups

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

subgroupindex
#365592
#349412
#331972
#323542
#315302
#299862
#292562
#292102
#286892
#285282
#282802
#278142
#258392
#243272
#228212
#221452
C4×#4592