Finite Group Databasebeta

← #6852641

Subgroups — #6852641

Order 512.

1661 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic, complemented
2#34929C211yesmaximal, characteristic, complemented
3#33197C211yesmaximal, characteristic, complemented
4#32368C214yesmaximal, complemented
5#31542C214yesmaximal, complemented
6#29986C212yesmaximal, complemented
7#29258C212yesmaximal, complemented
8#29212C212yesmaximal, complemented
9#28687C211yesmaximal, characteristic, complemented
10#28541C212yesmaximal, complemented
11#28279C211yesmaximal, characteristic, complemented
12#27821C212yesmaximal, complemented
13#25835C212yesmaximal, complemented
14#24327C212yesmaximal, complemented
15#22825C211yesmaximal, characteristic, complemented
16#22151C212yesmaximal
17C4×#459C211yesmaximal, characteristic
18C2×#554C2212yes—
19C2×#533—21no—
20C2×#498C2212yes—
21C2×#476C2212yes—
22C2×#474C2212yes—
23C2×#459C2211yescharacteristic
24C2×C4×D8C2212yes—
25#2868C2211yescharacteristic
26#2859C2212yes—
27#2566C2214yes—
28#2563C2212yes—
29#2556—22no—
30#2551C2211yescharacteristic
31#2546C2211yescharacteristic
32#2542C2211yescharacteristic
33#2538—21no—
34#2538—21no—
35#2538C2211yescharacteristic
36#2506C2212yes—
37#2492—21no—
38#2452—22no—
39#2452—22no—
40#2452C2212yes—
41#2429—21no—
42#2396—22no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×#127C2212yes—
48C4×#127C2211yescharacteristic
49C4×C22⋊Q8—21no—
50#2293C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2101C2214yes—
54#2094C2214yes—
55#2092C2214yes—
56#2084C2214yes—
57#2067C2214yes—
58#2059C2214yes—
59#2059C2214yes—
60#2059C2214yes—
61#2025C2214yes—
62#2023C2214yes—
63#1993C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1982C2212yes—
69#1972C2214yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1947C2211yescharacteristic
75#1947C2212yes—
76#1936C2214yes—
77#1925C2212yes—
78#1917C2214yes—
79#1879C2214yes—
80#1879C2212yes—
81#1864C2214yes—
82#1864C2214yes—
83#1864C2214yes—
84#1847C2214yes—
85#1842C2211yescharacteristic
86#1831C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×D8C2312yes—
96#554—44no—
97#554—44no—
98#533—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#125—22no—
108C2×#125C2314yes—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—24no—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2314yes—
129C2×#117—22no—
130C22×C4⋊C4—22no—
131C22×C42—21no—
132#498—44no—
133#476—44no—
134#474—44no—
135#459Q818yescomplemented
136C4×D8—44no—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#400—22no—
157#399—22no—
158#398—22no—
159#398—22no—
160#396—24no—
161#396—24no—
162#396—22no—
163#396—22no—
164#395—22no—
165#393—24no—
166#393—22no—
167#391—21no—
168#391—21no—
169#391—22no—
170#390—21no—
171#390—21no—
172#390—22no—
173#390—22no—
174#390C2314yes—
175#390C2312yes—
176#390C2312yes—
177#390C2312yes—
178#389—21no—
179#389—22no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#388—22no—
191#388—22no—
192#387—22no—
193#385—24no—
194#385—24no—
195#385—22no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2312yes—
203#382C2314yes—
204#380—22no—
205#380—22no—
206#380—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C43C2312yes—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8C2×D411yescharacteristic
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—24no—
249C2×D8—42no—
250C2×D8—42no—
251C2×D8C2×Q818yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—42no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264C4⋊Q8—82no—
265#127—22no—
266#127C2×Q814yes—
267#125—44no—
268#125—44no—
269#125—48no—
270C22⋊Q8—44no—
271C22⋊Q8—48no—
272C22⋊Q8—44no—
273C22⋊Q8—416no—
274C22⋊Q8—44no—
275C22⋊Q8—48no—
276C4×Q8—24no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—42no—
280C4×Q8—48no—
281C4×Q8—44no—
282C4×D4—22no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—41no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—84no—
294C4×D4—84no—
295C4×D4C2×Q814yes—
296#117—48no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—44no—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—42no—
363#107—44no—
364#106—44no—
365#106—44no—
366#106—44no—
367C4⋊C8—24no—
368C4⋊C8—22no—
369C4⋊C8—24no—
370C4⋊C8—42no—
371C4⋊C8C2×Q814yes—
372#102—24no—
373#102—44no—
374#102—42no—
375#102—42no—
376#102—44no—
377#102—44no—
378#102C2×Q818yes—
379#97—44no—
380C4×C8—24no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95—24no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4C22×Q814yes—
426C2×D4C22×Q812yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—81no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460D8—48no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8C22×Q814yes—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—81no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4C22×Q812yes—
509C4⋊C42−1+412yes—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C42—24no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—21no—
530C42—22no—
531C42—22no—
532C42—21no—
533C42—42no—
534C42—42no—
535C42—41no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42AGL(1,ℤ/8)14yes—
541C42C22×Q812yes—
542C42H(5,𝔽2)12yes—
543C42H(5,𝔽2)12yes—
544C422−1+412yes—
545C23—22no—
546C23—24no—
547C23—42no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—42no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23C23×D411yescharacteristic
560Q8—44no—
561Q8—88no—
562Q8—84no—
563Q8—164no—
564D4—22no—
565D4—24no—
566D4—22no—
567D4—24no—
568D4—48no—
569D4—48no—
570D4—41no—
571D4—42no—
572D4—41no—
573D4—42no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—84no—
618C2×C4—88no—
619C2×C4—84no—
620C2×C4—168no—
621C2×C4#53512yes—
622C2×C4#53512yes—
623C2×C4D4×Q812yes—
624C2×C4D4×Q812yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4#57512yes—
629C2×C4#57712yes—
630C2×C4C23×Q811yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C2×C4#58511yescharacteristic
636C8—24no—
637C8—42no—
638C8—44no—
639C22—42no—
640C22—44no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—42no—
645C22—82no—
646C22—81no—
647C22—81no—
648C22—82no—
649C22—82no—
650C22—81no—
651C22—81no—
652C22—82no—
653C22C2×#53511yescharacteristic
654C22C2×D4×Q811yescharacteristic
655C22#348911yescharacteristic
656C22C22×AGL(1,ℤ/8)11yescharacteristic
657C22C2×#57511yescharacteristic
658C22C2×#57711yescharacteristic
659C22#358511yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—82no—
671C4—81no—
672C4—82no—
673C4—82no—
674C4—84no—
675C4#321414yes—
676C4#338414yes—
677C4#350111yescharacteristic
678C4#350311yescharacteristic
679C4#355011yescharacteristic
680C4#355411yescharacteristic
681C2—82no—
682C2—84no—
683C2#3959611yescharacteristic
684C2C2×#321411yescharacteristic
685C2#6083311yescharacteristic
686C2C2×#338411yescharacteristic
687C2#6136711yescharacteristic
688C2#6211011yescharacteristic
689C2#6214511yescharacteristic
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. #34929C2semidirect
3. #33197C2semidirect
4. #32368C2semidirect
5. #31542C2semidirect
6. #29986C2semidirect
7. #29258C2semidirect
8. #29212C2semidirect
9. #28687C2semidirect
10. #28541C2semidirect
11. #28279C2semidirect
12. #27821C2semidirect
13. #25835C2semidirect
14. #24327C2semidirect
15. #22825C2semidirect
135. #459Q8semidirect

Maximal subgroups

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

subgroupindex
#365592
#349292
#331972
#323682
#315422
#299862
#292582
#292122
#286872
#285412
#282792
#278212
#258352
#243272
#228252
#221512
C4×#4592