Finite Group Databasebeta

← #6986635

Subgroups — #6986635

Order 512.

1856 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36473C211yesmaximal, characteristic, complemented
2#35317C211yesmaximal, characteristic, complemented
3#32834C212yesmaximal, complemented
4#32278C214yesmaximal, complemented
5#32046C214yesmaximal, complemented
6#30336C212yesmaximal, complemented
7#30059C212yesmaximal, complemented
8#29687C211yesmaximal, characteristic, complemented
9#29208C212yesmaximal, complemented
10#28527C212yesmaximal, complemented
11#27813C212yesmaximal, complemented
12#25813C211yesmaximal, characteristic, complemented
13#25449C212yesmaximal, complemented
14#24676C212yesmaximal, complemented
15#24298C211yesmaximal, characteristic, complemented
16#23567C211yesmaximal, characteristic
17#22639C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#536—21no—
20C2×#536—21no—
21C2×#532C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×#500C2212yes—
24C2×#474C2212yes—
25C2×#474C2212yes—
26C2×C4×D8C2212yes—
27#2868C2212yes—
28#2860C2212yes—
29#2858C2212yes—
30#2857C2211yescharacteristic
31#2658—22no—
32#2604—22no—
33#2551C2211yescharacteristic
34#2542—21no—
35#2542—21no—
36#2542C2212yes—
37#2542C2211yescharacteristic
38#2542C2212yes—
39#2521—22no—
40#2501—22no—
41#2501C2214yes—
42#2496C2212yes—
43#2492—21no—
44#2491C2212yes—
45#2415—21no—
46#2365C2211yescharacteristic
47#2362—21no—
48#2354C2211yescharacteristic
49C4×C4⋊Q8C2212yes—
50C4×#121C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2099C2214yes—
53#2093C2214yes—
54#2093C2214yes—
55#2083C2214yes—
56#2071C2214yes—
57#2067C2214yes—
58#2057C2211yescharacteristic
59#2057C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2057C2212yes—
63#2054C2214yes—
64#2041C2211yescharacteristic
65#2041C2212yes—
66#2033C2212yes—
67#2033C2214yes—
68#2023C2211yescharacteristic
69#2023C2214yes—
70#2023C2212yes—
71#2011C2211yescharacteristic
72#1999C2212yes—
73#1990C2214yes—
74#1990C2212yes—
75#1988C2212yes—
76#1970C2214yes—
77#1947C2214yes—
78#1947C2214yes—
79#1939C2212yes—
80#1934C2214yes—
81#1925C2214yes—
82#1925C2214yes—
83#1923C2214yes—
84#1874C2212yes—
85#1852C2212yes—
86#1852C2214yes—
87#1845C2214yes—
88#1839C2212yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41—21no—
96C23×Q8—21no—
97C22×D8C2312yes—
98#556—48no—
99#536—48no—
100#536—48no—
101#532—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—44no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127C2311yescharacteristic
114C2×#126—22no—
115C2×#126—22no—
116C2×#124—22no—
117C2×#124—22no—
118C2×#124—22no—
119C2×#124—21no—
120C2×#123—24no—
121C2×#123—24no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8—22no—
128C2×C22⋊Q8—21no—
129C2×#121—21no—
130C2×#121C2311yescharacteristic
131C2×#121C2312yes—
132C2×C4×Q8—21no—
133C2×C4×Q8—22no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4C2312yes—
137C2×C4×D4C2311yescharacteristic
138C2×#117—21no—
139C2×#117D411yescharacteristic
140C2×#117D411yescharacteristic
141C22×C4⋊C4—21no—
142C22×C4⋊C4C2312yes—
143C22×C42C2311yescharacteristic
144#500—44no—
145#474—44no—
146#474—44no—
147C4×D8—44no—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#107C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2314yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164C2×C4×C8C2312yes—
165C2×C4×C8C2314yes—
166#400—22no—
167#399—22no—
168#399—24no—
169#399—24no—
170#399—24no—
171#399—24no—
172#397—22no—
173#395—22no—
174#394—22no—
175#393—22no—
176#391—21no—
177#391—22no—
178#391—21no—
179#390—22no—
180#390—22no—
181#390C2312yes—
182#390C2314yes—
183#390C2314yes—
184#390C2311yescharacteristic
185#390C2312yes—
186#389—21no—
187#389—22no—
188#389—21no—
189#389C2311yescharacteristic
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#388—21no—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#387—22no—
200#387—22no—
201#386—21no—
202#384—22no—
203#384—22no—
204#384C2312yes—
205#384C2311yescharacteristic
206#384C2314yes—
207#384C2314yes—
208#380—21no—
209#380—21no—
210#380C2312yes—
211#380C2311yescharacteristic
212C4×C4⋊C4—22no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223#376D412yes—
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2314yes—
236#336C2314yes—
237C2×#41—44no—
238C2×#41—44no—
239C2×#41—44no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×D4—22no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C2412yes—
253C22×D4C2411yescharacteristic
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—22no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×D8—24no—
261C2×D8—42no—
262C2×D8—42no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—42no—
272C4⋊Q8—42no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—48no—
276C4⋊Q8—44no—
277C4⋊Q8—48no—
278#127—42no—
279#126—48no—
280#126—48no—
281#124—44no—
282#124—48no—
283#124—48no—
284#124—48no—
285#123—416no—
286#123—416no—
287C22⋊Q8—48no—
288C22⋊Q8—44no—
289C22⋊Q8—48no—
290C22⋊Q8—48no—
291C22⋊Q8—44no—
292C22⋊Q8—44no—
293C22⋊Q8—48no—
294#121—22no—
295#121—41no—
296#121—42no—
297#121—42no—
298#121—44no—
299C4×Q8—44no—
300C4×Q8—48no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—44no—
305C4×D4—44no—
306#117—24no—
307#117—44no—
308#117D818yescomplemented
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42C2×D412yes—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372#107—42no—
373#107—44no—
374#107—44no—
375#107—44no—
376#107—44no—
377C4⋊C8—28no—
378C4⋊C8—24no—
379#102—24no—
380#102—24no—
381#102—44no—
382#102—42no—
383#102—42no—
384#102—44no—
385#102—44no—
386C22⋊C8—24no—
387C22⋊C8—42no—
388C4×C8—24no—
389C4×C8—44no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403C24—21no—
404C24—22no—
405C24C22×D411yescharacteristic
406#41—816no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—44no—
420C2×Q8—42no—
421C2×Q8—82no—
422C2×Q8—816no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—81no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—41no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4C2×D814yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C2511yescharacteristic
481D8—48no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C82−1+414yes—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—81no—
520C4⋊C4C2×D814yes—
521C4⋊C4C2×D814yes—
522C4⋊C4AGL(1,ℤ/8)14yes—
523C4⋊C42−1+412yes—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—48no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—48no—
539C22⋊C4—42no—
540C22⋊C4—82no—
541C22⋊C4C2×D818yes—
542C42—22no—
543C42—22no—
544C42—24no—
545C42—24no—
546C42—22no—
547C42—24no—
548C42—22no—
549C42—22no—
550C42—42no—
551C42—44no—
552C42—42no—
553C42—44no—
554C42C2×D818yes—
555C42H(5,𝔽2)14yes—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—41no—
575C23—42no—
576C23C22×D812yes—
577C23C23×D411yescharacteristic
578Q8—88no—
579Q8—88no—
580Q8—84no—
581Q8—84no—
582Q8—88no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—84no—
590D4—84no—
591D4—84no—
592D4—88no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—28no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—88no—
637C2×C4—84no—
638C2×C4—88no—
639C2×C4#53512yes—
640C2×C4#53512yes—
641C2×C4C22×D814yes—
642C2×C4C22×D812yes—
643C2×C4C22×D814yes—
644C2×C4C22×D812yes—
645C2×C4C2×AGL(1,ℤ/8)14yes—
646C2×C4C2×AGL(1,ℤ/8)12yes—
647C2×C4#57812yes—
648C2×C4C2×H(5,𝔽2)12yes—
649C2×C4C2×2−1+412yes—
650C2×C4C2×2−1+411yescharacteristic
651C2×C4#58511yescharacteristic
652C8—24no—
653C8—44no—
654C8—42no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—81no—
663C22—82no—
664C22—81no—
665C22—82no—
666C22—82no—
667C22—81no—
668C22—82no—
669C22—81no—
670C22—81no—
671C22—82no—
672C22—81no—
673C22—82no—
674C22—81no—
675C22—82no—
676C22C2×#53511yescharacteristic
677C22#348911yescharacteristic
678C22#349911yescharacteristic
679C22C23×D811yescharacteristic
680C22C22×AGL(1,ℤ/8)11yescharacteristic
681C22C2×#57811yescharacteristic
682C22#358611yescharacteristic
683C4—24no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—24no—
688C4—24no—
689C4—42no—
690C4—42no—
691C4—41no—
692C4—81no—
693C4—81no—
694C4—82no—
695C4—84no—
696C4—82no—
697C4—82no—
698C4#320514yes—
699C4#354112yes—
700C4#355012yes—
701C2—42no—
702C2—162no—
703C2—161no—
704C2#4291111yescharacteristic
705C2C2×#320511yescharacteristic
706C2#6081411yescharacteristic
707C2#6209511yescharacteristic
708C2#6213211yescharacteristic
709C2#6247111yescharacteristic
710C2#6250911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23567C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36473C2semidirect
2. #35317C2semidirect
3. #32834C2semidirect
4. #32278C2semidirect
5. #32046C2semidirect
6. #30336C2semidirect
7. #30059C2semidirect
8. #29687C2semidirect
9. #29208C2semidirect
10. #28527C2semidirect
11. #27813C2semidirect
12. #25813C2semidirect
13. #25449C2semidirect
14. #24676C2semidirect
15. #24298C2semidirect
17. #22639C2semidirect
308. #117D8semidirect

Maximal subgroups

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

subgroupindex
#364732
#353172
#328342
#322782
#320462
#303362
#300592
#296872
#292082
#285272
#278132
#258132
#254492
#246762
#242982
#235672
#226392