Finite Group Databasebeta

← #6993539

Subgroups — #6993539

Order 512.

1861 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#36272C211yesmaximal, characteristic, complemented
2#35317C211yesmaximal, characteristic, complemented
3#32818C212yesmaximal, complemented
4#32278C214yesmaximal, complemented
5#31930C214yesmaximal, complemented
6#30304C212yesmaximal, complemented
7#29980C212yesmaximal, complemented
8#29687C211yesmaximal, characteristic, complemented
9#29552C212yesmaximal, complemented
10#28527C212yesmaximal, complemented
11#27727C212yesmaximal, complemented
12#25813C211yesmaximal, characteristic, complemented
13#25449C212yesmaximal, complemented
14#24811C212yesmaximal, complemented
15#24298C211yesmaximal, characteristic, complemented
16#23567C211yesmaximal, characteristic
17#22632C211yesmaximal, characteristic, complemented
18C2×#552—21no—
19C2×#536—21no—
20C2×#536—21no—
21C2×#532C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×#500C2212yes—
24C2×#474C2212yes—
25C2×#474C2212yes—
26C2×C4×D8C2212yes—
27#2868C2212yes—
28#2859C2212yes—
29#2858C2212yes—
30#2857C2211yescharacteristic
31#2658—22no—
32#2611—22no—
33#2553—22no—
34#2552C2211yescharacteristic
35#2542—21no—
36#2542—21no—
37#2542C2212yes—
38#2542C2212yes—
39#2542C2211yescharacteristic
40#2501—22no—
41#2501C2214yes—
42#2496C2212yes—
43#2489—21no—
44#2422C2212yes—
45#2415—21no—
46#2365C2211yescharacteristic
47#2360—21no—
48D4×C4⋊C4C2211yescharacteristic
49C4×#121C2211yescharacteristic
50#2293C2212yes—
51C4×#117C2211yescharacteristic
52#2099C2214yes—
53#2093C2214yes—
54#2093C2214yes—
55#2083C2214yes—
56#2083C2214yes—
57#2071C2214yes—
58#2067C2214yes—
59#2067C2214yes—
60#2057C2211yescharacteristic
61#2057C2212yes—
62#2057C2212yes—
63#2057C2212yes—
64#2057C2212yes—
65#2041C2211yescharacteristic
66#2041C2212yes—
67#2033C2212yes—
68#2033C2214yes—
69#2023C2211yescharacteristic
70#2023C2212yes—
71#2011C2211yescharacteristic
72#1999C2212yes—
73#1990C2212yes—
74#1988C2212yes—
75#1970C2214yes—
76#1947C2214yes—
77#1939C2212yes—
78#1934C2214yes—
79#1925C2214yes—
80#1923C2214yes—
81#1879C2214yes—
82#1874C2212yes—
83#1852C2212yes—
84#1852C2214yes—
85#1845C2214yes—
86#1834C2214yes—
87#1831C2212yes—
88#1802C2214yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C22×#41—21no—
96C23×Q8—21no—
97C22×D8C2312yes—
98#552—48no—
99#536—48no—
100#536—48no—
101#532—44no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—44no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×#127C2311yescharacteristic
112C2×#126—22no—
113C2×#125—21no—
114C2×#125—22no—
115C2×#124—22no—
116C2×#124—22no—
117C2×#124—21no—
118C2×#123—22no—
119C2×#123—24no—
120C2×#123—24no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8—21no—
128C2×#121—21no—
129C2×#121C2311yescharacteristic
130C2×#121C2312yes—
131C2×C4×Q8—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4C2312yes—
135C2×C4×D4C2311yescharacteristic
136C2×C4×D4C2312yes—
137C2×#117—22no—
138C2×#117D411yescharacteristic
139C2×#117D411yescharacteristic
140C22×C4⋊C4—21no—
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×#107C2312yes—
151C2×#107C2314yes—
152C2×#107C2314yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2314yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164C2×C4×C8C2314yes—
165C2×C4×C8C2312yes—
166#400—22no—
167#399—22no—
168#399—24no—
169#399—24no—
170#399—24no—
171#399—24no—
172#398—22no—
173#397—22no—
174#397—22no—
175#394—22no—
176#391—21no—
177#391—21no—
178#390—22no—
179#390—22no—
180#390C2314yes—
181#390C2314yes—
182#390C2312yes—
183#390C2311yescharacteristic
184#389—21no—
185#389—21no—
186#389—22no—
187#389—22no—
188#389C2312yes—
189#389C2311yescharacteristic
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#388—21no—
194#388—22no—
195#388—22no—
196#387—22no—
197#387—22no—
198#386—21no—
199#385—22no—
200#385—22no—
201#384—22no—
202#384—22no—
203#384C2311yescharacteristic
204#384C2314yes—
205#384C2312yes—
206#384C2314yes—
207#384C2312yes—
208#382—22no—
209#380—21no—
210#380—21no—
211#380C2312yes—
212#380C2311yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223#376D412yes—
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
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—44no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2412yes—
252C22×D4C2411yescharacteristic
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×D8—24no—
261C2×D8—24no—
262C2×D8—42no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—44no—
270C4⋊Q8—42no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—48no—
274C4⋊Q8—48no—
275#127—42no—
276#126—48no—
277#125—44no—
278#125—48no—
279#124—44no—
280#124—48no—
281#124—48no—
282#123—416no—
283#123—416no—
284#123—48no—
285C22⋊Q8—48no—
286C22⋊Q8—44no—
287C22⋊Q8—48no—
288C22⋊Q8—48no—
289C22⋊Q8—44no—
290C22⋊Q8—44no—
291C22⋊Q8—48no—
292#121—22no—
293#121—41no—
294#121—42no—
295#121—42no—
296#121—44no—
297C4×Q8—44no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—42no—
303C4×D4—44no—
304C4×D4—44no—
305#117—24no—
306#117—48no—
307#117D818yescomplemented
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
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—22no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42C2×D412yes—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372#107—42no—
373#107—44no—
374#107—44no—
375#107—44no—
376#107—44no—
377C4⋊C8—24no—
378C4⋊C8—44no—
379#102—24no—
380#102—44no—
381#102—42no—
382#102—42no—
383#102—42no—
384#102—44no—
385#102—44no—
386C22⋊C8—24no—
387C22⋊C8—42no—
388C4×C8—28no—
389C4×C8—24no—
390#95—22no—
391#95—21no—
392#95—21no—
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—42no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—816no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—81no—
441C2×D42−1+412yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—41no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—82no—
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—24no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C82−1+414yes—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—82no—
522C4⋊C4—81no—
523C4⋊C4C2×D814yes—
524C4⋊C4C2×D814yes—
525C4⋊C4AGL(1,ℤ/8)14yes—
526C22⋊C4—24no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—48no—
531C22⋊C4—42no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—42no—
536C22⋊C4—44no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—44no—
540C22⋊C4—48no—
541C22⋊C4—42no—
542C22⋊C4C2×D818yes—
543C42—22no—
544C42—22no—
545C42—24no—
546C42—24no—
547C42—22no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—42no—
552C42—44no—
553C42—42no—
554C42C2×D818yes—
555C42H(5,𝔽2)14yes—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23C22×D812yes—
577C23C23×D411yescharacteristic
578Q8—88no—
579Q8—84no—
580Q8—84no—
581Q8—88no—
582D4—24no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—84no—
589D4—84no—
590D4—84no—
591D4—88no—
592C2×C4—21no—
593C2×C4—22no—
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—21no—
604C2×C4—28no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—88no—
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+411yescharacteristic
650C2×C4C2×2−1+412yes—
651C2×C4#58511yescharacteristic
652C8—24no—
653C8—42no—
654C8—44no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22C2×#53511yescharacteristic
677C22#348911yescharacteristic
678C22#349911yescharacteristic
679C22C23×D811yescharacteristic
680C22C22×AGL(1,ℤ/8)11yescharacteristic
681C22C2×#57811yescharacteristic
682C22#358611yescharacteristic
683C4—24no—
684C4—24no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—24no—
689C4—42no—
690C4—41no—
691C4—81no—
692C4—81no—
693C4—82no—
694C4—84no—
695C4—82no—
696C4—82no—
697C4—81no—
698C4#320514yes—
699C4#353812yes—
700C4#354712yes—
701C2—42no—
702C2—82no—
703C2—162no—
704C2#4299411yescharacteristic
705C2C2×#320511yescharacteristic
706C2#6081411yescharacteristic
707C2#6209511yescharacteristic
708C2#6214811yescharacteristic
709C2#6247111yescharacteristic
710C2#6250311yescharacteristic
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. #36272C2semidirect
2. #35317C2semidirect
3. #32818C2semidirect
4. #32278C2semidirect
5. #31930C2semidirect
6. #30304C2semidirect
7. #29980C2semidirect
8. #29687C2semidirect
9. #29552C2semidirect
10. #28527C2semidirect
11. #27727C2semidirect
12. #25813C2semidirect
13. #25449C2semidirect
14. #24811C2semidirect
15. #24298C2semidirect
17. #22632C2semidirect
307. #117D8semidirect

Maximal subgroups

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

subgroupindex
#362722
#353172
#328182
#322782
#319302
#303042
#299802
#296872
#295522
#285272
#277272
#258132
#254492
#248112
#242982
#235672
#226322