Finite Group Databasebeta

← #6484438

Subgroups — #6484438

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic, complemented
2#34275C211yesmaximal, characteristic
3#32965C212yesmaximal, complemented
4#32962C212yesmaximal, complemented
5#29954C212yesmaximal, complemented
6#29952C212yesmaximal, complemented
7#29260C212yesmaximal, complemented
8#29258C212yesmaximal, complemented
9#29204C212yesmaximal, complemented
10#29199C212yesmaximal, complemented
11#29145C211yesmaximal, characteristic, complemented
12#29142C211yesmaximal, characteristic, complemented
13#28520C211yesmaximal, characteristic, complemented
14#28509C211yesmaximal, characteristic, complemented
15#28290C212yesmaximal, complemented
16#28289C212yesmaximal, complemented
17#21990C212yesmaximal, complemented
18#21813C212yesmaximal
19#20813C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C2×#475C2212yescomplemented
24C2×#476C2212yes—
25C2×#476C2212yescomplemented
26C2×#474C2212yes—
27C2×#433C2212yes—
28#2868C2211yescharacteristic, complemented
29#2862C2211yescharacteristic, complemented
30#2857C2212yes—
31#2647C2212yescomplemented
32#2628C2212yes—
33#2511—22no—
34#2492—22no—
35#2451—22no—
36#2423—22no—
37#2363—22no—
38#2361—22no—
39#2358—22no—
40#2358—22no—
41#2331C2212yescomplemented
42#2328C2211yescharacteristic
43#2328C2211yescharacteristic
44#2326C2211yescharacteristic
45#2326C2211yescharacteristic
46#2317C2211yescharacteristic
47#2314C2212yes—
48C4×C4⋊Q8C2212yescomplemented
49C4×#127C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C4×#124C2211yescharacteristic
52C4×#124C2211yescharacteristic
53#2297C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C4×#117C2211yescharacteristic
56#2095C2212yescomplemented
57#2094C2212yes—
58#2094C2212yes—
59#2094C2212yescomplemented
60#2067C2214yescomplemented
61#2066C2214yescomplemented
62#2065C2214yes—
63#2065C2214yes—
64#2024C2212yescomplemented
65#2023C2212yes—
66#2023C2212yes—
67#2023C2212yescomplemented
68#1983C2214yes—
69#1982C2214yes—
70#1961C2212yes—
71#1959C2212yes—
72#1958C2212yes—
73#1957C2212yes—
74#1949C2214yescomplemented
75#1948C2214yescomplemented
76#1865C2214yescomplemented
77#1864C2214yes—
78#1864C2214yescomplemented
79#1862C2214yes—
80#1854C2214yes—
81#1853C2214yes—
82#1847C2212yescomplemented
83#1844C2212yescomplemented
84#1843C2212yescomplemented
85#1836C2212yescomplemented
86#1835C2212yescomplemented
87#1831C2212yescomplemented
88#1805C2212yescomplemented
89#1804C2212yes—
90#1804C2212yes—
91#1804C2212yescomplemented
92#1798C2212yes—
93#1797C2212yes—
94#1788C2214yes—
95#1786C2214yes—
96#1768C2212yescomplemented
97#1767C2211yescharacteristic, complemented
98C4×C4⋊C8C2211yescharacteristic, complemented
99C22×#41C2311yescharacteristic
100C23×D4C2311yescharacteristic
101#535—24no—
102#535Q818yescomplemented
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#124C2311yescharacteristic
109C2×#124C2311yescharacteristic
110C2×C22⋊Q8—22no—
111C2×#121—22no—
112C2×#120—22no—
113#520—44no—
114#518—44no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117—21no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—22no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C42C2311yescharacteristic
140#476—44no—
141#476—44no—
142#475—44no—
143#474—44no—
144#433—28no—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×C22⋊C8C2314yes—
162C2×#97C2312yes—
163C2×C4×C8C2312yes—
164#400—22no—
165#400—24no—
166#399—22no—
167#398—24no—
168#395—22no—
169#393—22no—
170#391—22no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391—24no—
175#391C2311yescharacteristic
176#391C2311yescharacteristic
177#390—22no—
178#390—22no—
179#390—22no—
180#390C2311yescharacteristic
181#390C2311yescharacteristic
182#390C2311yescharacteristic
183#390C2311yescharacteristic
184#390C2311yescharacteristic
185#390C2311yescharacteristic
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2314yes—
190#389C2314yes—
191#388—24no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#387—24no—
197#387—24no—
198#386—22no—
199#386—22no—
200#386—22no—
201#386—22no—
202#386—22no—
203#385—22no—
204#385—22no—
205#385—22no—
206#384—22no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2312yes—
210#384C2312yes—
211#382C2312yes—
212#382C2312yes—
213#382C2311yescharacteristic
214#382C2311yescharacteristic
215#380—22no—
216#380C2314yes—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D412yes—
220C4×C4⋊C4C2314yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226#376C2311yescharacteristic
227#376C2311yescharacteristic
228C43C2311yescharacteristic
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C25—21no—
238C2×#41—22no—
239C2×#41—42no—
240C2×#41—42no—
241C2×#41C2×Q814yes—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8C2411yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—42no—
251C22×D4—42no—
252C22×D4C2×Q814yes—
253C22×D4C2411yescharacteristic
254C22×D4C2411yescharacteristic
255C22×D4C2411yescharacteristic
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C22×C8C2414yes—
267C22×C8C2414yes—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—84no—
272#127—22no—
273#127—22no—
274#127C2×Q814yes—
275#127C2×Q814yes—
276#124—22no—
277#124—22no—
278#124C2×Q814yes—
279#124C2×Q814yes—
280C22⋊Q8—84no—
281#121—216no—
282#120—216no—
283C4×Q8—24no—
284C4×Q8—42no—
285C4×Q8—84no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—42no—
290C4×D4—42no—
291C4×D4—42no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—84no—
297C4×D4—84no—
298#117—24no—
299#117—42no—
300#117—44no—
301#117C2×Q814yes—
302#117#4114yes—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2414yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—84no—
345C2×C22⋊C4C2×D411yescharacteristic
346C2×C22⋊C4C2×D411yescharacteristic
347C2×C22⋊C4C2×D412yes—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—24no—
352C2×C42—42no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—44no—
365#106—44no—
366#106—44no—
367#106—44no—
368C4⋊C8—28no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371C4⋊C8—44no—
372#103—44no—
373#103—44no—
374#102—44no—
375#102—44no—
376#102—44no—
377#102—44no—
378#102—44no—
379#102—44no—
380C22⋊C8—28no—
381#97—24no—
382C4×C8—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—24no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—21no—
399C24—21no—
400C24—21no—
401C24—24no—
402C24—42no—
403C24—42no—
404C24—82no—
405C24—82no—
406C24C22×D411yescharacteristic
407#41—48no—
408#41—48no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—84no—
417C2×Q8C22×Q812yes—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—48no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—48no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4C22×Q812yes—
442C2×D4C22×Q812yes—
443C2×D4C22×Q812yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×Q812yes—
483C22×C4C2×#4112yes—
484C22×C4C2511yescharacteristic
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4C22⋊Q818yes—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—84no—
532C22⋊C4C22⋊Q818yes—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—42no—
539C42—42no—
540C42—42no—
541C42—42no—
542C42—42no—
543C42—42no—
544C42—84no—
545C42C22×Q812yes—
546C42C22×Q812yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C42H(5,𝔽2)14yes—
550C23—22no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—48no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—88no—
571C23—164no—
572C23—162no—
573C23—162no—
574C23C2×C22⋊Q812yes—
575C23C23×D411yescharacteristic
576Q8—24no—
577Q8—42no—
578Q8—88no—
579Q8—164no—
580D4—24no—
581D4—24no—
582D4—24no—
583D4—48no—
584D4—48no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—88no—
589D4—88no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—48no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—84no—
628C2×C4—162no—
629C2×C4—162no—
630C2×C4—162no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4—162no—
634C2×C4—162no—
635C2×C4C2×C22⋊Q814yes—
636C2×C4C2×C22⋊Q814yes—
637C2×C4C2×C22⋊Q812yes—
638C2×C4#53512yes—
639C2×C4#53512yes—
640C2×C4#57714yes—
641C2×C4C23×Q811yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×H(5,𝔽2)12yes—
644C2×C4#58512yes—
645C8—44no—
646C8—44no—
647C22—24no—
648C22—22no—
649C22—42no—
650C22—41no—
651C22—41no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—168no—
668C22C22×C22⋊Q811yescharacteristic
669C22C2×#53511yescharacteristic
670C22#344911yescharacteristic
671C22C2×#57712yes—
672C22#358512yes—
673C4—24no—
674C4—24no—
675C4—24no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—42no—
680C4—42no—
681C4—41no—
682C4—41no—
683C4—84no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4—84no—
689C4#350112yes—
690C4#350612yes—
691C2—42no—
692C2—82no—
693C2—82no—
694C2—82no—
695C2#3517211yescharacteristic
696C2#6049512yes—
697C2#6081811yescharacteristic
698C2#6082511yescharacteristic
699C2#6098012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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. #35283C2semidirect
3. #32965C2semidirect
4. #32962C2semidirect
5. #29954C2semidirect
6. #29952C2semidirect
7. #29260C2semidirect
8. #29258C2semidirect
9. #29204C2semidirect
10. #29199C2semidirect
11. #29145C2semidirect
12. #29142C2semidirect
13. #28520C2semidirect
14. #28509C2semidirect
15. #28290C2semidirect
16. #28289C2semidirect
17. #21990C2semidirect
19. #20813C2semidirect
23. C2×#475C22semidirect
25. C2×#476C22semidirect
28. #2868C22semidirect
29. #2862C22semidirect
31. #2647C22semidirect
41. #2331C22semidirect
48. C4×C4⋊Q8C22semidirect
56. #2095C22semidirect
59. #2094C22semidirect
60. #2067C22semidirect
61. #2066C22semidirect
64. #2024C22semidirect
67. #2023C22semidirect
74. #1949C22semidirect
75. #1948C22semidirect
76. #1865C22semidirect
78. #1864C22semidirect
82. #1847C22semidirect
83. #1844C22semidirect
84. #1843C22semidirect
85. #1836C22semidirect
86. #1835C22semidirect
87. #1831C22semidirect
88. #1805C22semidirect
91. #1804C22semidirect
96. #1768C22semidirect
97. #1767C22semidirect
98. C4×C4⋊C8C22semidirect
102. #535Q8semidirect

Maximal subgroups

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

subgroupindex
#352832
#342752
#329652
#329622
#299542
#299522
#292602
#292582
#292042
#291992
#291452
#291422
#285202
#285092
#282902
#282892
#219902
#218132
#208132