Finite Group Databasebeta

← #6476743

Subgroups — #6476743

Order 512.

2001 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic, complemented
2#34269C211yesmaximal, characteristic
3#33119C212yesmaximal, complemented
4#32914C212yesmaximal, complemented
5#30113C212yesmaximal, complemented
6#30106C212yesmaximal, complemented
7#29338C211yesmaximal, characteristic, complemented
8#29332C211yesmaximal, characteristic, complemented
9#28311C211yesmaximal, characteristic, complemented
10#28302C211yesmaximal, characteristic, complemented
11#27683C212yesmaximal, complemented
12#24666C214yesmaximal, complemented
13#24209C212yesmaximal, complemented
14#24014C212yesmaximal, complemented
15#23926C212yesmaximal
16#22058C214yesmaximal, complemented
17#20720C211yesmaximal, characteristic, complemented
18C2×#536C2212yes—
19C2×#535C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic, complemented
21C2×#518C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#484C2212yes—
24C2×#483C2212yes—
25C2×#461C2214yes—
26C2×#448C2214yes—
27C2×#450C2212yes—
28C2×#450C2212yes—
29C2×#433C2212yescomplemented
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic, complemented
33#2857C2211yescharacteristic, complemented
34#2621C2212yes—
35#2621C2212yescomplemented
36#2547—22no—
37#2546—22no—
38#2486—22no—
39#2486—22no—
40#2367—22no—
41#2366—22no—
42#2358—22no—
43#2358—22no—
44#2324C2212yes—
45#2324C2212yes—
46#2311C2211yescharacteristic
47#2310C2212yescomplemented
48C4×#121C2212yes—
49C4×#121C2212yes—
50#2296C2211yescharacteristic
51C2×#390C2211yescharacteristic
52#2285C2211yescharacteristic, complemented
53#2274C2211yescharacteristic
54#2100C2212yescomplemented
55#2099C2212yescomplemented
56#2094C2212yes—
57#2093C2212yes—
58#2043C2212yescomplemented
59#2043C2212yescomplemented
60#2036C2214yescomplemented
61#2035C2214yescomplemented
62#2013C2212yes—
63#2013C2212yes—
64#2000C2214yes—
65#2000C2214yes—
66#1981C2214yescomplemented
67#1973C2214yes—
68#1968C2214yes—
69#1949C2214yescomplemented
70#1935C2214yescomplemented
71#1933C2214yescomplemented
72#1928C2214yescomplemented
73#1927C2214yescomplemented
74#1877C2214yes—
75#1877C2214yes—
76#1875C2214yes—
77#1873C2212yes—
78#1873C2212yes—
79#1873C2212yes—
80#1873C2212yes—
81#1854C2214yescomplemented
82#1817C2212yescomplemented
83#1804C2214yes—
84#1798C2212yes—
85#1797C2212yescomplemented
86#1787C2214yescomplemented
87#1780C2212yes—
88#1750C2212yescomplemented
89#1749C2211yescharacteristic
90C4×C22⋊C8C2211yescharacteristic
91C22×#41C2311yescharacteristic
92C23×Q8C2312yes—
93C23×D4C2311yescharacteristic
94C22×SD16C2314yes—
95C22×SD16C2314yes—
96#536—44no—
97#536—44no—
98#535—28no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127—21no—
106C2×#124—21no—
107C2×#124—22no—
108C2×#123—22no—
109C2×#123—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×#121C2312yes—
115C2×#121C2312yes—
116C2×#120—21no—
117C2×#120—21no—
118#518—44no—
119C4×#41—44no—
120C2×C4×Q8D412yes—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4D412yes—
125C2×C4×D4D412yes—
126C2×C4×D4D412yes—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×#117C2311yescharacteristic
130C2×#117C2312yes—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—22no—
138C22×C22⋊C4—22no—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#484—44no—
143#483—44no—
144#461—48no—
145#450—44no—
146#450—44no—
147#448—216no—
148#433—44no—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2314yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2314yes—
164C2×#97C2312yes—
165C2×C4×C8C2312yes—
166#398—24no—
167#398—24no—
168#397—24no—
169#397—24no—
170#391—21no—
171#391—24no—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#390—21no—
177#390—22no—
178#390—22no—
179#390—22no—
180#390C2312yes—
181#390C2312yes—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#386—21no—
193#386—21no—
194#386—22no—
195#386—22no—
196#386—22no—
197#386—22no—
198#385—22no—
199#385—22no—
200#385—22no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#382C2312yes—
204#381—24no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380C2312yes—
211#380C2312yes—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#336C2312yes—
225#336C2312yes—
226#336C2314yes—
227C25—21no—
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—42no—
231C22×Q8—22no—
232C22×Q8—28no—
233C22×Q8—44no—
234C22×Q8—42no—
235C22×Q8—42no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2412yes—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—24no—
244C22×D4—42no—
245C22×D4C2×D411yescharacteristic
246C22×D4C2×D411yescharacteristic
247C22×D4C2×D411yescharacteristic
248C22×D4C2412yes—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×SD16—28no—
260C2×SD16—28no—
261C2×SD16—44no—
262C2×SD16—44no—
263C2×SD16—44no—
264C2×SD16—44no—
265C22×C8C2414yes—
266C22×C8C2414yes—
267C4⋊Q8—28no—
268C4⋊Q8—42no—
269C4⋊Q8—42no—
270C4⋊Q8—48no—
271C4⋊Q8—48no—
272#127—28no—
273#124—28no—
274#124—48no—
275#123—48no—
276#123—48no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—42no—
282C22⋊Q8—42no—
283C22⋊Q8—42no—
284C22⋊Q8—42no—
285#121—24no—
286#121—24no—
287#121—24no—
288#121#4118yes—
289#120—28no—
290#120—28no—
291C4×Q8—44no—
292C4×D4—28no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—48no—
298C4×D4—84no—
299C4×D4—84no—
300#117—24no—
301#117—24no—
302#117—24no—
303#117—42no—
304#117—42no—
305#117—42no—
306#117—42no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4—84no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—21no—
352C2×C42—24no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—42no—
358C2×C42—42no—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—44no—
367#106—44no—
368C4⋊C8—28no—
369#103—28no—
370#103—28no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375#102—44no—
376#102—44no—
377C22⋊C8—28no—
378C22⋊C8—24no—
379C22⋊C8—24no—
380C22⋊C8—44no—
381#97—42no—
382C4×C8—42no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—22no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394C24—24no—
395C24—22no—
396C24—42no—
397C24—42no—
398C24—82no—
399C24—82no—
400C24C22×D411yescharacteristic
401C24C22×D411yescharacteristic
402C24C22×D411yescharacteristic
403#41—416no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—41no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—88no—
419C2×Q8—88no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—28no—
426C2×D4—28no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—416no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4C2×#4114yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C2×#4114yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C4C2511yescharacteristic
487SD16—88no—
488SD16—88no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—82no—
517C4⋊C4C2×#4114yes—
518C22⋊C4—24no—
519C22⋊C4—28no—
520C22⋊C4—28no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—44no—
547C42—82no—
548C42—82no—
549C42H(5,𝔽2)14yes—
550C23—22no—
551C23—22no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—22no—
564C23—48no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—88no—
572C23—162no—
573C23—162no—
574C23—164no—
575C23#52512yes—
576C23#53512yes—
577C23C23×D411yescharacteristic
578Q8—44no—
579Q8—44no—
580Q8—44no—
581Q8—44no—
582Q8—44no—
583Q8—88no—
584D4—28no—
585D4—44no—
586D4—44no—
587D4—48no—
588D4—44no—
589D4—44no—
590D4—88no—
591D4—88no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4—164no—
636C2×C4—164no—
637C2×C4—162no—
638C2×C4#52512yes—
639C2×C4#53414yes—
640C2×C4#53512yes—
641C2×C4#57714yes—
642C2×C4C22×#4112yes—
643C2×C4C2×H(5,𝔽2)12yes—
644C2×C4C2×H(5,𝔽2)12yes—
645C8—44no—
646C8—44no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—24no—
652C22—42no—
653C22—42no—
654C22—41no—
655C22—41no—
656C22—42no—
657C22—41no—
658C22—44no—
659C22—42no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22—44no—
664C22—44no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—168no—
670C22C2×#52511yescharacteristic
671C22C2×#53411yescharacteristic
672C22C2×#53511yescharacteristic
673C22C2×#57712yes—
674C22#358612yes—
675C4—22no—
676C4—22no—
677C4—24no—
678C4—24no—
679C4—41no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#351914yes—
690C2—42no—
691C2—42no—
692C2—81no—
693C2—84no—
694C2#3507611yescharacteristic
695C2#6059112yes—
696C2#6075812yes—
697C2#6082011yescharacteristic
698C2#6082811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35316C2semidirect
3. #33119C2semidirect
4. #32914C2semidirect
5. #30113C2semidirect
6. #30106C2semidirect
7. #29338C2semidirect
8. #29332C2semidirect
9. #28311C2semidirect
10. #28302C2semidirect
11. #27683C2semidirect
12. #24666C2semidirect
13. #24209C2semidirect
14. #24014C2semidirect
16. #22058C2semidirect
17. #20720C2semidirect
20. C22×C4⋊Q8C22semidirect
29. C2×#433C22semidirect
32. #2857C22semidirect
33. #2857C22semidirect
35. #2621C22semidirect
47. #2310C22semidirect
52. #2285C22semidirect
54. #2100C22semidirect
55. #2099C22semidirect
58. #2043C22semidirect
59. #2043C22semidirect
60. #2036C22semidirect
61. #2035C22semidirect
66. #1981C22semidirect
69. #1949C22semidirect
70. #1935C22semidirect
71. #1933C22semidirect
72. #1928C22semidirect
73. #1927C22semidirect
81. #1854C22semidirect
82. #1817C22semidirect
85. #1797C22semidirect
86. #1787C22semidirect
88. #1750C22semidirect

Maximal subgroups

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

subgroupindex
#353162
#342692
#331192
#329142
#301132
#301062
#293382
#293322
#283112
#283022
#276832
#246662
#242092
#240142
#239262
#220582
#207202