Finite Group Databasebeta

← #6434286

Subgroups — #6434286

Order 512.

1755 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35312C211yesmaximal, characteristic, complemented
2#35266C211yesmaximal, characteristic, complemented
3#30355C214yesmaximal, complemented
4#30348C214yesmaximal, complemented
5#30346C214yesmaximal, complemented
6#30345C214yesmaximal, complemented
7#23615C211yesmaximal, characteristic, complemented
8#23611C211yesmaximal, characteristic, complemented
9#23555C211yesmaximal, characteristic, complemented
10#23544C211yesmaximal, characteristic
11#23541C211yesmaximal, characteristic, complemented
12#23510C211yesmaximal, characteristic, complemented
13#23456C211yesmaximal, characteristic, complemented
14#23424C211yesmaximal, characteristic
15#21236C211yesmaximal, characteristic
16C2×#1973C211yesmaximal, characteristic, complemented
17#21094C211yesmaximal, characteristic, complemented
18#21086C211yesmaximal, characteristic, complemented
19#20427C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C2×#536C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×C4×M16C2211yescharacteristic
37C22×#97C2211yescharacteristic
38#2862C2212yescomplemented
39#2861C2212yescomplemented
40#2860C2214yes—
41#2644C2214yes—
42#2642C2212yescomplemented
43#2640C2212yescomplemented
44#2513—22no—
45#2512—22no—
46#2492—22no—
47#2489—22no—
48#2488—24no—
49#2480—24no—
50#2317C2211yescharacteristic
51#2316C2211yescharacteristic
52#2311C2211yescharacteristic
53#2310C2211yescharacteristic
54#2309C2211yescharacteristic
55#2309C2211yescharacteristic
56#2297C2211yescharacteristic
57#2296C2211yescharacteristic
58C2×#391C2211yescharacteristic
59C2×#390C2211yescharacteristic
60C4×#117C2211yescharacteristic
61#2068C2214yescomplemented
62#2068C2214yescomplemented
63#2067C2214yescomplemented
64#2066C2214yescomplemented
65#2059C2214yescomplemented
66#2057C2214yescomplemented
67#2055C2214yes—
68#2055C2214yes—
69#2055C2214yes—
70#2054C2214yes—
71#2043C2214yescomplemented
72#2042C2214yescomplemented
73#2025C2214yes—
74#2023C2214yes—
75#2013C2214yes—
76#2012C2214yes—
77#1973C2212yes—
78#1973C2214yes—
79#1973C2214yes—
80#1972C2212yes—
81#1970C2212yes—
82#1968C2212yes—
83#1936C2212yescomplemented
84#1936C2212yescomplemented
85#1935C2212yescomplemented
86#1935C2212yescomplemented
87#1935C2214yescomplemented
88#1933C2214yescomplemented
89#1768C2214yescomplemented
90#1767C2214yescomplemented
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C2×AGL(1,ℤ/8)—24no—
101C2×AGL(1,ℤ/8)—42no—
102C2×AGL(1,ℤ/8)—42no—
103C22×SD16—24no—
104C22×SD16—42no—
105C22×SD16—42no—
106C22×SD16C2312yes—
107C22×SD16C2314yes—
108C22×D8C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#544—42no—
114#544—42no—
115#536—42no—
116#536—42no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2312yes—
120C2×#126—24no—
121C2×#125—21no—
122C2×#125C2312yes—
123C2×#124C2312yes—
124C2×#123—22no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—22no—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137#428—42no—
138C2×#107—42no—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×#106C2312yes—
142#424—24no—
143#424—24no—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2314yes—
146#419—24no—
147#419—42no—
148#418—42no—
149#417—42no—
150#416—24no—
151#416—42no—
152C2×#103—42no—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#102—42no—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C4×M16—42no—
167C2×#97—42no—
168C2×#97C2312yes—
169C2×#97C2312yes—
170C2×#97C2312yes—
171C2×C4×C8C2312yes—
172#400—24no—
173#400—24no—
174#397—24no—
175#397—24no—
176#397—24no—
177#397—24no—
178#396—24no—
179#393—24no—
180#391—22no—
181#391—24no—
182#391—22no—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#391C2311yescharacteristic
188#391C2312yes—
189#391C2311yescharacteristic
190#390—21no—
191#390—22no—
192#390—21no—
193#390—22no—
194#390C2314yes—
195#390C2314yes—
196#390C2312yes—
197#390C2312yes—
198#390C2311yescharacteristic
199#390C2311yescharacteristic
200#390C2312yes—
201#389—22no—
202#389C2314yes—
203#389C2314yes—
204#389C2312yes—
205#389C2312yes—
206#388—24no—
207#388—24no—
208#388—24no—
209#388—24no—
210#388—24no—
211#386—22no—
212#386—22no—
213#385—22no—
214#385—22no—
215#384—21no—
216#382—21no—
217#382—24no—
218#381—22no—
219#380—22no—
220#380—22no—
221#379—22no—
222#379—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4D412yes—
225C4×C4⋊C4D412yes—
226C4×C22⋊C4—22no—
227C4×C22⋊C4D414yes—
228#376C2312yes—
229C2×#95—22no—
230C43C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2314yes—
236#340C2312yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8C2412yes—
255C22×Q8C2411yescharacteristic
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4C2411yescharacteristic
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4C2×D412yes—
271C23×C4C2411yescharacteristic
272AGL(1,ℤ/8)—416no—
273C2×SD16—28no—
274C2×SD16—24no—
275C2×SD16—44no—
276C2×SD16—44no—
277C2×SD16—42no—
278C2×SD16—42no—
279C2×SD16—416no—
280C2×D8—24no—
281C2×D8—42no—
282C2×D8—42no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16#4114yes—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8#4114yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—42no—
300C4⋊Q8—42no—
301C4⋊Q8—42no—
302C4⋊Q8—42no—
303C4⋊Q8—44no—
304#126—88no—
305#125—42no—
306#125—42no—
307#125—44no—
308#124—42no—
309#124—42no—
310#123—48no—
311C22⋊Q8—48no—
312C22⋊Q8—44no—
313C22⋊Q8—44no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—84no—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42C2×D412yes—
371C2×C42C2×D414yes—
372C2×C42C2×D414yes—
373C2×C42C2412yes—
374C2×C42C2412yes—
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377#107—44no—
378#107—42no—
379#106—42no—
380C4⋊C8—28no—
381C4⋊C8—28no—
382#103—24no—
383#103—28no—
384#103—44no—
385#103—42no—
386#103—42no—
387#103—44no—
388#102—24no—
389#102—44no—
390#102—42no—
391#102—42no—
392#102—42no—
393#102—42no—
394#97—42no—
395#97—42no—
396#97—42no—
397C4×C8—42no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95C2×D414yes—
410#95C2×D414yes—
411C24—21no—
412C24—22no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—82no—
417C24—82no—
418C24C22×D411yescharacteristic
419#41—44no—
420#41—44no—
421#41—44no—
422#41—44no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—84no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—44no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4C22×D412yes—
488C22×C4C22×D414yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C42−1+412yes—
499C22×C4C2511yescharacteristic
500SD16—416no—
501SD16—48no—
502D8—48no—
503M16—28no—
504M16—44no—
505C2×C8—28no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—44no—
517C2×C8C2×#4114yes—
518C2×C8C2×#4114yes—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—82no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—44no—
548C22⋊C4—48no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—84no—
555C22⋊C4—82no—
556C22⋊C4—84no—
557C22⋊C4—84no—
558C22⋊C4—84no—
559C22⋊C4—84no—
560C22⋊C4—84no—
561C42—24no—
562C42—22no—
563C42—22no—
564C42—21no—
565C42—21no—
566C42—22no—
567C42—22no—
568C42—21no—
569C42—21no—
570C42—44no—
571C42—44no—
572C42—42no—
573C42—42no—
574C42—42no—
575C23—22no—
576C23—22no—
577C23—44no—
578C23—42no—
579C23—42no—
580C23—44no—
581C23—41no—
582C23—42no—
583C23—41no—
584C23—42no—
585C23—41no—
586C23—42no—
587C23—88no—
588C23—84no—
589C23—164no—
590C23—162no—
591C23—162no—
592C23#53212yes—
593C23#53512yes—
594C23#53712yes—
595C23C23×D411yescharacteristic
596Q8—42no—
597Q8—41no—
598Q8—42no—
599Q8—41no—
600Q8—42no—
601Q8—42no—
602Q8—41no—
603Q8—41no—
604Q8—84no—
605Q8—164no—
606D4—41no—
607D4—42no—
608D4—42no—
609D4—41no—
610D4—42no—
611D4—42no—
612D4—41no—
613D4—41no—
614D4—84no—
615D4—84no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—162no—
653C2×C4—162no—
654C2×C4#53212yes—
655C2×C4#53512yes—
656C2×C4#53712yes—
657C2×C4#53814yes—
658C2×C4#53814yes—
659C2×C4#54214yes—
660C2×C4#54214yes—
661C2×C4C22×#4111yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×2−1+411yescharacteristic
667C8—24no—
668C8—24no—
669C8—42no—
670C8—42no—
671C22—24no—
672C22—24no—
673C22—22no—
674C22—22no—
675C22—82no—
676C22—84no—
677C22—82no—
678C22—84no—
679C22—81no—
680C22—82no—
681C22—81no—
682C22—82no—
683C22—81no—
684C22—82no—
685C22—168no—
686C22#266214yes—
687C22C2×#53211yescharacteristic
688C22C2×#53511yescharacteristic
689C22C2×#53711yescharacteristic
690C22#358512yes—
691C22#358612yes—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—42no—
696C4—42no—
697C4—82no—
698C4—82no—
699C4—81no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—81no—
704C4—81no—
705C4#266214yes—
706C4C2×#53811yescharacteristic
707C4C2×#53811yescharacteristic
708C4C2×#54211yescharacteristic
709C4C2×#54211yescharacteristic
710C2—24no—
711C2—161no—
712C2—162no—
713C2C2×#266211yescharacteristic
714C2#6069712yes—
715C2#6082511yescharacteristic
716C2#6082811yescharacteristic
717C2#6093812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
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. #35312C2semidirect
2. #35266C2semidirect
3. #30355C2semidirect
4. #30348C2semidirect
5. #30346C2semidirect
6. #30345C2semidirect
7. #23615C2semidirect
8. #23611C2semidirect
9. #23555C2semidirect
11. #23541C2semidirect
12. #23510C2semidirect
13. #23456C2semidirect
16. C2×#1973C2semidirect
17. #21094C2semidirect
18. #21086C2semidirect
19. #20427C2semidirect
38. #2862C22semidirect
39. #2861C22semidirect
42. #2642C22semidirect
43. #2640C22semidirect
61. #2068C22semidirect
62. #2068C22semidirect
63. #2067C22semidirect
64. #2066C22semidirect
65. #2059C22semidirect
66. #2057C22semidirect
71. #2043C22semidirect
72. #2042C22semidirect
83. #1936C22semidirect
84. #1936C22semidirect
85. #1935C22semidirect
86. #1935C22semidirect
87. #1935C22semidirect
88. #1933C22semidirect
89. #1768C22semidirect
90. #1767C22semidirect

Maximal subgroups

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

subgroupindex
#353122
#352662
#303552
#303482
#303462
#303452
#236152
#236112
#235552
#235442
#235412
#235102
#234562
#234242
#212362
C2×#19732
#210942
#210862
#204272