Finite Group Databasebeta

← #6434298

Subgroups — #6434298

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34637C211yesmaximal, characteristic, complemented
2C2×#2644C211yesmaximal, characteristic, complemented
3#30355C214yesmaximal, complemented
4#30354C214yesmaximal, complemented
5#30349C214yesmaximal, complemented
6#30348C214yesmaximal, complemented
7#23617C212yesmaximal, complemented
8#23554C211yesmaximal, characteristic, complemented
9#23546C212yesmaximal, complemented
10#23508C211yesmaximal, characteristic, complemented
11#23454C211yesmaximal, characteristic, complemented
12#23428C211yesmaximal, characteristic
13#21236C212yesmaximal, complemented
14#21093C212yesmaximal, complemented
15#20429C211yesmaximal, characteristic, complemented
16C22×AGL(1,ℤ/8)C2212yes—
17C2×#533C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C2×#428C2212yes—
20C2×#423C2212yes—
21C2×#417C2212yes—
22C2×#417C2212yes—
23C2×#417C2212yes—
24C2×#418C2212yes—
25C2×C4×M16C2212yes—
26#2644—22no—
27#2644—22no—
28#2644C2214yes—
29#2644C2214yes—
30#2642C2212yescomplemented
31#2641C2212yescomplemented
32#2640C2212yescomplemented
33#2640C2212yescomplemented
34#2639—21no—
35#2639—21no—
36#2639—21no—
37#2639—21no—
38#2398—22no—
39#2398—22no—
40C2×#397—21no—
41C2×#397—21no—
42C2×#397—21no—
43C2×#396—21no—
44#2297C2212yes—
45#2297C2212yes—
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#376C2211yescharacteristic
53#2068C2214yescomplemented
54#2067C2214yescomplemented
55#2066C2214yescomplemented
56#2065C2214yescomplemented
57#2059C2214yescomplemented
58#2058C2212yescomplemented
59#2057C2212yescomplemented
60#2055C2214yes—
61#2055C2214yes—
62#2054C2214yes—
63#2054C2214yes—
64#2043C2212yescomplemented
65#2042C2214yescomplemented
66#2041C2212yescomplemented
67#2025C2214yes—
68#2023C2212yes—
69#2023C2212yes—
70#2013C2212yes—
71#2012C2214yes—
72#2011C2212yes—
73#1973C2214yes—
74#1972C2214yes—
75#1970C2214yes—
76#1968C2214yes—
77#1936C2214yescomplemented
78#1935C2214yescomplemented
79#1934C2214yescomplemented
80#1933C2214yescomplemented
81#1768C2214yescomplemented
82#1768C2214yescomplemented
83#1736C2212yes—
84#1736C2212yes—
85#1728C2211yescharacteristic
86#1728C2211yescharacteristic
87C22×#41C2312yes—
88C23×D4C2311yescharacteristic
89C2×AGL(1,ℤ/8)—28no—
90C2×AGL(1,ℤ/8)—44no—
91C2×AGL(1,ℤ/8)—44no—
92C22×SD16C2314yes—
93C22×D8C2314yes—
94C22×M16C2312yes—
95C22×M16C2312yes—
96#533—44no—
97#532—44no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127C2311yescharacteristic
102C2×#125C2312yes—
103C2×#124C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×#121—21no—
107C2×#121—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×#117C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#428—44no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131#423—28no—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2314yes—
134#418—44no—
135#417—28no—
136#417—44no—
137#417—44no—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C4×M16—44no—
147C2×#97C2314yes—
148C2×C4×C8C2314yes—
149#400—22no—
150#400—22no—
151#400—22no—
152#400—22no—
153#397—22no—
154#397—22no—
155#397—22no—
156#397—22no—
157#397—24no—
158#397—24no—
159#397—24no—
160#397—24no—
161#397—24no—
162#397—24no—
163#396—24no—
164#396—24no—
165#394—22no—
166#394—22no—
167#394—22no—
168#394—22no—
169#393—22no—
170#393—22no—
171#393—22no—
172#393—22no—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#390—21no—
178#390—21no—
179#390—22no—
180#390—22no—
181#390C2314yes—
182#390C2312yes—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#390C2314yes—
187#390C2312yes—
188#390C2312yes—
189#389—21no—
190#389—21no—
191#389—22no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#389C2314yes—
196#389C2314yes—
197#386—22no—
198#386—22no—
199#386—22no—
200#386—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205#376D412yes—
206#376D412yes—
207#376C2312yes—
208#376C2312yes—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214C2×#95D411yescharacteristic
215C2×#95D411yescharacteristic
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2312yes—
221#336C2312yes—
222#336C2312yes—
223#336C2312yes—
224C25—21no—
225C2×#41—24no—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—24no—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8C2412yes—
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—24no—
239C22×D4—22no—
240C22×D4—24no—
241C22×D4—22no—
242C22×D4C2412yes—
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255AGL(1,ℤ/8)—432no—
256C2×SD16—28no—
257C2×SD16—44no—
258C2×SD16—44no—
259C2×D8—28no—
260C2×D8—44no—
261C2×D8—44no—
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16#4118yes—
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272C4⋊Q8—42no—
273#127—42no—
274#125—42no—
275#125—42no—
276#124—42no—
277#124—42no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280#121—44no—
281#121—44no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×D4—44no—
285C4×D4—44no—
286#117—24no—
287#117—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4—82no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4—82no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2412yes—
351#107—44no—
352#106—44no—
353C4⋊C8—28no—
354C4⋊C8—28no—
355#103—28no—
356#103—44no—
357#103—44no—
358#102—28no—
359#102—44no—
360#102—44no—
361#102—44no—
362#102—44no—
363#97—44no—
364C4×C8—44no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—22no—
386C24—22no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24C22×D411yescharacteristic
395#41—48no—
396#41—48no—
397#41—48no—
398#41—48no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—81no—
408C2×Q8—81no—
409C2×Q8—81no—
410C2×Q8—81no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—81no—
432C2×D4—81no—
433C2×D4—81no—
434C2×D4—81no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4114yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C42−1+412yes—
484C22×C4C2511yescharacteristic
485SD16—416no—
486D8—416no—
487M16—216no—
488M16—48no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8C2×#4118yes—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—44no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23#53212yes—
572C23#53512yes—
573C23#53712yes—
574C23C23×D411yescharacteristic
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—42no—
579Q8—84no—
580Q8—84no—
581D4—42no—
582D4—44no—
583D4—42no—
584D4—44no—
585D4—42no—
586D4—42no—
587D4—84no—
588D4—84no—
589D4—88no—
590D4—84no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—42no—
616C2×C4—42no—
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—42no—
632C2×C4—42no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—88no—
643C2×C4#53212yes—
644C2×C4#53512yes—
645C2×C4#53712yes—
646C2×C4#53818yes—
647C2×C4#54218yes—
648C2×C4C22×#4112yes—
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×H(5,𝔽2)12yes—
651C2×C4C2×2−1+411yescharacteristic
652C8—28no—
653C8—44no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—168no—
675C22#266214yes—
676C22C2×#53211yescharacteristic
677C22C2×#53511yescharacteristic
678C22C2×#53711yescharacteristic
679C22#358511yescharacteristic
680C22#358511yescharacteristic
681C22#358511yescharacteristic
682C22#358611yescharacteristic
683C4—42no—
684C4—42no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—42no—
690C4—42no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4#266214yes—
698C4C2×#53812yes—
699C4C2×#54212yes—
700C2—24no—
701C2—162no—
702C2—162no—
703C2C2×#266211yescharacteristic
704C2#6070311yescharacteristic
705C2#6070411yescharacteristic
706C2#6080811yescharacteristic
707C2#6081011yescharacteristic
708C2#6091911yescharacteristic
709C2#6092011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23428C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34637C2semidirect
2. C2×#2644C2semidirect
3. #30355C2semidirect
4. #30354C2semidirect
5. #30349C2semidirect
6. #30348C2semidirect
7. #23617C2semidirect
8. #23554C2semidirect
9. #23546C2semidirect
10. #23508C2semidirect
11. #23454C2semidirect
13. #21236C2semidirect
14. #21093C2semidirect
15. #20429C2semidirect
30. #2642C22semidirect
31. #2641C22semidirect
32. #2640C22semidirect
33. #2640C22semidirect
53. #2068C22semidirect
54. #2067C22semidirect
55. #2066C22semidirect
56. #2065C22semidirect
57. #2059C22semidirect
58. #2058C22semidirect
59. #2057C22semidirect
64. #2043C22semidirect
65. #2042C22semidirect
66. #2041C22semidirect
77. #1936C22semidirect
78. #1935C22semidirect
79. #1934C22semidirect
80. #1933C22semidirect
81. #1768C22semidirect
82. #1768C22semidirect

Maximal subgroups

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

subgroupindex
#346372
C2×#26442
#303552
#303542
#303492
#303482
#236172
#235542
#235462
#235082
#234542
#234282
#212362
#210932
#204292