Finite Group Databasebeta

← #6484529

Subgroups — #6484529

Order 512.

2132 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic, complemented
2C2×#2366C211yesmaximal, characteristic
3#32959C212yesmaximal, complemented
4#32956C211yesmaximal, characteristic, complemented
5#32953C211yesmaximal, characteristic, complemented
6#29956C212yesmaximal, complemented
7#29952C211yesmaximal, characteristic, complemented
8#29950C211yesmaximal, characteristic, complemented
9#26541C214yesmaximal, complemented
10#26538C214yesmaximal, complemented
11#26475C214yesmaximal, complemented
12#26472C214yesmaximal, complemented
13#21955C211yesmaximal, characteristic, complemented
14#21810C212yesmaximal
15#21363C212yesmaximal, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4—21no—
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#478C2214yescomplemented
21C2×#476C2212yescomplemented
22C2×#474C2212yescomplemented
23C2×#432C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2851C2212yes—
27#2627—21no—
28#2627—21no—
29#2606—22no—
30#2605—22no—
31#2570C2212yescomplemented
32#2570C2212yescomplemented
33#2550C2212yescomplemented
34#2550C2212yescomplemented
35#2366—22no—
36#2366—22no—
37#2366C2214yes—
38#2366C2214yes—
39#2329—21no—
40#2329—21no—
41C2×#390—21no—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#386—21no—
45C2×#386—21no—
46#2283C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2094C2212yescomplemented
50#2093C2212yescomplemented
51#2092C2214yescomplemented
52#2067C2214yescomplemented
53#2067C2214yescomplemented
54#2065C2214yescomplemented
55#2065C2214yescomplemented
56#2025C2214yescomplemented
57#2023C2212yescomplemented
58#2023C2212yescomplemented
59#1982C2214yes—
60#1982C2214yes—
61#1961C2214yes—
62#1961C2214yes—
63#1949C2214yescomplemented
64#1949C2214yescomplemented
65#1940C2214yescomplemented
66#1940C2214yescomplemented
67#1864C2214yescomplemented
68#1864C2214yescomplemented
69#1862C2214yescomplemented
70#1862C2214yescomplemented
71#1853C2212yes—
72#1853C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1804C2214yescomplemented
76#1802C2212yescomplemented
77#1802C2212yescomplemented
78#1797C2211yescharacteristic
79#1797C2212yes—
80#1797C2211yescharacteristic
81#1786C2212yes—
82#1786C2212yes—
83#1786C2212yes—
84#1786C2212yes—
85#1776C2214yescomplemented
86#1776C2214yescomplemented
87C23×D4—21no—
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C24×C4—21no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127—22no—
96C2×#127—24no—
97C2×#127Q818yescomplemented
98C2×#127C2314yes—
99C2×#125C2312yes—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—21no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—22no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—44no—
108C2×C4×D4—44no—
109C2×C4×D4—48no—
110C2×C4×D4C2314yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#478—48no—
128#476—44no—
129#474—44no—
130#432—28no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×#97C2314yes—
150#400—22no—
151#400—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#398—22no—
156#395—22no—
157#395—22no—
158#391—21no—
159#391—21no—
160#391—22no—
161#391—22no—
162#390—24no—
163#390—24no—
164#390—22no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2314yes—
169#390C2312yes—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#386—24no—
185#386—24no—
186#386—24no—
187#386—24no—
188#385—22no—
189#385—22no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#383—22no—
193#382C2312yes—
194#382C2312yes—
195#382C2312yes—
196#381—22no—
197#380—21no—
198#380—21no—
199#380C2312yes—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—22no—
210#376—22no—
211C2×#95—21no—
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2312yes—
220#336C2312yes—
221C25—21no—
222C25—21no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×D4—28no—
226C22×D4—28no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4—24no—
230C22×D4—24no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—42no—
234C22×D4—42no—
235C22×D4—42no—
236C22×D4—42no—
237C22×D4—44no—
238C22×D4—44no—
239C22×D4—82no—
240C22×D4—82no—
241C22×D4C2×Q814yes—
242C22×D4C2×Q814yes—
243C22×D4C2412yes—
244C22×D4C2412yes—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—24no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—44no—
254C23×C4—82no—
255C23×C4—82no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C22×C8C2414yes—
264C22×C8C2414yes—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267#127—216no—
268#127—28no—
269#127C2×Q8116yes—
270#125—44no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—28no—
276C4×D4—44no—
277C4×D4—42no—
278C4×D4—42no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—48no—
282C4×D4—416no—
283C4×D4—416no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—44no—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×Q814yes—
340C2×C42#4114yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345#107—28no—
346#107—44no—
347#106—28no—
348#106—44no—
349C4⋊C8—28no—
350C4⋊C8—42no—
351C4⋊C8—42no—
352C4⋊C8—44no—
353#102—44no—
354#102—44no—
355#102—44no—
356#102—44no—
357#102—44no—
358#102—44no—
359#102—44no—
360#102—44no—
361C22⋊C8—24no—
362C22⋊C8—24no—
363#97—28no—
364#95—24no—
365#95—22no—
366#95—22no—
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#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—24no—
384C24—24no—
385C24—22no—
386C24—22no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24C22×D411yescharacteristic
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×Q8—81no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—216no—
409C2×D4—216no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—432no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—84no—
430C2×D4—84no—
431C2×D4—816no—
432C2×D4C22×Q814yes—
433C2×D4C22×Q814yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
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—22no—
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—44no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—84no—
472C22×C4—816no—
473C22×C4—816no—
474C22×C4C22⋊Q814yes—
475C22×C4C22⋊Q814yes—
476C22×C4AGL(1,ℤ/8)14yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×Q812yes—
487C22×C4C2×#4112yes—
488C22×C4C2511yescharacteristic
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—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—81no—
512C4⋊C4—81no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4H(5,𝔽2)14yes—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—84no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—41no—
535C42—41no—
536C42—42no—
537C42—48no—
538C42—82no—
539C42—82no—
540C42C22⋊Q818yes—
541C42C22×Q814yes—
542C42C2×#4114yes—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—24no—
548C23—24no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—48no—
558C23—48no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23C2×C22⋊Q812yes—
576C23C2×AGL(1,ℤ/8)12yes—
577C23#57512yes—
578C23C23×D411yescharacteristic
579Q8—162no—
580Q8—162no—
581D4—24no—
582D4—24no—
583D4—28no—
584D4—432no—
585D4—48no—
586D4—48no—
587D4—42no—
588D4—42no—
589D4—44no—
590D4—816no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—28no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—48no—
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—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—88no—
639C2×C4—88no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—816no—
643C2×C4—84no—
644C2×C4C2×C22⋊Q812yes—
645C2×C4C2×C22⋊Q814yes—
646C2×C4C2×C22⋊Q814yes—
647C2×C4C2×AGL(1,ℤ/8)12yes—
648C2×C4C2×AGL(1,ℤ/8)12yes—
649C2×C4C23×Q811yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×H(5,𝔽2)12yes—
652C2×C4#58512yes—
653C8—44no—
654C8—44no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—816no—
668C22—816no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22#307714yes—
678C22C22×C22⋊Q811yescharacteristic
679C22#348411yescharacteristic
680C22#348711yescharacteristic
681C22C22×AGL(1,ℤ/8)11yescharacteristic
682C22C2×#57511yescharacteristic
683C22#358511yescharacteristic
684C22#358511yescharacteristic
685C4—24no—
686C4—24no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—42no—
691C4—44no—
692C4—41no—
693C4—42no—
694C4—41no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4#346012yes—
701C4#350112yes—
702C2—24no—
703C2—84no—
704C2—84no—
705C2#3517411yescharacteristic
706C2C2×#307711yescharacteristic
707C2#6049011yescharacteristic
708C2#6157111yescharacteristic
709C2#6157711yescharacteristic
710C2#6189011yescharacteristic
711C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35104C2semidirect
3. #32959C2semidirect
4. #32956C2semidirect
5. #32953C2semidirect
6. #29956C2semidirect
7. #29952C2semidirect
8. #29950C2semidirect
9. #26541C2semidirect
10. #26538C2semidirect
11. #26475C2semidirect
12. #26472C2semidirect
13. #21955C2semidirect
15. #21363C2semidirect
20. C2×#478C22semidirect
21. C2×#476C22semidirect
22. C2×#474C22semidirect
31. #2570C22semidirect
32. #2570C22semidirect
33. #2550C22semidirect
34. #2550C22semidirect
49. #2094C22semidirect
50. #2093C22semidirect
51. #2092C22semidirect
52. #2067C22semidirect
53. #2067C22semidirect
54. #2065C22semidirect
55. #2065C22semidirect
56. #2025C22semidirect
57. #2023C22semidirect
58. #2023C22semidirect
63. #1949C22semidirect
64. #1949C22semidirect
65. #1940C22semidirect
66. #1940C22semidirect
67. #1864C22semidirect
68. #1864C22semidirect
69. #1862C22semidirect
70. #1862C22semidirect
75. #1804C22semidirect
76. #1802C22semidirect
77. #1802C22semidirect
85. #1776C22semidirect
86. #1776C22semidirect
97. C2×#127Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#23662
#329592
#329562
#329532
#299562
#299522
#299502
#265412
#265382
#264752
#264722
#219552
#218102
#213632