Finite Group Databasebeta

← #6555341

Subgroups — #6555341

Order 512.

1919 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34310C211yesmaximal, characteristic
3#29980C211yesmaximal, characteristic, complemented
4#29950C212yesmaximal, complemented
5#29258C211yesmaximal, characteristic, complemented
6#29163C212yesmaximal, complemented
7#28508C211yesmaximal, characteristic, complemented
8#27683C212yesmaximal, complemented
9#26529C212yesmaximal, complemented
10#26352C214yesmaximal, complemented
11#26087C212yesmaximal, complemented
12#25966C214yesmaximal, complemented
13#23775C211yesmaximal, characteristic, complemented
14#23731C212yesmaximal, complemented
15#22700C211yesmaximal, characteristic, complemented
16#22665C212yesmaximal, complemented
17#22428C212yesmaximal
18C2×D42C2211yescharacteristic
19C2×#518—21no—
20C2×#474C2214yescomplemented
21C2×#436C2212yes—
22C2×#428C2214yescomplemented
23C2×#418C2214yes—
24#2868C2211yescharacteristic, complemented
25#2857C2212yescomplemented
26#2571C2212yescomplemented
27#2563—21no—
28#2538—21no—
29#2497—22no—
30#2492—22no—
31#2439C2214yescomplemented
32#2369—21no—
33#2366C2211yescharacteristic
34#2363—21no—
35#2361C2212yes—
36#2358—22no—
37#2355—21no—
38#2354C2211yescharacteristic
39#2354C2212yes—
40D4×C4⋊C4C2211yescharacteristic
41D4×C22⋊C4—21no—
42D4×C22⋊C4C2212yes—
43#2324—21no—
44#2321—22no—
45C4×C4⋊Q8C2211yescharacteristic, complemented
46C4×#127C2211yescharacteristic
47C4×#121C2212yes—
48#2293C2211yescharacteristic
49C2×#390C2212yes—
50#2285C2212yescomplemented
51C42×D4C2211yescharacteristic
52#2100C2211yescharacteristic, complemented
53#2099C2212yescomplemented
54#2094C2211yescharacteristic
55#2094C2212yescomplemented
56#2093C2212yes—
57#2093C2212yescomplemented
58#2093C2212yescomplemented
59#2092C2214yes—
60#2092C2214yescomplemented
61#2088C2212yescomplemented
62#2067C2214yes—
63#2067C2214yescomplemented
64#2065C2212yes—
65#2058C2211yescharacteristic, complemented
66#2057C2212yescomplemented
67#2023C2212yes—
68#2023C2211yescharacteristic
69#2023C2212yescomplemented
70#1982C2212yes—
71#1957C2212yes—
72#1949C2214yescomplemented
73#1948C2212yescomplemented
74#1947C2212yescomplemented
75#1939C2212yescomplemented
76#1921C2214yescomplemented
77#1895C2214yes—
78#1875C2214yes—
79#1864C2214yescomplemented
80#1864C2214yescomplemented
81#1854C2214yescomplemented
82#1852C2214yescomplemented
83#1851C2214yescomplemented
84#1845C2212yescomplemented
85#1831C2211yescharacteristic, complemented
86#1809C2214yescomplemented
87#1804C2214yescomplemented
88#1802C2212yes—
89#1797C2212yescomplemented
90#1786C2214yes—
91C4×#106C2212yescomplemented
92C4×C4⋊C8C2211yescharacteristic, complemented
93C4×#102C2212yes—
94#1728C2212yescomplemented
95C23×D4—21no—
96C23×D4C2312yes—
97C22×M16C2314yes—
98D42—24no—
99D42Q818yescomplemented
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×C22⋊Q8—22no—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×#120—22no—
109#518—48no—
110C2×C4×Q8—21no—
111C2×C4×D4—21no—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C2×#117—21no—
125C2×#117C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C22⋊C4—22no—
128C22×C22⋊C4—22no—
129C22×C22⋊C4—22no—
130C22×C42C2312yes—
131#474—48no—
132#436—28no—
133#428—48no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143#418—48no—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2312yes—
153#400—24no—
154#396—24no—
155#393—24no—
156#391—21no—
157#391—22no—
158#391—24no—
159#390—22no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390—22no—
164#390—22no—
165#390C2312yes—
166#390C2311yescharacteristic
167#390C2312yes—
168#390C2312yes—
169#390C2311yescharacteristic
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389C2312yes—
175#389C2311yescharacteristic
176#389C2311yescharacteristic
177#389C2314yes—
178#388—24no—
179#388—21no—
180#388—22no—
181#388—21no—
182#388—22no—
183#388—22no—
184#387—24no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199#382C2314yes—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—21no—
204#380C2312yes—
205#380C2314yes—
206#379—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222C43C2311yescharacteristic
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2314yes—
235C25—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×D4—22no—
239C22×D4—28no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—24no—
244C22×D4—44no—
245C22×D4—44no—
246C22×D4—44no—
247C22×D4—44no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2×Q818yes—
250C22×D4C2412yes—
251C22×D4C2411yescharacteristic
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C2×M16—28no—
260C2×M16—28no—
261C2×M16—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—42no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—82no—
269#127—22no—
270#127C2×Q814yes—
271C22⋊Q8—84no—
272#121—24no—
273#121—28no—
274#121C2×Q818yes—
275#120—216no—
276C4×Q8—44no—
277C4×D4—24no—
278C4×D4—22no—
279C4×D4—42no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—48no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—82no—
288C4×D4C2×Q814yes—
289C4×D4C2×Q814yes—
290C4×D4#4114yes—
291#117—44no—
292#117—44no—
293#117—44no—
294#117—44no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—48no—
339C2×C22⋊C4—48no—
340C2×C22⋊C4—84no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—44no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—28no—
360#107—44no—
361#107—44no—
362#106—24no—
363#106—42no—
364#106—44no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367C4⋊C8—44no—
368#102—24no—
369#102—28no—
370#102—44no—
371#102—42no—
372#102—44no—
373#102—44no—
374#102—44no—
375C22⋊C8—28no—
376C4×C8—24no—
377#95—21no—
378#95—22no—
379#95—21no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—24no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—21no—
392C24—22no—
393C24—28no—
394C24—44no—
395C24—44no—
396C24—82no—
397C24—82no—
398C24—82no—
399C24—82no—
400C24C22×D412yes—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—416no—
420C2×D4—432no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—88no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4C22⋊Q814yes—
431C2×D4C22×Q812yes—
432C2×D4C22×Q814yes—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×Q814yes—
470C22×C4C2×#4114yes—
471C22×C4C2511yescharacteristic
472M16—416no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8H(5,𝔽2)14yes—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—82no—
505C4⋊C4—81no—
506C4⋊C4—84no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4C22⋊Q814yes—
510C4⋊C4C22×Q812yes—
511C22⋊C4—24no—
512C22⋊C4—28no—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—81no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4C22⋊Q818yes—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—24no—
535C42—22no—
536C42—42no—
537C42—42no—
538C42—41no—
539C42—44no—
540C42—42no—
541C42—44no—
542C42—84no—
543C42AGL(1,ℤ/8)14yes—
544C42C22×Q812yes—
545C42C2×#4112yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—416no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23—41no—
566C23—42no—
567C23—88no—
568C23—88no—
569C23—164no—
570C23—164no—
571C23—164no—
572C23—164no—
573C23C2×C22⋊Q814yes—
574C23C23×D411yescharacteristic
575Q8—84no—
576Q8—164no—
577D4—28no—
578D4—24no—
579D4—28no—
580D4—416no—
581D4—48no—
582D4—44no—
583D4—42no—
584D4—88no—
585D4—84no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—48no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—164no—
632C2×C4—164no—
633C2×C4—164no—
634C2×C4C2×C22⋊Q812yes—
635C2×C4C2×C22⋊Q814yes—
636C2×C4C2×C22⋊Q812yes—
637C2×C4C2×AGL(1,ℤ/8)12yes—
638C2×C4C2×AGL(1,ℤ/8)12yes—
639C2×C4C2×AGL(1,ℤ/8)12yes—
640C2×C4#57512yes—
641C2×C4#57612yes—
642C2×C4C23×Q811yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×H(5,𝔽2)12yes—
645C2×C4#58512yes—
646C8—24no—
647C8—42no—
648C8—44no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—44no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22—82no—
663C22—81no—
664C22—82no—
665C22—81no—
666C22—82no—
667C22—81no—
668C22—82no—
669C22—81no—
670C22—1616no—
671C22C22×C22⋊Q811yescharacteristic
672C22#348511yescharacteristic
673C22#349811yescharacteristic
674C22C22×AGL(1,ℤ/8)11yescharacteristic
675C22C2×#57511yescharacteristic
676C22C2×#57611yescharacteristic
677C22#358611yescharacteristic
678C4—22no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—24no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—41no—
687C4—81no—
688C4—82no—
689C4—82no—
690C4—81no—
691C4—84no—
692C4—82no—
693C4#307714yes—
694C4#350112yes—
695C4#351312yes—
696C2—44no—
697C2—84no—
698C2—82no—
699C2#3583611yescharacteristic
700C2C2×#307711yescharacteristic
701C2#6049411yescharacteristic
702C2#6172211yescharacteristic
703C2#6174011yescharacteristic
704C2#6242311yescharacteristic
705C2#6245611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #29980C2semidirect
4. #29950C2semidirect
5. #29258C2semidirect
6. #29163C2semidirect
7. #28508C2semidirect
8. #27683C2semidirect
9. #26529C2semidirect
10. #26352C2semidirect
11. #26087C2semidirect
12. #25966C2semidirect
13. #23775C2semidirect
14. #23731C2semidirect
15. #22700C2semidirect
16. #22665C2semidirect
20. C2×#474C22semidirect
22. C2×#428C22semidirect
24. #2868C22semidirect
25. #2857C22semidirect
26. #2571C22semidirect
31. #2439C22semidirect
45. C4×C4⋊Q8C22semidirect
50. #2285C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
55. #2094C22semidirect
57. #2093C22semidirect
58. #2093C22semidirect
60. #2092C22semidirect
61. #2088C22semidirect
63. #2067C22semidirect
65. #2058C22semidirect
66. #2057C22semidirect
69. #2023C22semidirect
72. #1949C22semidirect
73. #1948C22semidirect
74. #1947C22semidirect
75. #1939C22semidirect
76. #1921C22semidirect
79. #1864C22semidirect
80. #1864C22semidirect
81. #1854C22semidirect
82. #1852C22semidirect
83. #1851C22semidirect
84. #1845C22semidirect
85. #1831C22semidirect
86. #1809C22semidirect
87. #1804C22semidirect
89. #1797C22semidirect
91. C4×#106C22semidirect
92. C4×C4⋊C8C22semidirect
94. #1728C22semidirect
99. D42Q8semidirect

Maximal subgroups

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

subgroupindex
#361942
#343102
#299802
#299502
#292582
#291632
#285082
#276832
#265292
#263522
#260872
#259662
#237752
#237312
#227002
#226652
#224282