Finite Group Databasebeta

← #6852333

Subgroups — #6852333

Order 512.

1881 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36552C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3#32953C211yesmaximal, characteristic
4#32364C214yesmaximal, complemented
5#31540C214yesmaximal, complemented
6#29956C212yesmaximal
7#28540C212yesmaximal, complemented
8#27953C212yesmaximal, complemented
9#27822C212yesmaximal, complemented
10#27681C212yesmaximal, complemented
11#25813C211yesmaximal, characteristic, complemented
12#25449C212yesmaximal, complemented
13#24813C212yesmaximal, complemented
14#24301C211yesmaximal, characteristic, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22049C212yesmaximal
17C4×#450C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8—21no—
21C22×C4×D4C2211yescharacteristic
22C2×#500C2212yescomplemented
23C2×#475C2212yescomplemented
24C2×#474C2212yes—
25C2×#450C2211yescharacteristic
26C2×C4×SD16C2212yes—
27#2857C2211yescharacteristic
28#2851C2212yes—
29#2590—22no—
30#2568C2211yescharacteristic, complemented
31#2552C2211yescharacteristic, complemented
32#2542—22no—
33#2542—21no—
34#2542—21no—
35#2542C2214yes—
36#2542C2212yes—
37#2542C2212yes—
38#2511—21no—
39#2508C2212yescomplemented
40#2505—22no—
41#2502—21no—
42#2453C2212yescomplemented
43C2×#399—22no—
44#2365C2211yescharacteristic, complemented
45#2360—21no—
46#2355—21no—
47Q8×C4⋊C4C2211yescharacteristic, complemented
48C4×C22⋊Q8C2212yes—
49C4×C22⋊Q8C2211yescharacteristic
50C2×#390C2212yes—
51C2×#384C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2100C2214yescomplemented
54#2095C2214yescomplemented
55#2092C2214yes—
56#2084C2214yescomplemented
57#2083C2214yescomplemented
58#2067C2214yes—
59#2065C2214yes—
60#2058C2212yescomplemented
61#2057C2214yescomplemented
62#2025C2214yes—
63#2024C2212yescomplemented
64#1992C2214yescomplemented
65#1990C2212yescomplemented
66#1988C2212yescomplemented
67#1973C2214yes—
68#1947C2214yescomplemented
69#1939C2212yescomplemented
70#1936C2214yes—
71#1925C2214yes—
72#1917C2214yescomplemented
73#1880C2214yes—
74#1877C2212yes—
75#1872C2212yes—
76#1872C2211yescharacteristic
77#1864C2212yes—
78#1862C2214yes—
79#1853C2212yes—
80#1853C2211yescharacteristic
81#1852C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1847C2214yescomplemented
85#1803C2214yescomplemented
86#1802C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1786C2211yescharacteristic
90#1786C2212yes—
91#1770C2212yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125C2312yes—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—48no—
114C2×C22⋊Q8—48no—
115C2×C22⋊Q8C2311yescharacteristic
116C2×C4×Q8—21no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—44no—
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#500—44no—
137#475—44no—
138#474—44no—
139#450Q818yescomplemented
140C4×SD16—44no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×#97C2314yes—
159C2×C4×C8C2312yes—
160#400—22no—
161#400—22no—
162#400—22no—
163#399—24no—
164#399—24no—
165#399—24no—
166#399—24no—
167#398—22no—
168#398—22no—
169#398—22no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391C2312yes—
175#391C2311yescharacteristic
176#390—22no—
177#390—22no—
178#390C2314yes—
179#390C2314yes—
180#389—22no—
181#389—21no—
182#389—21no—
183#389—22no—
184#389C2312yes—
185#389C2311yescharacteristic
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#387—22no—
190#387—21no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—21no—
200#384—21no—
201#384—21no—
202#384—22no—
203#384—22no—
204#384C2311yescharacteristic
205#384C2314yes—
206#384C2314yes—
207#382—22no—
208#382C2312yes—
209#380—21no—
210#380C2311yescharacteristic
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C2×#95D412yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236#336C2314yes—
237C25—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8C2411yescharacteristic
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—42no—
248C22×D4—42no—
249C22×D4C2×Q814yes—
250C22×D4C2412yes—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263C2×SD16—42no—
264C2×SD16—42no—
265C2×SD16C2×Q818yescomplemented
266C22×C8C2412yes—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275#125—44no—
276C22⋊Q8—22no—
277C22⋊Q8—416no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C22⋊Q8—48no—
281C22⋊Q8—416no—
282C22⋊Q8—44no—
283C22⋊Q8—48no—
284C22⋊Q8C2×Q814yes—
285C4×Q8—24no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—42no—
291C4×D4—42no—
292C4×D4—44no—
293C4×D4—44no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—48no—
346C2×C22⋊C4—48no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361#107—44no—
362#107—44no—
363#107—44no—
364#106—42no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—44no—
368#103—42no—
369#103—44no—
370#102—44no—
371#102—42no—
372#102—42no—
373#102—44no—
374#102C2×Q818yescomplemented
375C22⋊C8—22no—
376C22⋊C8—24no—
377C22⋊C8C2×Q814yes—
378#97—44no—
379C4×C8—24no—
380#95—21no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—24no—
400C24—22no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24C22×D411yescharacteristic
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—816no—
421C2×Q8C22×Q812yes—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—48no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—48no—
432C2×D4C22×Q814yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—816no—
467C22×C4AGL(1,ℤ/8)14yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×Q812yes—
478C22×C42−1+412yes—
479C22×C4C2511yescharacteristic
480SD16—48no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8C22×Q814yes—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—42no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4AGL(1,ℤ/8)14yes—
526C4⋊C4C22×Q812yes—
527C22⋊C4—24no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—48no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—24no—
540C42—24no—
541C42—42no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42—82no—
547C42H(5,𝔽2)14yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—24no—
553C23—48no—
554C23—48no—
555C23—42no—
556C23—48no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23D4×Q812yes—
567C23C2×AGL(1,ℤ/8)12yes—
568C23#57812yes—
569C23C23×D411yescharacteristic
570Q8—24no—
571Q8—42no—
572Q8—88no—
573Q8—84no—
574Q8—88no—
575D4—24no—
576D4—24no—
577D4—48no—
578D4—42no—
579D4—42no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—88no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4—88no—
634C2×C4#53512yes—
635C2×C4#53512yes—
636C2×C4D4×Q812yes—
637C2×C4C2×AGL(1,ℤ/8)14yes—
638C2×C4C2×AGL(1,ℤ/8)12yes—
639C2×C4C2×AGL(1,ℤ/8)12yes—
640C2×C4C2×AGL(1,ℤ/8)12yes—
641C2×C4C23×Q811yescharacteristic
642C2×C4C2×H(5,𝔽2)12yes—
643C2×C4C2×2−1+411yescharacteristic
644C2×C4#58512yes—
645C8—24no—
646C8—42no—
647C8—44no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—816no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22#338614yes—
663C22C2×#53511yescharacteristic
664C22C2×D4×Q811yescharacteristic
665C22#348511yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22C22×AGL(1,ℤ/8)11yescharacteristic
668C22C2×#57811yescharacteristic
669C22#358611yescharacteristic
670C4—24no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—41no—
678C4—42no—
679C4—42no—
680C4—82no—
681C4—81no—
682C4—82no—
683C4—84no—
684C4—82no—
685C4—81no—
686C4#321714yes—
687C4#350112yes—
688C4#354112yes—
689C2—24no—
690C2—84no—
691C2#3959411yescharacteristic
692C2C2×#321711yescharacteristic
693C2#6081411yescharacteristic
694C2C2×#338611yescharacteristic
695C2Q8×AGL(1,ℤ/8)11yescharacteristic
696C2#6171711yescharacteristic
697C2#6173711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #36552C2semidirect
4. #32364C2semidirect
5. #31540C2semidirect
7. #28540C2semidirect
8. #27953C2semidirect
9. #27822C2semidirect
10. #27681C2semidirect
11. #25813C2semidirect
12. #25449C2semidirect
13. #24813C2semidirect
14. #24301C2semidirect
15. #22629C2semidirect
22. C2×#500C22semidirect
23. C2×#475C22semidirect
30. #2568C22semidirect
31. #2552C22semidirect
39. #2508C22semidirect
42. #2453C22semidirect
44. #2365C22semidirect
47. Q8×C4⋊C4C22semidirect
53. #2100C22semidirect
54. #2095C22semidirect
56. #2084C22semidirect
57. #2083C22semidirect
60. #2058C22semidirect
61. #2057C22semidirect
63. #2024C22semidirect
64. #1992C22semidirect
65. #1990C22semidirect
66. #1988C22semidirect
68. #1947C22semidirect
69. #1939C22semidirect
72. #1917C22semidirect
84. #1847C22semidirect
85. #1803C22semidirect
91. #1770C22semidirect
139. #450Q8semidirect
265. C2×SD16C2×Q8semidirect
374. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365522
C2×#25422
#329532
#323642
#315402
#299562
#285402
#279532
#278222
#276812
#258132
#254492
#248132
#243012
#226292
#220492
C4×#4502