Finite Group Databasebeta

← #6624980

Subgroups — #6624980

Order 512.

2235 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic, complemented
2C2×#2366C211yesmaximal, characteristic
3#33149C211yesmaximal, characteristic, complemented
4#32953C212yesmaximal, complemented
5#29109C212yesmaximal, complemented
6#28123C212yesmaximal, complemented
7#26912C214yesmaximal, complemented
8#26771C214yesmaximal, complemented
9#26460C212yesmaximal, complemented
10#25855C211yesmaximal, characteristic
11#25322C212yesmaximal, complemented
12#24211C211yesmaximal, characteristic
13#23223C212yesmaximal, complemented
14#23003C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16C4×#484C211yesmaximal, characteristic, complemented
17#21453C212yesmaximal, complemented
18C22×#127C2211yescharacteristic
19C22×C4×D4—21no—
20C22×C4×D4C2212yes—
21C2×#486C2212yes—
22C2×#484C2211yescharacteristic
23C2×#474C2214yescomplemented
24C2×#460C2212yes—
25C2×#450C2212yes—
26C2×#440C2214yescomplemented
27#2857C2211yescharacteristic
28#2857C2212yes—
29#2586—22no—
30#2567C2211yescharacteristic, complemented
31#2551C2211yescharacteristic, complemented
32#2549—22no—
33#2507—21no—
34#2507C2212yescomplemented
35#2451C2212yescomplemented
36#2440—21no—
37#2369—21no—
38#2369C2211yescharacteristic, complemented
39#2366—22no—
40#2366—21no—
41#2366—21no—
42#2366C2214yes—
43#2366C2212yes—
44#2366C2212yes—
45Q8×C4⋊C4C2211yescharacteristic, complemented
46#2345—21no—
47C4×C22⋊Q8C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49C2×#390—21no—
50C2×#390C2212yes—
51C2×#386—22no—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2071C2214yescomplemented
55#2067C2214yescomplemented
56#2055C2214yescomplemented
57#2041C2214yes—
58#2041C2212yes—
59#2023C2214yescomplemented
60#2014C2212yes—
61#1987C2212yescomplemented
62#1982C2212yes—
63#1957C2212yes—
64#1947C2212yescomplemented
65#1939C2212yescomplemented
66#1936C2214yescomplemented
67#1936C2214yescomplemented
68#1928C2212yescomplemented
69#1925C2214yescomplemented
70#1921C2214yescomplemented
71#1921C2212yescomplemented
72#1917C2214yescomplemented
73#1880C2214yescomplemented
74#1877C2214yescomplemented
75#1872C2211yescharacteristic
76#1862C2214yescomplemented
77#1862C2214yescomplemented
78#1862C2214yescomplemented
79#1852C2211yescharacteristic
80#1852C2212yes—
81#1824C2211yescharacteristic
82#1824C2212yes—
83#1802C2214yescomplemented
84#1797C2211yescharacteristic
85#1797C2212yes—
86#1786C2212yes—
87#1786C2212yes—
88#1786C2212yes—
89C4×#107C2212yescomplemented
90#1770C2212yescomplemented
91#1762C2214yescomplemented
92C4×#103C2212yescomplemented
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C24×C4—22no—
99C22×SD16C2314yes—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127—28no—
104C2×#127—21no—
105C2×#127—21no—
106C2×#127C2312yes—
107C2×#127C2312yes—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4—24no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—48no—
119C2×C4×D4—48no—
120C2×C4×D4C2314yes—
121C2×C4×D4C2314yes—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C22⋊C4—22no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#486—28no—
133#484—24no—
134#474—48no—
135#460—28no—
136#450—28no—
137#440—48no—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#398—22no—
158#398—22no—
159#398—22no—
160#395—22no—
161#395—22no—
162#395—22no—
163#391—21no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391C2312yes—
168#391C2311yescharacteristic
169#390—22no—
170#390—24no—
171#390—22no—
172#390—22no—
173#390—22no—
174#390C2314yes—
175#390C2314yes—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2311yescharacteristic
189#387—22no—
190#387—22no—
191#387—21no—
192#387—22no—
193#386—24no—
194#386—24no—
195#386—24no—
196#386—24no—
197#385—22no—
198#385—21no—
199#384—21no—
200#384C2312yes—
201#384C2311yescharacteristic
202#380—21no—
203#380—22no—
204#380C2312yes—
205#380C2311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C2×#95—21no—
225C2×#95D412yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236#336C2312yes—
237C25—22no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8C2411yescharacteristic
241C22×D4—28no—
242C22×D4—28no—
243C22×D4—28no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—24no—
250C22×D4—24no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4—44no—
254C22×D4—82no—
255C22×D4—82no—
256C22×D4C2414yes—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—24no—
260C23×C4—24no—
261C23×C4—48no—
262C23×C4—48no—
263C23×C4—84no—
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D412yes—
268C23×C4C2411yescharacteristic
269C2×SD16—28no—
270C2×SD16—28no—
271C2×SD16—44no—
272C22×C8C2414yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C4⋊Q8—24no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#127—28no—
279#127—28no—
280#127—28no—
281#127—24no—
282#127#4118yes—
283C22⋊Q8—22no—
284C22⋊Q8—84no—
285C22⋊Q8—82no—
286C22⋊Q8#4114yes—
287C4×Q8—24no—
288C4×Q8—44no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—48no—
294C4×D4—48no—
295C4×D4—416no—
296C4×D4—48no—
297C4×D4—48no—
298C2×C4⋊C4—24no—
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—21no—
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—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—48no—
342C2×C22⋊C4—48no—
343C2×C22⋊C4—84no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—44no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—24no—
362#107—28no—
363#107—24no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366#103—24no—
367#103—24no—
368#102—28no—
369#102—28no—
370#102—28no—
371#102—28no—
372#102—44no—
373#102—44no—
374#102—44no—
375C22⋊C8—24no—
376C22⋊C8—22no—
377C22⋊C8—22no—
378#97—44no—
379C4×C8—24no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—21no—
394#95—22no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—28no—
400C24—24no—
401C24—44no—
402C24—44no—
403C24—44no—
404C24—44no—
405C24—44no—
406C24—44no—
407C24C22×D411yescharacteristic
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—82no—
414C2×Q8—81no—
415C2×Q8—82no—
416C2×Q8—81no—
417C2×Q8C2×#4112yes—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4—28no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—416no—
429C2×D4—416no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—432no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—816no—
442C2×D4C2×#4118yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—48no—
474C22×C4—84no—
475C22×C4—84no—
476C22×C4—832no—
477C22×C4#13714yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4112yes—
488C22×C42−1+412yes—
489C22×C4C2511yescharacteristic
490SD16—416no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—82no—
521C4⋊C4—81no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—81no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4AGL(1,ℤ/8)14yes—
530C4⋊C4C2×#4112yes—
531C4⋊C4H(5,𝔽2)14yes—
532C22⋊C4—24no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—48no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—24no—
545C42—24no—
546C42—42no—
547C42—42no—
548C42—44no—
549C42—42no—
550C42—44no—
551C42—42no—
552C42—82no—
553C42—82no—
554C42C2×#4114yes—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—28no—
560C23—416no—
561C23—416no—
562C23—416no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—88no—
570C23—88no—
571C23—88no—
572C23—88no—
573C23#54812yes—
574C23C2×#13712yes—
575C23#57612yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—42no—
579Q8—164no—
580Q8—162no—
581D4—28no—
582D4—28no—
583D4—416no—
584D4—416no—
585D4—416no—
586D4—44no—
587D4—44no—
588D4—88no—
589D4—88no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—48no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—816no—
641C2×C4—84no—
642C2×C4—816no—
643C2×C4—84no—
644C2×C4#52512yes—
645C2×C4#52512yes—
646C2×C4#54812yes—
647C2×C4C2×AGL(1,ℤ/8)14yes—
648C2×C4C2×AGL(1,ℤ/8)12yes—
649C2×C4C2×#13712yes—
650C2×C4C2×#13712yes—
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4112yes—
653C2×C4C2×H(5,𝔽2)12yes—
654C2×C4C2×2−1+411yescharacteristic
655C8—42no—
656C8—44no—
657C8—42no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—48no—
665C22—48no—
666C22—48no—
667C22—832no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22#337514yes—
673C22C2×#52511yescharacteristic
674C22C2×#54811yescharacteristic
675C22#348411yescharacteristic
676C22C22×AGL(1,ℤ/8)11yescharacteristic
677C22C22×#13711yescharacteristic
678C22C2×#57611yescharacteristic
679C22#358511yescharacteristic
680C4—22no—
681C4—24no—
682C4—22no—
683C4—22no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—44no—
689C4—41no—
690C4—42no—
691C4—82no—
692C4—82no—
693C4—81no—
694C4—81no—
695C4—82no—
696C4#311014yes—
697C4#348112yes—
698C4#352012yes—
699C2—24no—
700C2—88no—
701C2#3652811yescharacteristic
702C2C2×#311011yescharacteristic
703C2#6057811yescharacteristic
704C2C2×#337511yescharacteristic
705C2#6132111yescharacteristic
706C2#6158811yescharacteristic
707C2#6159111yescharacteristic
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. #36564C2semidirect
3. #33149C2semidirect
4. #32953C2semidirect
5. #29109C2semidirect
6. #28123C2semidirect
7. #26912C2semidirect
8. #26771C2semidirect
9. #26460C2semidirect
11. #25322C2semidirect
13. #23223C2semidirect
14. #23003C2semidirect
15. #22629C2semidirect
16. C4×#484C2semidirect
17. #21453C2semidirect
23. C2×#474C22semidirect
26. C2×#440C22semidirect
30. #2567C22semidirect
31. #2551C22semidirect
34. #2507C22semidirect
35. #2451C22semidirect
38. #2369C22semidirect
45. Q8×C4⋊C4C22semidirect
54. #2071C22semidirect
55. #2067C22semidirect
56. #2055C22semidirect
59. #2023C22semidirect
61. #1987C22semidirect
64. #1947C22semidirect
65. #1939C22semidirect
66. #1936C22semidirect
67. #1936C22semidirect
68. #1928C22semidirect
69. #1925C22semidirect
70. #1921C22semidirect
71. #1921C22semidirect
72. #1917C22semidirect
73. #1880C22semidirect
74. #1877C22semidirect
76. #1862C22semidirect
77. #1862C22semidirect
78. #1862C22semidirect
83. #1802C22semidirect
89. C4×#107C22semidirect
90. #1770C22semidirect
91. #1762C22semidirect
92. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×#23662
#331492
#329532
#291092
#281232
#269122
#267712
#264602
#258552
#253222
#242112
#232232
#230032
#226292
C4×#4842
#214532