Finite Group Databasebeta

← #6624346

Subgroups — #6624346

Order 512.

1955 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic, complemented
3#33149C212yesmaximal, complemented
4#32953C211yesmaximal, characteristic, complemented
5#29109C212yesmaximal, complemented
6#28117C212yesmaximal
7#27961C211yesmaximal, characteristic, complemented
8#27645C211yesmaximal, characteristic, complemented
9#27443C214yesmaximal, complemented
10#27379C214yesmaximal, complemented
11#26024C212yesmaximal, complemented
12#24676C212yesmaximal, complemented
13#23066C212yesmaximal, complemented
14#22758C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21580C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C2×#554—21no—
19C22×C4×D4C2211yescharacteristic
20C22×#117—21no—
21C2×#481C2212yes—
22C2×#474C2212yes—
23C2×#447C2211yescharacteristic
24C2×C4×D8C2212yes—
25#2857C2211yescharacteristic
26#2857C2212yes—
27#2564—22no—
28#2556—22no—
29#2551C2211yescharacteristic, complemented
30#2542C2211yescharacteristic, complemented
31#2506C2212yescomplemented
32#2492—21no—
33#2452C2212yescomplemented
34#2417—21no—
35#2365C2211yescharacteristic, complemented
36#2360—21no—
37#2354—21no—
38#2354—21no—
39#2354—22no—
40#2354C2212yes—
41#2354C2214yes—
42#2354C2212yes—
43D4×C4⋊C4C2211yescharacteristic, complemented
44C4×#121C2212yes—
45C4×#121C2211yescharacteristic
46C2×#390C2212yes—
47C2×#388—22no—
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2214yescomplemented
53#2084C2212yescomplemented
54#2071C2214yes—
55#2071C2212yescomplemented
56#2065C2214yes—
57#2054C2214yes—
58#2023C2214yes—
59#1990C2212yescomplemented
60#1987C2214yescomplemented
61#1948C2212yescomplemented
62#1947C2214yescomplemented
63#1939C2212yescomplemented
64#1936C2214yes—
65#1936C2214yes—
66#1926C2214yes—
67#1925C2214yescomplemented
68#1921C2214yescomplemented
69#1921C2214yescomplemented
70#1920C2214yescomplemented
71#1917C2214yescomplemented
72#1879C2212yes—
73#1879C2214yes—
74#1874C2211yescharacteristic
75#1862C2214yes—
76#1862C2212yes—
77#1852C2212yes—
78#1852C2211yescharacteristic
79#1834C2212yescomplemented
80#1824C2212yes—
81#1824C2212yes—
82#1824C2212yes—
83#1802C2212yes—
84#1802C2212yescomplemented
85#1797C2212yes—
86#1797C2211yescharacteristic
87#1786C2211yescharacteristic
88#1786C2212yes—
89#1776C2214yescomplemented
90#1770C2212yescomplemented
91#1762C2214yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×D8C2312yes—
98#554—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#125—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—21no—
108C2×#121C2311yescharacteristic
109C2×C4×Q8—21no—
110C2×C4×D4—28no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×#117—22no—
120C2×#117—24no—
121C2×#117—22no—
122C2×#117—48no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C42—22no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#481—28no—
137#474—28no—
138#447—24no—
139C4×D8—28no—
140C2×#107C2312yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2311yescharacteristic
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#399—22no—
159#398—22no—
160#396—22no—
161#395—22no—
162#393—22no—
163#391—21no—
164#391—21no—
165#391—22no—
166#390—22no—
167#390—22no—
168#390C2312yes—
169#390C2314yes—
170#390C2311yescharacteristic
171#390C2314yes—
172#389—21no—
173#389—22no—
174#389—21no—
175#389C2312yes—
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#388—22no—
181#388—21no—
182#388—24no—
183#388—24no—
184#388—24no—
185#388—24no—
186#387—22no—
187#387—22no—
188#386—21no—
189#386—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#384C2312yes—
198#380—21no—
199#380—22no—
200#380—21no—
201#380—22no—
202#380—24no—
203#380—22no—
204#380C2312yes—
205#380C2311yescharacteristic
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C2×#95—22no—
221C2×#95D412yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C25—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×D4—21no—
239C22×D4—24no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—42no—
244C22×D4C2411yescharacteristic
245C22×D4C2412yes—
246C23×C4—28no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C2×D8—24no—
261C2×D8—24no—
262C2×D8—42no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C4⋊Q8—24no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269#125—48no—
270C22⋊Q8—44no—
271C22⋊Q8—48no—
272C22⋊Q8—48no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275#121—22no—
276#121#4114yes—
277C4×Q8—44no—
278C4×D4—28no—
279C4×D4—28no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4#4118yes—
288#117—48no—
289#117—416no—
290#117—48no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—28no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—48no—
346C2×C22⋊C4—84no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—44no—
353C2×C42—48no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362#107—24no—
363#107—28no—
364#106—28no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367#102—24no—
368#102—28no—
369#102—28no—
370#102—24no—
371#102—24no—
372#102—44no—
373#102—42no—
374C22⋊C8—22no—
375C22⋊C8—24no—
376C22⋊C8—22no—
377#97—44no—
378C4×C8—24no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—24no—
395C24—21no—
396C24—22no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24C22×D411yescharacteristic
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4C2×#4112yes—
427C2×D4C2×#4114yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—42no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—432no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—82no—
466C22×C4—82no—
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×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C2×#4112yes—
478C22×C42−1+412yes—
479C22×C4C2511yescharacteristic
480D8—48no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4AGL(1,ℤ/8)14yes—
521C4⋊C4C2×#4112yes—
522C4⋊C4C2×#4114yes—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—24no—
526C22⋊C4—48no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—48no—
532C22⋊C4—42no—
533C22⋊C4—44no—
534C22⋊C4—81no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—82no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—24no—
542C42—24no—
543C42—42no—
544C42—42no—
545C42—48no—
546C42—44no—
547C42—48no—
548C42—42no—
549C42C2×#4114yes—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—24no—
555C23—22no—
556C23—48no—
557C23—48no—
558C23—48no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23—41no—
570C23—42no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23#54812yes—
576C23C2×AGL(1,ℤ/8)12yes—
577C23#57812yes—
578C23C23×D411yescharacteristic
579Q8—84no—
580Q8—164no—
581D4—24no—
582D4—24no—
583D4—24no—
584D4—48no—
585D4—48no—
586D4—42no—
587D4—42no—
588D4—42no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—48no—
615C2×C4—44no—
616C2×C4—416no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—416no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—84no—
639C2×C4—88no—
640C2×C4—84no—
641C2×C4—164no—
642C2×C4#52512yes—
643C2×C4#52512yes—
644C2×C4#54812yes—
645C2×C4C2×AGL(1,ℤ/8)14yes—
646C2×C4C2×AGL(1,ℤ/8)12yes—
647C2×C4C2×AGL(1,ℤ/8)12yes—
648C2×C4C2×AGL(1,ℤ/8)12yes—
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4112yes—
651C2×C4C2×2−1+411yescharacteristic
652C2×C4#58512yes—
653C8—44no—
654C8—42no—
655C8—42no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—42no—
663C22—44no—
664C22—42no—
665C22—44no—
666C22—42no—
667C22—44no—
668C22—816no—
669C22—81no—
670C22—82no—
671C22—81no—
672C22—82no—
673C22—82no—
674C22—81no—
675C22—82no—
676C22—81no—
677C22#336714yes—
678C22C2×#52511yescharacteristic
679C22C2×#54811yescharacteristic
680C22#348511yescharacteristic
681C22C22×AGL(1,ℤ/8)11yescharacteristic
682C22C22×AGL(1,ℤ/8)11yescharacteristic
683C22C2×#57811yescharacteristic
684C22#358511yescharacteristic
685C4—24no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—44no—
694C4—41no—
695C4—82no—
696C4—81no—
697C4—81no—
698C4—82no—
699C4—84no—
700C4#310914yes—
701C4#347512yes—
702C4#349512yes—
703C2—24no—
704C2—82no—
705C2—84no—
706C2#3651611yescharacteristic
707C2C2×#310911yescharacteristic
708C2#6058911yescharacteristic
709C2C2×#336711yescharacteristic
710C2#6132111yescharacteristic
711C2#6172011yescharacteristic
712C2#6173711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
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. #36561C2semidirect
2. C2×#2354C2semidirect
3. #33149C2semidirect
4. #32953C2semidirect
5. #29109C2semidirect
7. #27961C2semidirect
8. #27645C2semidirect
9. #27443C2semidirect
10. #27379C2semidirect
11. #26024C2semidirect
12. #24676C2semidirect
13. #23066C2semidirect
14. #22758C2semidirect
15. #22629C2semidirect
16. #21580C2semidirect
29. #2551C22semidirect
30. #2542C22semidirect
31. #2506C22semidirect
33. #2452C22semidirect
35. #2365C22semidirect
43. D4×C4⋊C4C22semidirect
52. #2100C22semidirect
53. #2084C22semidirect
55. #2071C22semidirect
59. #1990C22semidirect
60. #1987C22semidirect
61. #1948C22semidirect
62. #1947C22semidirect
63. #1939C22semidirect
67. #1925C22semidirect
68. #1921C22semidirect
69. #1921C22semidirect
70. #1920C22semidirect
71. #1917C22semidirect
79. #1834C22semidirect
84. #1802C22semidirect
89. #1776C22semidirect
90. #1770C22semidirect
91. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#365612
C2×#23542
#331492
#329532
#291092
#281172
#279612
#276452
#274432
#273792
#260242
#246762
#230662
#227582
#226292
#215802
C4×#4472