Finite Group Databasebeta

← #6560006

Subgroups — #6560006

Order 512.

2052 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36289C211yesmaximal, characteristic
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#30203C211yesmaximal, characteristic, complemented
4#30188C212yesmaximal, complemented
5#29354C211yesmaximal, characteristic, complemented
6#29148C211yesmaximal, characteristic, complemented
7#29109C212yesmaximal, complemented
8#27821C212yesmaximal, complemented
9#27778C212yesmaximal, complemented
10#27711C214yesmaximal, complemented
11#27262C214yesmaximal, complemented
12#25908C212yesmaximal, complemented
13#23817C211yesmaximal, characteristic, complemented
14C2×#2023C212yesmaximal, complemented
15#22487C212yesmaximal, complemented
16D4×#102C212yesmaximal
17#20558C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C2×D4×Q8—21no—
20C2×D42C2211yescharacteristic
21C22×C4⋊Q8C2212yescomplemented
22C2×#489C2214yescomplemented
23C2×C8×D4C2212yes—
24C22×#102C2214yes—
25#2868C2211yescharacteristic, complemented
26#2858C2211yescharacteristic
27#2857C2212yescomplemented
28#2850C2212yes—
29#2563C2211yescharacteristic
30#2551—21no—
31#2551C2212yescomplemented
32#2548C2211yescharacteristic
33#2546—21no—
34#2504C2212yes—
35#2490—21no—
36#2490—21no—
37#2472—22no—
38#2472—22no—
39#2440—22no—
40#2440C2214yescomplemented
41#2435C2212yes—
42#2434—22no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2212yes—
45D4×C22⋊C4—21no—
46D4×C22⋊C4—21no—
47C4×C4⋊Q8C2211yescharacteristic, complemented
48C4×#124C2211yescharacteristic
49C2×#390C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2212yescomplemented
52#2101C2214yescomplemented
53#2072C2211yescharacteristic, complemented
54#2071C2211yescharacteristic
55#2071C2214yescomplemented
56#2071C2212yescomplemented
57#2042C2212yes—
58#2042C2214yescomplemented
59#2024C2211yescharacteristic, complemented
60#2024C2212yescomplemented
61#2023C2211yescharacteristic
62#2023C2214yes—
63#2023C2212yes—
64#2023C2212yescomplemented
65#2023C2212yescomplemented
66#2023C2214yescomplemented
67#2023C2212yescomplemented
68#1993C2214yescomplemented
69#1990C2212yescomplemented
70#1987C2212yescomplemented
71#1982C2212yes—
72#1959C2212yes—
73#1957C2212yes—
74#1936C2214yes—
75#1936C2214yescomplemented
76#1936C2214yescomplemented
77#1893C2214yes—
78#1892C2214yescomplemented
79#1874C2214yes—
80#1872C2214yescomplemented
81#1836C2212yescomplemented
82#1824C2214yescomplemented
83#1803C2212yescomplemented
84#1802C2212yes—
85#1786C2214yes—
86#1770C2212yescomplemented
87#1767C2211yescharacteristic, complemented
88C4×#103C2212yescomplemented
89C4×#102C2212yes—
90#1752C2214yescomplemented
91#1749C2212yescomplemented
92#1744C2211yescharacteristic, complemented
93C2×#340C2214yescomplemented
94C2×#336C2212yescomplemented
95C23×Q8—22no—
96C23×D4—21no—
97C23×D4C2312yes—
98C23×C8C2314yes—
99#554—28no—
100D4×Q8—48no—
101D42—28no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—48no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2312yes—
112C2×#127C2311yescharacteristic
113C2×#125—21no—
114C2×#124—21no—
115C2×#124—21no—
116C2×#124C2311yescharacteristic
117C2×#124C2312yes—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8C2312yes—
122C2×#121—21no—
123C2×#121C2312yes—
124C2×#120—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—21no—
128C2×C4×D4—22no—
129C2×C4×D4D411yescharacteristic
130C2×C4×D4D411yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C22×C4⋊C4—22no—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2314yes—
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C42C2312yes—
139#489—48no—
140C8×D4—28no—
141C2×C4⋊C8C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#102—48no—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2314yes—
154C2×#97C2312yes—
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#398—24no—
158#397—22no—
159#397—22no—
160#395—24no—
161#395—24no—
162#394—22no—
163#394—22no—
164#394—22no—
165#393—22no—
166#391—21no—
167#391—22no—
168#391C2311yescharacteristic
169#390—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—22no—
174#390—22no—
175#390C2312yes—
176#390C2311yescharacteristic
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#389—22no—
181#389—22no—
182#389—21no—
183#389—22no—
184#389C2312yes—
185#389C2314yes—
186#388—22no—
187#388—22no—
188#388—21no—
189#388—21no—
190#386—22no—
191#386—22no—
192#386—22no—
193#386—22no—
194#385—24no—
195#385—24no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382C2312yes—
202#382C2311yescharacteristic
203#380—22no—
204#380—21no—
205#380—21no—
206#380C2314yes—
207#379—22no—
208#379—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2314yes—
235C25—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—24no—
239C22×Q8—21no—
240C22×Q8—48no—
241C22×Q8—48no—
242C22×Q8—84no—
243C22×Q8C2411yescharacteristic
244C22×D4—24no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—28no—
250C22×D4—44no—
251C22×D4—42no—
252C22×D4—44no—
253C22×D4—44no—
254C22×D4—44no—
255C22×D4C2×D411yescharacteristic
256C22×D4C2412yes—
257C22×D4C2411yescharacteristic
258C23×C4—24no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2412yes—
264C22×C8—28no—
265C22×C8—44no—
266C22×C8—44no—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—28no—
272C4⋊Q8—24no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C4⋊Q8—48no—
277C4⋊Q8—48no—
278C4⋊Q8—48no—
279C4⋊Q8—82no—
280#127—24no—
281#125—44no—
282#124—22no—
283#124—41no—
284#124—44no—
285#124—82no—
286#124—82no—
287C22⋊Q8—28no—
288C22⋊Q8—416no—
289C22⋊Q8—48no—
290C22⋊Q8—48no—
291#121—28no—
292#121—44no—
293#120—48no—
294C4×Q8—24no—
295C4×Q8—44no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—44no—
300C4×D4—48no—
301C4×D4#4118yes—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—84no—
350C2×C22⋊C4—84no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C4⋊C8—24no—
368#103—24no—
369#103—42no—
370#103—44no—
371#102—24no—
372#102—24no—
373#102—28no—
374#102—44no—
375#102—44no—
376#102—44no—
377#102—44no—
378C22⋊C8—28no—
379#97—42no—
380#97—44no—
381C4×C8—24no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95—24no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24—22no—
397C24—44no—
398C24—44no—
399C24—44no—
400C24—44no—
401C24—82no—
402C24—82no—
403C24—82no—
404C24C22×D412yes—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—48no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—48no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—84no—
419C2×Q8—81no—
420C2×Q8—81no—
421C2×Q8—832no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—416no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—88no—
440C2×D4—82no—
441C2×D4—81no—
442C2×D4—81no—
443C2×D4—816no—
444C2×D4#12414yes—
445C2×D4AGL(1,ℤ/8)14yes—
446C2×D4#13714yes—
447C2×D42−1+412yes—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—84no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C2×#4118yes—
483C22×C4C2511yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—88no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4#12414yes—
521C4⋊C4H(5,𝔽2)14yes—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—48no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4#12418yes—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—24no—
548C42—22no—
549C42—21no—
550C42—21no—
551C42—42no—
552C42—42no—
553C42—42no—
554C42—44no—
555C42C2×#4114yes—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—88no—
575C23—88no—
576C23—88no—
577C23—88no—
578C23—164no—
579C23—164no—
580C23—164no—
581C23C2×#12414yes—
582C23C2×AGL(1,ℤ/8)14yes—
583C23C2×#13714yes—
584C23C23×D411yescharacteristic
585Q8—48no—
586Q8—42no—
587Q8—42no—
588Q8—88no—
589Q8—816no—
590Q8—162no—
591D4—24no—
592D4—42no—
593D4—44no—
594D4—44no—
595D4—88no—
596D4—88no—
597D4—84no—
598D4—162no—
599D4#489116yescomplemented
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—816no—
646C2×C4—84no—
647C2×C4—164no—
648C2×C4C2×#12412yes—
649C2×C4C2×#12414yes—
650C2×C4C2×#12412yes—
651C2×C4C2×AGL(1,ℤ/8)12yes—
652C2×C4C2×#13712yes—
653C2×C4#57612yes—
654C2×C4#57712yes—
655C2×C4C22×#4112yes—
656C2×C4C2×H(5,𝔽2)12yes—
657C2×C4C2×2−1+411yescharacteristic
658C2×C4#58511yescharacteristic
659C8—44no—
660C8—42no—
661C8—42no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—1616no—
683C22C2×#48918yes—
684C22C22×#12411yescharacteristic
685C22#348411yescharacteristic
686C22#348911yescharacteristic
687C22C22×AGL(1,ℤ/8)11yescharacteristic
688C22C22×#13711yescharacteristic
689C22C2×#57611yescharacteristic
690C22C2×#57711yescharacteristic
691C4—24no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—42no—
696C4—44no—
697C4—42no—
698C4—41no—
699C4—41no—
700C4—81no—
701C4—84no—
702C4—82no—
703C4—81no—
704C4—81no—
705C4—82no—
706C4C2×#48914yes—
707C4#351412yes—
708C4#352012yes—
709C2—28no—
710C2—82no—
711C2—162no—
712C2#3593911yescharacteristic
713C2C22×#48911yescharacteristic
714C2#6062211yescharacteristic
715C2#6159111yescharacteristic
716C2#6159311yescharacteristic
717C2#6212711yescharacteristic
718C2#6212811yescharacteristic
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
2. D4×C4⋊Q8C2semidirect
3. #30203C2semidirect
4. #30188C2semidirect
5. #29354C2semidirect
6. #29148C2semidirect
7. #29109C2semidirect
8. #27821C2semidirect
9. #27778C2semidirect
10. #27711C2semidirect
11. #27262C2semidirect
12. #25908C2semidirect
13. #23817C2semidirect
14. C2×#2023C2semidirect
15. #22487C2semidirect
17. #20558C2semidirect
21. C22×C4⋊Q8C22semidirect
22. C2×#489C22semidirect
25. #2868C22semidirect
27. #2857C22semidirect
31. #2551C22semidirect
40. #2440C22semidirect
47. C4×C4⋊Q8C22semidirect
51. #2101C22semidirect
52. #2101C22semidirect
53. #2072C22semidirect
55. #2071C22semidirect
56. #2071C22semidirect
58. #2042C22semidirect
59. #2024C22semidirect
60. #2024C22semidirect
64. #2023C22semidirect
65. #2023C22semidirect
66. #2023C22semidirect
67. #2023C22semidirect
68. #1993C22semidirect
69. #1990C22semidirect
70. #1987C22semidirect
75. #1936C22semidirect
76. #1936C22semidirect
78. #1892C22semidirect
80. #1872C22semidirect
81. #1836C22semidirect
82. #1824C22semidirect
83. #1803C22semidirect
86. #1770C22semidirect
87. #1767C22semidirect
88. C4×#103C22semidirect
90. #1752C22semidirect
91. #1749C22semidirect
92. #1744C22semidirect
93. C2×#340C22semidirect
94. C2×#336C22semidirect
599. D4#489semidirect

Maximal subgroups

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

subgroupindex
#362892
D4×C4⋊Q82
#302032
#301882
#293542
#291482
#291092
#278212
#277782
#277112
#272622
#259082
#238172
C2×#20232
#224872
D4×#1022
#205582