Finite Group Databasebeta

← #6557133

Subgroups — #6557133

Order 512.

1934 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36261C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#30220C211yesmaximal, characteristic, complemented
4#30199C212yesmaximal, complemented
5#29335C212yesmaximal, complemented
6#29121C211yesmaximal, characteristic, complemented
7#28748C211yesmaximal, characteristic, complemented
8#27776C212yesmaximal, complemented
9#27756C212yesmaximal, complemented
10#27278C214yesmaximal, complemented
11#27274C214yesmaximal, complemented
12#25863C212yesmaximal, complemented
13#23815C211yesmaximal, characteristic, complemented
14#23508C212yesmaximal, complemented
15#22431C212yesmaximal, complemented
16#22431C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×#547C2212yes—
20C2×#532C2212yes—
21C2×#489C2214yes—
22C2×#487C2212yes—
23C2×#436C2212yes—
24C2×#417C2214yes—
25C2×#417C2214yes—
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2857C2212yes—
29#2857C2212yes—
30#2584—21no—
31#2583—22no—
32#2552C2211yescharacteristic
33#2551C2211yescharacteristic
34#2550—21no—
35#2549—22no—
36#2547C2212yes—
37#2537—21no—
38#2537—21no—
39#2440C2212yes—
40#2440C2212yes—
41#2432C2214yes—
42#2422—22no—
43#2422—22no—
44#2349—21no—
45D4×C4⋊C4—21no—
46D4×C4⋊C4—21no—
47D4×C4⋊C4C2211yescharacteristic
48D4×C4⋊C4C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C2×#389C2212yes—
52C42×D4C2211yescharacteristic
53#2099C2211yescharacteristic
54#2099C2211yescharacteristic
55#2099C2212yes—
56#2072C2214yes—
57#2071C2212yes—
58#2043C2212yes—
59#2043C2212yes—
60#2043C2214yes—
61#2043C2212yes—
62#2043C2212yes—
63#2042C2214yes—
64#2042C2214yes—
65#2041C2211yescharacteristic
66#2041C2211yescharacteristic
67#2041C2212yes—
68#2024C2214yes—
69#2023C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1959C2214yes—
73#1956C2212yes—
74#1956C2212yes—
75#1956C2212yes—
76#1935C2214yes—
77#1935C2214yes—
78#1934C2214yes—
79#1894C2214yes—
80#1894C2214yes—
81#1872C2214yes—
82#1872C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1788C2214yes—
86#1788C2214yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×#102C2212yes—
90C4×#102C2212yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×#41C2314yes—
96C22×M16C2314yes—
97#554—48no—
98#547—28no—
99#547—28no—
100#532—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#127C2311yescharacteristic
108C2×#125—21no—
109C2×#125—21no—
110C2×#124C2314yes—
111C2×#123—24no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8C2314yes—
116C2×#121—22no—
117C2×#121—22no—
118C2×#121—22no—
119C2×C4×Q8—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2312yes—
129C2×C4×D4C2311yescharacteristic
130C22×C4⋊C4—22no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C42C2312yes—
135#489—48no—
136#487—44no—
137#436—28no—
138C2×C4⋊C8C2312yes—
139#417—48no—
140#417—48no—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2314yes—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2312yes—
159#398—22no—
160#398—22no—
161#398—22no—
162#397—24no—
163#395—22no—
164#395—22no—
165#395—22no—
166#394—24no—
167#394—24no—
168#391—21no—
169#391—22no—
170#391C2314yes—
171#390—21no—
172#390—21no—
173#390C2312yes—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—21no—
185#389—22no—
186#389C2311yescharacteristic
187#389C2311yescharacteristic
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#385—24no—
197#385—24no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384—21no—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2312yes—
207#382—21no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380—22no—
212#380—22no—
213#380C2312yes—
214#380C2312yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C2×#95—22no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2311yescharacteristic
237C2×#41—28no—
238C2×#41—28no—
239C2×#41—28no—
240C2×#41—44no—
241C2×#41—44no—
242C2×#41—44no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8C2414yes—
246C22×D4—24no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2×D411yescharacteristic
250C22×D4C2412yes—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—24no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C2×M16—28no—
259C2×M16—44no—
260C2×M16—44no—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C4⋊Q8—22no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—41no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272C4⋊Q8—44no—
273#127—24no—
274#125—44no—
275#125—44no—
276#124—28no—
277#124—44no—
278#123—416no—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283C22⋊Q8—48no—
284#121—48no—
285#121—48no—
286#121—48no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—48no—
293C4×D4—48no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4#4118yes—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4—48no—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—44no—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C4⋊C8—24no—
366#103—28no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374#102—24no—
375#102—28no—
376#102—24no—
377#102—24no—
378#102—42no—
379#102—44no—
380C22⋊C8—28no—
381#97—44no—
382C4×C8—24no—
383C4×C8—42no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—22no—
400C24C22×D412yes—
401#41—416no—
402#41—416no—
403#41—88no—
404#41—88no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—81no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—84no—
433C2×D4#12414yes—
434C2×D4H(5,𝔽2)14yes—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—84no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4118yes—
474C22×C4C2511yescharacteristic
475M16—88no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4#12414yes—
525C4⋊C4H(5,𝔽2)12yes—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4#12418yes—
537C42—21no—
538C42—24no—
539C42—22no—
540C42—21no—
541C42—42no—
542C42—42no—
543C42—41no—
544C42—41no—
545C42—42no—
546C42—42no—
547C42AGL(1,ℤ/8)14yes—
548C42#13714yes—
549C42C2×#4114yes—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23C2×#12414yes—
564C23C23×D411yescharacteristic
565Q8—44no—
566Q8—44no—
567Q8—44no—
568Q8—44no—
569Q8—84no—
570Q8—164no—
571D4—28no—
572D4—24no—
573D4—24no—
574D4—48no—
575D4—48no—
576D4—42no—
577D4—42no—
578D4—816no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—816no—
626C2×C4—88no—
627C2×C4—88no—
628C2×C4—164no—
629C2×C4C2×#12412yes—
630C2×C4C2×#12414yes—
631C2×C4C2×#12412yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C2×#13712yes—
636C2×C4C2×#13712yes—
637C2×C4C2×#13712yes—
638C2×C4C22×#4112yes—
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4#58511yescharacteristic
642C8—42no—
643C8—44no—
644C8—42no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22C22×#12411yescharacteristic
660C22#348411yescharacteristic
661C22#348811yescharacteristic
662C22C22×AGL(1,ℤ/8)11yescharacteristic
663C22C22×#13711yescharacteristic
664C22#358611yescharacteristic
665C22#358611yescharacteristic
666C4—22no—
667C4—22no—
668C4—24no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—82no—
676C4—81no—
677C4—84no—
678C4—81no—
679C4—82no—
680C4—81no—
681C4—82no—
682C4#314114yes—
683C4#351312yes—
684C4#351912yes—
685C2—44no—
686C2—84no—
687C2#3590311yescharacteristic
688C2C2×#314111yescharacteristic
689C2#6062711yescharacteristic
690C2#6158511yescharacteristic
691C2#6158611yescharacteristic
692C2#6196211yescharacteristic
693C2#6196611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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. #36261C2semidirect
3. #30220C2semidirect
4. #30199C2semidirect
5. #29335C2semidirect
6. #29121C2semidirect
7. #28748C2semidirect
8. #27776C2semidirect
9. #27756C2semidirect
10. #27278C2semidirect
11. #27274C2semidirect
12. #25863C2semidirect
13. #23815C2semidirect
14. #23508C2semidirect
15. #22431C2semidirect
16. #22431C2semidirect
17. #20568C2semidirect

Maximal subgroups

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

subgroupindex
#362612
#361272
#302202
#301992
#293352
#291212
#287482
#277762
#277562
#272782
#272742
#258632
#238152
#235082
#224312
#205682