Finite Group Databasebeta

← #6592119

Subgroups — #6592119

Order 512.

1952 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36160C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#32980C212yesmaximal, complemented
5#27960C211yesmaximal, characteristic, complemented
6#27644C211yesmaximal, characteristic, complemented
7#27602C212yesmaximal, complemented
8#26299C212yesmaximal, complemented
9#26209C214yesmaximal, complemented
10#24968C212yesmaximal, complemented
11#24942C214yesmaximal, complemented
12#24550C212yesmaximal, complemented
13#22707C212yesmaximal, complemented
14#22489C212yesmaximal, complemented
15D4×C22⋊C8C211yesmaximal, characteristic, complemented
16C4×#481C211yesmaximal, characteristic, complemented
17#20699C212yesmaximal, complemented
18C2×#555C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21C2×#481C2211yescharacteristic
22C2×#480C2212yes—
23C2×C8×D4C2212yes—
24C22×C22⋊C8C2212yes—
25#2857C2211yescharacteristic
26#2847C2212yes—
27#2563C2211yescharacteristic
28#2537C2211yescharacteristic
29#2485C2212yes—
30#2472—22no—
31#2459—22no—
32#2420—21no—
33#2414C2212yes—
34#2408—21no—
35#2363—21no—
36#2354—21no—
37#2354—21no—
38#2354—22no—
39#2354C2211yescharacteristic
40#2354C2214yes—
41#2354C2212yes—
42#2354C2212yes—
43#2352—21no—
44C4×#121C2211yescharacteristic
45C2×#390—21no—
46C2×#388—22no—
47C2×#384C2211yescharacteristic
48#2280C2212yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2214yescomplemented
53#2083C2212yescomplemented
54#2077C2214yescomplemented
55#2076C2214yescomplemented
56#2075C2214yescomplemented
57#2071C2212yescomplemented
58#2033C2214yescomplemented
59#1990C2212yescomplemented
60#1987C2212yescomplemented
61#1954C2212yes—
62#1947C2212yescomplemented
63#1947C2212yescomplemented
64#1926C2212yescomplemented
65#1921C2214yescomplemented
66#1919C2212yescomplemented
67#1893C2214yescomplemented
68#1875C2214yescomplemented
69#1875C2214yescomplemented
70#1874C2211yescharacteristic
71#1869C2214yes—
72#1854C2214yescomplemented
73#1854C2214yescomplemented
74#1854C2214yescomplemented
75#1853C2211yescharacteristic
76#1852C2212yes—
77#1834C2212yescomplemented
78#1825C2214yescomplemented
79#1825C2214yescomplemented
80#1825C2214yescomplemented
81#1824C2211yescharacteristic
82#1824C2212yes—
83#1823C2212yes—
84#1823C2212yes—
85#1823C2212yes—
86#1808C2214yescomplemented
87#1802C2212yescomplemented
88#1797C2211yescharacteristic
89#1779C2212yes—
90C4×#106C2212yescomplemented
91C4×#102C2212yescomplemented
92C4×C22⋊C8C2211yescharacteristic
93#1735C2214yescomplemented
94#1735C2214yescomplemented
95C23×D4D411yescharacteristic
96C24×C4C2312yes—
97C23×C8C2314yes—
98#555—28no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#124—22no—
101C2×C22⋊Q8—21no—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2311yescharacteristic
105C2×#121C2312yes—
106C2×#121C2312yes—
107C2×#120—22no—
108C2×C4×D4—28no—
109C2×C4×D4—21no—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D412yes—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×#117—22no—
118C2×#117—22no—
119C2×#117—21no—
120C2×#117—28no—
121C2×#117—44no—
122C2×#117C2312yes—
123C2×#117C2314yes—
124C2×#117C2314yes—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4D412yes—
135C22×C42—22no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138#481—24no—
139#480—44no—
140C8×D4—28no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2312yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×C22⋊C8—44no—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C4×C8C2312yes—
157#397—22no—
158#394—22no—
159#394—22no—
160#394—22no—
161#393—22no—
162#392—22no—
163#391—21no—
164#390—21no—
165#390—22no—
166#390—24no—
167#390—22no—
168#390—21no—
169#390—22no—
170#390C2311yescharacteristic
171#389—21no—
172#389C2311yescharacteristic
173#389C2312yes—
174#388—22no—
175#388—24no—
176#388—24no—
177#388—24no—
178#388—24no—
179#388—22no—
180#388—22no—
181#388—21no—
182#387—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—21no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#382—21no—
193#381—22no—
194#380—21no—
195#380—22no—
196#380—22no—
197#380—21no—
198#380C2312yes—
199#380C2312yes—
200#380C2312yes—
201#380C2314yes—
202#380C2314yes—
203#380C2312yes—
204#380C2312yes—
205#380C2311yescharacteristic
206#379—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—24no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C2×#95—22no—
220C2×#95—21no—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236C25C2×D412yes—
237C22×Q8—21no—
238C22×D4—24no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—24no—
243C22×D4—42no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2×D412yes—
246C22×D4C2×D411yescharacteristic
247C22×D4C2411yescharacteristic
248C23×C4—21no—
249C23×C4—24no—
250C23×C4—28no—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—44no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2412yes—
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C22×C8—28no—
265C22×C8—44no—
266C22×C8—44no—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271#124—48no—
272C22⋊Q8—82no—
273#121—28no—
274#121—22no—
275#121—22no—
276#121—44no—
277#121—44no—
278#121—44no—
279#120—84no—
280C4×D4—24no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—44no—
287C4×D4#4118yes—
288#117—28no—
289#117—28no—
290#117—28no—
291#117—44no—
292#117—44no—
293#117—48no—
294#117—48no—
295#117—44no—
296#117—44no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4#4114yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—48no—
348C2×C22⋊C4—84no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—48no—
362C2×C42—44no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—44no—
368#107—44no—
369#106—24no—
370#106—42no—
371C4⋊C8—24no—
372#102—24no—
373#102—42no—
374#102—44no—
375C22⋊C8—28no—
376C22⋊C8—22no—
377C22⋊C8—22no—
378C22⋊C8—24no—
379C22⋊C8—44no—
380C22⋊C8—44no—
381C4×C8—24no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—22no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—24no—
399C24—21no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24C22×D412yes—
404C24C22×D412yes—
405C24C22×D411yescharacteristic
406C2×Q8—42no—
407C2×Q8—81no—
408C2×Q8—81no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—416no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4#12314yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—41no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4#13514yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4118yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4C2511yescharacteristic
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—88no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—81no—
523C4⋊C4—82no—
524C4⋊C4#12314yes—
525C4⋊C4C2×#4114yes—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—81no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4#12318yes—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—48no—
552C42—48no—
553C42—44no—
554C42—44no—
555C42—42no—
556C42—44no—
557C42—82no—
558C42C2×#4114yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—48no—
577C23—48no—
578C23—48no—
579C23—48no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23C2×#12314yes—
591C23#54612yes—
592C23C2×#13512yes—
593C23#57612yes—
594C23C23×D411yescharacteristic
595Q8—162no—
596D4—28no—
597D4—48no—
598D4—44no—
599D4—42no—
600D4—42no—
601D4—84no—
602D4—164no—
603D4—164no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4—48no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—44no—
654C2×C4—84no—
655C2×C4—88no—
656C2×C4—84no—
657C2×C4C2×#12312yes—
658C2×C4C2×#12314yes—
659C2×C4C2×#12312yes—
660C2×C4#54612yes—
661C2×C4C2×#13512yes—
662C2×C4C2×#13512yes—
663C2×C4#57512yes—
664C2×C4C22×#4111yescharacteristic
665C2×C4C22×#4112yes—
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C2×C4#58512yes—
668C8—42no—
669C8—44no—
670C8—42no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—816no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22#330514yes—
700C22C22×#12311yescharacteristic
701C22C2×#54611yescharacteristic
702C22#348511yescharacteristic
703C22C22×#13511yescharacteristic
704C22C2×#57511yescharacteristic
705C22C2×#57611yescharacteristic
706C22#358611yescharacteristic
707C4—22no—
708C4—24no—
709C4—22no—
710C4—22no—
711C4—42no—
712C4—41no—
713C4—44no—
714C4—81no—
715C4—82no—
716C4—82no—
717C4—81no—
718C4—82no—
719C4—84no—
720C4—81no—
721C4#309114yes—
722C4#347512yes—
723C4#351312yes—
724C2—24no—
725C2—44no—
726C2—82no—
727C2—161no—
728C2#3615511yescharacteristic
729C2C2×#309111yescharacteristic
730C2#6052111yescharacteristic
731C2C2×#330511yescharacteristic
732C2#6115911yescharacteristic
733C2#6170111yescharacteristic
734C2#6174011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36160C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2354C2semidirect
3. #33149C2semidirect
4. #32980C2semidirect
5. #27960C2semidirect
6. #27644C2semidirect
7. #27602C2semidirect
8. #26299C2semidirect
9. #26209C2semidirect
10. #24968C2semidirect
11. #24942C2semidirect
12. #24550C2semidirect
13. #22707C2semidirect
14. #22489C2semidirect
15. D4×C22⋊C8C2semidirect
16. C4×#481C2semidirect
17. #20699C2semidirect
52. #2099C22semidirect
53. #2083C22semidirect
54. #2077C22semidirect
55. #2076C22semidirect
56. #2075C22semidirect
57. #2071C22semidirect
58. #2033C22semidirect
59. #1990C22semidirect
60. #1987C22semidirect
62. #1947C22semidirect
63. #1947C22semidirect
64. #1926C22semidirect
65. #1921C22semidirect
66. #1919C22semidirect
67. #1893C22semidirect
68. #1875C22semidirect
69. #1875C22semidirect
72. #1854C22semidirect
73. #1854C22semidirect
74. #1854C22semidirect
77. #1834C22semidirect
78. #1825C22semidirect
79. #1825C22semidirect
80. #1825C22semidirect
86. #1808C22semidirect
87. #1802C22semidirect
90. C4×#106C22semidirect
91. C4×#102C22semidirect
93. #1735C22semidirect
94. #1735C22semidirect

Maximal subgroups

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

subgroupindex
#361602
C2×#23542
#331492
#329802
#279602
#276442
#276022
#262992
#262092
#249682
#249422
#245502
#227072
#224892
D4×C22⋊C82
C4×#4812
#206992