Finite Group Databasebeta

← #6566082

Subgroups — #6566082

Order 512.

1911 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3341C212yesmaximal, complemented
2#36187C211yesmaximal, characteristic, complemented
3#34321C211yesmaximal, characteristic, complemented
4#29868C211yesmaximal, characteristic, complemented
5#29758C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28572C212yesmaximal, complemented
8#28202C211yesmaximal, characteristic, complemented
9#27650C212yesmaximal, complemented
10#25450C212yesmaximal, complemented
11#25209C214yesmaximal, complemented
12#24812C212yesmaximal, complemented
13#24585C214yesmaximal, complemented
14#23676C211yesmaximal, characteristic, complemented
15#23656C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22489C212yesmaximal, complemented
18C2×#555C2212yes—
19C2×#555C2211yescharacteristic
20#3341—216no—
21C2×#500C2212yes—
22C2×#482C2214yes—
23C2×#482C2214yes—
24C2×#476C2212yes—
25C2×#476C2212yes—
26C2×#466C2214yes—
27C2×C4×D8C2212yes—
28C2×C8×D4C2212yes—
29C2×#427C2214yes—
30#2868C2211yescharacteristic
31#2856C2212yes—
32#2563C2212yes—
33#2537—21no—
34#2537—21no—
35#2500C2214yes—
36#2491—22no—
37#2491—22no—
38#2369C2211yescharacteristic
39#2367—22no—
40#2363C2212yes—
41#2361—21no—
42#2354—21no—
43#2354—21no—
44#2354C2211yescharacteristic
45D4×C4⋊C4C2211yescharacteristic
46#2347—21no—
47#2347C2212yes—
48#2337—22no—
49#2324—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×#121C2212yes—
53#2296C2212yes—
54#2293C2211yescharacteristic
55#2285C2212yes—
56#2274—21no—
57C42×D4C2211yescharacteristic
58#2085C2214yes—
59#2085C2214yes—
60#2084C2212yes—
61#2084C2212yes—
62#2084C2212yes—
63#2083C2212yes—
64#2083C2212yes—
65#2083C2211yescharacteristic
66#2071C2211yescharacteristic
67#2071C2212yes—
68#2071C2212yes—
69#2067C2212yes—
70#2067C2214yes—
71#2065C2214yes—
72#2054C2212yes—
73#2041C2212yes—
74#2041C2211yescharacteristic
75#2011C2211yescharacteristic
76#1990C2212yes—
77#1987C2212yes—
78#1949C2214yes—
79#1947C2214yes—
80#1939C2212yes—
81#1926C2214yes—
82#1925C2214yes—
83#1893C2214yes—
84#1879C2214yes—
85#1875C2214yes—
86#1850C2214yes—
87#1831C2211yescharacteristic
88#1825C2214yes—
89#1804C2214yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1735C2214yes—
95C2×#336C2212yes—
96C22×D8C2312yes—
97C23×C8C2314yes—
98#555—24no—
99#555—28no—
100#555—28no—
101#555D418yescomplemented
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#124—22no—
108C2×#124—24no—
109C2×#121—21no—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121C2314yes—
114C2×#121C2312yes—
115C2×C4×Q8—21no—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C2×#117—21no—
126C2×#117—22no—
127C2×#117—21no—
128C2×#117C2314yes—
129C2×#117C2312yes—
130C2×#117C2312yes—
131C22×C4⋊C4—21no—
132C22×C42C2312yes—
133#500—28no—
134#482—216no—
135#482—216no—
136#476—28no—
137#476—28no—
138#466—216no—
139C4×D8—28no—
140C8×D4—28no—
141#427—216no—
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C4×C8C2312yes—
159#394—24no—
160#394—24no—
161#393—24no—
162#391—21no—
163#391—22no—
164#390—21no—
165#390—22no—
166#390—21no—
167#390—21no—
168#390—22no—
169#390—22no—
170#390C2312yes—
171#390C2314yes—
172#390C2311yescharacteristic
173#390C2311yescharacteristic
174#389—21no—
175#389—21no—
176#389C2312yes—
177#389C2312yes—
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2311yescharacteristic
181#388—22no—
182#388—22no—
183#388—22no—
184#388—21no—
185#388—22no—
186#388—21no—
187#388—22no—
188#388—24no—
189#388—24no—
190#386—22no—
191#386—22no—
192#385—22no—
193#385—22no—
194#385—22no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#383—22no—
202#382C2314yes—
203#380—22no—
204#380—22no—
205#380—22no—
206#380—22no—
207#380—21no—
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229C43C2311yescharacteristic
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2312yes—
240#336C2311yescharacteristic
241#336C2314yes—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×D4—22no—
245C22×D4—24no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2412yes—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C2×D8—24no—
258C2×D8—24no—
259C2×D8—42no—
260C22×C8—28no—
261C22×C8—28no—
262C22×C8—44no—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267C4⋊Q8—22no—
268C4⋊Q8—24no—
269C4⋊Q8—82no—
270C4⋊Q8—82no—
271C4⋊Q8C2×D414yes—
272#127—42no—
273#124—216no—
274#124—416no—
275#121—28no—
276#121—24no—
277#121—28no—
278#121—28no—
279#121—48no—
280#121—84no—
281#121—84no—
282#121C2×D418yes—
283C4×Q8—82no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—24no—
288C4×D4—42no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—84no—
293C4×D4C2×D414yes—
294C4×D4C2×D414yes—
295C4×D4#4114yes—
296#117—24no—
297#117—28no—
298#117—28no—
299#117—28no—
300#117—48no—
301#117—44no—
302#117—44no—
303#117C2×D418yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—22no—
351C2×C42—24no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—24no—
368#107—28no—
369#107—28no—
370#106—28no—
371#106—28no—
372C4⋊C8—24no—
373C4⋊C8—24no—
374C4⋊C8—44no—
375#102—24no—
376#102—28no—
377#102—24no—
378#102—24no—
379#102—28no—
380#102—42no—
381#102—44no—
382C22⋊C8—28no—
383C4×C8—24no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—21no—
399C24—22no—
400C24C22×D412yes—
401C2×Q8—24no—
402C2×Q8—44no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×D4—28no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—48no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4—84no—
429C2×D4—84no—
430C2×D4#12114yes—
431C2×D4C22×D412yes—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D414yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4114yes—
467C22×C4C2511yescharacteristic
468D8—48no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—416no—
478C2×C82−1+414yes—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—82no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4#12114yes—
506C4⋊C4C22×D412yes—
507C4⋊C4C22×D414yes—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—81no—
522C22⋊C4—82no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4#12118yes—
528C42—28no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—22no—
534C42—42no—
535C42—42no—
536C42—41no—
537C42—48no—
538C42—42no—
539C42—82no—
540C42—82no—
541C42—82no—
542C42#13514yes—
543C42C22×D412yes—
544C42C2×#4112yes—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23C2×#12114yes—
566C23C23×D411yescharacteristic
567Q8—84no—
568Q8—164no—
569D4—24no—
570D4—28no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—84no—
577D4—84no—
578D4—88no—
579D4—168no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—88no—
626C2×C4—164no—
627C2×C4C2×#12114yes—
628C2×C4C2×#12112yes—
629C2×C4C2×#12112yes—
630C2×C4C2×#13512yes—
631C2×C4C2×#13512yes—
632C2×C4C2×#13512yes—
633C2×C4#57512yes—
634C2×C4#57612yes—
635C2×C4C23×D411yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×2−1+412yes—
638C2×C4#58512yes—
639C8—24no—
640C8—42no—
641C8—44no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22C22×#12111yescharacteristic
664C22#348511yescharacteristic
665C22#349911yescharacteristic
666C22C22×#13511yescharacteristic
667C22C2×#57511yescharacteristic
668C22C2×#57611yescharacteristic
669C22#358511yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—24no—
675C4—42no—
676C4—44no—
677C4—42no—
678C4—41no—
679C4—42no—
680C4—81no—
681C4—81no—
682C4—84no—
683C4—82no—
684C4—82no—
685C4#303514yes—
686C4#348912yes—
687C4#351512yes—
688C2—44no—
689C2—82no—
690C2—162no—
691C2#3602011yescharacteristic
692C2C2×#303511yescharacteristic
693C2#6042811yescharacteristic
694C2#6170011yescharacteristic
695C2#6174011yescharacteristic
696C2#6247411yescharacteristic
697C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. C2×#3341C2semidirect
2. #36187C2semidirect
3. #34321C2semidirect
4. #29868C2semidirect
5. #29758C2semidirect
6. #28689C2semidirect
7. #28572C2semidirect
8. #28202C2semidirect
9. #27650C2semidirect
10. #25450C2semidirect
11. #25209C2semidirect
12. #24812C2semidirect
13. #24585C2semidirect
14. #23676C2semidirect
15. #23656C2semidirect
16. #22696C2semidirect
17. #22489C2semidirect
101. #555D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33412
#361872
#343212
#298682
#297582
#286892
#285722
#282022
#276502
#254502
#252092
#248122
#245852
#236762
#236562
#226962
#224892