Finite Group Databasebeta

← #6809593

Subgroups — #6809593

Order 512.

1829 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3232C212yesmaximal, complemented
2#36117C211yesmaximal, characteristic
3C2×#2552C211yesmaximal, characteristic, complemented
4#33106C211yesmaximal, characteristic
5#30951C214yesmaximal, complemented
6#30810C214yesmaximal, complemented
7#29640C212yesmaximal, complemented
8#28811C212yesmaximal, complemented
9#27906C211yesmaximal, characteristic
10#27770C212yesmaximal, complemented
11#27648C211yesmaximal, characteristic, complemented
12#26463C212yesmaximal, complemented
13#26043C212yesmaximal, complemented
14#24603C212yesmaximal, complemented
15#24067C212yesmaximal, complemented
16#22599C211yesmaximal, characteristic, complemented
17C4×#483C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C22×#125C2212yes—
20#3232—216no—
21C2×#499C2214yes—
22C2×#488C2214yes—
23C2×#484C2212yes—
24C2×#481C2212yes—
25C2×#481C2212yes—
26C2×#483C2211yescharacteristic
27C2×#480C2212yes—
28C2×#479C2214yes—
29C2×#478C2214yes—
30C2×#431C2212yes—
31#2856C2211yescharacteristic
32#2854C2212yes—
33#2552—21no—
34#2552—22no—
35#2552—21no—
36#2552C2211yescharacteristic
37#2552C2214yes—
38#2552C2212yes—
39#2552C2212yes—
40#2548C2211yescharacteristic
41#2512—21no—
42#2501—21no—
43#2479—22no—
44#2454C2212yes—
45#2448C2212yes—
46#2425—22no—
47C2×#398—22no—
48#2366C2211yescharacteristic
49#2361—21no—
50#2354—21no—
51Q8×C4⋊C4C2211yescharacteristic
52C4×C22⋊Q8C2211yescharacteristic
53C2×#389—21no—
54C2×#389C2212yes—
55C2×#384—21no—
56C2×#384C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#2094C2214yes—
60#2088C2214yes—
61#2083C2212yes—
62#2072C2212yes—
63#2065C2214yes—
64#2059C2214yes—
65#2059C2214yes—
66#2059C2214yes—
67#2042C2214yes—
68#2025C2214yes—
69#2025C2214yes—
70#1982C2212yes—
71#1957C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1941C2214yes—
75#1939C2212yes—
76#1934C2214yes—
77#1927C2212yes—
78#1925C2214yes—
79#1920C2212yes—
80#1916C2214yes—
81#1874C2212yes—
82#1872C2211yescharacteristic
83#1865C2214yes—
84#1862C2214yes—
85#1853C2211yescharacteristic
86#1844C2212yes—
87#1824C2212yes—
88#1823C2211yescharacteristic
89#1817C2212yes—
90#1803C2212yes—
91#1797C2211yescharacteristic
92C4×#106C2212yes—
93C8×C4⋊C4C2212yes—
94C4×#103C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C2×#127C2311yescharacteristic
97#529—28no—
98#529—44no—
99C2×#126—22no—
100C2×#125—28no—
101C2×#125—22no—
102C2×#125—22no—
103C2×#125—44no—
104C2×#125C2314yes—
105C2×#125C2312yes—
106C2×#125C2314yes—
107C2×#124C2312yes—
108C2×#123—22no—
109C2×#123—22no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×#121—21no—
112C2×#121—21no—
113C2×#121C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C2×#117—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#499—216no—
134#488—216no—
135#484—28no—
136#483D418yescomplemented
137#481—28no—
138#481—44no—
139#480—28no—
140#479—216no—
141#478—216no—
142#431—28no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2312yes—
147C2×#106C2314yes—
148C2×#106C2312yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2312yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2311yescharacteristic
161C2×C4×C8C2312yes—
162C2×C4×C8C2314yes—
163#399—22no—
164#399—22no—
165#398—24no—
166#398—24no—
167#398—24no—
168#398—24no—
169#398—22no—
170#397—22no—
171#396—22no—
172#394—22no—
173#391C2311yescharacteristic
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390C2312yes—
180#390C2312yes—
181#390C2312yes—
182#389—21no—
183#389—22no—
184#389—22no—
185#389—24no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2314yes—
192#389C2314yes—
193#389C2311yescharacteristic
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#387—21no—
199#386—22no—
200#386—22no—
201#385—21no—
202#385—22no—
203#384—21no—
204#384—21no—
205#384—22no—
206#384—22no—
207#384—24no—
208#384—21no—
209#384—21no—
210#384C2312yes—
211#384C2312yes—
212#384C2311yescharacteristic
213#380—22no—
214#380C2311yescharacteristic
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D412yes—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—21no—
229C2×#95—21no—
230C2×#95—22no—
231C2×#95D412yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#336C2312yes—
240#336C2314yes—
241#336C2314yes—
242#336C2312yes—
243C22×Q8C2411yescharacteristic
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2412yes—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260#127—24no—
261#126—48no—
262#125—28no—
263#125—28no—
264#125—24no—
265#125—44no—
266#125—44no—
267#125—42no—
268#125—48no—
269#125—48no—
270#124—24no—
271#124—42no—
272#123—48no—
273#123—84no—
274C22⋊Q8—22no—
275C22⋊Q8C2×D414yes—
276#121—24no—
277#121—24no—
278#121—44no—
279#121—82no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—42no—
283C4×D4—24no—
284C4×D4—42no—
285C4×D4—48no—
286C4×D4—44no—
287#117—48no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
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—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—44no—
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—28no—
366#107—28no—
367#106—28no—
368#106—24no—
369#106—44no—
370#106C2×D418yes—
371C4⋊C8—28no—
372C4⋊C8—24no—
373#103—28no—
374#103—24no—
375#103C2×D418yes—
376#102—28no—
377#102—28no—
378#102—28no—
379#102—44no—
380C22⋊C8—24no—
381C22⋊C8—22no—
382C22⋊C8C2×D414yes—
383C4×C8—28no—
384C4×C8—24no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—22no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—22no—
401C24C22×D411yescharacteristic
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C22×D412yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—81no—
418C2×D4—81no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4#13514yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4H(5,𝔽2)12yes—
464C22×C4C2511yescharacteristic
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8C22×D414yes—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—42no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—84no—
507C4⋊C4—82no—
508C4⋊C4—84no—
509C4⋊C4—81no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4C22×D414yes—
513C4⋊C4C22×D412yes—
514C4⋊C42−1+414yes—
515C22⋊C4—24no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C42—22no—
528C42—22no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—44no—
537C42—42no—
538C42—44no—
539C42—44no—
540C42—42no—
541C422−1+414yes—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23D4212yes—
554C23C2×#13512yes—
555C23#57712yes—
556C23C23×D411yescharacteristic
557Q8—24no—
558Q8—42no—
559D4—44no—
560D4—44no—
561D4—88no—
562D4—84no—
563D4—162no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—84no—
617C2×C4—88no—
618C2×C4—88no—
619C2×C4#53612yes—
620C2×C4#53612yes—
621C2×C4D4212yes—
622C2×C4C2×#13512yes—
623C2×C4C2×#13512yes—
624C2×C4#57512yes—
625C2×C4C23×D411yescharacteristic
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×2−1+412yes—
628C2×C4C2×2−1+412yes—
629C8—24no—
630C8—44no—
631C8—42no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22#328414yes—
646C22C2×#53611yescharacteristic
647C22C2×D4211yescharacteristic
648C22#348611yescharacteristic
649C22C22×#13511yescharacteristic
650C22C2×#57511yescharacteristic
651C22C2×#57711yescharacteristic
652C22#358511yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—44no—
659C4—44no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—41no—
665C4—82no—
666C4—81no—
667C4—84no—
668C4—81no—
669C4#323214yescomplemented
670C4#349912yes—
671C4#354412yes—
672C2—24no—
673C2—162no—
674C2#3933711yescharacteristic
675C2C2×#323211yescharacteristic
676C2#6087311yescharacteristic
677C2C2×#328411yescharacteristic
678C2#6107011yescharacteristic
679C2#6181911yescharacteristic
680C2#6183611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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. C2×#3232C2semidirect
3. C2×#2552C2semidirect
5. #30951C2semidirect
6. #30810C2semidirect
7. #29640C2semidirect
8. #28811C2semidirect
10. #27770C2semidirect
11. #27648C2semidirect
12. #26463C2semidirect
13. #26043C2semidirect
14. #24603C2semidirect
15. #24067C2semidirect
16. #22599C2semidirect
17. C4×#483C2semidirect
136. #483D4semidirect
669. C4#3232semidirect

Maximal subgroups

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

subgroupindex
C2×#32322
#361172
C2×#25522
#331062
#309512
#308102
#296402
#288112
#279062
#277702
#276482
#264632
#260432
#246032
#240672
#225992
C4×#4832