Finite Group Databasebeta

← #6990255

Subgroups — #6990255

Order 512.

1727 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36314C211yesmaximal, characteristic, complemented
2#35285C211yesmaximal, characteristic
3#32823C212yesmaximal, complemented
4#31294C214yesmaximal, complemented
5#30996C214yesmaximal, complemented
6#30221C212yesmaximal, complemented
7#29801C212yesmaximal, complemented
8#29688C211yesmaximal, characteristic, complemented
9#29555C212yesmaximal, complemented
10#28402C212yesmaximal, complemented
11#27750C212yesmaximal, complemented
12#25855C211yesmaximal, characteristic
13#25279C212yesmaximal, complemented
14#24733C212yesmaximal, complemented
15#24211C211yesmaximal, characteristic
16#23571C211yesmaximal, characteristic, complemented
17#22632C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×#127C2211yescharacteristic
20C2×#486C2212yes—
21C2×#475C2212yescomplemented
22C2×#464C2214yescomplemented
23C2×#460C2212yes—
24C2×#450C2212yes—
25C2×C4×Q16C2212yescomplemented
26#2864C2212yescomplemented
27#2858C2212yes—
28#2858C2212yes—
29#2857C2211yescharacteristic
30#2641—22no—
31#2571C2211yescharacteristic, complemented
32#2568C2211yescharacteristic, complemented
33#2548—21no—
34#2548—21no—
35#2548C2212yes—
36#2548C2212yes—
37#2533—22no—
38#2530—22no—
39#2511—21no—
40#2499—21no—
41#2495C2212yes—
42#2435—22no—
43#2435C2214yes—
44#2421C2212yes—
45#2365C2211yescharacteristic, complemented
46#2362—21no—
47#2356—21no—
48Q8×C4⋊C4C2211yescharacteristic, complemented
49#2309—21no—
50#2309—21no—
51C4×C22⋊Q8C2211yescharacteristic
52#2295C2212yescomplemented
53C4×#117C2211yescharacteristic
54#2092C2214yescomplemented
55#2073C2214yescomplemented
56#2067C2214yescomplemented
57#2065C2214yescomplemented
58#2058C2212yescomplemented
59#2057C2211yescharacteristic, complemented
60#2043C2211yescharacteristic, complemented
61#2043C2212yescomplemented
62#2042C2214yescomplemented
63#2042C2214yescomplemented
64#2042C2214yescomplemented
65#2041C2212yes—
66#2041C2212yes—
67#2041C2212yes—
68#2036C2212yes—
69#2035C2214yes—
70#2024C2211yescharacteristic, complemented
71#2016C2211yescharacteristic, complemented
72#2015C2214yescomplemented
73#2014C2212yes—
74#2001C2212yes—
75#1983C2214yescomplemented
76#1982C2212yes—
77#1968C2214yescomplemented
78#1958C2214yescomplemented
79#1958C2214yescomplemented
80#1957C2212yes—
81#1939C2212yescomplemented
82#1934C2214yescomplemented
83#1926C2214yescomplemented
84#1923C2214yes—
85#1880C2214yescomplemented
86#1831C2212yescomplemented
87#1803C2212yescomplemented
88#1797C2211yescharacteristic
89#1786C2214yes—
90#1786C2212yes—
91C8×C4⋊C4C2212yescomplemented
92C4×#103C2212yescomplemented
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yescomplemented
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×Q16C2312yes—
99C22×SD16C2314yes—
100#533—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127—21no—
105C2×#127—21no—
106C2×#127—44no—
107C2×#127C2312yes—
108C2×#127C2312yes—
109C2×#125—21no—
110C2×#124—22no—
111C2×#124—22no—
112C2×#124C2314yes—
113C2×#124C2314yes—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8C2311yescharacteristic
117C2×C22⋊Q8C2312yes—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×#117—21no—
124C2×#117D411yescharacteristic
125C2×#117D411yescharacteristic
126C22×C42C2311yescharacteristic
127#486—44no—
128#475—44no—
129#464—48no—
130#460—44no—
131#450—44no—
132C4×Q16—44no—
133C2×#107C2312yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#400—22no—
155#400—22no—
156#400—22no—
157#400—22no—
158#397—22no—
159#397—22no—
160#394—24no—
161#394—24no—
162#394—24no—
163#394—24no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#391C2311yescharacteristic
170#390—22no—
171#390—22no—
172#390—22no—
173#390—22no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390—22no—
180#390C2314yes—
181#390C2314yes—
182#389—21no—
183#389—21no—
184#389—22no—
185#389C2311yescharacteristic
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#388—22no—
191#388—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—21no—
199#387—22no—
200#387—22no—
201#385—21no—
202#384—21no—
203#384C2311yescharacteristic
204#383—22no—
205#382—21no—
206#382C2312yes—
207#382C2312yes—
208#380—21no—
209#380C2312yes—
210#380C2311yescharacteristic
211#379—24no—
212#379—24no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225#376—22no—
226#376D412yes—
227C43C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235#336C2311yescharacteristic
236C25—22no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8C2412yes—
240C22×Q8C2411yescharacteristic
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—44no—
247C22×D4—44no—
248C22×D4—42no—
249C22×D4—42no—
250C22×D4—44no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4—44no—
254C22×D4—82no—
255C22×D4—82no—
256C22×D4C2414yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×Q16—24no—
261C2×Q16—42no—
262C2×Q16—42no—
263C2×SD16—28no—
264C2×SD16—44no—
265C2×SD16—44no—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272#127—24no—
273#127—24no—
274#127—44no—
275#127—44no—
276#127—44no—
277#125—44no—
278#124—44no—
279#124—44no—
280#124—44no—
281#124—44no—
282#124—84no—
283#124—84no—
284C22⋊Q8—22no—
285C22⋊Q8—41no—
286C22⋊Q8—42no—
287C22⋊Q8—42no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—42no—
295C4×Q8—44no—
296C4×Q8—44no—
297#117—24no—
298#117—42no—
299#117—44no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—42no—
357#106—44no—
358C4⋊C8—24no—
359C4⋊C8—44no—
360#103—42no—
361#103—42no—
362#103—42no—
363#103—42no—
364#103—44no—
365#103—44no—
366#102—44no—
367#102—44no—
368#102—44no—
369#102—44no—
370#102—44no—
371#102—44no—
372#102—44no—
373C22⋊C8—42no—
374C22⋊C8—42no—
375C4×C8—24no—
376C4×C8—44no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—24no—
399C24—44no—
400C24—44no—
401C24—44no—
402C24—82no—
403C24—82no—
404C24—82no—
405C24—82no—
406C24C22×D411yescharacteristic
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8H(5,𝔽2)12yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4—816no—
445C2×D4—88no—
446C2×D4—88no—
447C2×D4—816no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4#13514yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2511yescharacteristic
476Q16—48no—
477SD16—88no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C22⋊C4—24no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—81no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—24no—
537C42—22no—
538C42—42no—
539C42—42no—
540C42—44no—
541C42—44no—
542C42—44no—
543C42—42no—
544C42—42no—
545C42H(5,𝔽2)14yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—88no—
560C23—88no—
561C23—88no—
562C23—164no—
563C23—164no—
564C23—164no—
565C23—164no—
566C23C2×#13512yes—
567C23C23×D411yescharacteristic
568Q8—24no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—84no—
575Q8—84no—
576D4—44no—
577D4—44no—
578D4—44no—
579D4—44no—
580D4—88no—
581D4—88no—
582D4—88no—
583D4—88no—
584D4—88no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4#53512yes—
635C2×C4#53512yes—
636C2×C4C2×#13512yes—
637C2×C4C2×#13512yes—
638C2×C4#57512yes—
639C2×C4#57712yes—
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×H(5,𝔽2)12yes—
642C2×C4#58512yes—
643C2×C4#58511yescharacteristic
644C8—44no—
645C8—42no—
646C8—42no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—1616no—
662C22C2×#53511yescharacteristic
663C22#348811yescharacteristic
664C22#350011yescharacteristic
665C22C22×#13511yescharacteristic
666C22C2×#57511yescharacteristic
667C22C2×#57711yescharacteristic
668C22#358511yescharacteristic
669C4—22no—
670C4—22no—
671C4—22no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—41no—
677C4—42no—
678C4—82no—
679C4—81no—
680C4—82no—
681C4—81no—
682C4—81no—
683C4—82no—
684C4—82no—
685C4#321114yes—
686C4#354212yes—
687C4#355112yes—
688C2—42no—
689C2—164no—
690C2#4296211yescharacteristic
691C2C2×#321111yescharacteristic
692C2#6081211yescharacteristic
693C2#6195311yescharacteristic
694C2#6199711yescharacteristic
695C2#6253111yescharacteristic
696C2#6258011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36314C2semidirect
3. #32823C2semidirect
4. #31294C2semidirect
5. #30996C2semidirect
6. #30221C2semidirect
7. #29801C2semidirect
8. #29688C2semidirect
9. #29555C2semidirect
10. #28402C2semidirect
11. #27750C2semidirect
13. #25279C2semidirect
14. #24733C2semidirect
16. #23571C2semidirect
17. #22632C2semidirect
21. C2×#475C22semidirect
22. C2×#464C22semidirect
25. C2×C4×Q16C22semidirect
26. #2864C22semidirect
31. #2571C22semidirect
32. #2568C22semidirect
45. #2365C22semidirect
48. Q8×C4⋊C4C22semidirect
52. #2295C22semidirect
54. #2092C22semidirect
55. #2073C22semidirect
56. #2067C22semidirect
57. #2065C22semidirect
58. #2058C22semidirect
59. #2057C22semidirect
60. #2043C22semidirect
61. #2043C22semidirect
62. #2042C22semidirect
63. #2042C22semidirect
64. #2042C22semidirect
70. #2024C22semidirect
71. #2016C22semidirect
72. #2015C22semidirect
75. #1983C22semidirect
77. #1968C22semidirect
78. #1958C22semidirect
79. #1958C22semidirect
81. #1939C22semidirect
82. #1934C22semidirect
83. #1926C22semidirect
85. #1880C22semidirect
86. #1831C22semidirect
87. #1803C22semidirect
91. C8×C4⋊C4C22semidirect
92. C4×#103C22semidirect
94. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#363142
#352852
#328232
#312942
#309962
#302212
#298012
#296882
#295552
#284022
#277502
#258552
#252792
#247332
#242112
#235712
#226322