Finite Group Databasebeta

← #6490691

Subgroups — #6490691

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2C2×C4×#125C211yesmaximal, characteristic
3#33151C212yesmaximal
4#33150C212yesmaximal
5#30097C214yesmaximal, complemented
6#29279C214yesmaximal, complemented
7#29227C212yesmaximal, complemented
8#28695C212yesmaximal, complemented
9#27969C211yesmaximal, characteristic
10#27969C211yesmaximal, characteristic
11#27617C212yesmaximal
12#27616C212yesmaximal
13#25819C211yesmaximal, characteristic
14#25819C211yesmaximal, characteristic
15#22251C212yesmaximal, complemented
16#22250C212yesmaximal, complemented
17#20778C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C2×#501C2212yes—
20C2×#501C2212yes—
21#2864C2212yes—
22#2863—21no—
23#2863—21no—
24#2862C2211yescharacteristic
25#2859C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2855—21no—
29#2855—21no—
30#2851C2212yes—
31C4×C4⋊Q8C2211yescharacteristic
32C4×C4⋊Q8C2211yescharacteristic
33C4×#125—21no—
34C4×#125—21no—
35C4×#125—21no—
36C4×#125—21no—
37C4×#125C2212yes—
38C4×#125C2212yes—
39C4×#125C2212yes—
40C4×#125C2212yes—
41C4×#125C2212yes—
42C4×C22⋊Q8—21no—
43C4×C22⋊Q8—21no—
44#2295—21no—
45#2295—21no—
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#387—21no—
49C2×#387—21no—
50#2283C2212yes—
51C2×#382C2211yescharacteristic
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4—21no—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C43C2211yescharacteristic
57#2101C2214yes—
58#2101C2214yes—
59#2101C2214yes—
60#2101C2214yes—
61#2085C2214yes—
62#2085C2214yes—
63#2059C2214yes—
64#2059C2214yes—
65#1993C2212yes—
66#1993C2212yes—
67#1992C2212yes—
68#1992C2212yes—
69#1989C2214yes—
70#1989C2214yes—
71#1949C2214yes—
72#1949C2214yes—
73#1921C2214yes—
74#1921C2214yes—
75#1920C2214yes—
76#1920C2214yes—
77#1919C2214yes—
78#1919C2214yes—
79#1918C2214yes—
80#1918C2214yes—
81#1853C2212yes—
82#1853C2212yes—
83#1852C2212yes—
84#1852C2212yes—
85#1839C2212yes—
86#1836C2212yes—
87#1836C2212yes—
88#1831C2212yes—
89#1824C2212yes—
90#1824C2212yes—
91#1823C2212yes—
92#1823C2212yes—
93#1817C2212yes—
94#1797C2212yes—
95#1767C2212yes—
96#1749C2211yescharacteristic
97#1749C2211yescharacteristic
98#1745C2212yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125—28no—
102C2×#125—21no—
103C2×#125—21no—
104C2×#125C2312yes—
105C2×#125C2312yes—
106C2×#125C2312yes—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C42—21no—
120C22×C42—21no—
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#501—28no—
129#501—28no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2312yes—
138C2×#97C2312yes—
139#391—21no—
140#391—21no—
141#391—21no—
142#391—21no—
143#391—21no—
144#391—21no—
145#391—21no—
146#391—21no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#385—22no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#383—22no—
188#382—21no—
189#382—21no—
190#382C2312yes—
191#382C2312yes—
192#382C2312yes—
193#382C2312yes—
194#382C2312yes—
195#381—22no—
196#380—21no—
197#380—21no—
198#380C2312yes—
199#380C2312yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—22no—
223#376—22no—
224C2×#95—21no—
225C2×#95—21no—
226C2×#95—21no—
227C2×#95—21no—
228C2×#95—21no—
229C2×#95—21no—
230C43D411yescharacteristic
231C43D411yescharacteristic
232C43D411yescharacteristic
233C43D411yescharacteristic
234C43C2311yescharacteristic
235C43C2311yescharacteristic
236C43C2311yescharacteristic
237C43C2311yescharacteristic
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#336C2314yes—
251#336C2312yes—
252#336C2312yes—
253#336C2314yes—
254#336C2314yes—
255C22×Q8—21no—
256C22×Q8—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272#125—28no—
273#125—28no—
274#125—24no—
275#125—24no—
276#125—24no—
277#125—28no—
278#125#4118yes—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C4×Q8—44no—
282C4×Q8—44no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—44no—
339C2×C42—44no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—28no—
369#107—28no—
370#106—28no—
371#106—28no—
372C4⋊C8—28no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375#97—24no—
376#97—24no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403C24C22×D411yescharacteristic
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C42−1+414yes—
464C22×C4C2511yescharacteristic
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
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—44no—
502C4⋊C4—44no—
503C4⋊C4C2×#4114yes—
504C4⋊C4C2×#4114yes—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—44no—
534C42—44no—
535C42—44no—
536C42—44no—
537C42—44no—
538C42—44no—
539C42C2×#4114yes—
540C422−1+414yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23#53612yes—
546C23#57512yes—
547C23#57512yes—
548C23C23×D411yescharacteristic
549Q8—84no—
550Q8—84no—
551C2×C4—28no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
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—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4#52512yes—
619C2×C4#52512yes—
620C2×C4#53612yes—
621C2×C4#53612yes—
622C2×C4#53612yes—
623C2×C4#57512yes—
624C2×C4#57512yes—
625C2×C4C22×#4112yes—
626C2×C4C2×2−1+412yes—
627C2×C4C2×2−1+412yes—
628C8—44no—
629C8—44no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22#324014yes—
637C22C2×#52511yescharacteristic
638C22C2×#53611yescharacteristic
639C22C2×#53611yescharacteristic
640C22C2×#57511yescharacteristic
641C22C2×#57511yescharacteristic
642C22C2×#57511yescharacteristic
643C22C2×#57511yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—41no—
652C4—41no—
653C4—44no—
654C4—44no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4#352314yes—
666C2—24no—
667C2#3524811yescharacteristic
668C2#6056311yescharacteristic
669C2#6056311yescharacteristic
670C2C2×#324011yescharacteristic
671C2#6085411yescharacteristic
672C2#6086111yescharacteristic
673C2#6086611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #30097C2semidirect
6. #29279C2semidirect
7. #29227C2semidirect
8. #28695C2semidirect
15. #22251C2semidirect
16. #22250C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×C4×#1252
#331512
#331502
#300972
#292792
#292272
#286952
#279692
#276172
#276162
#258192
#222512
#222502
#207782