Finite Group Databasebeta

← #6615695

Subgroups — #6615695

Order 512.

1773 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2C4×#554C211yesmaximal, characteristic, complemented
3#33196C211yesmaximal, characteristic, complemented
4#33107C212yesmaximal, complemented
5#29206C211yesmaximal, characteristic, complemented
6#28280C211yesmaximal, characteristic, complemented
7#27628C212yesmaximal, complemented
8#27330C214yesmaximal, complemented
9#26788C214yesmaximal, complemented
10#25012C212yesmaximal, complemented
11#24653C212yesmaximal, complemented
12#24118C212yesmaximal, complemented
13#23061C212yesmaximal, complemented
14#22805C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16C4×#460C211yesmaximal, characteristic, complemented
17#21359C212yesmaximal, complemented
18C2×#555—21no—
19C2×#554C2211yescharacteristic
20C2×#481C2214yes—
21C2×#475C2212yes—
22C2×#476C2212yes—
23C2×#460C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#431C2212yes—
26#2868C2211yescharacteristic
27#2856C2212yes—
28#2563—21no—
29#2537—21no—
30#2537C2212yes—
31#2500—22no—
32#2491—22no—
33#2491C2214yes—
34#2370—21no—
35#2369C2211yescharacteristic
36#2368—22no—
37#2360C2212yes—
38#2355C2211yescharacteristic
39#2354—21no—
40#2354C2212yes—
41#2349—21no—
42D4×C4⋊C4C2211yescharacteristic
43#2345C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#125—21no—
47C4×C22⋊Q8—22no—
48C4×C22⋊Q8C2212yes—
49#2285C2212yes—
50C42×Q8C2211yescharacteristic
51C42×D4—21no—
52C42×D4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54#2077C2214yes—
55#2065C2214yes—
56#2058C2214yes—
57#2035C2214yes—
58#2034C2214yes—
59#2000C2214yes—
60#1990C2212yes—
61#1988C2212yes—
62#1982C2212yes—
63#1982C2211yescharacteristic
64#1968C2214yes—
65#1954C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1934C2214yes—
73#1927C2214yes—
74#1926C2214yes—
75#1926C2214yes—
76#1926C2212yes—
77#1923C2214yes—
78#1920C2214yes—
79#1880C2212yes—
80#1874C2214yes—
81#1853C2214yes—
82#1853C2214yes—
83#1844C2212yes—
84#1842C2211yescharacteristic
85#1803C2212yes—
86#1803C2212yes—
87#1803C2211yescharacteristic
88#1802C2212yes—
89#1797C2212yes—
90C4×#106C2214yes—
91C4×#106C2214yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×SD16C2312yes—
96#555—48no—
97#554—24no—
98#554—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#125—21no—
105C2×#124—24no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×#121—22no—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121C2314yes—
112C2×C4×Q8—21no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C2×#117—22no—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C42—22no—
129C22×C42C2312yes—
130#481—48no—
131#476—28no—
132#475—44no—
133#460—24no—
134C4×SD16—28no—
135#431—28no—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×#106C2312yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2311yescharacteristic
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2314yes—
157#394—24no—
158#394—24no—
159#393—24no—
160#391—21no—
161#391—22no—
162#391C2312yes—
163#390—21no—
164#390—21no—
165#390—22no—
166#390—22no—
167#390C2312yes—
168#390C2314yes—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#388—24no—
176#388—24no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—21no—
192#382—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380C2312yes—
198#380C2314yes—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225C2×#95—22no—
226C43—21no—
227C43C2311yescharacteristic
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2411yescharacteristic
240C22×D4—24no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2412yes—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C2×SD16—24no—
252C2×SD16—24no—
253C2×SD16—42no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—22no—
258C4⋊Q8—42no—
259C4⋊Q8—42no—
260C4⋊Q8—44no—
261C4⋊Q8—82no—
262C4⋊Q8#4114yes—
263#127—22no—
264#127—22no—
265#125—28no—
266#124—416no—
267C22⋊Q8—216no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270#121—48no—
271#121—48no—
272#121—44no—
273#121—44no—
274#121—84no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—41no—
280C4×Q8—82no—
281C4×Q8#4114yes—
282C4×D4—28no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—42no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4SD1618yescomplemented
292#117—48no—
293#117—48no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
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—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—21no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—84no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#106—24no—
362#106—28no—
363#106—28no—
364#106—44no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—22no—
369C4⋊C8—42no—
370C4⋊C8#4114yes—
371#103—24no—
372#103—44no—
373#102—24no—
374#102—24no—
375#102—28no—
376#102—24no—
377#102—44no—
378C22⋊C8—28no—
379C22⋊C8—44no—
380C4×C8—28no—
381C4×C8—24no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—22no—
399C24C22×D412yes—
400C2×Q8—24no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—44no—
407C2×Q8—81no—
408C2×Q8—81no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8C2×#4112yes—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—84no—
428C2×D4—84no—
429C2×D4C2×SD1614yes—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4C2×SD1618yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2511yescharacteristic
469SD16—48no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8C2×#4114yes—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—82no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4C2×SD1614yes—
511C4⋊C4C2×#4112yes—
512C4⋊C4C2×#4114yes—
513C4⋊C4H(5,𝔽2)12yes—
514C22⋊C4—28no—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4C2×SD1618yes—
526C42—24no—
527C42—24no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—21no—
533C42—22no—
534C42—21no—
535C42—41no—
536C42—42no—
537C42—42no—
538C42—48no—
539C42—42no—
540C42—42no—
541C42—44no—
542C42—81no—
543C42—82no—
544C42—82no—
545C42C2×SD1614yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C422−1+412yes—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23C22×SD1614yes—
564C23C23×D411yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—84no—
570Q8—164no—
571D4—28no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—84no—
577D4—84no—
578D4—168no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—216no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—84no—
626C2×C4—88no—
627C2×C4—88no—
628C2×C4—88no—
629C2×C4#52512yes—
630C2×C4#52512yes—
631C2×C4#54712yes—
632C2×C4#54712yes—
633C2×C4C22×SD1614yes—
634C2×C4C22×SD1612yes—
635C2×C4C22×SD1612yes—
636C2×C4C22×SD1612yes—
637C2×C4#57512yes—
638C2×C4#57712yes—
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C2×C4#58511yescharacteristic
645C8—24no—
646C8—44no—
647C8—42no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22C2×#52511yescharacteristic
663C22C2×#54711yescharacteristic
664C22#348811yescharacteristic
665C22C23×SD1611yescharacteristic
666C22C2×#57511yescharacteristic
667C22C2×#57711yescharacteristic
668C22#358511yescharacteristic
669C4—24no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—41no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—82no—
681C4—81no—
682C4—82no—
683C4—84no—
684C4#310314yes—
685C4#333514yes—
686C4#347511yescharacteristic
687C4#348311yescharacteristic
688C4#351111yescharacteristic
689C4#351811yescharacteristic
690C2—44no—
691C2—162no—
692C2#3641111yescharacteristic
693C2C2×#310311yescharacteristic
694C2#6059711yescharacteristic
695C2C2×#333511yescharacteristic
696C2#6125611yescharacteristic
697C2#6195111yescharacteristic
698C2#6199611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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
2. C4×#554C2semidirect
3. #33196C2semidirect
4. #33107C2semidirect
5. #29206C2semidirect
6. #28280C2semidirect
7. #27628C2semidirect
8. #27330C2semidirect
9. #26788C2semidirect
10. #25012C2semidirect
11. #24653C2semidirect
12. #24118C2semidirect
13. #23061C2semidirect
14. #22805C2semidirect
15. #22795C2semidirect
16. C4×#460C2semidirect
17. #21359C2semidirect
291. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#361872
C4×#5542
#331962
#331072
#292062
#282802
#276282
#273302
#267882
#250122
#246532
#241182
#230612
#228052
#227952
C4×#4602
#213592