Finite Group Databasebeta

← #6513713

Subgroups — #6513713

Order 512.

1974 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34978C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic
3#29258C212yesmaximal, complemented
4#29256C212yesmaximal, complemented
5#28286C212yesmaximal, complemented
6#28279C212yesmaximal, complemented
7#26530C212yesmaximal, complemented
8#26529C212yesmaximal, complemented
9#26467C212yesmaximal, complemented
10#26460C212yesmaximal, complemented
11#25976C214yesmaximal, complemented
12#25966C214yesmaximal, complemented
13#21974C211yesmaximal, characteristic, complemented
14#21964C211yesmaximal, characteristic, complemented
15#21803C211yesmaximal, characteristic
16#21362C211yesmaximal, characteristic, complemented
17#21260C211yesmaximal, characteristic
18C22×#127C2211yescharacteristic
19C2×#518—21no—
20C2×#478C2214yescomplemented
21C2×#474C2214yescomplemented
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#418C2212yes—
25C2×#418C2212yes—
26#2612—22no—
27#2571C2211yescharacteristic, complemented
28#2552C2211yescharacteristic, complemented
29#2552C2211yescharacteristic, complemented
30#2551C2211yescharacteristic, complemented
31#2441—21no—
32#2441C2212yes—
33#2439—21no—
34#2439C2212yes—
35#2396—22no—
36#2366—21no—
37#2366—21no—
38#2366C2212yes—
39#2366C2212yes—
40D4×C22⋊C4—22no—
41D4×C22⋊C4C2214yes—
42#2328—22no—
43#2309—21no—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8C2212yescomplemented
46C4×#127C2212yes—
47C4×#127C2212yes—
48C4×#125—21no—
49C4×#125C2212yescomplemented
50#2285C2211yescharacteristic
51#2285C2211yescharacteristic
52C4×#117C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2094C2214yescomplemented
55#2092C2214yescomplemented
56#2067C2214yescomplemented
57#2067C2214yescomplemented
58#2067C2214yescomplemented
59#2065C2214yescomplemented
60#2025C2214yescomplemented
61#2023C2214yescomplemented
62#1982C2212yes—
63#1982C2212yes—
64#1982C2212yes—
65#1982C2212yes—
66#1982C2212yes—
67#1982C2212yes—
68#1957C2212yes—
69#1957C2212yes—
70#1948C2211yescharacteristic, complemented
71#1948C2211yescharacteristic, complemented
72#1948C2212yescomplemented
73#1948C2212yescomplemented
74#1947C2212yescomplemented
75#1947C2212yescomplemented
76#1947C2211yescharacteristic, complemented
77#1947C2211yescharacteristic, complemented
78#1939C2212yescomplemented
79#1939C2212yescomplemented
80#1864C2214yescomplemented
81#1864C2214yescomplemented
82#1864C2214yescomplemented
83#1862C2214yescomplemented
84#1854C2212yes—
85#1854C2212yes—
86#1851C2214yes—
87#1842C2212yescomplemented
88#1831C2212yescomplemented
89#1809C2214yes—
90#1804C2214yescomplemented
91#1802C2214yescomplemented
92#1786C2214yes—
93C4×#107C2212yescomplemented
94C4×#106C2212yescomplemented
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×M16C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×#127—21no—
102C2×#127—21no—
103C2×#127—24no—
104C2×#127Q818yescomplemented
105C2×#127C2312yes—
106C2×#127C2312yes—
107C2×#125—21no—
108C2×#125C2312yes—
109#518—48no—
110C2×C4×Q8—22no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4C2314yes—
118C2×C4×D4C2314yes—
119C2×#117—21no—
120C2×#117—21no—
121C2×#117D411yescharacteristic
122C2×#117D411yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C22⋊C4—24no—
126C22×C42C2311yescharacteristic
127#478—48no—
128#474—48no—
129#431—24no—
130C2×#107C2312yes—
131C2×#107C2312yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136#423—24no—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143#418—44no—
144#418—44no—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C4×C8C2312yes—
155#400—22no—
156#399—22no—
157#398—22no—
158#398—22no—
159#396—22no—
160#396—22no—
161#395—22no—
162#393—22no—
163#391—22no—
164#391—22no—
165#390—22no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390—22no—
170#390—22no—
171#390C2314yes—
172#390C2314yes—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#386—24no—
188#386—24no—
189#386—24no—
190#386—24no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#382—21no—
198#382—21no—
199#382—21no—
200#382C2312yes—
201#380—21no—
202#380—21no—
203#380C2312yes—
204#380C2312yes—
205#379—24no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225#376—21no—
226#376D412yes—
227C43—21no—
228C43C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2314yes—
237C25—22no—
238C22×Q8—22no—
239C22×D4—28no—
240C22×D4—28no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—24no—
247C22×D4—42no—
248C22×D4—42no—
249C22×D4—42no—
250C22×D4—42no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4—82no—
254C22×D4—82no—
255C22×D4C2×Q818yes—
256C22×D4C2414yes—
257C23×C4—24no—
258C23×C4—24no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×M16—24no—
263C2×M16—42no—
264C2×M16—42no—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C4⋊Q8—24no—
269C4⋊Q8—42no—
270C4⋊Q8—44no—
271#127—28no—
272#127—28no—
273#127—24no—
274#127—24no—
275#127C2×Q818yes—
276#127C2×Q818yes—
277#125—24no—
278#125—42no—
279#125—44no—
280C4×Q8—84no—
281C4×D4—48no—
282C4×D4—48no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—48no—
289C4×D4—84no—
290C4×D4—84no—
291#117—24no—
292#117—22no—
293#117—22no—
294#117—42no—
295#117—41no—
296#117—41no—
297#117—44no—
298#117—82no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—416no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×Q814yes—
356C2×C42#4114yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—24no—
362#107—24no—
363#107—44no—
364#106—42no—
365#106—42no—
366#106—44no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369C4⋊C8—22no—
370C4⋊C8—22no—
371C4⋊C8—44no—
372C4⋊C8—44no—
373#102—28no—
374#102—28no—
375#102—44no—
376#102—44no—
377#102—44no—
378#102—44no—
379#102—44no—
380#102—44no—
381C22⋊C8—24no—
382C4×C8—24no—
383#95—21no—
384#95—21no—
385#95—21no—
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#95—22no—
395#95—22no—
396#95—22no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—24no—
402C24—28no—
403C24—44no—
404C24—44no—
405C24—82no—
406C24—82no—
407C24—82no—
408C24—82no—
409C24C22×D411yescharacteristic
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—82no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—432no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4—816no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4C22×Q818yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4C22⋊Q814yes—
463C22×C4C22⋊Q814yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×Q812yes—
474C22×C4C2×#4112yes—
475C22×C4C2511yescharacteristic
476M16—84no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4AGL(1,ℤ/8)14yes—
521C4⋊C4H(5,𝔽2)12yes—
522C4⋊C4H(5,𝔽2)12yes—
523C22⋊C4—24no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—82no—
535C42—24no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42—42no—
547C42—42no—
548C42—44no—
549C42—42no—
550C42—42no—
551C42C22⋊Q814yes—
552C42C22⋊Q814yes—
553C42C22×Q812yes—
554C42C22×Q812yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—28no—
564C23—416no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—88no—
572C23—88no—
573C23—164no—
574C23—164no—
575C23—164no—
576C23—164no—
577C23C2×C22⋊Q812yes—
578C23C23×D411yescharacteristic
579Q8—88no—
580D4—28no—
581D4—28no—
582D4—416no—
583D4—416no—
584D4—416no—
585D4—44no—
586D4—44no—
587D4—88no—
588D4—88no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—168no—
637C2×C4—168no—
638C2×C4C2×C22⋊Q812yes—
639C2×C4C2×C22⋊Q812yes—
640C2×C4C2×C22⋊Q812yes—
641C2×C4C2×C22⋊Q812yes—
642C2×C4C2×C22⋊Q812yes—
643C2×C4C2×AGL(1,ℤ/8)14yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4#57512yes—
646C2×C4#57612yes—
647C2×C4#57712yes—
648C2×C4C23×Q811yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4#58511yescharacteristic
653C2×C4#58511yescharacteristic
654C8—42no—
655C8—44no—
656C8—42no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—48no—
665C22—48no—
666C22—48no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—1616no—
672C22C22×C22⋊Q811yescharacteristic
673C22#348811yescharacteristic
674C22#348811yescharacteristic
675C22C22×AGL(1,ℤ/8)11yescharacteristic
676C22C2×#57511yescharacteristic
677C22C2×#57611yescharacteristic
678C22C2×#57711yescharacteristic
679C4—24no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—82no—
689C4—82no—
690C4—81no—
691C4—82no—
692C4—81no—
693C4—82no—
694C4#307714yes—
695C4#308114yes—
696C4#350111yescharacteristic
697C4#350111yescharacteristic
698C4#350811yescharacteristic
699C4#351011yescharacteristic
700C2—42no—
701C2—88no—
702C2#3551511yescharacteristic
703C2C2×#307711yescharacteristic
704C2C2×#308111yescharacteristic
705C2#6197911yescharacteristic
706C2#6198811yescharacteristic
707C2#6199711yescharacteristic
708C2#6199911yescharacteristic
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. #34978C2semidirect
3. #29258C2semidirect
4. #29256C2semidirect
5. #28286C2semidirect
6. #28279C2semidirect
7. #26530C2semidirect
8. #26529C2semidirect
9. #26467C2semidirect
10. #26460C2semidirect
11. #25976C2semidirect
12. #25966C2semidirect
13. #21974C2semidirect
14. #21964C2semidirect
16. #21362C2semidirect
20. C2×#478C22semidirect
21. C2×#474C22semidirect
27. #2571C22semidirect
28. #2552C22semidirect
29. #2552C22semidirect
30. #2551C22semidirect
45. C4×C4⋊Q8C22semidirect
49. C4×#125C22semidirect
54. #2094C22semidirect
55. #2092C22semidirect
56. #2067C22semidirect
57. #2067C22semidirect
58. #2067C22semidirect
59. #2065C22semidirect
60. #2025C22semidirect
61. #2023C22semidirect
70. #1948C22semidirect
71. #1948C22semidirect
72. #1948C22semidirect
73. #1948C22semidirect
74. #1947C22semidirect
75. #1947C22semidirect
76. #1947C22semidirect
77. #1947C22semidirect
78. #1939C22semidirect
79. #1939C22semidirect
80. #1864C22semidirect
81. #1864C22semidirect
82. #1864C22semidirect
83. #1862C22semidirect
87. #1842C22semidirect
88. #1831C22semidirect
90. #1804C22semidirect
91. #1802C22semidirect
93. C4×#107C22semidirect
94. C4×#106C22semidirect
104. C2×#127Q8semidirect

Maximal subgroups

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

subgroupindex
#349782
#342102
#292582
#292562
#282862
#282792
#265302
#265292
#264672
#264602
#259762
#259662
#219742
#219642
#218032
#213622
#212602