Finite Group Databasebeta

← #6848568

Subgroups — #6848568

Order 512.

1728 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#36176C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3#32957C211yesmaximal, characteristic, complemented
4#32182C214yesmaximal, complemented
5#32100C214yesmaximal, complemented
6#29624C212yesmaximal
7#28416C212yesmaximal, complemented
8#27757C212yesmaximal, complemented
9#27589C212yesmaximal, complemented
10#25856C211yesmaximal, characteristic, complemented
11#25278C212yesmaximal, complemented
12#24630C212yesmaximal, complemented
13#24509C212yesmaximal
14#24209C211yesmaximal, characteristic
15#22603C211yesmaximal, characteristic, complemented
16#21811C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#487C2212yes—
22C2×#477C2212yes—
23C2×#461C2212yes—
24C2×#448C2211yescharacteristic
25C2×#448C2212yes—
26C2×C4×SD16C2212yes—
27C2×#432C2212yes—
28#2857C2211yescharacteristic
29#2854C2212yes—
30#2571C2211yescharacteristic
31#2563C2211yescharacteristic
32#2546—21no—
33#2546—21no—
34#2546—22no—
35#2546C2212yes—
36#2546C2212yes—
37#2546C2214yes—
38#2532—22no—
39#2513C2212yes—
40#2508C2212yes—
41#2495—21no—
42#2488—22no—
43#2486—21no—
44C2×#398—22no—
45#2369C2211yescharacteristic
46#2354C2211yescharacteristic
47#2352—21no—
48#2346—21no—
49C4×#121C2211yescharacteristic
50C2×#391C2212yes—
51C2×#389—21no—
52C2×#389—21no—
53#2283C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2097C2214yes—
56#2074C2214yes—
57#2068C2214yes—
58#2055C2214yes—
59#2043C2214yes—
60#2043C2212yes—
61#2043C2214yes—
62#2043C2214yes—
63#2013C2214yes—
64#2013C2214yes—
65#2013C2212yes—
66#1981C2214yes—
67#1981C2214yes—
68#1981C2212yes—
69#1973C2214yes—
70#1960C2214yes—
71#1960C2214yes—
72#1956C2212yes—
73#1939C2212yes—
74#1935C2214yes—
75#1928C2214yes—
76#1927C2214yes—
77#1917C2214yes—
78#1880C2212yes—
79#1874C2211yescharacteristic
80#1865C2212yes—
81#1863C2214yes—
82#1853C2211yescharacteristic
83#1852C2212yes—
84#1817C2212yes—
85#1803C2212yes—
86#1797C2211yescharacteristic
87#1787C2212yes—
88#1787C2212yes—
89#1787C2211yescharacteristic
90#1787C2212yes—
91#1787C2212yes—
92#1770C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C23×Q8C2311yescharacteristic
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×#123—22no—
112C2×C22⋊Q8—21no—
113C2×#121C2311yescharacteristic
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—44no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×#117—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#487—44no—
134#477—44no—
135#461—44no—
136#448—24no—
137#448—44no—
138C4×SD16—44no—
139#432—44no—
140C2×#107C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#102C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×C22⋊C8C2311yescharacteristic
160C2×#97C2314yes—
161C2×C4×C8C2312yes—
162#400—22no—
163#400—22no—
164#398—22no—
165#398—24no—
166#398—24no—
167#398—24no—
168#398—24no—
169#397—22no—
170#397—22no—
171#393—22no—
172#391—21no—
173#391—22no—
174#391—22no—
175#391—22no—
176#391C2314yes—
177#391C2314yes—
178#390C2312yes—
179#390C2312yes—
180#390C2311yescharacteristic
181#389—21no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—24no—
187#389—24no—
188#389—22no—
189#389—21no—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2311yescharacteristic
194#388—22no—
195#388—22no—
196#388—22no—
197#388—21no—
198#388—22no—
199#388—22no—
200#387—22no—
201#386—22no—
202#386—21no—
203#384C2311yescharacteristic
204#383—22no—
205#382—21no—
206#382—22no—
207#382C2312yes—
208#382C2312yes—
209#381—22no—
210#380—21no—
211#380—21no—
212#380—21no—
213#380—21no—
214#380C2312yes—
215#380C2311yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D412yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—21no—
228#376—22no—
229C2×#95—22no—
230C2×#95—22no—
231C2×#95D412yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—24no—
244C22×Q8—24no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8—44no—
248C22×Q8—44no—
249C22×Q8#4114yes—
250C22×Q8C2414yes—
251C22×Q8C2412yes—
252C22×D4—21no—
253C22×D4C2411yescharacteristic
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×SD16—24no—
264C2×SD16—42no—
265C2×SD16—42no—
266C2×SD16—44no—
267C2×SD16—44no—
268C2×SD16—44no—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C4⋊Q8—28no—
273C4⋊Q8—28no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—48no—
281C4⋊Q8—48no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284#123—84no—
285C22⋊Q8—82no—
286#121—22no—
287#121—22no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—42no—
291C4×Q8—42no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×D4—24no—
295C4×D4—44no—
296#117—44no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
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⋊C4—22no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—44no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362#107—44no—
363#106—42no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366#103—24no—
367#103—42no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—42no—
373#103—44no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378#102—42no—
379C22⋊C8—22no—
380C22⋊C8—22no—
381C22⋊C8—24no—
382#97—44no—
383C4×C8—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95—22no—
400#95—22no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405C24—21no—
406C24C22×D411yescharacteristic
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—48no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—48no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—42no—
426C2×Q8—44no—
427C2×Q8—81no—
428C2×Q8—81no—
429C2×Q8—88no—
430C2×Q8—88no—
431C2×Q8C2×#4114yes—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—42no—
460C22×C4—41no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4#13714yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C2×#4112yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4C2511yescharacteristic
485SD16—48no—
486SD16—88no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—81no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4#13714yes—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—81no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—44no—
548C42—44no—
549C42—44no—
550C42—42no—
551C42—82no—
552C42H(5,𝔽2)14yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23#54612yes—
565C23C2×#13712yes—
566C23#57712yes—
567C23C23×D411yescharacteristic
568Q8—24no—
569Q8—24no—
570Q8—48no—
571Q8—42no—
572Q8—42no—
573Q8—44no—
574Q8—44no—
575Q8—44no—
576Q8—44no—
577Q8—88no—
578Q8—162no—
579D4—42no—
580D4—42no—
581D4—84no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4—84no—
636C2×C4#53512yes—
637C2×C4#53512yes—
638C2×C4#54612yes—
639C2×C4C2×#13714yes—
640C2×C4C2×#13712yes—
641C2×C4C2×#13712yes—
642C2×C4C2×#13712yes—
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×H(5,𝔽2)12yes—
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4#58512yes—
647C8—42no—
648C8—42no—
649C8—44no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22#332914yes—
664C22C2×#53511yescharacteristic
665C22C2×#54611yescharacteristic
666C22#348711yescharacteristic
667C22C22×#13711yescharacteristic
668C22C22×#13711yescharacteristic
669C22C2×#57711yescharacteristic
670C22#358611yescharacteristic
671C4—22no—
672C4—22no—
673C4—22no—
674C4—24no—
675C4—44no—
676C4—41no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—84no—
681C4—81no—
682C4—82no—
683C4—82no—
684C4—81no—
685C4—82no—
686C4—81no—
687C4#321814yes—
688C4#350512yes—
689C4#354212yes—
690C2—24no—
691C2—161no—
692C2#3946711yescharacteristic
693C2C2×#321811yescharacteristic
694C2#6081311yescharacteristic
695C2C2×#332911yescharacteristic
696C2#6115711yescharacteristic
697C2#6189311yescharacteristic
698C2#6190611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2546C2semidirect
3. #32957C2semidirect
4. #32182C2semidirect
5. #32100C2semidirect
7. #28416C2semidirect
8. #27757C2semidirect
9. #27589C2semidirect
10. #25856C2semidirect
11. #25278C2semidirect
12. #24630C2semidirect
15. #22603C2semidirect
16. #21811C2semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×#25462
#329572
#321822
#321002
#296242
#284162
#277572
#275892
#258562
#252782
#246302
#245092
#242092
#226032
#218112
C4×#4482