Finite Group Databasebeta

← #6492066

Subgroups — #6492066

Order 512.

1696 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#34129C211yesmaximal, characteristic
3#33114C212yesmaximal, complemented
4#33114C212yesmaximal, complemented
5#29923C212yesmaximal, complemented
6#29922C212yesmaximal, complemented
7#26322C212yesmaximal, complemented
8#26320C212yesmaximal, complemented
9#26233C212yesmaximal, complemented
10#26231C212yesmaximal, complemented
11#24892C214yesmaximal, complemented
12#24888C214yesmaximal, complemented
13#21964C211yesmaximal, characteristic, complemented
14#21849C211yesmaximal, characteristic
15#21804C211yesmaximal, characteristic, complemented
16#21362C211yesmaximal, characteristic, complemented
17#21259C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#485C2214yes—
21C2×#485C2214yes—
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#419C2212yes—
25C2×#419C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2628—22no—
29#2568C2212yes—
30#2567—21no—
31#2563—21no—
32#2542C2212yes—
33#2487—22no—
34#2409C2214yes—
35#2354C2212yes—
36#2354C2212yes—
37#2354C2211yescharacteristic
38#2354C2211yescharacteristic
39Q8×C4⋊C4C2211yescharacteristic
40Q8×C4⋊C4C2211yescharacteristic
41#2346—21no—
42#2346—22no—
43#2341—21no—
44#2317—21no—
45#2311—21no—
46#2293—21no—
47#2293—22no—
48#2285C2211yescharacteristic
49#2285C2212yes—
50#2285C2211yescharacteristic
51C42×Q8—21no—
52C4×#117C2211yescharacteristic
53C4×#117C2212yes—
54#2096C2214yes—
55#2095C2214yes—
56#2072C2214yes—
57#2072C2214yes—
58#2036C2214yes—
59#2036C2214yes—
60#2035C2214yes—
61#2035C2214yes—
62#1984C2212yes—
63#1981C2212yes—
64#1981C2212yes—
65#1981C2212yes—
66#1981C2212yes—
67#1981C2212yes—
68#1981C2212yes—
69#1956C2212yes—
70#1956C2212yes—
71#1954C2212yes—
72#1948C2211yescharacteristic
73#1948C2212yes—
74#1948C2211yescharacteristic
75#1948C2212yes—
76#1947C2211yescharacteristic
77#1947C2212yes—
78#1947C2211yescharacteristic
79#1947C2212yes—
80#1928C2214yes—
81#1928C2214yes—
82#1927C2214yes—
83#1927C2214yes—
84#1873C2214yes—
85#1873C2214yes—
86#1870C2214yes—
87#1854C2212yes—
88#1854C2212yes—
89#1809C2214yes—
90#1797C2212yes—
91#1797C2212yes—
92#1787C2214yes—
93C4×#107C2212yes—
94C4×#106C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8C2312yes—
97C22×M16C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C22⋊Q8C2314yes—
105C2×C22⋊Q8C2314yes—
106C4×#41—24no—
107C4×#41—24no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D412yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2311yescharacteristic
116C2×#117—21no—
117C2×#117—22no—
118C2×#117—21no—
119C2×#117D411yescharacteristic
120C2×#117D412yes—
121C2×#117C2312yes—
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#485—48no—
130#485—48no—
131#431—24no—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#107C2312yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138#423—24no—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143#419—44no—
144#419—44no—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C4×C8C2312yes—
157#400—24no—
158#399—24no—
159#395—24no—
160#393—24no—
161#391—22no—
162#391C2314yes—
163#391C2314yes—
164#390—22no—
165#390—21no—
166#390—21no—
167#390—22no—
168#390—22no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#386—24no—
184#385—24no—
185#384—22no—
186#384C2312yes—
187#384C2312yes—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#382—22no—
193#382—22no—
194#382—21no—
195#382—21no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—21no—
202#380—21no—
203#380C2312yes—
204#380C2312yes—
205#380C2312yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226#376—22no—
227#376—21no—
228C43—22no—
229C43—21no—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2314yes—
238C2×#41—24no—
239C2×#41—24no—
240C2×#41—42no—
241C22×Q8—22no—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—42no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8—42no—
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2414yes—
250C22×D4C2×D411yescharacteristic
251C22×D4C2×D412yes—
252C23×C4—22no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C2×M16—24no—
260C2×M16—42no—
261C2×M16—42no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C4×Q8—24no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286#117—24no—
287#117—28no—
288#117—22no—
289#117—22no—
290#117—24no—
291#117—24no—
292#117—42no—
293#117—42no—
294#117—41no—
295#117—41no—
296#117—44no—
297#117—82no—
298#117—84no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42#4114yes—
363C2×C42#4114yes—
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—24no—
368#107—42no—
369#107—44no—
370#106—24no—
371#106—42no—
372#106—44no—
373C4⋊C8—24no—
374C4⋊C8—24no—
375C4⋊C8—22no—
376C4⋊C8—22no—
377#103—28no—
378#103—28no—
379#103—44no—
380#103—44no—
381#103—44no—
382#103—44no—
383#103—44no—
384#103—44no—
385C22⋊C8—24no—
386C22⋊C8—44no—
387C22⋊C8—44no—
388C4×C8—24no—
389#95—21no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405C24C22×D412yes—
406C24C22×D411yescharacteristic
407#41—416no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—48no—
417C2×Q8—48no—
418C2×Q8—44no—
419C2×Q8—82no—
420C2×Q8—82no—
421C2×Q8—84no—
422C2×Q8—84no—
423C2×Q8#13714yes—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—42no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4#12314yes—
463C22×C4#12314yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2511yescharacteristic
475M16—84no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
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—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
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—84no—
520C4⋊C4—84no—
521C4⋊C4H(5,𝔽2)12yes—
522C4⋊C4H(5,𝔽2)12yes—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—42no—
541C42—42no—
542C42—44no—
543C42—44no—
544C42—44no—
545C42—44no—
546C42—42no—
547C42—44no—
548C42—42no—
549C42—44no—
550C42—42no—
551C42—42no—
552C42#12314yes—
553C42#12314yes—
554C42C2×#4112yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C42C2×#4112yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23C2×#12312yes—
572C23C23×D411yescharacteristic
573Q8—28no—
574Q8—44no—
575Q8—44no—
576Q8—44no—
577Q8—44no—
578Q8—88no—
579D4—48no—
580D4—44no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—48no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—88no—
635C2×C4C2×#12312yes—
636C2×C4C2×#12312yes—
637C2×C4C2×#12312yes—
638C2×C4C2×#12312yes—
639C2×C4C2×#12312yes—
640C2×C4C2×#13714yes—
641C2×C4C2×#13712yes—
642C2×C4#57512yes—
643C2×C4#57712yes—
644C2×C4#57812yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4#58511yescharacteristic
650C2×C4#58511yescharacteristic
651C8—42no—
652C8—42no—
653C8—44no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22C22×#12311yescharacteristic
669C22#348811yescharacteristic
670C22#348811yescharacteristic
671C22C22×#13711yescharacteristic
672C22C2×#57511yescharacteristic
673C22C2×#57711yescharacteristic
674C22C2×#57811yescharacteristic
675C4—22no—
676C4—24no—
677C4—22no—
678C4—22no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—82no—
684C4—81no—
685C4—82no—
686C4—82no—
687C4—81no—
688C4—82no—
689C4—84no—
690C4#309514yes—
691C4#309714yes—
692C4#346011yescharacteristic
693C4#346011yescharacteristic
694C4#350811yescharacteristic
695C4#351011yescharacteristic
696C2—42no—
697C2—82no—
698C2#3525411yescharacteristic
699C2C2×#309511yescharacteristic
700C2C2×#309711yescharacteristic
701C2#6198511yescharacteristic
702C2#6198711yescharacteristic
703C2#6200011yescharacteristic
704C2#6200211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33114C2semidirect
4. #33114C2semidirect
5. #29923C2semidirect
6. #29922C2semidirect
7. #26322C2semidirect
8. #26320C2semidirect
9. #26233C2semidirect
10. #26231C2semidirect
11. #24892C2semidirect
12. #24888C2semidirect
13. #21964C2semidirect
15. #21804C2semidirect
16. #21362C2semidirect
17. #21259C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#341292
#331142
#299232
#299222
#263222
#263202
#262332
#262312
#248922
#248882
#219642
#218492
#218042
#213622
#212592