Finite Group Databasebeta

← #6435269

Subgroups — #6435269

Order 512.

1807 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35308C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#29935C214yesmaximal, complemented
4#29930C214yesmaximal, complemented
5#29928C212yesmaximal, complemented
6#29927C212yesmaximal, complemented
7#29923C212yesmaximal, complemented
8#29922C212yesmaximal, complemented
9#23732C211yesmaximal, characteristic, complemented
10#23730C211yesmaximal, characteristic, complemented
11#23630C211yesmaximal, characteristic, complemented
12C2×#2072C211yesmaximal, characteristic, complemented
13#23487C212yesmaximal, complemented
14#23485C212yesmaximal
15#21262C212yesmaximal, complemented
16#21175C212yesmaximal, complemented
17#20630C211yesmaximal, characteristic, complemented
18C2×#537C2211yescharacteristic
19C2×#536C2211yescharacteristic
20C2×#536C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2212yes—
23C2×#427C2212yes—
24C2×#424C2212yes—
25C2×#417C2212yes—
26C2×#419C2212yes—
27C2×#416C2212yes—
28C22×#103C2212yes—
29C2×#408C2212yes—
30#2857C2212yes—
31#2850C2212yes—
32#2621C2212yes—
33#2620C2212yes—
34#2429—21no—
35#2429—21no—
36#2427—21no—
37#2427—21no—
38#2425—22no—
39#2425—22no—
40#2355—21no—
41#2355—21no—
42#2354—21no—
43#2354—21no—
44#2352—22no—
45#2352—22no—
46#2317C2211yescharacteristic
47#2314C2214yes—
48#2311C2212yes—
49#2297C2211yescharacteristic
50C2×#391C2211yescharacteristic
51#2285C2212yes—
52#2285C2212yes—
53#2285C2211yescharacteristic
54C2×#384C2211yescharacteristic
55#2274C2212yes—
56#2097C2212yes—
57#2097C2214yes—
58#2096C2211yescharacteristic
59#2096C2212yes—
60#2095C2211yescharacteristic
61#2095C2212yes—
62#2094C2211yescharacteristic
63#2093C2211yescharacteristic
64#2092C2212yes—
65#2072C2212yes—
66#2072C2212yes—
67#2072C2214yes—
68#2072C2212yes—
69#2072C2212yes—
70#2071C2212yes—
71#2071C2212yes—
72#2036C2214yes—
73#2036C2214yes—
74#2036C2214yes—
75#2035C2214yes—
76#2035C2214yes—
77#2035C2214yes—
78#2034C2214yes—
79#2033C2214yes—
80#1983C2214yes—
81#1983C2214yes—
82#1981C2214yes—
83#1981C2214yes—
84#1949C2214yes—
85#1949C2214yes—
86#1948C2214yes—
87#1947C2214yes—
88#1798C2214yes—
89#1797C2212yes—
90#1797C2212yes—
91#1736C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C23×Q8C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#537—44no—
101#536—28no—
102#536—44no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×#124—21no—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×#123—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×#121—22no—
116C2×#121—21no—
117C2×#121C2312yes—
118C4×#41—28no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D411yescharacteristic
121C2×C4×D4D412yes—
122C2×C4×D4D412yes—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×#117—22no—
126C2×#117—22no—
127C2×#117C2312yes—
128C2×#117C2311yescharacteristic
129C2×#117C2312yes—
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#428—44no—
138#427—44no—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#107C2314yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×#106C2314yes—
145#424—28no—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148#419—44no—
149#417—44no—
150#416—44no—
151C2×#103—44no—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162#408—44no—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165#398—22no—
166#398—22no—
167#397—22no—
168#397—22no—
169#396—22no—
170#396—22no—
171#394—22no—
172#394—22no—
173#391—22no—
174#391—21no—
175#391—21no—
176#391C2312yes—
177#391C2311yescharacteristic
178#391C2312yes—
179#391C2311yescharacteristic
180#390—22no—
181#390C2311yescharacteristic
182#390C2311yescharacteristic
183#389—21no—
184#389—21no—
185#389—22no—
186#389—22no—
187#388—24no—
188#388—24no—
189#388—22no—
190#388—22no—
191#387—24no—
192#387—24no—
193#386—24no—
194#386—22no—
195#386—22no—
196#386—21no—
197#386—21no—
198#385—22no—
199#384—21no—
200#384—21no—
201#384C2312yes—
202#384C2312yes—
203#384C2312yes—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#382—22no—
207#382—22no—
208#382C2312yes—
209#382C2312yes—
210#380—22no—
211#380—22no—
212#380—21no—
213#380—21no—
214#380—22no—
215#380—22no—
216#380C2314yes—
217#380C2314yes—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4C2314yes—
225C4×C4⋊C4C2314yes—
226C4×C22⋊C4—24no—
227C4×C22⋊C4—21no—
228C4×C22⋊C4—21no—
229C2×#95—21no—
230C2×#95—21no—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#336C2312yes—
236#336C2311yescharacteristic
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2312yes—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—24no—
245C2×#41—24no—
246C2×#41—22no—
247C2×#41—22no—
248C22×Q8—21no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8C2×D411yescharacteristic
254C22×Q8C2412yes—
255C22×Q8C2411yescharacteristic
256C22×D4—21no—
257C22×D4—22no—
258C22×D4C2×D412yes—
259C22×D4C2×D411yescharacteristic
260C22×D4C2411yescharacteristic
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D412yes—
268C23×C4C2×D412yes—
269C23×C4C2412yes—
270C23×C4C2411yescharacteristic
271C2×M16—24no—
272C2×M16—24no—
273C2×M16—24no—
274C22×C8—24no—
275C22×C8—24no—
276C22×C8—24no—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2414yes—
280C4⋊Q8—28no—
281C4⋊Q8—44no—
282#124—28no—
283#124—44no—
284#123—48no—
285#123—48no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C22⋊Q8—44no—
291C22⋊Q8—44no—
292#121—44no—
293#121—48no—
294#121—44no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—28no—
300C4×D4—28no—
301#117—28no—
302#117—24no—
303#117—24no—
304#117—24no—
305#117—84no—
306#117—84no—
307#117#4118yes—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—24no—
358C2×C42—28no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370#107—42no—
371#107—42no—
372#107—44no—
373#106—42no—
374#106—42no—
375#106—44no—
376C4⋊C8—28no—
377C4⋊C8—28no—
378#103—44no—
379#103—44no—
380#103—42no—
381#103—42no—
382#103—44no—
383#103—44no—
384#103—44no—
385#102—42no—
386#102—42no—
387#102—44no—
388C22⋊C8—44no—
389C22⋊C8—44no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—22no—
405#95—22no—
406C24—21no—
407C24C22×D412yes—
408C24C22×D411yescharacteristic
409#41—48no—
410#41—48no—
411#41—48no—
412#41—48no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—48no—
428C2×Q8—48no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4—82no—
444C2×D4—82no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—41no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4114yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C42−1+412yes—
493C22×C4C2511yescharacteristic
494M16—48no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—48no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4#12314yes—
534C4⋊C4#12314yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—81no—
548C22⋊C4—81no—
549C22⋊C4—84no—
550C22⋊C4—84no—
551C22⋊C4#12318yes—
552C42—22no—
553C42—22no—
554C42—24no—
555C42—24no—
556C42—22no—
557C42—22no—
558C42—44no—
559C42—42no—
560C42—84no—
561C42—84no—
562C42C2×#4118yes—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23C2×#12312yes—
582C23#53512yes—
583C23#53712yes—
584C23C23×D411yescharacteristic
585Q8—44no—
586Q8—44no—
587Q8—42no—
588Q8—44no—
589Q8—42no—
590Q8—84no—
591D4—48no—
592D4—44no—
593D4—42no—
594D4—42no—
595D4—84no—
596D4—164no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—162no—
647C2×C4—164no—
648C2×C4C2×#12314yes—
649C2×C4C2×#12314yes—
650C2×C4C2×#12312yes—
651C2×C4#53512yes—
652C2×C4#53712yes—
653C2×C4#54018yes—
654C2×C4#54218yes—
655C2×C4#57712yes—
656C2×C4#57812yes—
657C2×C4C22×#4112yes—
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4C2×H(5,𝔽2)12yes—
660C2×C4C2×2−1+411yescharacteristic
661C8—42no—
662C8—42no—
663C8—44no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22#262414yes—
685C22C22×#12311yescharacteristic
686C22C2×#53511yescharacteristic
687C22C2×#53711yescharacteristic
688C22C2×#57711yescharacteristic
689C22C2×#57811yescharacteristic
690C22#358511yescharacteristic
691C22#358611yescharacteristic
692C4—22no—
693C4—24no—
694C4—22no—
695C4—24no—
696C4—42no—
697C4—41no—
698C4—41no—
699C4—81no—
700C4—82no—
701C4—81no—
702C4—82no—
703C4—82no—
704C4—81no—
705C4—82no—
706C4—84no—
707C4#262414yes—
708C4C2×#54012yes—
709C4C2×#54212yes—
710C2—24no—
711C2—82no—
712C2—161no—
713C2C2×#262411yescharacteristic
714C2#6052211yescharacteristic
715C2#6052611yescharacteristic
716C2#6082411yescharacteristic
717C2#6082911yescharacteristic
718C2#6092411yescharacteristic
719C2#6092511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
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
2. #34188C2semidirect
3. #29935C2semidirect
4. #29930C2semidirect
5. #29928C2semidirect
6. #29927C2semidirect
7. #29923C2semidirect
8. #29922C2semidirect
9. #23732C2semidirect
10. #23730C2semidirect
11. #23630C2semidirect
12. C2×#2072C2semidirect
13. #23487C2semidirect
15. #21262C2semidirect
16. #21175C2semidirect
17. #20630C2semidirect

Maximal subgroups

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

subgroupindex
#353082
#341882
#299352
#299302
#299282
#299272
#299232
#299222
#237322
#237302
#236302
C2×#20722
#234872
#234852
#212622
#211752
#206302