Finite Group Databasebeta

← #6434315

Subgroups — #6434315

Order 512.

1800 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35338C211yesmaximal, characteristic, complemented
2#35266C211yesmaximal, characteristic, complemented
3#30354C214yesmaximal, complemented
4#30348C214yesmaximal, complemented
5#30346C214yesmaximal, complemented
6#30340C214yesmaximal, complemented
7#23613C211yesmaximal, characteristic, complemented
8#23612C211yesmaximal, characteristic, complemented
9#23551C211yesmaximal, characteristic, complemented
10#23541C211yesmaximal, characteristic, complemented
11#23540C211yesmaximal, characteristic
12#23506C211yesmaximal, characteristic, complemented
13#23454C211yesmaximal, characteristic, complemented
14C2×#2012C211yesmaximal, characteristic
15#21233C211yesmaximal, characteristic
16#21231C211yesmaximal, characteristic, complemented
17#21089C211yesmaximal, characteristic, complemented
18#21088C211yesmaximal, characteristic, complemented
19#20430C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C23×D8C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×#405C2211yescharacteristic
37C2×#405C2211yescharacteristic
38#2861C2214yescomplemented
39#2860C2214yescomplemented
40#2644C2214yescomplemented
41#2640C2214yescomplemented
42#2592—22no—
43#2592—22no—
44#2561—24no—
45#2513—22no—
46#2512—22no—
47#2480—24no—
48#2317C2211yescharacteristic
49#2316C2211yescharacteristic
50#2310C2211yescharacteristic
51#2310C2211yescharacteristic
52#2309C2211yescharacteristic
53#2309C2211yescharacteristic
54#2309C2211yescharacteristic
55#2309C2211yescharacteristic
56#2296C2211yescharacteristic
57C2×#390C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2068C2214yescomplemented
60#2067C2214yescomplemented
61#2066C2214yescomplemented
62#2065C2214yescomplemented
63#2059C2212yescomplemented
64#2059C2212yescomplemented
65#2058C2212yescomplemented
66#2057C2212yescomplemented
67#2055C2214yescomplemented
68#2055C2214yescomplemented
69#2054C2214yes—
70#2054C2214yes—
71#2042C2214yescomplemented
72#2042C2214yescomplemented
73#2025C2214yescomplemented
74#2023C2214yescomplemented
75#2012C2212yes—
76#2012C2212yes—
77#2012C2212yes—
78#2012C2212yes—
79#1973C2214yescomplemented
80#1972C2214yes—
81#1970C2214yes—
82#1968C2214yescomplemented
83#1936C2214yescomplemented
84#1935C2214yescomplemented
85#1934C2214yescomplemented
86#1933C2214yescomplemented
87#1768C2212yescomplemented
88#1768C2212yescomplemented
89#1767C2212yescomplemented
90C4×C4⋊C8C2212yescomplemented
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—42no—
102C22×SD16—42no—
103C22×SD16C2312yes—
104C22×SD16C2312yes—
105C22×D8—24no—
106C22×D8—42no—
107C22×D8—42no—
108C22×D8C2312yes—
109C22×D8C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#533—42no—
115#533—42no—
116C2×C4⋊Q8C2312yes—
117C2×#126—24no—
118C2×#125C2312yes—
119C2×#124—22no—
120C2×#124—42no—
121C2×#124—42no—
122C2×#124C2312yes—
123C2×#124C2312yes—
124C2×C22⋊Q8—22no—
125C2×C4×Q8—22no—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C42C2311yescharacteristic
136#427—42no—
137#427—42no—
138C2×#107C2312yes—
139C2×#107C2312yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142#424—24no—
143#424—24no—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148#419—42no—
149#419—42no—
150#418—42no—
151#418—42no—
152#418—42no—
153#418—42no—
154C2×#103—24no—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#102—24no—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166#405—42no—
167#405—42no—
168C2×#97C2312yes—
169C2×#97C2312yes—
170C2×C4×C8C2312yes—
171C2×C4×C8C2312yes—
172#397—24no—
173#397—24no—
174#397—24no—
175#397—24no—
176#397—24no—
177#396—24no—
178#393—24no—
179#393—24no—
180#391—21no—
181#391—21no—
182#391—22no—
183#391C2314yes—
184#391C2314yes—
185#390—21no—
186#390—21no—
187#390—22no—
188#390—21no—
189#390—22no—
190#390—21no—
191#390—22no—
192#390C2314yes—
193#390C2314yes—
194#390C2314yes—
195#390C2314yes—
196#390C2312yes—
197#390C2312yes—
198#389—22no—
199#389C2312yes—
200#389C2312yes—
201#389C2314yes—
202#389C2314yes—
203#388—24no—
204#388—24no—
205#388—24no—
206#388—24no—
207#388—24no—
208#388—24no—
209#388—24no—
210#385—22no—
211#385—22no—
212#384—21no—
213#382—21no—
214#381—22no—
215#381—22no—
216#380—22no—
217#379—22no—
218#379—22no—
219#379—22no—
220#379—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4D412yes—
223C4×C4⋊C4D412yes—
224C4×C22⋊C4D414yes—
225#376C2312yes—
226C43C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2311yescharacteristic
236#336C2311yescharacteristic
237#336C2312yes—
238#336C2312yes—
239C25—21no—
240C25—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8C2412yes—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4C2412yes—
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4C2×D412yes—
269C23×C4C2411yescharacteristic
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16—42no—
273C2×SD16—42no—
274C2×SD16—42no—
275C2×SD16—42no—
276C2×SD16—416no—
277C2×D8—24no—
278C2×D8—24no—
279C2×D8—42no—
280C2×D8—42no—
281C2×D8—42no—
282C2×D8—42no—
283C2×D8—416no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8#4114yes—
295C22×C8#4114yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2411yescharacteristic
299C22×C8C2411yescharacteristic
300C22×C8C2411yescharacteristic
301C22×C8C2411yescharacteristic
302C4⋊Q8—42no—
303C4⋊Q8—42no—
304#126—88no—
305#125—42no—
306#125—42no—
307#124—42no—
308#124—42no—
309#124—42no—
310#124—42no—
311#124—48no—
312C22⋊Q8—48no—
313C4×Q8—48no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—48no—
361C2×C22⋊C4—84no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42C2×D412yes—
371C2×C42C2×D414yes—
372C2×C42C2×D414yes—
373C2×C42C2412yes—
374C2×C42C2412yes—
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377#107—42no—
378#107—42no—
379#106—42no—
380#106—42no—
381C4⋊C8—24no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384C4⋊C8—24no—
385#103—24no—
386#103—24no—
387#103—44no—
388#103—44no—
389#102—24no—
390#102—24no—
391#102—44no—
392#102—44no—
393#102—44no—
394#102—44no—
395#97—42no—
396#97—42no—
397C4×C8—42no—
398C4×C8—42no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95C2×D414yes—
408#95C2×D414yes—
409C24—22no—
410C24—22no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—82no—
418C24—82no—
419C24—82no—
420C24—82no—
421C24C22×D411yescharacteristic
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—82no—
437C2×Q8—82no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—44no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—82no—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+412yes—
490C22×C4C2511yescharacteristic
491SD16—48no—
492SD16—48no—
493D8—48no—
494D8—48no—
495M16—44no—
496M16—44no—
497C2×C8—28no—
498C2×C8—28no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8C2×#4112yes—
514C2×C8C2×#4112yes—
515C2×C8C2×#4112yes—
516C2×C8C2×#4112yes—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—82no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—24no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—22no—
565C42—44no—
566C42—44no—
567C42—42no—
568C42—42no—
569C42—42no—
570C42—42no—
571C23—22no—
572C23—22no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—88no—
584C23—88no—
585C23—84no—
586C23—84no—
587C23—162no—
588C23—162no—
589C23—162no—
590C23—162no—
591C23—164no—
592C23—164no—
593C23#53212yes—
594C23#53512yes—
595C23#53712yes—
596C23C23×D411yescharacteristic
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—88no—
602Q8—84no—
603D4—42no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—42no—
611D4—84no—
612D4—84no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—162no—
646C2×C4#53212yes—
647C2×C4#53512yes—
648C2×C4#53712yes—
649C2×C4#53814yes—
650C2×C4#53814yes—
651C2×C4#54214yes—
652C2×C4#54214yes—
653C2×C4C22×#4111yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4C2×2−1+411yescharacteristic
659C8—22no—
660C8—22no—
661C8—22no—
662C8—22no—
663C8—42no—
664C8—42no—
665C22—22no—
666C22—22no—
667C22—24no—
668C22—24no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—168no—
680C22—168no—
681C22#266214yes—
682C22C2×#53211yescharacteristic
683C22C2×#53511yescharacteristic
684C22C2×#53711yescharacteristic
685C22#358412yes—
686C22#358512yes—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—44no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4#266214yes—
699C4C2×#53811yescharacteristic
700C4C2×#53811yescharacteristic
701C4C2×#54211yescharacteristic
702C4C2×#54211yescharacteristic
703C2—24no—
704C2—162no—
705C2—162no—
706C2C2×#266211yescharacteristic
707C2#6071311yescharacteristic
708C2#6071911yescharacteristic
709C2#6081512yes—
710C2#6092612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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. #35338C2semidirect
2. #35266C2semidirect
3. #30354C2semidirect
4. #30348C2semidirect
5. #30346C2semidirect
6. #30340C2semidirect
7. #23613C2semidirect
8. #23612C2semidirect
9. #23551C2semidirect
10. #23541C2semidirect
12. #23506C2semidirect
13. #23454C2semidirect
16. #21231C2semidirect
17. #21089C2semidirect
18. #21088C2semidirect
19. #20430C2semidirect
38. #2861C22semidirect
39. #2860C22semidirect
40. #2644C22semidirect
41. #2640C22semidirect
59. #2068C22semidirect
60. #2067C22semidirect
61. #2066C22semidirect
62. #2065C22semidirect
63. #2059C22semidirect
64. #2059C22semidirect
65. #2058C22semidirect
66. #2057C22semidirect
67. #2055C22semidirect
68. #2055C22semidirect
71. #2042C22semidirect
72. #2042C22semidirect
73. #2025C22semidirect
74. #2023C22semidirect
79. #1973C22semidirect
82. #1968C22semidirect
83. #1936C22semidirect
84. #1935C22semidirect
85. #1934C22semidirect
86. #1933C22semidirect
87. #1768C22semidirect
88. #1768C22semidirect
89. #1767C22semidirect
90. C4×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#353382
#352662
#303542
#303482
#303462
#303402
#236132
#236122
#235512
#235412
#235402
#235062
#234542
C2×#20122
#212332
#212312
#210892
#210882
#204302