Finite Group Databasebeta

← #6451363

Subgroups — #6451363

Order 512.

1932 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35674C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic
3#27220C214yesmaximal, complemented
4#27213C214yesmaximal, complemented
5#27201C214yesmaximal, complemented
6#27192C214yesmaximal, complemented
7#23736C211yesmaximal, characteristic, complemented
8#23633C211yesmaximal, characteristic, complemented
9#23618C211yesmaximal, characteristic
10C2×#2013C211yesmaximal, characteristic
11#21259C211yesmaximal, characteristic, complemented
12#21231C211yesmaximal, characteristic
13#21064C211yesmaximal, characteristic, complemented
14#21032C211yesmaximal, characteristic, complemented
15#21027C211yesmaximal, characteristic, complemented
16#20936C211yesmaximal, characteristic
17#20933C211yesmaximal, characteristic
18#20933C211yesmaximal, characteristic
19#20417C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2484—24no—
38#2446—22no—
39#2446—22no—
40#2446C2214yescomplemented
41#2446C2214yescomplemented
42#2357C2214yes—
43#2354C2214yes—
44#2342—24no—
45#2319—22no—
46#2317C2211yescharacteristic
47#2310C2211yescharacteristic
48#2310C2211yescharacteristic
49#2310C2211yescharacteristic
50#2309C2211yescharacteristic
51#2296C2211yescharacteristic
52#2296C2211yescharacteristic
53#2293—22no—
54C2×#390C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yescomplemented
58#2097C2212yescomplemented
59#2096C2212yescomplemented
60#2095C2212yescomplemented
61#2074C2214yescomplemented
62#2072C2214yescomplemented
63#2067C2214yes—
64#2065C2214yes—
65#2013C2214yes—
66#2013C2214yes—
67#1981C2214yescomplemented
68#1981C2214yescomplemented
69#1973C2212yes—
70#1973C2212yes—
71#1968C2212yes—
72#1968C2212yes—
73#1928C2214yescomplemented
74#1927C2214yescomplemented
75#1921C2212yescomplemented
76#1920C2212yescomplemented
77#1919C2212yescomplemented
78#1918C2212yescomplemented
79#1917C2214yescomplemented
80#1916C2214yescomplemented
81#1880C2214yes—
82#1880C2214yes—
83#1877C2214yes—
84#1877C2214yes—
85#1877C2214yes—
86#1877C2214yes—
87#1763C2214yescomplemented
88C4×#103C2214yescomplemented
89#1736C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—24no—
102C22×SD16—24no—
103C22×SD16—24no—
104C22×SD16—42no—
105C22×SD16—42no—
106C22×SD16C2314yes—
107C22×SD16C2312yes—
108C22×SD16C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C2×C4⋊Q8—24no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—42no—
117C2×C4⋊Q8C2314yes—
118#519—24no—
119#519—42no—
120#518—24no—
121#518—24no—
122C2×C4×Q8—22no—
123C2×C4×Q8C2314yes—
124C2×C4×D4—22no—
125C2×C4×D4C2314yes—
126C2×#117—24no—
127C2×#117—22no—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—22no—
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C42C2311yescharacteristic
139#428—42no—
140#427—24no—
141C2×#107C2311yescharacteristic
142C2×#107C2311yescharacteristic
143C2×#107C2312yes—
144C2×#106C2311yescharacteristic
145C2×#106C2311yescharacteristic
146C2×#106C2312yes—
147#423—42no—
148C2×C4⋊C8C2314yes—
149#419—24no—
150#419—24no—
151#419—42no—
152#419—42no—
153#418—42no—
154#418—42no—
155C2×#103—24no—
156C2×#103—42no—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#102C2312yes—
167C2×#102C2314yes—
168C2×#102C2312yes—
169#405—42no—
170C2×#97C2312yes—
171C2×C4×C8C2312yes—
172#400—24no—
173#400—24no—
174#398—24no—
175#398—24no—
176#391—22no—
177#391—22no—
178#391—22no—
179#391C2314yes—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#390—22no—
185#390—21no—
186#390—21no—
187#390C2312yes—
188#390C2312yes—
189#390C2314yes—
190#389—22no—
191#389D412yes—
192#389D412yes—
193#389C2312yes—
194#389C2312yes—
195#388—24no—
196#388—24no—
197#388—24no—
198#388—24no—
199#387—24no—
200#387—24no—
201#387—24no—
202#387—24no—
203#385—22no—
204#385—22no—
205#385D414yes—
206#384C2312yes—
207#382C2312yes—
208#381—22no—
209#381—22no—
210#381—22no—
211#381—24no—
212#380—24no—
213#379—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2314yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—24no—
224#376—21no—
225C43—21no—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2312yes—
238#336C2312yes—
239C25—21no—
240C22×Q8—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×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8#4114yes—
254C22×Q8C2412yes—
255C22×Q8C2412yes—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4#4114yes—
263C22×D4C2412yes—
264C23×C4—21no—
265C23×C4—24no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4C2×D412yes—
269C23×C4C2411yescharacteristic
270C2×SD16—28no—
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—28no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—42no—
277C2×SD16—42no—
278C2×SD16—44no—
279C2×SD16—416no—
280C2×SD16—416no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8#4114yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2411yescharacteristic
297C22×C8C2411yescharacteristic
298C4⋊Q8—28no—
299C4⋊Q8—44no—
300C4⋊Q8—44no—
301C4⋊Q8—44no—
302C4×Q8—28no—
303C4×Q8—44no—
304C4×Q8—48no—
305C4×D4—216no—
306C4×D4—28no—
307C4×D4—28no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—48no—
320#117—88no—
321#117#4114yes—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—48no—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C22⋊C4—216no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—84no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—24no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2412yes—
369C2×C42C2412yes—
370C2×C42C2412yes—
371#107—22no—
372#107—22no—
373#107—42no—
374#106—22no—
375#106—22no—
376#106—42no—
377C4⋊C8—44no—
378#103—24no—
379#103—24no—
380#103—28no—
381#103—28no—
382#103—44no—
383#103—42no—
384#103—42no—
385#103—42no—
386#103—42no—
387#102—42no—
388#102—42no—
389#102—44no—
390#97—42no—
391C4×C8—42no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95C2×D414yes—
405#95C2×D414yes—
406C24—22no—
407C24—24no—
408C24—42no—
409C24—42no—
410C24—82no—
411C24—82no—
412C24C22×D411yescharacteristic
413C2×Q8—24no—
414C2×Q8—22no—
415C2×Q8—28no—
416C2×Q8—28no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—44no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8C2×#4114yes—
439C2×D4—24no—
440C2×D4—28no—
441C2×D4—28no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4C2×#4114yes—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—216no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—82no—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4C2511yescharacteristic
487SD16—48no—
488SD16—48no—
489SD16—416no—
490M16—44no—
491M16—44no—
492C2×C8—28no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—44no—
507C2×C8C2×#4112yes—
508C2×C8C2×#4112yes—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—28no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—84no—
533C4⋊C4C2×#4114yes—
534C22⋊C4—216no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—48no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—48no—
555C42—42no—
556C42—42no—
557C42—42no—
558C42—44no—
559C42—82no—
560C42—84no—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—48no—
569C23—88no—
570C23—164no—
571C23—162no—
572C23—162no—
573C23#52512yes—
574C23#54612yes—
575C23#55012yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—24no—
579Q8—42no—
580Q8—42no—
581Q8—48no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—84no—
587D4—24no—
588D4—24no—
589D4—42no—
590D4—42no—
591D4—48no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—816no—
628C2×C4—84no—
629C2×C4—162no—
630C2×C4#52512yes—
631C2×C4#52914yes—
632C2×C4#54014yes—
633C2×C4#54612yes—
634C2×C4#55012yes—
635C2×C4#55114yes—
636C2×C4#55114yes—
637C2×C4#57714yes—
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C8—22no—
645C8—22no—
646C8—42no—
647C8—42no—
648C8—42no—
649C22—24no—
650C22—24no—
651C22—22no—
652C22—22no—
653C22—44no—
654C22—48no—
655C22—84no—
656C22—82no—
657C22—82no—
658C22—168no—
659C22#247114yes—
660C22C2×#52511yescharacteristic
661C22C2×#54611yescharacteristic
662C22C2×#55011yescharacteristic
663C22C2×#57712yes—
664C22#358612yes—
665C4—24no—
666C4—44no—
667C4—42no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—82no—
672C4—82no—
673C4—81no—
674C4—81no—
675C4—82no—
676C4—81no—
677C4—81no—
678C4#247114yes—
679C4C2×#52911yescharacteristic
680C4C2×#54011yescharacteristic
681C4C2×#55111yescharacteristic
682C4C2×#55111yescharacteristic
683C2—24no—
684C2—84no—
685C2C2×#247111yescharacteristic
686C2#6059112yes—
687C2#6118512yes—
688C2#6141011yescharacteristic
689C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35674C2semidirect
3. #27220C2semidirect
4. #27213C2semidirect
5. #27201C2semidirect
6. #27192C2semidirect
7. #23736C2semidirect
8. #23633C2semidirect
11. #21259C2semidirect
13. #21064C2semidirect
14. #21032C2semidirect
15. #21027C2semidirect
19. #20417C2semidirect
40. #2446C22semidirect
41. #2446C22semidirect
57. #2097C22semidirect
58. #2097C22semidirect
59. #2096C22semidirect
60. #2095C22semidirect
61. #2074C22semidirect
62. #2072C22semidirect
67. #1981C22semidirect
68. #1981C22semidirect
73. #1928C22semidirect
74. #1927C22semidirect
75. #1921C22semidirect
76. #1920C22semidirect
77. #1919C22semidirect
78. #1918C22semidirect
79. #1917C22semidirect
80. #1916C22semidirect
87. #1763C22semidirect
88. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#356742
#342552
#272202
#272132
#272012
#271922
#237362
#236332
#236182
C2×#20132
#212592
#212312
#210642
#210322
#210272
#209362
#209332
#204172