Finite Group Databasebeta

← #6460367

Subgroups — #6460367

Order 512.

1854 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35673C211yesmaximal, characteristic
2#35628C211yesmaximal, characteristic
3#29002C214yesmaximal, complemented
4#29001C214yesmaximal, complemented
5#28978C214yesmaximal, complemented
6#28977C214yesmaximal, complemented
7#23639C211yesmaximal, characteristic
8#23629C211yesmaximal, characteristic, complemented
9#23455C211yesmaximal, characteristic, complemented
10#23389C211yesmaximal, characteristic
11#21318C211yesmaximal, characteristic, complemented
12C2×#1969C211yesmaximal, characteristic, complemented
13#21088C211yesmaximal, characteristic, complemented
14#21037C211yesmaximal, characteristic, complemented
15#20974C211yesmaximal, characteristic
16#20974C211yesmaximal, characteristic
17#20935C211yesmaximal, characteristic, complemented
18#20417C211yesmaximal, characteristic, complemented
19#20383C211yesmaximal, characteristic
20C23×Q16C2211yescharacteristic
21C22×#131C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#536C2211yescharacteristic
24C2×#533C2211yescharacteristic
25C2×#519C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C2×#408C2211yescharacteristic
34C2×#405C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2592—22no—
37#2568—22no—
38#2498—24no—
39#2492C2214yes—
40#2489C2214yes—
41#2488—24no—
42#2448—22no—
43#2446—22no—
44#2423C2214yes—
45#2421C2214yes—
46#2316C2211yescharacteristic
47#2311C2211yescharacteristic
48#2311C2211yescharacteristic
49#2310C2211yescharacteristic
50#2297C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53#2285C2211yescharacteristic
54#2274C2211yescharacteristic
55#2076C2214yes—
56#2075C2214yes—
57#2074C2214yes—
58#2072C2214yes—
59#2026C2214yes—
60#2024C2214yes—
61#2001C2214yes—
62#2001C2214yes—
63#1991C2212yes—
64#1991C2212yes—
65#1990C2212yes—
66#1990C2212yes—
67#1969C2212yes—
68#1969C2212yes—
69#1969C2212yes—
70#1969C2212yes—
71#1935C2214yes—
72#1933C2214yes—
73#1921C2214yes—
74#1918C2214yes—
75#1894C2214yes—
76#1894C2214yes—
77#1892C2214yes—
78#1892C2214yes—
79#1878C2214yes—
80#1878C2214yes—
81#1763C2214yes—
82C4×#103C2214yes—
83#1753C2212yes—
84#1753C2212yes—
85C8×C22⋊C4C2212yes—
86C8×C22⋊C4C2212yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C22×Q16—24no—
101C22×Q16—24no—
102C22×Q16—42no—
103C22×Q16C2312yes—
104C22×Q16C2312yes—
105C2×#131—24no—
106C2×#131—42no—
107C2×#131—42no—
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C22×M16C2312yes—
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2312yes—
114#536—24no—
115#536—24no—
116#536—42no—
117#536—42no—
118#533—24no—
119#533—42no—
120C2×C4⋊Q8—21no—
121C2×C4⋊Q8—21no—
122C2×C4⋊Q8C2312yes—
123C2×#125C2312yes—
124C2×#124—21no—
125C2×#124—21no—
126C2×#123—22no—
127C2×#123—22no—
128C2×C22⋊Q8—22no—
129C2×C22⋊Q8C2314yes—
130C2×C22⋊Q8C2314yes—
131#519—24no—
132#519—42no—
133C2×C4×Q8—22no—
134C2×C4×Q8—22no—
135C2×C4×Q8C2314yes—
136C2×#117—22no—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C42C2311yescharacteristic
145C2×#107—24no—
146C2×#107C2312yes—
147C2×#107C2312yes—
148#419—24no—
149#419—42no—
150#419—42no—
151#419—42no—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162#408—24no—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2312yes—
165#405—42no—
166C2×#97C2312yes—
167C2×C4×C8—42no—
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2312yes—
173#400—24no—
174#400—24no—
175#400—24no—
176#399—24no—
177#398—24no—
178#397—24no—
179#397—24no—
180#393—24no—
181#391—21no—
182#391—21no—
183#391—22no—
184#391—22no—
185#391—22no—
186#391—24no—
187#391D412yes—
188#391C2314yes—
189#391C2312yes—
190#391C2312yes—
191#391C2312yes—
192#391C2312yes—
193#390—21no—
194#390—21no—
195#390D412yes—
196#389—22no—
197#389C2314yes—
198#388—24no—
199#388—24no—
200#388—24no—
201#388—24no—
202#387—24no—
203#387—24no—
204#386—22no—
205#386—22no—
206#386D414yes—
207#384C2312yes—
208#382—24no—
209#382C2312yes—
210#381—22no—
211#380—22no—
212#380—22no—
213#380C2314yes—
214#380C2314yes—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4C2314yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4C2312yes—
220C4×C22⋊C4C2312yes—
221C2×#95—22no—
222C2×#95—22no—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2312yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8C2×D411yescharacteristic
260C22×Q8#4114yes—
261C22×Q8C2412yes—
262C22×Q8C2412yes—
263C22×D4C2×D412yes—
264C22×D4C2×D411yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4C2×D412yes—
270C23×C4C2412yes—
271C23×C4C2411yescharacteristic
272C2×Q16—24no—
273C2×Q16—24no—
274C2×Q16—24no—
275C2×Q16—24no—
276C2×Q16—42no—
277C2×Q16—42no—
278C2×Q16—416no—
279#131—416no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—24no—
287C2×M16—42no—
288C2×M16—42no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—24no—
294C22×C8—42no—
295C22×C8—42no—
296C22×C8#4114yes—
297C22×C8#4114yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2411yescharacteristic
301C22×C8C2411yescharacteristic
302C22×C8C2411yescharacteristic
303C22×C8C2411yescharacteristic
304C4⋊Q8—24no—
305C4⋊Q8—42no—
306C4⋊Q8—44no—
307C4⋊Q8—44no—
308#125—24no—
309#125—42no—
310#124—44no—
311#124—44no—
312#123—48no—
313#123—48no—
314C22⋊Q8—28no—
315C22⋊Q8—28no—
316C22⋊Q8—48no—
317C22⋊Q8—44no—
318C22⋊Q8—44no—
319C4×Q8—28no—
320C4×Q8—48no—
321C4×Q8—44no—
322C4×Q8—48no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—48no—
333#117#4114yes—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D414yes—
366C2×C22⋊C4C2×D414yes—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—24no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—24no—
375C2×C42C2×D412yes—
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—24no—
380#107—24no—
381#103—24no—
382#103—24no—
383#103—24no—
384#103—24no—
385#103—28no—
386#103—44no—
387#103—44no—
388#103—44no—
389C22⋊C8—24no—
390C22⋊C8—24no—
391#97—42no—
392C4×C8—41no—
393C4×C8—41no—
394C4×C8—41no—
395C4×C8—41no—
396C4×C8—42no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95C2×D414yes—
410#95C2×D414yes—
411C24C22×D412yes—
412C24C22×D411yescharacteristic
413#41—44no—
414#41—44no—
415#41—44no—
416#41—44no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—28no—
423C2×Q8—28no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—44no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—82no—
448C2×Q8C2×#4114yes—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4C2511yescharacteristic
494Q16—48no—
495Q16—48no—
496M16—48no—
497M16—44no—
498M16—44no—
499C2×C8—28no—
500C2×C8—28no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—48no—
516C2×C8C2×#4112yes—
517C2×C8C2×#4112yes—
518C2×C8C2×#4112yes—
519C2×C8C2×#4112yes—
520C4⋊C4—24no—
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—22no—
531C4⋊C4—22no—
532C4⋊C4—48no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—82no—
542C4⋊C4—84no—
543C4⋊C4C2×#4114yes—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—28no—
547C22⋊C4—44no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C42—24no—
556C42—24no—
557C42—21no—
558C42—21no—
559C42—22no—
560C42—21no—
561C42—21no—
562C42—22no—
563C42—48no—
564C42—42no—
565C42—42no—
566C42—42no—
567C42—84no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23#53212yes—
576C23#54612yes—
577C23#54612yes—
578C23C23×D411yescharacteristic
579Q8—24no—
580Q8—24no—
581Q8—42no—
582Q8—42no—
583Q8—48no—
584Q8—44no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—42no—
589Q8—88no—
590Q8—84no—
591D4—48no—
592D4—44no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—84no—
633C2×C4—162no—
634C2×C4—162no—
635C2×C4#52914yes—
636C2×C4#53212yes—
637C2×C4#53814yes—
638C2×C4#54612yes—
639C2×C4#54612yes—
640C2×C4#55114yes—
641C2×C4#55914yes—
642C2×C4#57814yes—
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C8—22no—
650C8—22no—
651C8—22no—
652C8—22no—
653C8—42no—
654C8—42no—
655C22—24no—
656C22—22no—
657C22—22no—
658C22—24no—
659C22—42no—
660C22—44no—
661C22—42no—
662C22—42no—
663C22—44no—
664C22#255714yes—
665C22C2×#53211yescharacteristic
666C22C2×#54611yescharacteristic
667C22C2×#54611yescharacteristic
668C22C2×#57812yes—
669C22#358412yes—
670C4—24no—
671C4—42no—
672C4—44no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#255714yes—
683C4C2×#52911yescharacteristic
684C4C2×#53811yescharacteristic
685C4C2×#55111yescharacteristic
686C4C2×#55911yescharacteristic
687C2—24no—
688C2—82no—
689C2C2×#255711yescharacteristic
690C2#6071311yescharacteristic
691C2#6071511yescharacteristic
692C2#6117312yes—
693C2#6121412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29002C2semidirect
4. #29001C2semidirect
5. #28978C2semidirect
6. #28977C2semidirect
8. #23629C2semidirect
9. #23455C2semidirect
11. #21318C2semidirect
12. C2×#1969C2semidirect
13. #21088C2semidirect
14. #21037C2semidirect
17. #20935C2semidirect
18. #20417C2semidirect

Maximal subgroups

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

subgroupindex
#356732
#356282
#290022
#290012
#289782
#289772
#236392
#236292
#234552
#233892
#213182
C2×#19692
#210882
#210372
#209742
#209352
#204172
#203832