Finite Group Databasebeta

← #6464258

Subgroups — #6464258

Order 512.

1854 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35307C211yesmaximal, characteristic
2#34268C211yesmaximal, characteristic
3#29471C214yesmaximal, complemented
4#29470C214yesmaximal, complemented
5#29463C214yesmaximal, complemented
6#29462C214yesmaximal, complemented
7#23639C211yesmaximal, characteristic
8#23487C211yesmaximal, characteristic
9#23487C211yesmaximal, characteristic
10#23389C211yesmaximal, characteristic
11C2×#1969C211yesmaximal, characteristic, complemented
12#21151C211yesmaximal, characteristic, complemented
13#21094C211yesmaximal, characteristic, complemented
14#21094C211yesmaximal, characteristic, complemented
15#21066C211yesmaximal, characteristic, complemented
16#21037C211yesmaximal, characteristic, complemented
17#20935C211yesmaximal, characteristic, complemented
18#20880C211yesmaximal, characteristic, complemented
19C4×#408C211yesmaximal, characteristic
20C23×Q16C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×C4×#41C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C2×#408C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×C4×M16C2211yescharacteristic
37C22×C4×C8C2211yescharacteristic
38#2855C2214yes—
39#2625C2214yes—
40#2447—22no—
41#2446—22no—
42#2419—24no—
43#2367—22no—
44#2365—22no—
45#2359—24no—
46#2323C2214yes—
47#2316C2211yescharacteristic
48#2311C2211yescharacteristic
49#2311C2211yescharacteristic
50#2310C2211yescharacteristic
51C4×C22⋊Q8C2214yes—
52#2296C2211yescharacteristic
53#2296C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#391C2211yescharacteristic
56#2274C2211yescharacteristic
57#2076C2214yes—
58#2075C2214yes—
59#2036C2214yes—
60#2036C2214yes—
61#2035C2214yes—
62#2035C2214yes—
63#2001C2214yes—
64#2001C2214yes—
65#1969C2212yes—
66#1969C2212yes—
67#1969C2212yes—
68#1969C2212yes—
69#1942C2212yes—
70#1942C2212yes—
71#1941C2212yes—
72#1941C2212yes—
73#1935C2214yes—
74#1935C2214yes—
75#1933C2214yes—
76#1933C2214yes—
77#1928C2214yes—
78#1927C2214yes—
79#1921C2214yes—
80#1918C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1865C2214yes—
84#1863C2214yes—
85#1750C2212yes—
86#1750C2212yes—
87C4×C22⋊C8C2212yes—
88C4×C22⋊C8C2212yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C22×Q16—24no—
101C22×Q16—24no—
102C22×Q16—42no—
103C22×Q16C2312yes—
104C22×Q16C2312yes—
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#536—24no—
110#536—24no—
111#536—42no—
112#536—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×#124—21no—
116C2×#124—21no—
117C2×#123—22no—
118C2×#123—22no—
119C2×C22⋊Q8C2314yes—
120C2×C22⋊Q8C2314yes—
121#519—42no—
122#519—42no—
123C4×#41—42no—
124C4×#41—42no—
125C2×C4×Q8—22no—
126C2×C4×Q8D412yes—
127C2×C4×Q8C2314yes—
128C2×C4×D4D414yes—
129C2×C4×D4D412yes—
130C2×#117—22no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2312yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2312yes—
141#427—24no—
142#427—42no—
143C2×#107—24no—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#107C2312yes—
147C2×#107C2312yes—
148C2×#106C2312yes—
149C2×#106C2312yes—
150#419—24no—
151#419—24no—
152#419—42no—
153#419—42no—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164#408—24no—
165#408—24no—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2312yes—
170C4×M16—42no—
171C2×#97C2312yes—
172C2×C4×C8—42no—
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176C2×C4×C8C2311yescharacteristic
177C2×C4×C8C2312yes—
178#400—24no—
179#398—24no—
180#397—24no—
181#397—24no—
182#391—21no—
183#391—21no—
184#391—22no—
185#391—22no—
186#391—22no—
187#391C2314yes—
188#391C2312yes—
189#391C2312yes—
190#391C2312yes—
191#391C2312yes—
192#390—21no—
193#390—21no—
194#389—22no—
195#389—24no—
196#389C2314yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#388—24no—
202#388—24no—
203#387—24no—
204#387—24no—
205#387—24no—
206#387—24no—
207#386—22no—
208#386—22no—
209#385—24no—
210#385—22no—
211#385—24no—
212#385—22no—
213#381—24no—
214#381—22no—
215#380—22no—
216#380C2314yes—
217#380C2314yes—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—24no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2314yes—
223C4×C22⋊C4—24no—
224C4×C22⋊C4—22no—
225C2×#95—22no—
226C2×#95—22no—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
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×Q8C2×D411yescharacteristic
259C22×Q8#4114yes—
260C22×Q8C2412yes—
261C22×Q8C2412yes—
262C22×D4C2×D412yes—
263C22×D4C2×D411yescharacteristic
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4C2×D412yes—
269C23×C4C2×D414yes—
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—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8#4114yes—
290C22×C8#4114yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2411yescharacteristic
294C22×C8C2411yescharacteristic
295C22×C8C2411yescharacteristic
296C22×C8C2411yescharacteristic
297C4⋊Q8—28no—
298C4⋊Q8—44no—
299#124—28no—
300#124—44no—
301#123—216no—
302#123—48no—
303C22⋊Q8—28no—
304C22⋊Q8—28no—
305C22⋊Q8—44no—
306C22⋊Q8—44no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×Q8—48no—
311C4×D4—44no—
312C4×D4—48no—
313#117—22no—
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—42no—
323#117—42no—
324#117—48no—
325#117#4114yes—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D414yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D414yes—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—24no—
366C2×C42—42no—
367C2×C42—42no—
368C2×C42C2×D412yes—
369C2×C42C2×D414yes—
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—24no—
374#107—24no—
375#107—24no—
376#107—42no—
377#106—24no—
378#106—42no—
379#103—28no—
380#103—24no—
381#103—24no—
382#103—24no—
383#103—24no—
384#103—28no—
385#103—44no—
386#103—44no—
387C22⋊C8—24no—
388C22⋊C8—24no—
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—22no—
402#95—24no—
403#95—24no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—24no—
411C24C22×D412yes—
412C24C22×D411yescharacteristic
413#41—44no—
414#41—44no—
415#41—44no—
416#41—44no—
417C2×Q8—24no—
418C2×Q8—22no—
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—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8C2×#4114yes—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—28no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
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×D414yes—
484C22×C4C22×D412yes—
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—44no—
497M16—44no—
498C2×C8—28no—
499C2×C8—28no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8C2×#4112yes—
515C2×C8C2×#4112yes—
516C2×C8C2×#4112yes—
517C2×C8C2×#4112yes—
518C4⋊C4—216no—
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—22no—
531C4⋊C4—44no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—84no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4C2×#4114yes—
542C22⋊C4—216no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—44no—
546C22⋊C4—48no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C42—24no—
553C42—24no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—22no—
559C42—22no—
560C42—44no—
561C42—48no—
562C42—42no—
563C42—42no—
564C42—42no—
565C42—84no—
566C23—22no—
567C23—22no—
568C23—24no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23#52512yes—
574C23#53212yes—
575C23#53412yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—24no—
579Q8—48no—
580Q8—44no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—84no—
588D4—48no—
589D4—44no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
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—88no—
630C2×C4—84no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4#52512yes—
634C2×C4#53212yes—
635C2×C4#53412yes—
636C2×C4#55114yes—
637C2×C4#55314yes—
638C2×C4#55314yes—
639C2×C4#55914yes—
640C2×C4#57814yes—
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C8—22no—
648C8—22no—
649C8—22no—
650C8—22no—
651C8—42no—
652C8—42no—
653C22—22no—
654C22—22no—
655C22—24no—
656C22—24no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22#260214yes—
663C22C2×#52511yescharacteristic
664C22C2×#53211yescharacteristic
665C22C2×#53411yescharacteristic
666C22C2×#57812yes—
667C22#358412yes—
668C4—24no—
669C4—42no—
670C4—44no—
671C4—42no—
672C4—44no—
673C4—44no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#260214yes—
681C4C2×#55111yescharacteristic
682C4C2×#55311yescharacteristic
683C4C2×#55311yescharacteristic
684C4C2×#55911yescharacteristic
685C2—24no—
686C2—82no—
687C2C2×#260211yescharacteristic
688C2#6057312yes—
689C2#6071311yescharacteristic
690C2#6071511yescharacteristic
691C2#6077012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. #29471C2semidirect
4. #29470C2semidirect
5. #29463C2semidirect
6. #29462C2semidirect
11. C2×#1969C2semidirect
12. #21151C2semidirect
13. #21094C2semidirect
14. #21094C2semidirect
15. #21066C2semidirect
16. #21037C2semidirect
17. #20935C2semidirect
18. #20880C2semidirect

Maximal subgroups

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

subgroupindex
#353072
#342682
#294712
#294702
#294632
#294622
#236392
#234872
#233892
C2×#19692
#211512
#210942
#210662
#210372
#209352
#208802
C4×#4082