Finite Group Databasebeta

← #6460465

Subgroups — #6460465

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35677C211yesmaximal, characteristic
2#35629C211yesmaximal, characteristic, complemented
3#29069C214yesmaximal, complemented
4#29065C214yesmaximal, complemented
5#29046C214yesmaximal, complemented
6#29042C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23506C211yesmaximal, characteristic, complemented
9#23487C211yesmaximal, characteristic, complemented
10#23486C211yesmaximal, characteristic
11C2×#1958C211yesmaximal, characteristic, complemented
12#21151C211yesmaximal, characteristic, complemented
13#21089C211yesmaximal, characteristic, complemented
14#21065C211yesmaximal, characteristic, complemented
15#20975C211yesmaximal, characteristic
16#20974C211yesmaximal, characteristic, complemented
17#20880C211yesmaximal, characteristic, complemented
18#20417C211yesmaximal, characteristic, complemented
19#20383C211yesmaximal, characteristic, complemented
20C22×#131C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#535C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×#405C2211yescharacteristic
37C22×C4×C8C2211yescharacteristic
38#2548—22no—
39#2545—22no—
40#2514—22no—
41#2513—22no—
42#2510—24no—
43#2495C2214yescomplemented
44#2494C2214yescomplemented
45#2426C2214yes—
46#2426C2214yes—
47#2411—24no—
48#2316C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2211yescharacteristic
51#2309C2211yescharacteristic
52#2297C2211yescharacteristic
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#382C2211yescharacteristic
56#2274C2211yescharacteristic
57#2101C2214yescomplemented
58#2101C2214yescomplemented
59#2042C2214yes—
60#2042C2214yes—
61#2036C2214yescomplemented
62#2035C2214yescomplemented
63#2034C2214yes—
64#2033C2214yes—
65#1958C2212yes—
66#1958C2212yes—
67#1958C2212yes—
68#1958C2212yes—
69#1942C2212yescomplemented
70#1942C2212yescomplemented
71#1941C2212yescomplemented
72#1941C2212yescomplemented
73#1936C2214yes—
74#1934C2214yes—
75#1926C2214yes—
76#1925C2214yes—
77#1895C2214yes—
78#1894C2214yescomplemented
79#1893C2214yes—
80#1892C2214yescomplemented
81#1865C2214yescomplemented
82#1863C2214yescomplemented
83#1763C2214yescomplemented
84C4×#103C2214yescomplemented
85#1753C2212yescomplemented
86#1753C2212yescomplemented
87C8×C22⋊C4C2212yescomplemented
88C8×C22⋊C4C2212yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C2×#131—24no—
101C2×#131—42no—
102C2×#131—42no—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2312yes—
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2312yes—
109#536—24no—
110#536—42no—
111#535—24no—
112#535—42no—
113C2×C4⋊Q8—21no—
114C2×#127—21no—
115C2×#124—21no—
116C2×#124—21no—
117C2×#124—24no—
118C2×#124—22no—
119C2×#124—42no—
120C2×#124C2312yes—
121C2×#124C2312yes—
122C2×#123—22no—
123C2×C22⋊Q8C2314yes—
124C2×#121C2314yes—
125C2×#120—22no—
126#519—24no—
127#519—42no—
128C2×C4×Q8—22no—
129C2×C4×Q8C2314yes—
130C2×#117—22no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—22no—
139C22×C42C2311yescharacteristic
140#427—24no—
141#427—42no—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146#419—24no—
147#419—42no—
148#418—42no—
149#418—42no—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102—24no—
159C2×#102—24no—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166#408—24no—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2312yes—
169#405—42no—
170C2×#97C2312yes—
171C2×C4×C8—42no—
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176C2×C4×C8C2312yes—
177#401—24no—
178#400—24no—
179#397—24no—
180#397—24no—
181#396—24no—
182#396—24no—
183#394—24no—
184#394—24no—
185#391—21no—
186#391—21no—
187#391D412yes—
188#390—21no—
189#390—21no—
190#390—22no—
191#390—24no—
192#390—24no—
193#390D412yes—
194#390C2314yes—
195#389—22no—
196#389C2314yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#388—24no—
202#388—24no—
203#388—24no—
204#388—24no—
205#387—24no—
206#386—22no—
207#386—22no—
208#386D414yes—
209#385—22no—
210#385—22no—
211#383—24no—
212#382—22no—
213#382C2312yes—
214#382C2312yes—
215#380—22no—
216#380C2314yes—
217#380C2314yes—
218#379—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2314yes—
223C4×C22⋊C4—22no—
224C4×C22⋊C4C2312yes—
225C4×C22⋊C4C2312yes—
226C2×#95—22no—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2312yes—
239C25—21no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8C2×D411yescharacteristic
253C22×Q8#4114yes—
254C22×Q8C2412yes—
255C22×D4—24no—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4C2×D412yes—
264C22×D4C2×D411yescharacteristic
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4C2×D412yes—
270C23×C4C2412yes—
271C23×C4C2411yescharacteristic
272#131—416no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—24no—
280C2×M16—42no—
281C2×M16—42no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—24no—
287C22×C8—42no—
288C22×C8—42no—
289C22×C8#4114yes—
290C22×C8#4114yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2411yescharacteristic
294C22×C8C2411yescharacteristic
295C22×C8C2411yescharacteristic
296C22×C8C2411yescharacteristic
297C4⋊Q8—44no—
298#127—44no—
299#124—24no—
300#124—24no—
301#124—42no—
302#124—42no—
303#124—48no—
304#124—44no—
305#124—44no—
306#123—48no—
307C22⋊Q8—28no—
308C22⋊Q8—44no—
309#121—28no—
310#121—44no—
311#120—48no—
312C4×Q8—28no—
313C4×Q8—44no—
314C4×Q8—48no—
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—48no—
325#117#4114yes—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
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—48no—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D414yes—
361C2×C22⋊C4C2×D414yes—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—24no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—24no—
370C2×C42C2×D412yes—
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—24no—
375#107—42no—
376#106—24no—
377#106—42no—
378#103—28no—
379#103—24no—
380#103—24no—
381#103—24no—
382#103—24no—
383#103—44no—
384#102—24no—
385#102—24no—
386#102—24no—
387#102—24no—
388#102—44no—
389#102—44no—
390C22⋊C8—24no—
391C22⋊C8—24no—
392#97—42no—
393C4×C8—41no—
394C4×C8—41no—
395C4×C8—41no—
396C4×C8—41no—
397C4×C8—42no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95C2×D414yes—
409#95C2×D414yes—
410C24—22no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—82no—
415C24—82no—
416C24C22×D412yes—
417C24C22×D411yescharacteristic
418#41—44no—
419#41—44no—
420#41—44no—
421#41—44no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—28no—
428C2×Q8—28no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—42no—
434C2×Q8—44no—
435C2×Q8—42no—
436C2×Q8—44no—
437C2×Q8C2×#4114yes—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
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—48no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—84no—
459C2×D4—84no—
460C22×C4—24no—
461C22×C4—24no—
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—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C4C2511yescharacteristic
495M16—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×C8—48no—
515C2×C8C2×#4112yes—
516C2×C8C2×#4112yes—
517C2×C8C2×#4112yes—
518C2×C8C2×#4112yes—
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—48no—
532C4⋊C4—48no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—84no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4C2×#4114yes—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—24no—
557C42—24no—
558C42—21no—
559C42—21no—
560C42—22no—
561C42—21no—
562C42—21no—
563C42—22no—
564C42—48no—
565C42—42no—
566C42—42no—
567C42—42no—
568C42—84no—
569C23—22no—
570C23—22no—
571C23—24no—
572C23—24no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—44no—
577C23—44no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—88no—
582C23—84no—
583C23—164no—
584C23—162no—
585C23—162no—
586C23#53212yes—
587C23#54612yes—
588C23#54612yes—
589C23C23×D411yescharacteristic
590Q8—24no—
591Q8—24no—
592Q8—42no—
593Q8—42no—
594Q8—48no—
595Q8—44no—
596D4—48no—
597D4—42no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—44no—
602D4—84no—
603D4—168no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—48no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—162no—
642C2×C4—162no—
643C2×C4#52914yes—
644C2×C4#53212yes—
645C2×C4#53814yes—
646C2×C4#54612yes—
647C2×C4#54612yes—
648C2×C4#55314yes—
649C2×C4#55314yes—
650C2×C4C22×#4111yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C8—22no—
657C8—22no—
658C8—22no—
659C8—22no—
660C8—42no—
661C8—42no—
662C22—24no—
663C22—22no—
664C22—22no—
665C22—24no—
666C22—42no—
667C22—44no—
668C22—42no—
669C22—42no—
670C22—44no—
671C22—82no—
672C22—84no—
673C22—82no—
674C22—82no—
675C22—84no—
676C22—168no—
677C22#255914yes—
678C22C2×#53211yescharacteristic
679C22C2×#54611yescharacteristic
680C22C2×#54611yescharacteristic
681C22#358412yes—
682C22#358412yes—
683C4—42no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—42no—
688C4—42no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4#255914yes—
695C4C2×#52911yescharacteristic
696C4C2×#53811yescharacteristic
697C4C2×#55311yescharacteristic
698C4C2×#55311yescharacteristic
699C2—24no—
700C2—82no—
701C2—162no—
702C2C2×#255911yescharacteristic
703C2#6071311yescharacteristic
704C2#6071311yescharacteristic
705C2#6117012yes—
706C2#6121612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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
2. #35629C2semidirect
3. #29069C2semidirect
4. #29065C2semidirect
5. #29046C2semidirect
6. #29042C2semidirect
7. #23746C2semidirect
8. #23506C2semidirect
9. #23487C2semidirect
11. C2×#1958C2semidirect
12. #21151C2semidirect
13. #21089C2semidirect
14. #21065C2semidirect
16. #20974C2semidirect
17. #20880C2semidirect
18. #20417C2semidirect
19. #20383C2semidirect
43. #2495C22semidirect
44. #2494C22semidirect
57. #2101C22semidirect
58. #2101C22semidirect
61. #2036C22semidirect
62. #2035C22semidirect
69. #1942C22semidirect
70. #1942C22semidirect
71. #1941C22semidirect
72. #1941C22semidirect
78. #1894C22semidirect
80. #1892C22semidirect
81. #1865C22semidirect
82. #1863C22semidirect
83. #1763C22semidirect
84. C4×#103C22semidirect
85. #1753C22semidirect
86. #1753C22semidirect
87. C8×C22⋊C4C22semidirect
88. C8×C22⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#356772
#356292
#290692
#290652
#290462
#290422
#237462
#235062
#234872
#234862
C2×#19582
#211512
#210892
#210652
#209752
#209742
#208802
#204172
#203832