Finite Group Databasebeta

← #6435264

Subgroups — #6435264

Order 512.

1863 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35256C211yesmaximal, characteristic
2#34189C211yesmaximal, characteristic, complemented
3#29935C214yesmaximal, complemented
4#29929C214yesmaximal, complemented
5#29925C214yesmaximal, complemented
6#29924C214yesmaximal, complemented
7#23733C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic, complemented
9#23628C211yesmaximal, characteristic, complemented
10#23627C211yesmaximal, characteristic, complemented
11#23489C211yesmaximal, characteristic, complemented
12#23488C211yesmaximal, characteristic, complemented
13#23483C211yesmaximal, characteristic, complemented
14#23483C211yesmaximal, characteristic, complemented
15#21263C211yesmaximal, characteristic, complemented
16#21263C211yesmaximal, characteristic, complemented
17#21174C211yesmaximal, characteristic, complemented
18#21173C211yesmaximal, characteristic, complemented
19#20630C211yesmaximal, characteristic, complemented
20C2×#537C2211yescharacteristic
21C2×#535C2211yescharacteristic
22C2×#535C2211yescharacteristic
23C2×C4×#41C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#424C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#417C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C2×#416C2211yescharacteristic
35C2×#416C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C2×#408C2211yescharacteristic
39C2×#408C2211yescharacteristic
40#2856C2212yes—
41#2850C2212yes—
42#2620C2212yes—
43#2619C2212yes—
44#2428—22no—
45#2426—22no—
46#2424—24no—
47#2357—22no—
48#2356—22no—
49#2353—24no—
50#2317C2211yescharacteristic
51#2314C2214yescomplemented
52#2311C2211yescharacteristic
53#2311C2211yescharacteristic
54#2297C2211yescharacteristic
55C2×#390C2211yescharacteristic
56#2285C2211yescharacteristic
57#2283C2214yescomplemented
58C2×#382C2211yescharacteristic
59#2274C2211yescharacteristic
60#2274C2211yescharacteristic
61#2097C2212yescomplemented
62#2097C2212yescomplemented
63#2092C2214yes—
64#2092C2214yes—
65#2092C2212yescomplemented
66#2092C2212yescomplemented
67#2074C2212yescomplemented
68#2073C2214yes—
69#2073C2212yescomplemented
70#2072C2212yescomplemented
71#2071C2214yes—
72#2071C2212yescomplemented
73#2036C2214yescomplemented
74#2035C2214yescomplemented
75#2034C2214yes—
76#2034C2214yes—
77#2034C2214yescomplemented
78#2033C2214yes—
79#2033C2214yes—
80#2033C2214yescomplemented
81#1983C2214yes—
82#1983C2214yes—
83#1982C2214yes—
84#1982C2214yes—
85#1949C2214yescomplemented
86#1949C2214yescomplemented
87#1948C2214yescomplemented
88#1947C2214yescomplemented
89#1798C2214yescomplemented
90#1797C2214yescomplemented
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C22×#41C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C22×M16C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104#537—42no—
105#537—42no—
106#535—24no—
107#535—24no—
108#535—42no—
109#535—42no—
110C2×#127—21no—
111C2×#127C2312yes—
112C2×#124—21no—
113C2×#124C2312yes—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C2312yes—
116C2×#121—22no—
117C2×#121—21no—
118C2×#121—24no—
119C2×#121—22no—
120C2×#121C2312yes—
121C2×#121C2312yes—
122C2×#121C2312yes—
123C2×#120—22no—
124C2×#120—22no—
125C4×#41—24no—
126C4×#41—24no—
127C2×C4×Q8D412yes—
128C2×C4×D4D414yes—
129C2×C4×D4D412yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2312yes—
134C2×#117C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—22no—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141#428—42no—
142#428—42no—
143C2×#107—42no—
144C2×#107C2314yes—
145C2×#107C2312yes—
146C2×#107C2312yes—
147C2×#106—42no—
148C2×#106C2314yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151#424—24no—
152#424—24no—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2314yes—
155#418—42no—
156#418—42no—
157#417—42no—
158#417—42no—
159#416—42no—
160#416—42no—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#102—42no—
166C2×#102—42no—
167C2×#102C2314yes—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#102C2314yes—
172C2×#102C2314yes—
173C2×#102C2312yes—
174C2×#102C2312yes—
175#408—42no—
176#408—42no—
177C2×C22⋊C8C2314yes—
178C2×C22⋊C8C2314yes—
179#396—24no—
180#394—24no—
181#394—24no—
182#392—24no—
183#391—21no—
184#391—21no—
185#391C2311yescharacteristic
186#391C2311yescharacteristic
187#390—21no—
188#390—21no—
189#390—24no—
190#390—22no—
191#390C2312yes—
192#390C2311yescharacteristic
193#390C2312yes—
194#390C2311yescharacteristic
195#389—22no—
196#388—24no—
197#388—24no—
198#388—24no—
199#387—24no—
200#387—24no—
201#386—22no—
202#386—22no—
203#386—24no—
204#386—22no—
205#385—22no—
206#384C2312yes—
207#384C2312yes—
208#383—24no—
209#382—22no—
210#382C2312yes—
211#382C2312yes—
212#382C2312yes—
213#382C2312yes—
214#381—24no—
215#380—22no—
216#380C2314yes—
217#380C2314yes—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4C2314yes—
222C4×C4⋊C4C2314yes—
223C4×C22⋊C4—24no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227#376—24no—
228C2×#95—22no—
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—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41—22no—
252C22×Q8C2×D411yescharacteristic
253C22×Q8C2411yescharacteristic
254C22×D4—21no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—24no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4C2×D412yes—
267C22×D4C2×D411yescharacteristic
268C22×D4C2412yes—
269C22×D4C2411yescharacteristic
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4C2×D412yes—
275C23×C4C2×D414yes—
276C23×C4C2412yes—
277C23×C4C2411yescharacteristic
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294#127—24no—
295#127—24no—
296#127—44no—
297#124—24no—
298#124—24no—
299#124—44no—
300C22⋊Q8—42no—
301C22⋊Q8—42no—
302C22⋊Q8—44no—
303#121—42no—
304#121—42no—
305#121—42no—
306#121—42no—
307#121—42no—
308#121—42no—
309#121—48no—
310#121—44no—
311#121—48no—
312#121—88no—
313#120—48no—
314#120—48no—
315C4×Q8—28no—
316C4×D4—28no—
317C4×D4—216no—
318#117—24no—
319#117—24no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117#4114yes—
327#117#4114yes—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4C2×D414yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—84no—
365C2×C22⋊C4C2×D414yes—
366C2×C22⋊C4C2×D412yes—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—24no—
375C2×C42—24no—
376C2×C42C2×D414yes—
377C2×C42C2×D412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2412yes—
381#107—44no—
382#107—42no—
383#107—42no—
384#106—44no—
385#106—42no—
386#106—42no—
387C4⋊C8—28no—
388C4⋊C8—28no—
389#103—42no—
390#103—42no—
391#103—42no—
392#103—42no—
393#102—44no—
394#102—44no—
395#102—42no—
396#102—42no—
397#102—42no—
398#102—42no—
399#102—44no—
400#102—44no—
401C22⋊C8—44no—
402C22⋊C8—44no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—24no—
410#95—24no—
411#95—22no—
412#95—22no—
413#95—24no—
414C24—22no—
415C24—21no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—82no—
420C24—82no—
421C24C22×D412yes—
422C24C22×D411yescharacteristic
423#41—44no—
424#41—44no—
425#41—44no—
426#41—44no—
427#41—44no—
428#41—44no—
429#41—44no—
430#41—44no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—42no—
465C2×D4—48no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—84no—
473C2×D4—84no—
474C2×D4—84no—
475C2×D4—84no—
476C2×D4—84no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—28no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4—82no—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D414yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4H(5,𝔽2)12yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C42−1+412yes—
517C22×C4C2511yescharacteristic
518M16—44no—
519M16—44no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—44no—
533C2×C8—44no—
534C4⋊C4—28no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4—84no—
556C4⋊C4#12318yes—
557C22⋊C4—28no—
558C22⋊C4—48no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—42no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C22⋊C4—48no—
568C22⋊C4—82no—
569C22⋊C4—84no—
570C22⋊C4—84no—
571C22⋊C4—84no—
572C22⋊C4#12318yes—
573C42—24no—
574C42—28no—
575C42—21no—
576C42—21no—
577C42—21no—
578C42—21no—
579C42—42no—
580C42—42no—
581C42—42no—
582C42—84no—
583C42—84no—
584C42C2×#4114yes—
585C42C2×#4114yes—
586C23—22no—
587C23—22no—
588C23—24no—
589C23—24no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—42no—
594C23—44no—
595C23—44no—
596C23—42no—
597C23—42no—
598C23—41no—
599C23—41no—
600C23—42no—
601C23—41no—
602C23—42no—
603C23—88no—
604C23—84no—
605C23—164no—
606C23—162no—
607C23—162no—
608C23C2×#12312yes—
609C23#53512yes—
610C23#53712yes—
611C23C23×D411yescharacteristic
612Q8—44no—
613Q8—41no—
614Q8—41no—
615Q8—41no—
616Q8—41no—
617D4—48no—
618D4—44no—
619D4—42no—
620D4—42no—
621D4—41no—
622D4—41no—
623D4—42no—
624D4—42no—
625D4—41no—
626D4—41no—
627D4—84no—
628D4—84no—
629D4—168no—
630D4—164no—
631D4—164no—
632D4—164no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—88no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—162no—
671C2×C4—162no—
672C2×C4—162no—
673C2×C4C2×#12318yes—
674C2×C4C2×#12312yes—
675C2×C4#53512yes—
676C2×C4#53712yes—
677C2×C4#54014yes—
678C2×C4#54014yes—
679C2×C4#54214yes—
680C2×C4#54214yes—
681C2×C4C22×#4111yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4C2×H(5,𝔽2)11yescharacteristic
684C2×C4C2×H(5,𝔽2)11yescharacteristic
685C2×C4C2×H(5,𝔽2)11yescharacteristic
686C2×C4C2×2−1+411yescharacteristic
687C8—42no—
688C8—42no—
689C8—42no—
690C8—42no—
691C22—24no—
692C22—24no—
693C22—22no—
694C22—22no—
695C22—44no—
696C22—42no—
697C22—44no—
698C22—42no—
699C22—42no—
700C22—82no—
701C22—84no—
702C22—81no—
703C22—82no—
704C22—82no—
705C22—84no—
706C22—82no—
707C22—81no—
708C22—81no—
709C22—82no—
710C22—168no—
711C22#262414yes—
712C22C22×#12311yescharacteristic
713C22C2×#53511yescharacteristic
714C22C2×#53711yescharacteristic
715C22#358412yes—
716C22#358512yes—
717C4—28no—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—82no—
722C4—82no—
723C4—81no—
724C4—82no—
725C4—81no—
726C4—82no—
727C4—82no—
728C4—81no—
729C4#262414yes—
730C4C2×#54011yescharacteristic
731C4C2×#54011yescharacteristic
732C4C2×#54211yescharacteristic
733C4C2×#54211yescharacteristic
734C2—24no—
735C2—82no—
736C2—161no—
737C2—162no—
738C2C2×#262411yescharacteristic
739C2#6052312yes—
740C2#6081911yescharacteristic
741C2#6082711yescharacteristic
742C2#6093312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35256C2
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
2. #34189C2semidirect
3. #29935C2semidirect
4. #29929C2semidirect
5. #29925C2semidirect
6. #29924C2semidirect
7. #23733C2semidirect
8. C2×#2092C2semidirect
9. #23628C2semidirect
10. #23627C2semidirect
11. #23489C2semidirect
12. #23488C2semidirect
13. #23483C2semidirect
14. #23483C2semidirect
15. #21263C2semidirect
16. #21263C2semidirect
17. #21174C2semidirect
18. #21173C2semidirect
19. #20630C2semidirect
51. #2314C22semidirect
57. #2283C22semidirect
61. #2097C22semidirect
62. #2097C22semidirect
65. #2092C22semidirect
66. #2092C22semidirect
67. #2074C22semidirect
69. #2073C22semidirect
70. #2072C22semidirect
72. #2071C22semidirect
73. #2036C22semidirect
74. #2035C22semidirect
77. #2034C22semidirect
80. #2033C22semidirect
85. #1949C22semidirect
86. #1949C22semidirect
87. #1948C22semidirect
88. #1947C22semidirect
89. #1798C22semidirect
90. #1797C22semidirect

Maximal subgroups

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

subgroupindex
#352562
#341892
#299352
#299292
#299252
#299242
#237332
C2×#20922
#236282
#236272
#234892
#234882
#234832
#212632
#211742
#211732
#206302