Finite Group Databasebeta

← #6477843

Subgroups — #6477843

Order 512.

2274 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic, complemented
2C2×#2361C211yesmaximal, characteristic, complemented
3#32978C212yesmaximal, complemented
4#32953C211yesmaximal, characteristic, complemented
5#32902C211yesmaximal, characteristic, complemented
6#29966C211yesmaximal, characteristic, complemented
7#29754C211yesmaximal, characteristic, complemented
8#29592C212yesmaximal, complemented
9#27094C214yesmaximal, complemented
10#26926C214yesmaximal, complemented
11#26784C214yesmaximal, complemented
12#26656C214yesmaximal, complemented
13#22040C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic, complemented
15C4×#480C212yesmaximal, complemented
16C2×#538—21no—
17C22×#121C2211yescharacteristic
18C22×C4×D4—21no—
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#480C2212yes—
22C2×#480C2214yescomplemented
23C2×#474C2212yescomplemented
24C2×#459C2212yescomplemented
25C2×#447C2212yes—
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2847C2212yes—
29#2639—21no—
30#2617—21no—
31#2559—22no—
32#2557—22no—
33#2496C2212yescomplemented
34#2491C2212yescomplemented
35#2428C2212yescomplemented
36#2422C2212yescomplemented
37#2361—22no—
38#2361—22no—
39#2361C2214yes—
40#2361C2214yes—
41#2325—21no—
42C4×#121C2212yes—
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#388—21no—
46C2×#386—21no—
47C2×#385—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2099C2212yescomplemented
51#2086C2214yescomplemented
52#2083C2212yescomplemented
53#2067C2214yescomplemented
54#2057C2212yescomplemented
55#2041C2212yescomplemented
56#2033C2214yes—
57#2033C2214yes—
58#1999C2214yes—
59#1999C2214yes—
60#1970C2214yescomplemented
61#1934C2214yescomplemented
62#1925C2214yescomplemented
63#1925C2214yescomplemented
64#1923C2214yescomplemented
65#1923C2214yescomplemented
66#1918C2214yescomplemented
67#1879C2214yescomplemented
68#1879C2214yescomplemented
69#1874C2212yes—
70#1874C2212yes—
71#1874C2214yescomplemented
72#1862C2214yescomplemented
73#1852C2212yes—
74#1852C2212yes—
75#1852C2214yescomplemented
76#1823C2212yes—
77#1823C2214yescomplemented
78#1802C2212yescomplemented
79#1802C2212yescomplemented
80#1797C2211yescharacteristic
81#1797C2211yescharacteristic
82#1786C2212yes—
83#1779C2212yes—
84C4×#107C2214yescomplemented
85C4×#102C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4—21no—
91C24×C4C2311yescharacteristic
92C22×D8C2314yes—
93#538—48no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#126—22no—
97C2×#124—22no—
98C2×#123—22no—
99C2×C22⋊Q8—21no—
100C2×#121—24no—
101C2×#121—22no—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121—24no—
105C2×#121—22no—
106C2×#121C2312yes—
107C2×#121C2314yes—
108C2×#121C2312yes—
109C2×#121C2312yes—
110C2×#120—22no—
111C2×C4×D4—22no—
112C2×C4×D4—24no—
113C2×C4×D4—24no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—44no—
117C2×C4×D4—48no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—44no—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4D411yescharacteristic
134C22×C22⋊C4D411yescharacteristic
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#480—28no—
139#480—48no—
140#474—44no—
141#459—44no—
142#447—28no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C4×C8C2314yes—
159#400—22no—
160#397—22no—
161#397—22no—
162#394—22no—
163#394—22no—
164#394—22no—
165#393—22no—
166#392—22no—
167#391—21no—
168#390—21no—
169#390—21no—
170#390—22no—
171#390C2312yes—
172#390C2314yes—
173#390C2312yes—
174#389—21no—
175#389—21no—
176#389—21no—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—24no—
184#388—24no—
185#387—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—24no—
191#386—24no—
192#386—22no—
193#385—22no—
194#385—24no—
195#385—24no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#380—21no—
199#380C2312yes—
200#380C2312yes—
201#380C2312yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4D412yes—
207C4×C22⋊C4D412yes—
208C4×C22⋊C4C2312yes—
209C4×C22⋊C4C2312yes—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2312yes—
221#336C2312yes—
222#336C2314yes—
223C25—21no—
224C25C2×D411yescharacteristic
225C22×Q8—21no—
226C22×D4—28no—
227C22×D4—24no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—42no—
236C22×D4—42no—
237C22×D4—44no—
238C22×D4—44no—
239C22×D4#4114yes—
240C22×D4C2412yes—
241C22×D4C2412yes—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—24no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—42no—
252C23×C4—42no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—82no—
258C23×C4—82no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C2×D8—28no—
268C2×D8—44no—
269C2×D8—44no—
270C22×C8C2414yes—
271C22×C8C2414yes—
272C4⋊Q8—42no—
273#127—42no—
274#126—48no—
275#124—48no—
276#123—48no—
277C22⋊Q8—44no—
278#121—216no—
279#121—28no—
280#121—28no—
281#121—24no—
282#121—24no—
283#121—44no—
284#121—44no—
285#121—48no—
286#121—44no—
287#121—44no—
288#120—48no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—48no—
297C4×D4—416no—
298C4×D4—416no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4#4114yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—28no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—82no—
348C2×C22⋊C4D818yescomplemented
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—42no—
356C2×C42—42no—
357C2×C42—44no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2412yes—
364#107—28no—
365#107—28no—
366#107—44no—
367C4⋊C8—42no—
368C4⋊C8—42no—
369#102—28no—
370#102—28no—
371#102—28no—
372#102—44no—
373#102—44no—
374#102—44no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377C22⋊C8—44no—
378C22⋊C8—44no—
379C4×C8—44no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—22no—
394C24—22no—
395C24—24no—
396C24—22no—
397C24—22no—
398C24—41no—
399C24—41no—
400C24—41no—
401C24—41no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24C2×D814yes—
409C24C22×D412yes—
410C24C22×D411yescharacteristic
411C2×Q8—42no—
412C2×Q8—81no—
413C2×Q8—81no—
414C2×D4—216no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—48no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—48no—
429C2×D4—42no—
430C2×D4—432no—
431C2×D4—48no—
432C2×D4—48no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4—81no—
440C2×D4—81no—
441C2×D4—81no—
442C2×D4—81no—
443C2×D4—82no—
444C2×D4—82no—
445C2×D4—82no—
446C2×D4—82no—
447C2×D4C2×#4114yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—42no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4—84no—
499C22×C4—816no—
500C22×C4—816no—
501C22×C4C2×D814yes—
502C22×C4C2×D814yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D412yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4C2511yescharacteristic
514D8—88no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—48no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—81no—
540C4⋊C4—81no—
541C4⋊C4C2×#4114yes—
542C4⋊C4H(5,𝔽2)14yes—
543C22⋊C4—216no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—82no—
557C22⋊C4—84no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4C2×D818yes—
562C22⋊C4C2×D818yes—
563C42—22no—
564C42—22no—
565C42—24no—
566C42—42no—
567C42—41no—
568C42—41no—
569C42—48no—
570C42—44no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—22no—
577C23—28no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—24no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—48no—
590C23—48no—
591C23—42no—
592C23—42no—
593C23—48no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—84no—
606C23—84no—
607C23—84no—
608C23—84no—
609C23#52512yes—
610C23C22×D814yes—
611C23C22×D812yes—
612C23#57512yes—
613C23C23×D411yescharacteristic
614Q8—162no—
615D4—24no—
616D4—24no—
617D4—48no—
618D4—432no—
619D4—42no—
620D4—42no—
621D4—44no—
622D4—44no—
623D4—88no—
624D4—162no—
625D4—162no—
626D4—164no—
627D4—164no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—22no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—48no—
655C2×C4—44no—
656C2×C4—48no—
657C2×C4—42no—
658C2×C4—48no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—88no—
677C2×C4—88no—
678C2×C4—82no—
679C2×C4—82no—
680C2×C4—816no—
681C2×C4—84no—
682C2×C4—164no—
683C2×C4#52512yes—
684C2×C4C22×D814yes—
685C2×C4C22×D812yes—
686C2×C4C22×D812yes—
687C2×C4C22×#4111yescharacteristic
688C2×C4C22×#4111yescharacteristic
689C2×C4C2×H(5,𝔽2)12yes—
690C2×C4#58512yes—
691C8—44no—
692C8—44no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22—416no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—816no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22#310014yes—
723C22C2×#52511yescharacteristic
724C22#348411yescharacteristic
725C22#348511yescharacteristic
726C22C23×D811yescharacteristic
727C22C2×#57511yescharacteristic
728C22#358511yescharacteristic
729C22#358611yescharacteristic
730C4—22no—
731C4—22no—
732C4—22no—
733C4—24no—
734C4—42no—
735C4—42no—
736C4—44no—
737C4—44no—
738C4—41no—
739C4—41no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4#346012yes—
745C4#350512yes—
746C2—24no—
747C2—44no—
748C2—84no—
749C2—162no—
750C2#3509111yescharacteristic
751C2C2×#310011yescharacteristic
752C2#6056511yescharacteristic
753C2#6155911yescharacteristic
754C2#6158311yescharacteristic
755C2#6169411yescharacteristic
756C2#6171311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35005C2semidirect
2. C2×#2361C2semidirect
3. #32978C2semidirect
4. #32953C2semidirect
5. #32902C2semidirect
6. #29966C2semidirect
7. #29754C2semidirect
8. #29592C2semidirect
9. #27094C2semidirect
10. #26926C2semidirect
11. #26784C2semidirect
12. #26656C2semidirect
13. #22040C2semidirect
14. #21902C2semidirect
15. C4×#480C2semidirect
22. C2×#480C22semidirect
23. C2×#474C22semidirect
24. C2×#459C22semidirect
33. #2496C22semidirect
34. #2491C22semidirect
35. #2428C22semidirect
36. #2422C22semidirect
50. #2099C22semidirect
51. #2086C22semidirect
52. #2083C22semidirect
53. #2067C22semidirect
54. #2057C22semidirect
55. #2041C22semidirect
60. #1970C22semidirect
61. #1934C22semidirect
62. #1925C22semidirect
63. #1925C22semidirect
64. #1923C22semidirect
65. #1923C22semidirect
66. #1918C22semidirect
67. #1879C22semidirect
68. #1879C22semidirect
71. #1874C22semidirect
72. #1862C22semidirect
75. #1852C22semidirect
77. #1823C22semidirect
78. #1802C22semidirect
79. #1802C22semidirect
84. C4×#107C22semidirect
85. C4×#102C22semidirect
348. C2×C22⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#23612
#329782
#329532
#329022
#299662
#297542
#295922
#270942
#269262
#267842
#266562
#220402
#219022
C4×#4802