Finite Group Databasebeta

← #6500581

Subgroups — #6500581

Order 512.

1998 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic, complemented
2#34205C211yesmaximal, characteristic
3#29237C212yesmaximal, complemented
4#29235C212yesmaximal, complemented
5#27677C212yesmaximal, complemented
6#26436C214yesmaximal, complemented
7#26364C214yesmaximal, complemented
8#26012C214yesmaximal, complemented
9#25968C214yesmaximal, complemented
10#24305C212yesmaximal, complemented
11#21942C211yesmaximal, characteristic, complemented
12#21934C211yesmaximal, characteristic, complemented
13#21764C211yesmaximal, characteristic
14#20912C211yesmaximal, characteristic
15#20846C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic, complemented
17C22×#121C2211yescharacteristic
18C2×#518—21no—
19C22×#117C2211yescharacteristic
20C2×#478C2214yescomplemented
21C2×#428C2212yescomplemented
22C2×#418C2212yes—
23C2×#407C2211yescharacteristic
24C2×#407C2211yescharacteristic
25#2441—22no—
26#2441—22no—
27#2441C2214yes—
28#2441C2214yescomplemented
29C2×#399—21no—
30C2×#396—21no—
31#2364—22no—
32#2361C2212yes—
33#2361C2212yes—
34#2344C2212yes—
35#2342—22no—
36D4×C22⋊C4C2212yes—
37#2325—21no—
38#2325—21no—
39#2308—21no—
40C4×#121C2212yes—
41C4×#121C2212yes—
42C2×#389—21no—
43C2×#389C2211yescharacteristic, complemented
44C2×#389C2211yescharacteristic, complemented
45C2×#387—21no—
46C2×#384C2211yescharacteristic, complemented
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2099C2214yescomplemented
51#2093C2214yescomplemented
52#2088C2214yescomplemented
53#2087C2214yescomplemented
54#2065C2214yescomplemented
55#2065C2214yescomplemented
56#2059C2214yescomplemented
57#2025C2214yescomplemented
58#1989C2214yescomplemented
59#1947C2214yescomplemented
60#1920C2214yescomplemented
61#1919C2214yescomplemented
62#1895C2214yes—
63#1875C2214yes—
64#1874C2212yes—
65#1874C2212yes—
66#1874C2212yes—
67#1874C2212yes—
68#1864C2214yescomplemented
69#1862C2214yescomplemented
70#1854C2214yes—
71#1853C2212yescomplemented
72#1853C2212yescomplemented
73#1853C2212yescomplemented
74#1852C2212yescomplemented
75#1852C2212yescomplemented
76#1852C2212yescomplemented
77#1851C2214yes—
78#1851C2214yescomplemented
79#1824C2212yes—
80#1809C2214yes—
81#1802C2214yescomplemented
82#1797C2212yescomplemented
83#1797C2212yescomplemented
84#1786C2212yes—
85#1736C2212yes—
86#1736C2212yescomplemented
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C22×M16C2312yes—
91C22×M16C2312yes—
92C2×#125—22no—
93C2×#125—44no—
94C2×#125C2314yes—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121—24no—
98C2×#121Q818yes—
99C2×#121C2312yes—
100C2×#121C2312yes—
101#518—48no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117—21no—
109C2×#117—21no—
110C2×#117—22no—
111C2×#117—44no—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#478—48no—
138#428—44no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145#418—44no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150#407—24no—
151#407—24no—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156#399—24no—
157#399—24no—
158#396—24no—
159#396—24no—
160#390C2312yes—
161#390C2312yes—
162#389—22no—
163#389—22no—
164#389—24no—
165#389—24no—
166#389C2312yes—
167#389C2314yes—
168#389C2314yes—
169#389C2312yes—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#387—22no—
175#387—22no—
176#387—24no—
177#387—24no—
178#386—22no—
179#386—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#385—22no—
185#385—22no—
186#384—22no—
187#384C2314yes—
188#381—22no—
189#380—21no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#379—22no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4C2312yes—
205C4×C22⋊C4C2312yes—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C25C2×D411yescharacteristic
224C22×D4—28no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4—42no—
230C22×D4—42no—
231C22×D4—44no—
232C22×D4—44no—
233C22×D4C2×Q814yes—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×Q814yes—
254C23×C4#4114yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×M16—24no—
259C2×M16—24no—
260C2×M16—42no—
261C2×M16—42no—
262C2×M16—42no—
263C2×M16—42no—
264C22×C8C2414yes—
265C22×C8C2414yes—
266#125—28no—
267#125—44no—
268#125—48no—
269#121—28no—
270#121—28no—
271#121—24no—
272#121—24no—
273#121C2×Q818yes—
274#121C2×Q818yes—
275C4×D4—44no—
276C4×D4—44no—
277C4×D4—48no—
278C4×D4—48no—
279C4×D4—48no—
280C4×D4—48no—
281#117—24no—
282#117—24no—
283#117—48no—
284#117—42no—
285#117—42no—
286#117—44no—
287#117—44no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4C2×Q814yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
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—22no—
340C2×C22⋊C4—48no—
341C2×C22⋊C4—42no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—82no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—42no—
354C2×C42—42no—
355C2×C42—44no—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360#107—44no—
361#107—44no—
362#106—28no—
363#106—44no—
364C4⋊C8—44no—
365C4⋊C8—44no—
366#102—28no—
367#102—44no—
368#102—44no—
369#102—44no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—24no—
392C24—41no—
393C24—41no—
394C24—41no—
395C24—41no—
396C24—42no—
397C24—42no—
398C24—81no—
399C24—81no—
400C24—81no—
401C24—81no—
402C24C22⋊Q814yes—
403C24C22×D411yescharacteristic
404C24C22×D411yescharacteristic
405C24C22×D411yescharacteristic
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—48no—
413C2×D4—44no—
414C2×D4—432no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—84no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4C22×Q814yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4C22⋊Q814yes—
482C22×C4C22⋊Q814yes—
483C22×C4C22⋊Q814yes—
484C22×C4AGL(1,ℤ/8)14yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×Q812yes—
493C22×C4C22×Q812yes—
494C22×C4C22×Q812yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2511yescharacteristic
499M16—48no—
500M16—84no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—42no—
520C4⋊C4—48no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—81no—
527C4⋊C4—81no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4C22×Q814yes—
531C4⋊C4H(5,𝔽2)14yes—
532C22⋊C4—28no—
533C22⋊C4—28no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C42—24no—
551C42—24no—
552C42—42no—
553C42—42no—
554C42—44no—
555C42—44no—
556C42—44no—
557C42—44no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—28no—
575C23—24no—
576C23—44no—
577C23—44no—
578C23—48no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—82no—
586C23—82no—
587C23—82no—
588C23—82no—
589C23—84no—
590C23—84no—
591C23—162no—
592C23—162no—
593C23—162no—
594C23—162no—
595C23C2×C22⋊Q812yes—
596C23C2×C22⋊Q812yes—
597C23C2×C22⋊Q812yes—
598C23C2×AGL(1,ℤ/8)12yes—
599C23#57512yes—
600C23C23×D411yescharacteristic
601D4—28no—
602D4—416no—
603D4—416no—
604D4—44no—
605D4—84no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—48no—
637C2×C4—48no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—82no—
658C2×C4—82no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—164no—
663C2×C4—164no—
664C2×C4—162no—
665C2×C4—162no—
666C2×C4C2×C22⋊Q812yes—
667C2×C4C2×C22⋊Q812yes—
668C2×C4C2×C22⋊Q812yes—
669C2×C4C2×AGL(1,ℤ/8)14yes—
670C2×C4C23×Q811yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C2×H(5,𝔽2)12yes—
673C2×C4#58512yes—
674C8—44no—
675C8—44no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—88no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—168no—
705C22#307714yes—
706C22C22×C22⋊Q811yescharacteristic
707C22#348411yescharacteristic
708C22#348511yescharacteristic
709C22C22×AGL(1,ℤ/8)11yescharacteristic
710C22C2×#57511yescharacteristic
711C22#358511yescharacteristic
712C22#358611yescharacteristic
713C4—24no—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—42no—
718C4—44no—
719C4—42no—
720C4—44no—
721C4—42no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4#350112yes—
728C4#350512yes—
729C2—24no—
730C2—42no—
731C2—42no—
732C2—84no—
733C2#3534911yescharacteristic
734C2C2×#307711yescharacteristic
735C2#6049311yescharacteristic
736C2#6157511yescharacteristic
737C2#6158711yescharacteristic
738C2#6170411yescharacteristic
739C2#6171811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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. C2×#2441C2semidirect
3. #29237C2semidirect
4. #29235C2semidirect
5. #27677C2semidirect
6. #26436C2semidirect
7. #26364C2semidirect
8. #26012C2semidirect
9. #25968C2semidirect
10. #24305C2semidirect
11. #21942C2semidirect
12. #21934C2semidirect
15. #20846C2semidirect
16. C22×#125C22semidirect
20. C2×#478C22semidirect
21. C2×#428C22semidirect
28. #2441C22semidirect
43. C2×#389C22semidirect
44. C2×#389C22semidirect
46. C2×#384C22semidirect
50. #2099C22semidirect
51. #2093C22semidirect
52. #2088C22semidirect
53. #2087C22semidirect
54. #2065C22semidirect
55. #2065C22semidirect
56. #2059C22semidirect
57. #2025C22semidirect
58. #1989C22semidirect
59. #1947C22semidirect
60. #1920C22semidirect
61. #1919C22semidirect
68. #1864C22semidirect
69. #1862C22semidirect
71. #1853C22semidirect
72. #1853C22semidirect
73. #1853C22semidirect
74. #1852C22semidirect
75. #1852C22semidirect
76. #1852C22semidirect
78. #1851C22semidirect
81. #1802C22semidirect
82. #1797C22semidirect
83. #1797C22semidirect
86. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
#342052
#292372
#292352
#276772
#264362
#263642
#260122
#259682
#243052
#219422
#219342
#217642
#209122
#208462