Finite Group Databasebeta

← #6615707

Subgroups — #6615707

Order 512.

1729 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#36187C211yesmaximal, characteristic
2C4×#556C211yesmaximal, characteristic, complemented
3#33198C211yesmaximal, characteristic, complemented
4#33113C212yesmaximal, complemented
5#29206C211yesmaximal, characteristic, complemented
6#28279C211yesmaximal, characteristic, complemented
7#27630C212yesmaximal, complemented
8#27335C214yesmaximal, complemented
9#26793C214yesmaximal, complemented
10#25013C212yesmaximal, complemented
11#24653C212yesmaximal, complemented
12#24119C212yesmaximal, complemented
13#23060C212yesmaximal, complemented
14#22805C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16C4×#460C211yesmaximal, characteristic, complemented
17#21362C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C2×#555—21no—
20C2×#482C2214yes—
21C2×#477C2212yes—
22C2×#474C2212yes—
23C2×#460C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#431C2212yes—
26#2868C2211yescharacteristic
27#2856C2212yes—
28#2563—21no—
29#2537—21no—
30#2537C2212yes—
31#2500—22no—
32#2491—22no—
33#2491C2214yes—
34#2373—22no—
35#2367—21no—
36#2365C2211yescharacteristic
37#2362C2212yes—
38#2356—21no—
39#2355C2211yescharacteristic
40#2354—21no—
41#2354C2212yes—
42#2354C2211yescharacteristic
43#2347C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#126—22no—
47C4×#124—21no—
48C4×C22⋊Q8C2212yes—
49#2285C2212yes—
50C42×Q8—21no—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53C4×#117C2212yes—
54#2077C2214yes—
55#2067C2214yes—
56#2058C2214yes—
57#2036C2214yes—
58#2034C2214yes—
59#2000C2214yes—
60#1990C2212yes—
61#1987C2212yes—
62#1982C2212yes—
63#1982C2211yescharacteristic
64#1968C2214yes—
65#1954C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1948C2211yescharacteristic
69#1947C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1934C2214yes—
73#1927C2214yes—
74#1926C2214yes—
75#1926C2212yes—
76#1925C2214yes—
77#1923C2214yes—
78#1918C2214yes—
79#1880C2212yes—
80#1875C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1844C2212yes—
84#1842C2211yescharacteristic
85#1803C2212yes—
86#1803C2212yes—
87#1803C2211yescharacteristic
88#1802C2212yes—
89#1797C2212yes—
90C4×#107C2214yes—
91C4×#106C2214yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×SD16C2312yes—
96#556—24no—
97#556—24no—
98#555—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#126—22no—
105C2×#124—21no—
106C2×#124—24no—
107C2×C22⋊Q8C2312yes—
108C2×#121—22no—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121C2314yes—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C2×#117—22no—
126C2×#117C2312yes—
127C2×#117C2312yes—
128C22×C42C2312yes—
129#482—48no—
130#477—44no—
131#474—28no—
132#460—24no—
133C4×SD16—28no—
134#431—28no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2312yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2314yes—
156#394—24no—
157#394—24no—
158#393—24no—
159#391—21no—
160#391—21no—
161#391C2312yes—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—22no—
166#390—22no—
167#390C2312yes—
168#390C2314yes—
169#389—21no—
170#389C2312yes—
171#389C2312yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—24no—
178#388—24no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#383—22no—
191#382—21no—
192#382—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380C2312yes—
197#380C2314yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224#376—22no—
225C43—22no—
226C43—21no—
227C43C2311yescharacteristic
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2411yescharacteristic
240C22×D4—24no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2412yes—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×SD16—42no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—22no—
256C4⋊Q8—42no—
257C4⋊Q8—42no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8#4114yes—
261#127—22no—
262#127—22no—
263#126—216no—
264#124—28no—
265#124—416no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268#121—48no—
269#121—48no—
270#121—44no—
271#121—44no—
272#121—84no—
273C4×Q8—28no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—41no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×Q8#4114yes—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—44no—
289C4×D4—44no—
290#117—24no—
291#117—24no—
292#117—24no—
293#117—42no—
294#117—48no—
295#117—48no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—28no—
361#107—28no—
362#106—24no—
363#106—44no—
364#106—44no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367C4⋊C8—22no—
368C4⋊C8—42no—
369C4⋊C8#4114yes—
370#103—24no—
371#103—44no—
372#102—24no—
373#102—24no—
374#102—28no—
375#102—24no—
376#102—44no—
377C22⋊C8—28no—
378C22⋊C8—44no—
379C4×C8—28no—
380C4×C8—24no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—24no—
392#95—22no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—22no—
398C24C22×D412yes—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8C2×#4112yes—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—48no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—84no—
428C2×D4—84no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2511yescharacteristic
462SD16—48no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8C2×#4114yes—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4AGL(1,ℤ/8)14yes—
502C4⋊C4C2×#4112yes—
503C4⋊C4C2×#4114yes—
504C4⋊C4H(5,𝔽2)12yes—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—82no—
519C42—28no—
520C42—24no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—21no—
527C42—22no—
528C42—21no—
529C42—41no—
530C42—44no—
531C42—42no—
532C42—42no—
533C42—44no—
534C42—48no—
535C42—44no—
536C42—42no—
537C42—42no—
538C42—81no—
539C42—82no—
540C42#13514yes—
541C42C2×#4112yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C422−1+412yes—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23C23×D411yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—48no—
563Q8—41no—
564Q8—41no—
565Q8—164no—
566D4—44no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—84no—
572D4—84no—
573D4—168no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—48no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—88no—
622C2×C4#52512yes—
623C2×C4#52512yes—
624C2×C4#54712yes—
625C2×C4#54712yes—
626C2×C4C2×#13512yes—
627C2×C4C2×#13512yes—
628C2×C4C2×#13512yes—
629C2×C4C2×AGL(1,ℤ/8)14yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4#57712yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4#58511yescharacteristic
638C8—24no—
639C8—44no—
640C8—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22C2×#52511yescharacteristic
656C22C2×#54711yescharacteristic
657C22#348811yescharacteristic
658C22C22×#13511yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22C2×#57711yescharacteristic
661C22#358611yescharacteristic
662C4—22no—
663C4—22no—
664C4—24no—
665C4—22no—
666C4—42no—
667C4—41no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—82no—
674C4—84no—
675C4—81no—
676C4—82no—
677C4#310614yes—
678C4#334014yes—
679C4#347511yescharacteristic
680C4#348311yescharacteristic
681C4#351111yescharacteristic
682C4#351811yescharacteristic
683C2—82no—
684C2—162no—
685C2#3641111yescharacteristic
686C2C2×#310611yescharacteristic
687C2#6059911yescharacteristic
688C2C2×#334011yescharacteristic
689C2#6125211yescharacteristic
690C2#6195411yescharacteristic
691C2#6199911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#556C2semidirect
3. #33198C2semidirect
4. #33113C2semidirect
5. #29206C2semidirect
6. #28279C2semidirect
7. #27630C2semidirect
8. #27335C2semidirect
9. #26793C2semidirect
10. #25013C2semidirect
11. #24653C2semidirect
12. #24119C2semidirect
13. #23060C2semidirect
14. #22805C2semidirect
15. #22795C2semidirect
16. C4×#460C2semidirect
17. #21362C2semidirect

Maximal subgroups

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

subgroupindex
#361872
C4×#5562
#331982
#331132
#292062
#282792
#276302
#273352
#267932
#250132
#246532
#241192
#230602
#228052
#227952
C4×#4602
#213622