Finite Group Databasebeta

← #6591291

Subgroups — #6591291

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#36134C211yesmaximal, characteristic, complemented
3#33289C211yesmaximal, characteristic, complemented
4#33197C211yesmaximal, characteristic, complemented
5#33195C211yesmaximal, characteristic, complemented
6#28194C212yesmaximal
7#27736C212yesmaximal, complemented
8#27735C212yesmaximal, complemented
9#26513C212yesmaximal, complemented
10#26512C212yesmaximal, complemented
11#24993C212yesmaximal, complemented
12#24992C212yesmaximal, complemented
13#24677C212yesmaximal
14#24676C212yesmaximal
15#23067C212yesmaximal, complemented
16#23066C212yesmaximal, complemented
17#22743C212yesmaximal, complemented
18#21712C211yesmaximal, characteristic, complemented
19C42×D8C211yesmaximal, characteristic
20C2×#533—21no—
21C2×#476C2212yes—
22C2×#474C2212yes—
23C2×#446C2212yes—
24C2×C4×D8C2211yescharacteristic
25C2×C4×D8C2211yescharacteristic
26C2×C4×D8C2211yescharacteristic
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2857—21no—
30#2857—21no—
31#2571C2212yes—
32#2563C2212yes—
33#2552C2212yes—
34#2542C2212yes—
35#2538C2212yes—
36#2538C2212yes—
37#2497—22no—
38#2492—22no—
39#2441—22no—
40#2439—22no—
41#2429—22no—
42#2423—22no—
43#2369C2212yes—
44#2365C2212yes—
45#2355C2212yes—
46#2354C2212yes—
47#2354C2212yes—
48Q8×C4⋊C4C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50#2285—21no—
51C42×D4C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2101C2214yes—
54#2100C2214yes—
55#2087C2214yes—
56#2073C2214yes—
57#2071C2214yes—
58#2054C2214yes—
59#1990C2212yes—
60#1990C2212yes—
61#1990C2212yes—
62#1990C2212yes—
63#1988C2212yes—
64#1987C2212yes—
65#1984C2212yes—
66#1949C2214yes—
67#1948C2212yes—
68#1948C2212yes—
69#1947C2212yes—
70#1947C2214yes—
71#1947C2212yes—
72#1939C2212yes—
73#1939C2212yes—
74#1936C2214yes—
75#1936C2214yes—
76#1926C2214yes—
77#1925C2214yes—
78#1921C2214yes—
79#1919C2214yes—
80#1842C2211yescharacteristic
81#1836C2214yes—
82#1834C2212yes—
83#1834C2212yes—
84#1802C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1776C2214yes—
89C4×#107C2211yescharacteristic
90C4×#107C2211yescharacteristic
91C4×#107C2211yescharacteristic
92#1770C2212yes—
93#1770C2212yes—
94C4×#102C2211yescharacteristic
95C4×#102C2211yescharacteristic
96C4×#102C2211yescharacteristic
97C4×#102C2211yescharacteristic
98C4×#102C2211yescharacteristic
99C4×#102C2211yescharacteristic
100C42×C8C2211yescharacteristic
101C22×D8C2311yescharacteristic
102#533—48no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8D411yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#125—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C2×#117—22no—
126C22×C42—21no—
127C22×C42—21no—
128#476—28no—
129#474—28no—
130#446—28no—
131C4×D8—24no—
132C4×D8—24no—
133C4×D8—24no—
134C2×#107C2311yescharacteristic
135C2×#107C2311yescharacteristic
136C2×#107C2311yescharacteristic
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2311yescharacteristic
146C2×#102C2311yescharacteristic
147C2×#102C2311yescharacteristic
148C2×#102C2311yescharacteristic
149C2×#102C2311yescharacteristic
150C2×#102C2311yescharacteristic
151C2×#97C2314yes—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158#400—24no—
159#399—24no—
160#398—24no—
161#396—24no—
162#396—24no—
163#393—24no—
164#391—22no—
165#391—22no—
166#391—24no—
167#391D412yes—
168#390C2314yes—
169#390C2314yes—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—24no—
175#389D412yes—
176#389D412yes—
177#389C2314yes—
178#389C2314yes—
179#388—24no—
180#388—24no—
181#387—24no—
182#387—24no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384—21no—
187#384D411yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#384C2314yes—
193#382—22no—
194#382C2314yes—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2311yescharacteristic
229#340C2311yescharacteristic
230#340C2311yescharacteristic
231#340C2311yescharacteristic
232#340C2311yescharacteristic
233#340C2311yescharacteristic
234#336C2314yes—
235#336C2314yes—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8C2×D412yes—
239C22×D4C2411yescharacteristic
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C2×D8—22no—
248C2×D8—22no—
249C2×D8—22no—
250C22×C8C2411yescharacteristic
251C22×C8C2411yescharacteristic
252C22×C8C2414yes—
253C22×C8C2411yescharacteristic
254C22×C8C2411yescharacteristic
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—82no—
262C4⋊Q8—82no—
263C4⋊Q8—82no—
264#125—48no—
265C22⋊Q8—48no—
266C22⋊Q8—48no—
267C22⋊Q8—84no—
268C22⋊Q8—84no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—48no—
272C4×Q8—48no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4#4114yes—
284C4×D4#4114yes—
285#117—48no—
286#117—84no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C42—24no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—24no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2414yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—22no—
344#107—22no—
345#107—22no—
346#107—28no—
347#106—28no—
348C4⋊C8—28no—
349C4⋊C8—24no—
350C4⋊C8—24no—
351C4⋊C8—24no—
352#102—22no—
353#102—22no—
354#102—28no—
355#102—28no—
356#102—22no—
357#102—22no—
358#102—22no—
359#102—22no—
360#97—28no—
361C4×C8—22no—
362C4×C8—22no—
363C4×C8—22no—
364C4×C8—22no—
365C4×C8—22no—
366C4×C8#4114yes—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95C2×D414yes—
380#95C2×D414yes—
381C24—21no—
382C24—21no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4C2×#4112yes—
403C2×D4C2×#4112yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2511yescharacteristic
439D8—28no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8C2×#4112yes—
454C2×C8C2×#4112yes—
455C4⋊C4—24no—
456C4⋊C4—28no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4C2×#4112yes—
490C4⋊C4C2×#4112yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—48no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—24no—
506C42—48no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—84no—
514C42AGL(1,ℤ/8)14yes—
515C42C2×#4112yes—
516C42C2×#4112yes—
517C42C2×#4112yes—
518C422−1+412yes—
519C422−1+412yes—
520C422−1+412yes—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23C23×D411yescharacteristic
536Q8—48no—
537Q8—88no—
538Q8—164no—
539D4—22no—
540D4—22no—
541D4—22no—
542D4—22no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4#54712yes—
590C2×C4#54712yes—
591C2×C4#54812yes—
592C2×C4#54812yes—
593C2×C4#54812yes—
594C2×C4#54812yes—
595C2×C4C2×AGL(1,ℤ/8)12yes—
596C2×C4C2×AGL(1,ℤ/8)12yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4#57612yes—
599C2×C4#57712yes—
600C2×C4#57812yes—
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C2×2−1+411yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C2×C4C2×2−1+411yescharacteristic
607C8—22no—
608C8—22no—
609C8—44no—
610C8—41no—
611C8—41no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22C2×#54711yescharacteristic
627C22C2×#54811yescharacteristic
628C22C2×#54811yescharacteristic
629C22C22×AGL(1,ℤ/8)11yescharacteristic
630C22C2×#57611yescharacteristic
631C22C2×#57711yescharacteristic
632C22C2×#57811yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4—84no—
644C4—81no—
645C4—84no—
646C4#335414yes—
647C4#336614yes—
648C4#336714yes—
649C4#344411yescharacteristic
650C4#350911yescharacteristic
651C4#350911yescharacteristic
652C4#350911yescharacteristic
653C2—82no—
654C2—82no—
655C2#3615111yescharacteristic
656C2C2×#335411yescharacteristic
657C2#6128711yescharacteristic
658C2C2×#336611yescharacteristic
659C2C2×#336711yescharacteristic
660C2#6132911yescharacteristic
661C2#6133011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D8C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
2. #36134C2semidirect
3. #33289C2semidirect
4. #33197C2semidirect
5. #33195C2semidirect
7. #27736C2semidirect
8. #27735C2semidirect
9. #26513C2semidirect
10. #26512C2semidirect
11. #24993C2semidirect
12. #24992C2semidirect
15. #23067C2semidirect
16. #23066C2semidirect
17. #22743C2semidirect
18. #21712C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#361342
#332892
#331972
#331952
#281942
#277362
#277352
#265132
#265122
#249932
#249922
#246772
#246762
#230672
#230662
#227432
#217122
C42×D82