Finite Group Databasebeta

← #6803178

Subgroups — #6803178

Order 512.

1581 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2#34945C211yesmaximal, characteristic, complemented
3#33197C211yesmaximal, characteristic, complemented
4#32531C214yesmaximal, complemented
5#31910C214yesmaximal, complemented
6#29827C212yesmaximal, complemented
7#28511C211yesmaximal, characteristic, complemented
8#28371C212yesmaximal
9#28282C212yesmaximal, complemented
10#28263C212yesmaximal, complemented
11#28252C211yesmaximal, characteristic, complemented
12#26087C212yesmaximal, complemented
13#25293C212yesmaximal, complemented
14#24627C212yesmaximal, complemented
15Q8×C4⋊C8C211yesmaximal, characteristic, complemented
16#22113C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556—21no—
19C2×#477C2212yes—
20C2×#477C2212yes—
21C2×#476C2212yes—
22C2×#460C2212yes—
23C2×#461C2212yes—
24C2×#459C2211yescharacteristic
25C2×C8×Q8C2212yes—
26C2×C4×D8C2212yes—
27#2868C2211yescharacteristic
28#2858C2212yes—
29#2573—22no—
30#2571C2211yescharacteristic
31#2568C2211yescharacteristic
32#2547—21no—
33#2547C2211yescharacteristic
34#2545—22no—
35#2538—21no—
36#2538C2211yescharacteristic
37#2538C2212yes—
38#2514C2212yes—
39#2512—22no—
40#2512C2214yes—
41#2494—21no—
42#2447C2212yes—
43#2421—21no—
44#2401—22no—
45#2354—21no—
46#2354C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×#124C2212yes—
51C4×C22⋊Q8—21no—
52#2295C2212yes—
53#2283—21no—
54C42×Q8C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2095C2214yes—
57#2092C2214yes—
58#2067C2214yes—
59#2067C2214yes—
60#2067C2214yes—
61#2066C2214yes—
62#2055C2214yes—
63#2055C2214yes—
64#2055C2214yes—
65#2012C2214yes—
66#1987C2212yes—
67#1983C2214yes—
68#1982C2211yescharacteristic
69#1970C2214yes—
70#1968C2214yes—
71#1956C2212yes—
72#1949C2214yes—
73#1948C2212yes—
74#1948C2212yes—
75#1947C2211yescharacteristic
76#1942C2214yes—
77#1939C2214yes—
78#1939C2212yes—
79#1933C2214yes—
80#1928C2214yes—
81#1925C2212yes—
82#1921C2214yes—
83#1879C2212yes—
84#1845C2212yes—
85#1842C2211yescharacteristic
86#1842C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2211yescharacteristic
91#1802C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#102C2212yes—
95C22×SD16C2314yes—
96C22×D8C2312yes—
97#556—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#126—24no—
104C2×#124—22no—
105C2×#124—22no—
106C2×#124C2312yes—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C22×C42—21no—
122#477—28no—
123#477—44no—
124#476—44no—
125#461—44no—
126#460—44no—
127#459—24no—
128C8×Q8—44no—
129C4×D8—44no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2314yes—
152#401—22no—
153#400—22no—
154#400—22no—
155#400—22no—
156#397—22no—
157#397—22no—
158#397—22no—
159#397—24no—
160#396—22no—
161#396—24no—
162#396—24no—
163#391—21no—
164#391—22no—
165#391C2312yes—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—22no—
172#390—22no—
173#390C2312yes—
174#390C2314yes—
175#390C2314yes—
176#390C2312yes—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389C2314yes—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#388—24no—
187#388—24no—
188#387—22no—
189#387—22no—
190#387—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#383—22no—
199#382—21no—
200#382—21no—
201#382C2312yes—
202#381—22no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380—21no—
207#380—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—21no—
227#376—22no—
228C43C2311yescharacteristic
229C43C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2412yes—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C2×SD16—44no—
249C2×SD16—44no—
250C2×SD16—44no—
251C2×D8—24no—
252C2×D8—24no—
253C2×D8—42no—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—44no—
262C4⋊Q8—82no—
263#127—22no—
264#127—22no—
265#126—416no—
266#124—24no—
267#124—42no—
268#124—48no—
269#124—84no—
270C22⋊Q8—44no—
271C22⋊Q8—48no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—48no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—24no—
286C4×D4—41no—
287C4×D4—44no—
288C4×D4—44no—
289#117—48no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—28no—
351#107—42no—
352#107—44no—
353#106—44no—
354C4⋊C8—24no—
355C4⋊C8—24no—
356C4⋊C8—22no—
357C4⋊C8—42no—
358C4⋊C8—44no—
359C4⋊C8—44no—
360C4⋊C8#4114yes—
361#103—28no—
362#103—44no—
363#103—44no—
364#102—24no—
365#102—42no—
366#102—42no—
367#102—42no—
368#102—44no—
369#102—44no—
370C4×C8—24no—
371C4×C8—44no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—21no—
393C24—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8AGL(1,ℤ/8)14yes—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D42−1+412yes—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—81no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2511yescharacteristic
454SD16—88no—
455D8—48no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8C2×#4114yes—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—82no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C42—24no—
513C42—22no—
514C42—24no—
515C42—22no—
516C42—24no—
517C42—22no—
518C42—21no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—22no—
523C42—42no—
524C42—41no—
525C42—48no—
526C42—44no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42#13514yes—
531C42C2×#4112yes—
532C42H(5,𝔽2)12yes—
533C42H(5,𝔽2)12yes—
534C422−1+412yes—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23C23×D411yescharacteristic
550Q8—28no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—88no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—42no—
560D4—42no—
561D4—41no—
562D4—42no—
563D4—42no—
564D4—84no—
565D4—84no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—88no—
611C2×C4#53712yes—
612C2×C4#53712yes—
613C2×C4#55012yes—
614C2×C4#55012yes—
615C2×C4C2×#13512yes—
616C2×C4C2×#13512yes—
617C2×C4C2×#13512yes—
618C2×C4C2×AGL(1,ℤ/8)14yes—
619C2×C4C2×AGL(1,ℤ/8)12yes—
620C2×C4#57812yes—
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×2−1+411yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C2×C4#58511yescharacteristic
627C8—24no—
628C8—42no—
629C8—44no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22C2×#53711yescharacteristic
645C22C2×#55011yescharacteristic
646C22#348911yescharacteristic
647C22C22×#13511yescharacteristic
648C22C22×AGL(1,ℤ/8)11yescharacteristic
649C22C2×#57811yescharacteristic
650C22#358511yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—24no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—81no—
660C4—81no—
661C4—82no—
662C4—84no—
663C4—82no—
664C4—81no—
665C4—82no—
666C4#324714yes—
667C4#339314yes—
668C4#351811yescharacteristic
669C4#352011yescharacteristic
670C4#354611yescharacteristic
671C4#355211yescharacteristic
672C2—82no—
673C2—162no—
674C2#3913611yescharacteristic
675C2C2×#324711yescharacteristic
676C2#6093011yescharacteristic
677C2C2×#339311yescharacteristic
678C2#6140211yescharacteristic
679C2#6209811yescharacteristic
680C2#6214811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36594C2semidirect
2. #34945C2semidirect
3. #33197C2semidirect
4. #32531C2semidirect
5. #31910C2semidirect
6. #29827C2semidirect
7. #28511C2semidirect
9. #28282C2semidirect
10. #28263C2semidirect
11. #28252C2semidirect
12. #26087C2semidirect
13. #25293C2semidirect
14. #24627C2semidirect
15. Q8×C4⋊C8C2semidirect
16. #22113C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#349452
#331972
#325312
#319102
#298272
#285112
#283712
#282822
#282632
#282522
#260872
#252932
#246272
Q8×C4⋊C82
#221132
C4×#4592