Finite Group Databasebeta

← #6801228

Subgroups — #6801228

Order 512.

1547 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2#34934C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#32488C214yesmaximal, complemented
5#31978C214yesmaximal, complemented
6#29995C212yesmaximal, complemented
7#29130C212yesmaximal, complemented
8#28716C211yesmaximal, characteristic, complemented
9#28645C212yesmaximal, complemented
10#28279C211yesmaximal, characteristic, complemented
11#28279C212yesmaximal, complemented
12#26512C212yesmaximal, complemented
13#25477C212yesmaximal, complemented
14#25274C212yesmaximal
15#22722C211yesmaximal, characteristic, complemented
16#22116C212yesmaximal, complemented
17C4×#474C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#486C2212yes—
20C2×#476C2212yes—
21C2×#474C2212yes—
22C2×#474C2211yescharacteristic
23C2×#446C2212yes—
24#2868C2211yescharacteristic
25#2859C2212yes—
26#2582—21no—
27#2567C2212yes—
28#2554—22no—
29#2552—21no—
30#2552C2211yescharacteristic
31#2550—22no—
32#2548C2211yescharacteristic
33#2541C2211yescharacteristic
34#2538C2211yescharacteristic
35#2512C2212yes—
36#2507C2214yes—
37#2490—21no—
38#2451—22no—
39#2445C2212yes—
40#2426—21no—
41#2392—22no—
42#2369C2212yes—
43#2355—21no—
44Q8×C4⋊C4—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#127C2212yes—
49C4×#121—21no—
50#2293C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2100C2214yes—
54#2088C2214yes—
55#2087C2214yes—
56#2071C2214yes—
57#2067C2214yes—
58#2067C2214yes—
59#2067C2214yes—
60#2067C2214yes—
61#2065C2214yes—
62#2041C2214yes—
63#1990C2212yes—
64#1988C2211yescharacteristic
65#1987C2212yes—
66#1982C2212yes—
67#1982C2214yes—
68#1948C2212yes—
69#1948C2212yes—
70#1947C2211yescharacteristic
71#1947C2212yes—
72#1947C2214yes—
73#1943C2214yes—
74#1940C2214yes—
75#1939C2212yes—
76#1936C2214yes—
77#1936C2214yes—
78#1936C2214yes—
79#1926C2214yes—
80#1921C2214yes—
81#1918C2212yes—
82#1862C2212yes—
83#1842C2212yes—
84#1842C2211yescharacteristic
85#1802C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91C4×#107C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127C2311yescharacteristic
100C2×#127C2312yes—
101#529—48no—
102C2×#126—22no—
103C2×#125—21no—
104C2×#125—22no—
105C2×#124—21no—
106C2×#123—22no—
107C2×C22⋊Q8—21no—
108C2×#121—21no—
109C2×#121—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C22×C42—21no—
125#486—44no—
126#476—44no—
127#474—28no—
128#474—24no—
129#446—44no—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150#398—24no—
151#398—22no—
152#398—24no—
153#397—22no—
154#396—22no—
155#396—22no—
156#395—24no—
157#394—22no—
158#394—22no—
159#394—22no—
160#392—22no—
161#391—22no—
162#391—22no—
163#391—21no—
164#391—24no—
165#390—21no—
166#390—21no—
167#390—22no—
168#390—22no—
169#390—21no—
170#390C2312yes—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—24no—
182#389C2314yes—
183#389C2312yes—
184#389C2314yes—
185#389C2314yes—
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—24no—
192#386—22no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382—22no—
202#380—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C43C2312yes—
222C43C2311yescharacteristic
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4C2412yes—
242C22×D4C2411yescharacteristic
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—24no—
253C4⋊Q8—82no—
254C4⋊Q8—82no—
255C4⋊Q8#4114yes—
256#127—24no—
257#127—22no—
258#127—22no—
259#127—42no—
260#126—48no—
261#125—44no—
262#125—84no—
263#124—44no—
264#123—48no—
265C22⋊Q8—44no—
266#121—44no—
267#121—44no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—44no—
273C4×Q8—84no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—24no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—24no—
281C4×D4—41no—
282C4×D4—48no—
283C4×D4—44no—
284C4×D4—84no—
285C4×D4—82no—
286#117—48no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
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×C42—21no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—24no—
352#107—28no—
353#107—42no—
354#106—44no—
355C4⋊C8—22no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358C4⋊C8—22no—
359C4⋊C8—42no—
360C4⋊C8—44no—
361C4⋊C8—44no—
362#102—24no—
363#102—24no—
364#102—28no—
365#102—44no—
366#102—44no—
367#102—44no—
368#102—44no—
369#97—44no—
370C4×C8—24no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24—22no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—82no—
395C2×Q8—82no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—82no—
416C2×D4—81no—
417C2×D4—81no—
418C2×D4—82no—
419C2×D42−1+412yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—81no—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2511yescharacteristic
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—82no—
489C4⋊C4—81no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4#13714yes—
496C4⋊C4C2×#4114yes—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C42—24no—
512C42—22no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—22no—
519C42—22no—
520C42—21no—
521C42—22no—
522C42—42no—
523C42—41no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—44no—
528C42—84no—
529C42AGL(1,ℤ/8)14yes—
530C42C2×#4112yes—
531C42H(5,𝔽2)12yes—
532C422−1+412yes—
533C422−1+412yes—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23C23×D411yescharacteristic
549Q8—48no—
550Q8—44no—
551Q8—164no—
552D4—22no—
553D4—22no—
554D4—41no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—41no—
559D4—42no—
560D4—88no—
561D4—84no—
562D4—88no—
563D4—84no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—88no—
608C2×C4—84no—
609C2×C4#53712yes—
610C2×C4#53712yes—
611C2×C4#54812yes—
612C2×C4#54812yes—
613C2×C4C2×AGL(1,ℤ/8)12yes—
614C2×C4C2×AGL(1,ℤ/8)12yes—
615C2×C4C2×AGL(1,ℤ/8)12yes—
616C2×C4C2×#13714yes—
617C2×C4C2×#13712yes—
618C2×C4#57612yes—
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×2−1+411yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C2×C4#58511yescharacteristic
625C8—42no—
626C8—44no—
627C8—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22C2×#53711yescharacteristic
643C22C2×#54811yescharacteristic
644C22#348911yescharacteristic
645C22C22×AGL(1,ℤ/8)11yescharacteristic
646C22C22×#13711yescharacteristic
647C22C2×#57611yescharacteristic
648C22#358511yescharacteristic
649C4—22no—
650C4—22no—
651C4—24no—
652C4—22no—
653C4—44no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—42no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4—81no—
662C4—81no—
663C4—84no—
664C4#325414yes—
665C4#336614yes—
666C4#351711yescharacteristic
667C4#352311yescharacteristic
668C4#355211yescharacteristic
669C4#355511yescharacteristic
670C2—82no—
671C2—162no—
672C2#3912711yescharacteristic
673C2C2×#325411yescharacteristic
674C2#6092211yescharacteristic
675C2C2×#336611yescharacteristic
676C2#6131911yescharacteristic
677C2#6211011yescharacteristic
678C2#6212711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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. #36563C2semidirect
3. #33197C2semidirect
4. #32488C2semidirect
5. #31978C2semidirect
6. #29995C2semidirect
7. #29130C2semidirect
8. #28716C2semidirect
9. #28645C2semidirect
10. #28279C2semidirect
11. #28279C2semidirect
12. #26512C2semidirect
13. #25477C2semidirect
15. #22722C2semidirect
16. #22116C2semidirect
17. C4×#474C2semidirect

Maximal subgroups

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

subgroupindex
#365632
#349342
#331972
#324882
#319782
#299952
#291302
#287162
#286452
#282792
#265122
#254772
#252742
#227222
#221162
C4×#4742