Finite Group Databasebeta

← #6801062

Subgroups — #6801062

Order 512.

1597 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic, complemented
2#34945C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic, complemented
4#31827C214yesmaximal, complemented
5#31170C214yesmaximal, complemented
6#29999C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#29131C212yesmaximal, complemented
9#28253C212yesmaximal
10#28252C211yesmaximal, characteristic, complemented
11#28194C212yesmaximal, complemented
12#26111C212yesmaximal, complemented
13#25301C212yesmaximal, complemented
14#24727C212yesmaximal, complemented
15#22722C211yesmaximal, characteristic, complemented
16#22153C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#554—21no—
19C2×#474C2212yes—
20C2×#460C2212yes—
21C2×#463C2214yes—
22C2×#461C2212yes—
23C2×#459C2211yescharacteristic
24C2×#446C2212yes—
25C2×C4×D8C2212yes—
26#2868C2211yescharacteristic
27#2859C2212yes—
28#2573—22no—
29#2571C2211yescharacteristic
30#2568C2211yescharacteristic
31#2566—22no—
32#2563—21no—
33#2547C2211yescharacteristic
34#2538—21no—
35#2538C2211yescharacteristic
36#2538C2212yes—
37#2514C2212yes—
38#2494—21no—
39#2452—22no—
40#2452C2214yes—
41#2447C2212yes—
42#2421—21no—
43#2401—22no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#127C2212yes—
49C4×#124C2212yes—
50C4×C22⋊Q8—21no—
51#2295C2212yes—
52#2283—21no—
53C42×Q8C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2101C2214yes—
56#2087C2214yes—
57#2087C2214yes—
58#2068C2214yes—
59#2068C2214yes—
60#2055C2214yes—
61#2055C2214yes—
62#2054C2214yes—
63#2042C2214yes—
64#2024C2214yes—
65#1988C2212yes—
66#1982C2211yescharacteristic
67#1973C2214yes—
68#1956C2212yes—
69#1947C2211yescharacteristic
70#1947C2212yes—
71#1947C2212yes—
72#1940C2214yes—
73#1939C2212yes—
74#1936C2214yes—
75#1936C2214yes—
76#1935C2214yes—
77#1925C2212yes—
78#1879C2212yes—
79#1877C2214yes—
80#1862C2214yes—
81#1842C2212yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1804C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2211yescharacteristic
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×SD16C2314yes—
96C22×D8C2312yes—
97#554—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#125—22no—
104C2×#124C2312yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—24no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4—22no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2314yes—
121C22×C4⋊C4—22no—
122C22×C42—21no—
123#474—44no—
124#463—48no—
125#461—44no—
126#460—28no—
127#459—24no—
128#446—44no—
129C4×D8—44no—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150#401—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#397—22no—
155#397—22no—
156#397—22no—
157#396—24no—
158#396—24no—
159#396—22no—
160#393—24no—
161#391—21no—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#390—22no—
167#390—21no—
168#390—21no—
169#390—22no—
170#390C2314yes—
171#390C2312yes—
172#390C2312yes—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389C2312yes—
179#389C2314yes—
180#389C2314yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—24no—
189#385—24no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2312yes—
196#383—22no—
197#382—21no—
198#382—21no—
199#382—22no—
200#382C2312yes—
201#381—22no—
202#380—22no—
203#380—22no—
204#380—21no—
205#380—21no—
206#380—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224#376—22no—
225C43C2311yescharacteristic
226C43C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2412yes—
242C22×D4C2411yescharacteristic
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—24no—
248C2×SD16—28no—
249C2×SD16—44no—
250C2×SD16—44no—
251C2×D8—24no—
252C2×D8—24no—
253C2×D8—42no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8—44no—
262C4⋊Q8—82no—
263#127—22no—
264#127#4114yes—
265#125—48no—
266#124—24no—
267#124—42no—
268C22⋊Q8—416no—
269C22⋊Q8—44no—
270C22⋊Q8—48no—
271C4×Q8—24no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—42no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—41no—
285C4×D4—44no—
286C4×D4—48no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—84no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
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—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—44no—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—42no—
352#107—44no—
353C4⋊C8—22no—
354C4⋊C8—24no—
355C4⋊C8—24no—
356C4⋊C8—22no—
357C4⋊C8—44no—
358C4⋊C8—44no—
359C4⋊C8—42no—
360#103—44no—
361#103—44no—
362#103—44no—
363#102—24no—
364#102—28no—
365#102—28no—
366#102—42no—
367#102—42no—
368#102—42no—
369#97—44no—
370C4×C8—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—21no—
392C24—22no—
393C2×Q8—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—82no—
404C2×Q8—82no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4C2×#4114yes—
422C2×D4H(5,𝔽2)12yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—48no—
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×C4—84no—
444C22×C4—84no—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2511yescharacteristic
456SD16—88no—
457D8—48no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—42no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—82no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—22no—
519C42—21no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—42no—
525C42—42no—
526C42—41no—
527C42—44no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42AGL(1,ℤ/8)14yes—
532C42C2×#4112yes—
533C42H(5,𝔽2)12yes—
534C422−1+412yes—
535C422−1+412yes—
536C23—24no—
537C23—22no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—41no—
549C23—42no—
550C23C23×D411yescharacteristic
551Q8—44no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—164no—
557D4—24no—
558D4—22no—
559D4—24no—
560D4—22no—
561D4—48no—
562D4—48no—
563D4—41no—
564D4—42no—
565D4—41no—
566D4—42no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
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—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4—168no—
614C2×C4#53712yes—
615C2×C4#53712yes—
616C2×C4#54812yes—
617C2×C4#54812yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4C2×AGL(1,ℤ/8)12yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4#57512yes—
622C2×C4#57812yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4C2×2−1+411yescharacteristic
627C2×C4C2×2−1+411yescharacteristic
628C2×C4#58511yescharacteristic
629C8—42no—
630C8—44no—
631C8—42no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—82no—
639C22—81no—
640C22—82no—
641C22—82no—
642C22—81no—
643C22—81no—
644C22—82no—
645C22—81no—
646C22C2×#53711yescharacteristic
647C22C2×#54811yescharacteristic
648C22#348811yescharacteristic
649C22C22×AGL(1,ℤ/8)11yescharacteristic
650C22C2×#57511yescharacteristic
651C22C2×#57811yescharacteristic
652C22#358511yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—41no—
660C4—42no—
661C4—81no—
662C4—84no—
663C4—81no—
664C4—82no—
665C4—82no—
666C4—81no—
667C4—82no—
668C4#325114yes—
669C4#336714yes—
670C4#351511yescharacteristic
671C4#351911yescharacteristic
672C4#354411yescharacteristic
673C4#354711yescharacteristic
674C2—82no—
675C2—84no—
676C2#3912611yescharacteristic
677C2C2×#325111yescharacteristic
678C2#6093011yescharacteristic
679C2C2×#336711yescharacteristic
680C2#6131811yescharacteristic
681C2#6195911yescharacteristic
682C2#6198511yescharacteristic
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. #36559C2semidirect
2. #34945C2semidirect
3. #33195C2semidirect
4. #31827C2semidirect
5. #31170C2semidirect
6. #29999C2semidirect
7. #29141C2semidirect
8. #29131C2semidirect
10. #28252C2semidirect
11. #28194C2semidirect
12. #26111C2semidirect
13. #25301C2semidirect
14. #24727C2semidirect
15. #22722C2semidirect
16. #22153C2semidirect

Maximal subgroups

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

subgroupindex
#365592
#349452
#331952
#318272
#311702
#299992
#291412
#291312
#282532
#282522
#281942
#261112
#253012
#247272
#227222
#221532
C4×#4592