Finite Group Databasebeta

← #6803029

Subgroups — #6803029

Order 512.

1674 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#36579C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3#33109C211yesmaximal, characteristic, complemented
4#32676C214yesmaximal, complemented
5#32619C214yesmaximal, complemented
6#29719C212yesmaximal
7#28744C212yesmaximal, complemented
8#28315C212yesmaximal
9#28121C212yesmaximal, complemented
10#26319C212yesmaximal, complemented
11#25722C211yesmaximal, characteristic, complemented
12#25632C212yesmaximal, complemented
13#25301C212yesmaximal, complemented
14#24209C211yesmaximal, characteristic
15#22510C211yesmaximal, characteristic, complemented
16#22050C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#494C2212yes—
22C2×#484C2212yes—
23C2×#481C2211yescharacteristic
24C2×#461C2212yes—
25C2×#451C2212yes—
26C2×#451C2212yes—
27C2×#448C2212yes—
28C2×#446C2212yes—
29#2857C2211yescharacteristic
30#2850C2212yes—
31#2592C2212yes—
32#2581C2212yes—
33#2568—21no—
34#2568—21no—
35#2568—22no—
36#2568C2212yes—
37#2568C2211yescharacteristic
38#2568C2212yes—
39#2568C2214yes—
40#2538C2211yescharacteristic
41#2481—22no—
42#2469—22no—
43#2421—21no—
44#2409—21no—
45C2×#400—22no—
46#2362—21no—
47#2355C2211yescharacteristic
48#2352—21no—
49C4×C22⋊Q8C2212yes—
50C4×#121C2211yescharacteristic
51C2×#391—21no—
52C2×#391C2212yes—
53C2×#391C2212yes—
54C2×#382—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56#2097C2214yes—
57#2074C2214yes—
58#2068C2214yes—
59#2056C2214yes—
60#2056C2214yes—
61#2055C2214yes—
62#2043C2212yes—
63#2016C2214yes—
64#2015C2214yes—
65#2015C2214yes—
66#2013C2212yes—
67#1985C2214yes—
68#1984C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1973C2214yes—
72#1971C2214yes—
73#1971C2214yes—
74#1956C2212yes—
75#1947C2212yes—
76#1935C2214yes—
77#1928C2214yes—
78#1928C2214yes—
79#1927C2214yes—
80#1926C2212yes—
81#1919C2212yes—
82#1878C2214yes—
83#1878C2214yes—
84#1874C2211yescharacteristic
85#1872C2212yes—
86#1853C2211yescharacteristic
87#1824C2211yescharacteristic
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1787C2212yes—
91C4×#106C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×Q8C2312yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×SD16C2314yes—
100#556—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#126—22no—
108C2×#124C2312yes—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2312yes—
112C2×#121C2311yescharacteristic
113C2×C4×Q8—44no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D412yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×D4—21no—
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D412yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#494—44no—
134#484—44no—
135#481—24no—
136#461—44no—
137#451—28no—
138#451—44no—
139#448—44no—
140#446—44no—
141C2×#107C2314yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2311yescharacteristic
158C2×#97C2314yes—
159C2×C4×C8C2312yes—
160#400—24no—
161#400—24no—
162#400—24no—
163#400—24no—
164#400—22no—
165#400—22no—
166#397—22no—
167#396—22no—
168#395—22no—
169#393—22no—
170#391—22no—
171#391—22no—
172#391—24no—
173#391—21no—
174#391—21no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391—22no—
179#391C2314yes—
180#391C2312yes—
181#391C2312yes—
182#391C2314yes—
183#391C2314yes—
184#391C2314yes—
185#391C2312yes—
186#390C2312yes—
187#390C2311yescharacteristic
188#389—21no—
189#389—21no—
190#389C2312yes—
191#389C2311yescharacteristic
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—21no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#386—22no—
202#386—22no—
203#386—21no—
204#385—22no—
205#384C2311yescharacteristic
206#382—22no—
207#382—22no—
208#382—24no—
209#382—21no—
210#382C2312yes—
211#380—21no—
212#380—21no—
213#380—21no—
214#380C2311yescharacteristic
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225C2×#95—21no—
226C2×#95—22no—
227C2×#95—22no—
228C2×#95—22no—
229C2×#95—22no—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—24no—
240C22×Q8—22no—
241C22×Q8—24no—
242C22×Q8—24no—
243C22×Q8—42no—
244C22×Q8—42no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2×D412yes—
250C22×Q8C2412yes—
251C22×Q8C2414yes—
252C22×D4—21no—
253C22×D4C2411yescharacteristic
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263C2×Q16—28no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×Q16—44no—
267C2×Q16—44no—
268C2×Q16—44no—
269C2×SD16—44no—
270C2×SD16—44no—
271C2×SD16—44no—
272C22×C8C2414yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281#126—48no—
282#124—44no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287#121—22no—
288#121—22no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8—42no—
294C4×Q8—42no—
295C4×Q8—44no—
296C4×D4—24no—
297C4×D4—44no—
298#117—44no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42—44no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359#107—44no—
360#106—24no—
361#106—42no—
362C4⋊C8—24no—
363C4⋊C8—44no—
364#103—28no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371#102—24no—
372#102—42no—
373C22⋊C8—24no—
374C22⋊C8—22no—
375C22⋊C8#4114yes—
376#97—44no—
377C4×C8—24no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401C24—21no—
402C24C22×D411yescharacteristic
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—416no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—88no—
427C2×Q8—88no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4#13714yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4C2511yescharacteristic
481Q16—88no—
482Q16—88no—
483SD16—88no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8C2×#4114yes—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—81no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—81no—
533C22⋊C4—82no—
534C42—22no—
535C42—22no—
536C42—24no—
537C42—22no—
538C42—24no—
539C42—42no—
540C42—42no—
541C42—44no—
542C42—44no—
543C42—42no—
544C42—44no—
545C42—82no—
546C42H(5,𝔽2)14yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23#55012yes—
559C23C2×#13712yes—
560C23#57812yes—
561C23C23×D411yescharacteristic
562Q8—48no—
563Q8—44no—
564Q8—44no—
565Q8—44no—
566Q8—44no—
567Q8—44no—
568Q8—44no—
569Q8—44no—
570Q8—44no—
571Q8—88no—
572D4—42no—
573D4—42no—
574D4—84no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—88no—
629C2×C4—84no—
630C2×C4#53412yes—
631C2×C4#53412yes—
632C2×C4#55012yes—
633C2×C4C2×#13712yes—
634C2×C4C2×#13712yes—
635C2×C4#57512yes—
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×H(5,𝔽2)12yes—
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4#58512yes—
640C8—24no—
641C8—44no—
642C8—42no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22#339814yes—
657C22C2×#53411yescharacteristic
658C22C2×#55011yescharacteristic
659C22#348711yescharacteristic
660C22C22×#13711yescharacteristic
661C22C2×#57511yescharacteristic
662C22C2×#57811yescharacteristic
663C22#358611yescharacteristic
664C4—22no—
665C4—22no—
666C4—22no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4—82no—
678C4—84no—
679C4—82no—
680C4#319614yes—
681C4#351412yes—
682C4#355812yes—
683C2—24no—
684C2—161no—
685C2#3913511yescharacteristic
686C2C2×#319611yescharacteristic
687C2#6075911yescharacteristic
688C2C2×#339811yescharacteristic
689C2#6140311yescharacteristic
690C2#6189311yescharacteristic
691C2#6190511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2568C2semidirect
3. #33109C2semidirect
4. #32676C2semidirect
5. #32619C2semidirect
7. #28744C2semidirect
9. #28121C2semidirect
10. #26319C2semidirect
11. #25722C2semidirect
12. #25632C2semidirect
13. #25301C2semidirect
15. #22510C2semidirect
16. #22050C2semidirect

Maximal subgroups

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

subgroupindex
#365792
C2×#25682
#331092
#326762
#326192
#297192
#287442
#283152
#281212
#263192
#257222
#256322
#253012
#242092
#225102
#220502
C4×#4812