Finite Group Databasebeta

← #6651592

Subgroups — #6651592

Order 512.

1652 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35249C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#33096C212yesmaximal, complemented
4#32695C212yesmaximal, complemented
5#32684C212yesmaximal, complemented
6#32647C212yesmaximal, complemented
7#32629C212yesmaximal, complemented
8#32580C214yesmaximal, complemented
9#32573C214yesmaximal, complemented
10#29638C212yesmaximal, complemented
11#27616C212yesmaximal, complemented
12#24608C212yesmaximal, complemented
13#24600C212yesmaximal, complemented
14#24060C212yesmaximal
15#21820C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#480C2212yes—
20C2×#480C2212yes—
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2854C2212yes—
24#2854C2212yes—
25#2632—21no—
26#2632—21no—
27#2580C2212yes—
28#2580C2212yes—
29#2567—21no—
30#2567—21no—
31#2567—21no—
32#2567—21no—
33#2567C2212yes—
34#2567C2212yes—
35#2567C2212yes—
36#2567C2212yes—
37#2567C2211yescharacteristic
38#2567C2211yescharacteristic
39#2548C2211yescharacteristic
40#2548C2211yescharacteristic
41#2501—21no—
42#2436—21no—
43#2406—21no—
44#2406—21no—
45C2×#395—21no—
46C2×#395—21no—
47#2326—21no—
48#2326—21no—
49C2×#391—21no—
50C2×#389C2212yes—
51C2×#389C2212yes—
52C2×#384—21no—
53#2283C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55#2101C2214yes—
56#2087C2214yes—
57#2086C2214yes—
58#2085C2214yes—
59#2067C2214yes—
60#2065C2214yes—
61#2057C2214yes—
62#2057C2214yes—
63#2057C2214yes—
64#2041C2214yes—
65#2023C2214yes—
66#2023C2214yes—
67#1985C2212yes—
68#1985C2212yes—
69#1984C2212yes—
70#1984C2212yes—
71#1944C2214yes—
72#1942C2214yes—
73#1936C2214yes—
74#1934C2214yes—
75#1926C2214yes—
76#1925C2214yes—
77#1921C2214yes—
78#1921C2214yes—
79#1920C2214yes—
80#1920C2214yes—
81#1874C2212yes—
82#1874C2212yes—
83#1862C2214yes—
84#1862C2214yes—
85#1853C2212yes—
86#1852C2212yes—
87#1823C2212yes—
88#1823C2212yes—
89#1817C2212yes—
90#1817C2212yes—
91C22×#41—21no—
92C23×Q8—21no—
93C23×Q8D411yescharacteristic
94#532—42no—
95#532—42no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—42no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#124—21no—
105C2×#124—21no—
106C2×#124—21no—
107C2×#123—22no—
108C2×#123—22no—
109C2×#121—21no—
110C2×#121C2312yes—
111C2×C4×Q8—42no—
112C2×C4×Q8—42no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4C2312yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#480—44no—
136#480—44no—
137#432—42no—
138#431—42no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×#97C2312yes—
152C2×C4×C8C2312yes—
153#399—22no—
154#399—22no—
155#395—22no—
156#395—22no—
157#395—22no—
158#395—22no—
159#395—22no—
160#395—22no—
161#395—22no—
162#395—22no—
163#395—21no—
164#395—21no—
165#395—21no—
166#395—21no—
167#394—21no—
168#394—21no—
169#394—21no—
170#394—21no—
171#393—22no—
172#393—22no—
173#391—22no—
174#391—22no—
175#391—22no—
176#391—22no—
177#391—21no—
178#391—21no—
179#390—21no—
180#390—21no—
181#390C2312yes—
182#390C2312yes—
183#390C2312yes—
184#390C2312yes—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2314yes—
190#389C2314yes—
191#389C2314yes—
192#389C2314yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#386—22no—
202#386—22no—
203#385—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#384—22no—
208#384—22no—
209#384—22no—
210#384—22no—
211#384—21no—
212#384—21no—
213#383—21no—
214#382C2312yes—
215#381—21no—
216#380C2312yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—21no—
224#376—21no—
225C2×#95—21no—
226C2×#95—22no—
227C2×#95—22no—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239#336C2314yes—
240C2×#41—44no—
241C2×#41—84no—
242C22×Q8—24no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—42no—
247C22×Q8—42no—
248C22×Q8—44no—
249C22×Q8—84no—
250C22×Q8C2×D411yescharacteristic
251C22×Q8C2×D411yescharacteristic
252C22×Q8C2×D411yescharacteristic
253C22×Q8C2×D411yescharacteristic
254C22×D4—22no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4C2412yes—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272C4⋊Q8—42no—
273C4⋊Q8—42no—
274C4⋊Q8—41no—
275C4⋊Q8—41no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#127—41no—
279#127—41no—
280#124—44no—
281#124—44no—
282#124—44no—
283#123—84no—
284#123—84no—
285#121—24no—
286#121—24no—
287#121—44no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×Q8—42no—
291C4×Q8—42no—
292C4×Q8—41no—
293C4×Q8—41no—
294C4×Q8—41no—
295C4×Q8—41no—
296C4×Q8—41no—
297C4×Q8—41no—
298C4×Q8—41no—
299C4×Q8—41no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4—44no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—84no—
324C2×C4⋊C4—84no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2414yes—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—28no—
377#107—44no—
378#107—44no—
379#106—28no—
380C4⋊C8—42no—
381C4⋊C8—42no—
382#102—28no—
383#102—28no—
384#102—44no—
385#102—44no—
386C22⋊C8—24no—
387C22⋊C8—24no—
388#97—42no—
389C4×C8—42no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405C24—22no—
406C24C22×D411yescharacteristic
407#41—816no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—41no—
414C2×Q8—41no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8—416no—
420C2×Q8—42no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—84no—
424C2×Q8—84no—
425C2×Q8—82no—
426C2×Q8—816no—
427C2×Q8#13714yes—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4—82no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4C2511yescharacteristic
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C4⋊C4—28no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C42−1+418yes—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—44no—
555C42—42no—
556C42—42no—
557C42—44no—
558C23—22no—
559C23—21no—
560C23—21no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23#53412yes—
569C23#57614yes—
570C23C23×D411yescharacteristic
571Q8—44no—
572Q8—44no—
573Q8—44no—
574Q8—44no—
575Q8—88no—
576Q8—88no—
577Q8—84no—
578D4—44no—
579D4—44no—
580D4—84no—
581D4—84no—
582D4—88no—
583C2×C4—28no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—28no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—416no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—88no—
635C2×C4—88no—
636C2×C4#53412yes—
637C2×C4C2×#13712yes—
638C2×C4C2×#13712yes—
639C2×C4C2×#13712yes—
640C2×C4#57512yes—
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×2−1+414yes—
644C8—44no—
645C8—44no—
646C22—22no—
647C22—22no—
648C22—24no—
649C22—24no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22#319914yes—
658C22C2×#53411yescharacteristic
659C22#348612yes—
660C22C22×#13711yescharacteristic
661C22C2×#57511yescharacteristic
662C22C2×#57612yes—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—48no—
677C4—82no—
678C4—84no—
679C4—82no—
680C4—82no—
681C4#354412yes—
682C4#354412yes—
683C2—24no—
684C2—162no—
685C2#3725811yescharacteristic
686C2C2×#319911yescharacteristic
687C2#6075111yescharacteristic
688C2#6183512yes—
689C2#6184412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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×#2567C2semidirect
3. #33096C2semidirect
4. #32695C2semidirect
5. #32684C2semidirect
6. #32647C2semidirect
7. #32629C2semidirect
8. #32580C2semidirect
9. #32573C2semidirect
10. #29638C2semidirect
11. #27616C2semidirect
12. #24608C2semidirect
13. #24600C2semidirect
15. #21820C2semidirect

Maximal subgroups

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

subgroupindex
#352492
C2×#25672
#330962
#326952
#326842
#326472
#326292
#325802
#325732
#296382
#276162
#246082
#246002
#240602
#218202