Finite Group Databasebeta

← #6489424

Subgroups — #6489424

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#36635C211yesmaximal, characteristic
3#33209C211yesmaximal, characteristic
4#33207C211yesmaximal, characteristic
5#33043C211yesmaximal, characteristic
6#33043C211yesmaximal, characteristic
7#30380C214yesmaximal, complemented
8#29220C211yesmaximal, characteristic, complemented
9#29178C211yesmaximal, characteristic, complemented
10#29177C211yesmaximal, characteristic, complemented
11#28719C211yesmaximal, characteristic, complemented
12#28688C211yesmaximal, characteristic, complemented
13#28656C211yesmaximal, characteristic, complemented
14#28653C211yesmaximal, characteristic, complemented
15#28515C211yesmaximal, characteristic, complemented
16#27860C214yesmaximal
17#24840C214yesmaximal
18#22115C214yesmaximal, complemented
19C2×C4×C4⋊C8C211yesmaximal, characteristic
20C2×#500C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#479C2214yes—
25C22×C4⋊C8C2212yes—
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2864—21no—
29#2864C2212yes—
30#2863—21no—
31#2863—21no—
32#2863—21no—
33#2863C2212yes—
34#2862C2211yescharacteristic
35#2862C2211yescharacteristic
36#2859C2211yescharacteristic
37#2859C2211yescharacteristic
38#2857C2211yescharacteristic
39#2857C2211yescharacteristic
40#2857C2211yescharacteristic
41#2857C2211yescharacteristic
42#2855—21no—
43#2855—21no—
44#2855—21no—
45#2855C2212yes—
46#2851—21no—
47#2851C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#125—21no—
53C4×#125C2212yes—
54C4×C22⋊Q8—21no—
55C4×C22⋊Q8—21no—
56C4×C22⋊Q8—21no—
57C4×C22⋊Q8C2212yes—
58#2295—21no—
59#2295—21no—
60#2295—21no—
61#2295C2212yes—
62#2283—21no—
63#2283C2212yes—
64C2×C43C2211yescharacteristic
65#2101C2214yes—
66#2097C2214yes—
67#2088C2214yes—
68#2086C2214yes—
69#2085C2214yes—
70#2074C2214yes—
71#2068C2214yes—
72#2066C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1989C2214yes—
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1949C2214yes—
83#1942C2214yes—
84#1941C2214yes—
85#1933C2214yes—
86#1933C2214yes—
87#1928C2214yes—
88#1927C2214yes—
89#1919C2214yes—
90#1918C2214yes—
91#1892C2214yes—
92#1892C2214yes—
93#1845C2211yescharacteristic
94#1844C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1808C2214yes—
98#1807C2214yes—
99#1805C2214yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2212yes—
103C4×C4⋊C8C2212yes—
104C4×C4⋊C8C2211yescharacteristic
105C4×C4⋊C8C2211yescharacteristic
106C2×#336C2212yes—
107C23×C8C2312yes—
108C23×C8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#125—21no—
114C2×#125C2312yes—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8C2312yes—
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#500—42no—
131#498—42no—
132#487—42no—
133#487—42no—
134#479—216no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8—28no—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×C4×C8—42no—
149C2×C4×C8—42no—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2312yes—
155C2×C4×C8C2312yes—
156#391—21no—
157#391—21no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#391C2312yes—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#385—22no—
213#385—22no—
214#385—22no—
215#385—22no—
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#383—22no—
221#383—22no—
222#382—21no—
223#382—21no—
224#382—21no—
225#382C2312yes—
226#382C2312yes—
227#382C2312yes—
228#381—22no—
229#381—22no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380—21no—
234#380—21no—
235#380C2312yes—
236#380C2312yes—
237#380C2312yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4—22no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C22⋊C4—22no—
251C4×C22⋊C4—22no—
252#376—22no—
253#376—22no—
254C43D411yescharacteristic
255C43D411yescharacteristic
256C43D411yescharacteristic
257C43D411yescharacteristic
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8C2412yes—
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285C22×C8—24no—
286C22×C8—24no—
287C22×C8—24no—
288C22×C8—24no—
289C22×C8—42no—
290C22×C8—42no—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303#125—28no—
304#125—44no—
305C22⋊Q8—28no—
306C22⋊Q8—44no—
307C22⋊Q8—44no—
308C22⋊Q8—44no—
309C4×Q8—28no—
310C4×Q8—44no—
311C4×Q8—44no—
312C4×Q8—44no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—42no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389#107—28no—
390#107—44no—
391#106—28no—
392#106—44no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—24no—
396C4⋊C8#4118yes—
397#103—28no—
398#103—28no—
399#103—44no—
400#103—44no—
401C4×C8—22no—
402C4×C8—22no—
403C4×C8—22no—
404C4×C8—22no—
405C4×C8—42no—
406C4×C8—42no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435C24C22×D411yescharacteristic
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—22no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4#13514yes—
477C22×C4#13714yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C42−1+414yes—
492C22×C4C2511yescharacteristic
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—48no—
504C2×C8—48no—
505C2×C8C2×#4114yes—
506C2×C8C2×#4114yes—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—21no—
567C42—22no—
568C42—21no—
569C42—22no—
570C42—41no—
571C42—41no—
572C42—44no—
573C42—44no—
574C42—44no—
575C42—44no—
576C42#13514yes—
577C42C2×#4114yes—
578C42H(5,𝔽2)12yes—
579C422−1+412yes—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23#53612yes—
585C23C2×#13512yes—
586C23C2×#13712yes—
587C23C23×D411yescharacteristic
588Q8—44no—
589Q8—44no—
590Q8—84no—
591Q8—84no—
592Q8—84no—
593C2×C4—28no—
594C2×C4—21no—
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—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4#53212yes—
652C2×C4#53212yes—
653C2×C4#53612yes—
654C2×C4#53712yes—
655C2×C4#53712yes—
656C2×C4C2×#13512yes—
657C2×C4C2×#13512yes—
658C2×C4C2×#13512yes—
659C2×C4C2×#13512yes—
660C2×C4C2×#13512yes—
661C2×C4C2×#13712yes—
662C2×C4C2×#13712yes—
663C2×C4#57512yes—
664C2×C4C22×#4112yes—
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×2−1+412yes—
667C2×C4C2×2−1+411yescharacteristic
668C8—24no—
669C8—24no—
670C8—42no—
671C8—42no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22#323114yes—
679C22C2×#53211yescharacteristic
680C22C2×#53611yescharacteristic
681C22C2×#53711yescharacteristic
682C22C22×#13511yescharacteristic
683C22C22×#13511yescharacteristic
684C22C22×#13711yescharacteristic
685C22C2×#57511yescharacteristic
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4#324814yes—
706C4#325014yes—
707C4#352012yes—
708C4#352312yes—
709C2—24no—
710C2#3524111yescharacteristic
711C2#6067311yescharacteristic
712C2#6070111yescharacteristic
713C2C2×#323111yescharacteristic
714C2#6084611yescharacteristic
715C2C2×#324811yescharacteristic
716C2C2×#325011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #30380C2semidirect
8. #29220C2semidirect
9. #29178C2semidirect
10. #29177C2semidirect
11. #28719C2semidirect
12. #28688C2semidirect
13. #28656C2semidirect
14. #28653C2semidirect
15. #28515C2semidirect
18. #22115C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#332092
#332072
#330432
#303802
#292202
#291782
#291772
#287192
#286882
#286562
#286532
#285152
#278602
#248402
#221152
C2×C4×C4⋊C82