Finite Group Databasebeta

← #6495502

Subgroups — #6495502

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33311C211yesmaximal, characteristic, complemented
4#33290C211yesmaximal, characteristic, complemented
5#33227C211yesmaximal, characteristic, complemented
6#33226C211yesmaximal, characteristic, complemented
7#29845C211yesmaximal, characteristic, complemented
8#29844C211yesmaximal, characteristic, complemented
9#29843C211yesmaximal, characteristic
10#29842C211yesmaximal, characteristic
11#29194C214yesmaximal, complemented
12#28657C214yesmaximal, complemented
13#27893C214yesmaximal, complemented
14#25675C214yesmaximal, complemented
15#22221C211yesmaximal, characteristic, complemented
16#22220C211yesmaximal, characteristic, complemented
17#22219C211yesmaximal, characteristic, complemented
18#22218C211yesmaximal, characteristic, complemented
19C4×#431C211yesmaximal, characteristic, complemented
20C2×#529C2212yes—
21C2×#499C2214yes—
22C2×#488C2214yes—
23C2×#487C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#486C2211yescharacteristic
26C2×#486C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29#2864C2212yes—
30#2863—21no—
31#2863—21no—
32#2862C2211yescharacteristic
33#2861C2212yes—
34#2860—21no—
35#2860—21no—
36#2859C2211yescharacteristic
37#2859C2211yescharacteristic
38#2858C2211yescharacteristic
39#2857C2211yescharacteristic
40#2857C2211yescharacteristic
41#2857C2211yescharacteristic
42#2856C2211yescharacteristic
43#2855—21no—
44#2855—21no—
45#2854—21no—
46#2854—21no—
47#2851C2212yes—
48#2850C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#127C2211yescharacteristic
53C4×#125C2212yes—
54C4×#124C2212yes—
55C4×C22⋊Q8—21no—
56C4×C22⋊Q8—21no—
57C4×#121—21no—
58C4×#121—21no—
59#2295—21no—
60#2295—21no—
61#2293—21no—
62#2293—21no—
63#2283C2212yes—
64C2×C43C2211yescharacteristic
65#2068C2214yes—
66#2067C2214yes—
67#2066C2214yes—
68#2065C2214yes—
69#2036C2214yes—
70#2035C2214yes—
71#2034C2214yes—
72#2033C2214yes—
73#1991C2214yes—
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1958C2214yes—
79#1958C2214yes—
80#1958C2214yes—
81#1957C2211yescharacteristic
82#1957C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1928C2214yes—
86#1927C2214yes—
87#1926C2214yes—
88#1925C2214yes—
89#1865C2214yes—
90#1864C2214yes—
91#1863C2214yes—
92#1862C2214yes—
93#1846C2214yes—
94#1845C2211yescharacteristic
95#1845C2211yescharacteristic
96#1844C2211yescharacteristic
97#1844C2211yescharacteristic
98#1835C2214yes—
99C4×C4⋊C8C2212yes—
100C4×C4⋊C8C2212yes—
101C4×C22⋊C8C2212yes—
102C4×C22⋊C8C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C42×C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×#127C2311yescharacteristic
111#529—48no—
112C2×#126—22no—
113C2×#125C2312yes—
114C2×#125C2312yes—
115C2×#124C2312yes—
116C2×#124C2312yes—
117C2×#123—22no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8C2312yes—
121C2×#121—21no—
122C2×#121—21no—
123C2×#121C2312yes—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2312yes—
127C2×C4×D4—22no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×#117—22no—
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#499—48no—
140#488—48no—
141#487—24no—
142#487—24no—
143#486—24no—
144#486—24no—
145#431—24no—
146#431—24no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C4×C8C2312yes—
164C2×C4×C8C2312yes—
165C2×C4×C8C2312yes—
166C2×C4×C8C2312yes—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391C2312yes—
180#391C2312yes—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390C2312yes—
190#390C2312yes—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#388—22no—
212#388—22no—
213#388—22no—
214#388—22no—
215#388—22no—
216#388—22no—
217#388—22no—
218#388—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#387—22no—
224#387—22no—
225#387—22no—
226#387—22no—
227#386—22no—
228#386—22no—
229#385—22no—
230#385—22no—
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#384C2311yescharacteristic
234#384C2311yescharacteristic
235#383—22no—
236#382C2312yes—
237#382C2312yes—
238#382C2312yes—
239#382C2312yes—
240#381—22no—
241#380—21no—
242#380—21no—
243#380—21no—
244#380—21no—
245#380C2312yes—
246#380C2312yes—
247C4×C4⋊C4—22no—
248C4×C4⋊C4—22no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2312yes—
255C4×C22⋊C4—22no—
256C4×C22⋊C4—22no—
257C4×C22⋊C4—22no—
258#376—22no—
259C43D411yescharacteristic
260C43D411yescharacteristic
261C43D411yescharacteristic
262C43D411yescharacteristic
263C43C2311yescharacteristic
264C43C2311yescharacteristic
265C43C2311yescharacteristic
266C43C2311yescharacteristic
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8C2412yes—
278C22×D4—22no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4C2412yes—
282C23×C4—22no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2411yescharacteristic
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300#127—22no—
301#127—22no—
302#126—48no—
303#125—28no—
304#125—28no—
305#124—28no—
306#124—28no—
307#123—48no—
308C22⋊Q8—44no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311#121—44no—
312#121—44no—
313#121—44no—
314C4×Q8—44no—
315C4×Q8—44no—
316C4×Q8—44no—
317C4×D4—44no—
318C4×D4—44no—
319C4×D4—48no—
320C4×D4—44no—
321#117—48no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—42no—
373C2×C42—42no—
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×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402#107—44no—
403#107—44no—
404#106—44no—
405#106—44no—
406C4⋊C8—24no—
407C4⋊C8—24no—
408#103—28no—
409#103—28no—
410#103—44no—
411#103—44no—
412#102—28no—
413#102—28no—
414#102—44no—
415#102—44no—
416C22⋊C8—24no—
417C22⋊C8—24no—
418C4×C8—22no—
419C4×C8—22no—
420C4×C8—24no—
421C4×C8—24no—
422C4×C8—24no—
423C4×C8—24no—
424C4×C8#4114yes—
425C4×C8#4114yes—
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—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446C24—22no—
447C24C22×D411yescharacteristic
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—24no—
451C2×Q8—24no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4#13514yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C2511yescharacteristic
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8C2×#4114yes—
528C2×C8C2×#4114yes—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—21no—
590C42—21no—
591C42—22no—
592C42—22no—
593C42—41no—
594C42—41no—
595C42—41no—
596C42—41no—
597C42—44no—
598C42—44no—
599C42—44no—
600C42#13514yes—
601C42C2×#4112yes—
602C42C2×#4112yes—
603C42H(5,𝔽2)14yes—
604C422−1+414yes—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23C2×#13512yes—
619C23C23×D411yescharacteristic
620Q8—44no—
621Q8—44no—
622Q8—84no—
623Q8—84no—
624D4—44no—
625D4—44no—
626D4—88no—
627D4—84no—
628D4—84no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—28no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—88no—
685C2×C4#53212yes—
686C2×C4#53212yes—
687C2×C4#53512yes—
688C2×C4#53512yes—
689C2×C4#53612yes—
690C2×C4#53612yes—
691C2×C4C2×#13512yes—
692C2×C4C2×#13512yes—
693C2×C4C2×#13512yes—
694C2×C4C2×#13512yes—
695C2×C4C2×#13512yes—
696C2×C4#57512yes—
697C2×C4#57512yes—
698C2×C4C22×#4111yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C2×C4C2×H(5,𝔽2)12yes—
701C2×C4C2×2−1+412yes—
702C8—24no—
703C8—24no—
704C8—42no—
705C8—42no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22C2×#53211yescharacteristic
721C22C2×#53511yescharacteristic
722C22C2×#53611yescharacteristic
723C22C22×#13511yescharacteristic
724C22C22×#13511yescharacteristic
725C22C2×#57511yescharacteristic
726C22C2×#57511yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4#316814yes—
746C4#318314yes—
747C4#351112yes—
748C4#351112yes—
749C2—42no—
750C2—162no—
751C2#3529811yescharacteristic
752C2C2×#316811yescharacteristic
753C2C2×#318311yescharacteristic
754C2#6078211yescharacteristic
755C2#6078211yescharacteristic
756C2#6084611yescharacteristic
757C2#6084611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
1. #36635C2semidirect
3. #33311C2semidirect
4. #33290C2semidirect
5. #33227C2semidirect
6. #33226C2semidirect
7. #29845C2semidirect
8. #29844C2semidirect
11. #29194C2semidirect
12. #28657C2semidirect
13. #27893C2semidirect
14. #25675C2semidirect
15. #22221C2semidirect
16. #22220C2semidirect
17. #22219C2semidirect
18. #22218C2semidirect
19. C4×#431C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#366312
#333112
#332902
#332272
#332262
#298452
#298442
#298432
#298422
#291942
#286572
#278932
#256752
#222212
#222202
#222192
#222182
C4×#4312