Finite Group Databasebeta

← #6806465

Subgroups — #6806465

Order 512.

1486 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#36574C211yesmaximal, characteristic, complemented
3#33232C211yesmaximal, characteristic, complemented
4#32101C214yesmaximal, complemented
5#32099C214yesmaximal, complemented
6#30219C211yesmaximal, characteristic
7#30219C211yesmaximal, characteristic
8#29193C211yesmaximal, characteristic, complemented
9#29191C211yesmaximal, characteristic, complemented
10#28750C211yesmaximal, characteristic
11#28750C211yesmaximal, characteristic
12#28748C211yesmaximal, characteristic, complemented
13#28514C211yesmaximal, characteristic
14#25656C214yesmaximal, complemented
15#25289C214yesmaximal, complemented
16#22221C211yesmaximal, characteristic, complemented
17#22218C211yesmaximal, characteristic, complemented
18#21646C211yesmaximal, characteristic
19#21340C211yesmaximal, characteristic, complemented
20C2×#556C2212yes—
21C2×#491C2214yes—
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#478C2214yes—
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2858C2211yescharacteristic
32#2584—21no—
33#2584C2212yes—
34#2581—21no—
35#2547C2212yes—
36#2545C2212yes—
37#2545C2212yes—
38#2538—21no—
39#2538—21no—
40#2512—21no—
41#2512—21no—
42#2512—21no—
43#2512—21no—
44#2509—21no—
45#2509C2212yes—
46#2508—21no—
47#2508C2212yes—
48#2453—21no—
49#2453—21no—
50#2370—21no—
51#2370C2212yes—
52#2354—21no—
53#2354—21no—
54#2351—21no—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C42×Q8C2211yescharacteristic
62#2092C2214yes—
63#2068C2214yes—
64#2066C2214yes—
65#2043C2214yes—
66#2042C2214yes—
67#2042C2214yes—
68#2042C2214yes—
69#2042C2214yes—
70#2042C2214yes—
71#2042C2214yes—
72#1990C2211yescharacteristic
73#1985C2212yes—
74#1983C2214yes—
75#1983C2214yes—
76#1981C2214yes—
77#1960C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1940C2212yes—
88#1936C2214yes—
89#1928C2214yes—
90#1927C2214yes—
91#1917C2212yes—
92#1917C2212yes—
93#1865C2214yes—
94#1865C2214yes—
95#1863C2214yes—
96#1863C2214yes—
97#1845C2211yescharacteristic
98#1844C2211yescharacteristic
99#1834C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103#1762C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107#556—48no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127C2311yescharacteristic
114C2×#126—22no—
115C2×#126—22no—
116C2×#125—21no—
117C2×#125—21no—
118C2×#125C2312yes—
119C2×#124C2312yes—
120C2×#124C2312yes—
121C2×#124C2312yes—
122C2×#124C2312yes—
123C2×C22⋊Q8—22no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8D411yescharacteristic
128C2×C4×Q8D411yescharacteristic
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2311yescharacteristic
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4C2312yes—
135C2×#117—22no—
136#491—48no—
137#487—24no—
138#487—24no—
139#486—24no—
140#478—48no—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#97C2312yes—
159C2×#97C2312yes—
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164#400—22no—
165#400—22no—
166#400—22no—
167#398—22no—
168#398—22no—
169#397—22no—
170#397—22no—
171#396—22no—
172#396—22no—
173#396—22no—
174#396—22no—
175#395—22no—
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#390—21no—
190#390—21no—
191#390C2312yes—
192#389—22no—
193#389—22no—
194#389—22no—
195#389—22no—
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#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#389C2312yes—
215#389C2312yes—
216#389C2312yes—
217#389C2312yes—
218#388—22no—
219#388—22no—
220#388—22no—
221#388—22no—
222#387—22no—
223#387—22no—
224#387—22no—
225#387—22no—
226#384—21no—
227#384—21no—
228#384—21no—
229#384—21no—
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#384C2311yescharacteristic
234#382—21no—
235#382—21no—
236#382—22no—
237#382C2312yes—
238#380—22no—
239#380—22no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—22no—
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2311yescharacteristic
258C4×C4⋊C4C2311yescharacteristic
259C4×C4⋊C4C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8C2×D411yescharacteristic
274C22×Q8C2412yes—
275C22×Q8C2412yes—
276C22×D4—21no—
277C22×D4—21no—
278C22×D4C2412yes—
279C23×C4—22no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—28no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—44no—
291C4⋊Q8#4114yes—
292#127—22no—
293#127—22no—
294#126—48no—
295#126—48no—
296#125—28no—
297#125—44no—
298#125—82no—
299#124—44no—
300#124—44no—
301#124—44no—
302#124—44no—
303C22⋊Q8—48no—
304C4×Q8—24no—
305C4×Q8—42no—
306C4×Q8—42no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×Q8—82no—
311C4×Q8—82no—
312C4×D4—28no—
313C4×D4—44no—
314C4×D4—44no—
315#117—48no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
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—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
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⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
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×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393#107—44no—
394#106—44no—
395C4⋊C8—24no—
396C4⋊C8—42no—
397C4⋊C8—42no—
398#103—28no—
399#103—28no—
400#103—44no—
401#103—44no—
402#103—44no—
403#103—44no—
404#103—44no—
405#103—44no—
406#102—28no—
407#102—44no—
408#102—44no—
409#102—44no—
410#97—24no—
411#97—42no—
412C4×C8—22no—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8—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#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438C24—22no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—24no—
449C2×Q8—41no—
450C2×Q8—41no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
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×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C2511yescharacteristic
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4C2×#4114yes—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
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—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—44no—
572C42—44no—
573C42—44no—
574C42—82no—
575C42—82no—
576C42C2×#4112yes—
577C42H(5,𝔽2)14yes—
578C422−1+412yes—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23C23×D411yescharacteristic
587Q8—44no—
588Q8—48no—
589Q8—44no—
590Q8—44no—
591Q8—44no—
592Q8—44no—
593D4—44no—
594D4—44no—
595D4—84no—
596D4—84no—
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—24no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
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—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
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—42no—
652C2×C4—88no—
653C2×C4#53512yes—
654C2×C4#53512yes—
655C2×C4#54812yes—
656C2×C4#54812yes—
657C2×C4#57512yes—
658C2×C4#57512yes—
659C2×C4#57712yes—
660C2×C4#57812yes—
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×H(5,𝔽2)12yes—
663C2×C4C2×2−1+411yescharacteristic
664C2×C4#58512yes—
665C8—42no—
666C8—42no—
667C8—42no—
668C8—42no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22C2×#53511yescharacteristic
677C22C2×#54811yescharacteristic
678C22#350011yescharacteristic
679C22C2×#57511yescharacteristic
680C22C2×#57511yescharacteristic
681C22C2×#57711yescharacteristic
682C22C2×#57811yescharacteristic
683C4—22no—
684C4—22no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—44no—
694C4—41no—
695C4—41no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4#337614yes—
703C4#337714yes—
704C4#351712yes—
705C4#355412yes—
706C2—162no—
707C2#3917411yescharacteristic
708C2#6080111yescharacteristic
709C2#6082411yescharacteristic
710C2C2×#337611yescharacteristic
711C2C2×#337711yescharacteristic
712C2#6254311yescharacteristic
713C2#6254511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
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
2. #36574C2semidirect
3. #33232C2semidirect
4. #32101C2semidirect
5. #32099C2semidirect
8. #29193C2semidirect
9. #29191C2semidirect
12. #28748C2semidirect
14. #25656C2semidirect
15. #25289C2semidirect
16. #22221C2semidirect
17. #22218C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#365742
#332322
#321012
#320992
#302192
#291932
#291912
#287502
#287482
#285142
#256562
#252892
#222212
#222182
#216462
#213402