Finite Group Databasebeta

← #6621130

Subgroups — #6621130

Order 512.

1781 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2C4×#550C211yesmaximal, characteristic
3#33201C212yesmaximal, complemented
4#33197C211yesmaximal, characteristic, complemented
5#28521C212yesmaximal, complemented
6#28509C211yesmaximal, characteristic, complemented
7#28280C212yesmaximal, complemented
8#28279C211yesmaximal, characteristic, complemented
9#26538C214yesmaximal, complemented
10#26533C212yesmaximal, complemented
11#26099C214yesmaximal, complemented
12#26091C212yesmaximal, complemented
13#22729C212yesmaximal, complemented
14#22722C211yesmaximal, characteristic, complemented
15#22545C212yesmaximal
16#21614C212yesmaximal, complemented
17C4×#474C211yesmaximal, characteristic, complemented
18C2×#550C2211yescharacteristic
19C2×C4×#41—21no—
20C2×#476C2212yes—
21C2×#476C2212yes—
22C2×#476C2212yes—
23C2×#474C2211yescharacteristic
24C2×#474C2212yes—
25C2×#446C2212yes—
26#2868C2211yescharacteristic
27#2862C2212yes—
28#2582—21no—
29#2571C2212yes—
30#2570C2214yes—
31#2568—21no—
32#2511—22no—
33#2446—22no—
34#2366C2211yescharacteristic
35#2366C2212yes—
36#2365—21no—
37#2363—21no—
38#2363—22no—
39#2355—21no—
40#2355C2212yes—
41#2354C2211yescharacteristic
42#2354C2212yes—
43Q8×C4⋊C4C2211yescharacteristic
44#2341—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×#127C2211yescharacteristic
48C4×#127C2212yes—
49C4×#121—21no—
50C4×#121—22no—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53C42×D4C2212yes—
54#2094C2214yes—
55#2092C2214yes—
56#2092C2214yes—
57#2067C2214yes—
58#2065C2214yes—
59#2065C2214yes—
60#1988C2211yescharacteristic
61#1987C2212yes—
62#1982C2212yes—
63#1982C2214yes—
64#1961C2214yes—
65#1957C2212yes—
66#1949C2214yes—
67#1949C2214yes—
68#1948C2214yes—
69#1948C2212yes—
70#1948C2212yes—
71#1947C2211yescharacteristic
72#1947C2212yes—
73#1940C2214yes—
74#1939C2212yes—
75#1919C2214yes—
76#1918C2212yes—
77#1862C2212yes—
78#1862C2214yes—
79#1847C2214yes—
80#1844C2212yes—
81#1842C2211yescharacteristic
82#1842C2212yes—
83#1804C2214yes—
84#1804C2214yes—
85#1802C2212yes—
86#1802C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90C4×#107C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1762C2214yes—
94C4×#102C2212yes—
95C22×#41—21no—
96#550—24no—
97#550Q818yescomplemented
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#127C2312yes—
104C2×#121—21no—
105C2×#121—22no—
106C4×#41—48no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—21no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C2×#117—22no—
127C22×C42—22no—
128C22×C42—21no—
129#476—28no—
130#476—44no—
131#476—44no—
132#474—24no—
133#474—44no—
134#446—28no—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2311yescharacteristic
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#400—24no—
158#400—24no—
159#398—24no—
160#391—22no—
161#391—22no—
162#391—21no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#390—21no—
168#390—22no—
169#390—21no—
170#390—22no—
171#390—22no—
172#390—21no—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#389—22no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389—21no—
181#389C2312yes—
182#389C2314yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—24no—
187#387—24no—
188#387—24no—
189#386—22no—
190#386—22no—
191#386—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#382—22no—
200#382C2314yes—
201#382C2314yes—
202#380—21no—
203#380—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226C43C2312yes—
227C43C2311yescharacteristic
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237C2×#41—28no—
238C2×#41—44no—
239C2×#41—44no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8C2×D411yescharacteristic
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4C2411yescharacteristic
248C22×D4C2412yes—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—21no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264#127—22no—
265#127—24no—
266#127C2×Q814yes—
267#127C2×Q818yes—
268#121—216no—
269#121—28no—
270C4×Q8—24no—
271C4×Q8—42no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—84no—
275C4×Q8C2×Q814yes—
276C4×Q8#4114yes—
277C4×D4—22no—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—24no—
281C4×D4—22no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—41no—
286C4×D4—42no—
287C4×D4—48no—
288C4×D4—48no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—84no—
292C4×D4—82no—
293C4×D4C2×Q814yes—
294C4×D4C2×Q818yes—
295#117—44no—
296#117—48no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—24no—
343C2×C42—48no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—24no—
359#107—24no—
360#107—44no—
361#106—28no—
362#106—44no—
363#106—44no—
364C4⋊C8—24no—
365C4⋊C8—28no—
366C4⋊C8—24no—
367C4⋊C8—22no—
368C4⋊C8—42no—
369C4⋊C8—44no—
370C4⋊C8#4114yescomplemented
371#102—24no—
372#102—28no—
373#102—42no—
374#102—44no—
375#102—44no—
376#102—44no—
377#102—44no—
378#97—28no—
379C4×C8—24no—
380#95—21no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—21no—
397C24—22no—
398#41—432no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8C22⋊Q814yes—
408C2×D4—28no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—28no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—82no—
427C2×D4—81no—
428C2×D4—82no—
429C2×D4—81no—
430C2×D4C22×Q812yes—
431C2×D4C22×Q814yes—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—81no—
456C22×C4—82no—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2511yescharacteristic
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8C2×#4114yes—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—82no—
503C4⋊C4—81no—
504C4⋊C4—84no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4C22⋊Q818yes—
508C4⋊C4C22⋊Q814yes—
509C4⋊C4C22×Q812yes—
510C4⋊C42−1+412yes—
511C22⋊C4—24no—
512C22⋊C4—28no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C42—24no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—21no—
528C42—22no—
529C42—21no—
530C42—22no—
531C42—42no—
532C42—42no—
533C42—41no—
534C42—44no—
535C42—44no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42—84no—
541C42AGL(1,ℤ/8)14yes—
542C42C22×Q814yes—
543C42C22×Q812yes—
544C42C2×#4112yes—
545C42C2×#4114yes—
546C42C2×#4112yes—
547C42H(5,𝔽2)12yes—
548C23—24no—
549C23—22no—
550C23—41no—
551C23—42no—
552C23—41no—
553C23—42no—
554C23—41no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—41no—
560C23—42no—
561C23—41no—
562C23C23×D411yescharacteristic
563Q8—28no—
564Q8—88no—
565Q8—84no—
566D4—22no—
567D4—24no—
568D4—24no—
569D4—22no—
570D4—416no—
571D4—48no—
572D4—41no—
573D4—42no—
574D4—42no—
575D4—41no—
576D4—88no—
577D4—84no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—416no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—84no—
621C2×C4—88no—
622C2×C4—84no—
623C2×C4—88no—
624C2×C4C2×C22⋊Q812yes—
625C2×C4C2×C22⋊Q814yes—
626C2×C4C2×C22⋊Q814yes—
627C2×C4C2×C22⋊Q812yes—
628C2×C4#55012yes—
629C2×C4#55012yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4C2×AGL(1,ℤ/8)12yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4#57512yes—
634C2×C4#57712yes—
635C2×C4C23×Q811yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×2−1+411yescharacteristic
640C2×C4#58511yescharacteristic
641C8—24no—
642C8—42no—
643C8—44no—
644C22—44no—
645C22—42no—
646C22—44no—
647C22—42no—
648C22—44no—
649C22—42no—
650C22—81no—
651C22—82no—
652C22—82no—
653C22—81no—
654C22—81no—
655C22—82no—
656C22—82no—
657C22—81no—
658C22C22×C22⋊Q811yescharacteristic
659C22C2×#55011yescharacteristic
660C22#348911yescharacteristic
661C22C22×AGL(1,ℤ/8)11yescharacteristic
662C22C2×#57511yescharacteristic
663C22C2×#57711yescharacteristic
664C22#358511yescharacteristic
665C4—24no—
666C4—24no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—41no—
674C4—42no—
675C4—82no—
676C4—82no—
677C4—81no—
678C4—82no—
679C4—84no—
680C4#307714yes—
681C4#339714yes—
682C4Q8×#4111yescharacteristic
683C4#348211yescharacteristic
684C4#350311yescharacteristic
685C4#351711yescharacteristic
686C2—84no—
687C2—82no—
688C2#3644511yescharacteristic
689C2C2×#307711yescharacteristic
690C2#6049511yescharacteristic
691C2C2×#339711yescharacteristic
692C2#6140511yescharacteristic
693C2#6211011yescharacteristic
694C2#6213511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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
1. #36598C2semidirect
3. #33201C2semidirect
4. #33197C2semidirect
5. #28521C2semidirect
6. #28509C2semidirect
7. #28280C2semidirect
8. #28279C2semidirect
9. #26538C2semidirect
10. #26533C2semidirect
11. #26099C2semidirect
12. #26091C2semidirect
13. #22729C2semidirect
14. #22722C2semidirect
16. #21614C2semidirect
17. C4×#474C2semidirect
97. #550Q8semidirect
370. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#365982
C4×#5502
#332012
#331972
#285212
#285092
#282802
#282792
#265382
#265332
#260992
#260912
#227292
#227222
#225452
#216142
C4×#4742