Finite Group Databasebeta

← #6795517

Subgroups — #6795517

Order 512.

1851 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36179C211yesmaximal, characteristic
2#34941C211yesmaximal, characteristic, complemented
3#33196C211yesmaximal, characteristic, complemented
4#32282C214yesmaximal, complemented
5#31932C214yesmaximal, complemented
6#29921C212yesmaximal, complemented
7#28687C211yesmaximal, characteristic, complemented
8#28530C212yesmaximal, complemented
9#28280C211yesmaximal, characteristic, complemented
10#27727C212yesmaximal, complemented
11#27680C212yesmaximal, complemented
12#25699C212yesmaximal, complemented
13#24653C212yesmaximal, complemented
14#24120C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16#21981C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#547—21no—
20C2×#547C2212yes—
21C2×#498C2212yes—
22C2×#475C2212yes—
23C2×#476C2212yes—
24C2×#460C2211yescharacteristic
25C2×C4×SD16C2212yes—
26C2×#431C2212yes—
27#2868C2211yescharacteristic
28#2856C2212yes—
29#2574—22no—
30#2567C2211yescharacteristic
31#2563C2211yescharacteristic
32#2552C2211yescharacteristic
33#2542—21no—
34#2542—21no—
35#2542C2212yes—
36#2538C2211yescharacteristic
37#2508—21no—
38#2501—22no—
39#2501—22no—
40#2501C2214yes—
41#2497C2212yes—
42#2454—21no—
43#2423C2212yes—
44#2396—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#125—21no—
48C4×C22⋊Q8C2212yes—
49#2293—21no—
50#2285C2212yes—
51C2×#384C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2099C2214yes—
55#2096C2214yes—
56#2094C2214yes—
57#2084C2214yes—
58#2071C2214yes—
59#2065C2214yes—
60#2034C2214yes—
61#2034C2214yes—
62#2034C2214yes—
63#2000C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1982C2212yes—
67#1982C2211yescharacteristic
68#1968C2214yes—
69#1954C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1948C2211yescharacteristic
76#1934C2214yes—
77#1926C2212yes—
78#1923C2214yes—
79#1880C2212yes—
80#1872C2214yes—
81#1853C2214yes—
82#1853C2214yes—
83#1853C2214yes—
84#1844C2214yes—
85#1842C2211yescharacteristic
86#1834C2214yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×#41—22no—
96C22×SD16C2312yes—
97#547—44no—
98#547—44no—
99#547—48no—
100#547—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#123—24no—
108C2×#123—24no—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4C2314yes—
131C22×C42C2312yes—
132#498—44no—
133#476—44no—
134#475—44no—
135#460—24no—
136C4×SD16—44no—
137#431—28no—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2311yescharacteristic
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C4×C8C2314yes—
158C2×C4×C8C2312yes—
159#400—22no—
160#399—22no—
161#399—24no—
162#399—24no—
163#399—24no—
164#398—22no—
165#398—22no—
166#396—22no—
167#396—22no—
168#395—22no—
169#393—22no—
170#391—21no—
171#391C2312yes—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—22no—
177#390—22no—
178#390C2312yes—
179#390C2314yes—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389C2312yes—
186#389C2312yes—
187#388—22no—
188#388—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384—22no—
196#384—22no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#384C2312yes—
202#384C2312yes—
203#384C2314yes—
204#382—21no—
205#382—22no—
206#380—22no—
207#380C2312yes—
208#380C2312yes—
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227C43—21no—
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238C2×#41—48no—
239C2×#41—48no—
240C2×#41—48no—
241C22×Q8—22no—
242C22×Q8—21no—
243C22×Q8C2411yescharacteristic
244C22×D4—24no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2412yes—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266#127—22no—
267#127—22no—
268#125—44no—
269#123—416no—
270#123—416no—
271#123—416no—
272C22⋊Q8—24no—
273C22⋊Q8—42no—
274C22⋊Q8—48no—
275C22⋊Q8—48no—
276C22⋊Q8—48no—
277#121—44no—
278#121—44no—
279#121—48no—
280#121—48no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—41no—
287C4×Q8—82no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4SD1618yescomplemented
296#117—84no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4—48no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#106—42no—
367#106—44no—
368#106—44no—
369#106—44no—
370#106—44no—
371C4⋊C8—24no—
372C4⋊C8—24no—
373C4⋊C8—22no—
374C4⋊C8—42no—
375C4⋊C8#4114yescomplemented
376#103—42no—
377#103—44no—
378#102—24no—
379#102—42no—
380#102—44no—
381#102—42no—
382#102—44no—
383C22⋊C8—28no—
384C22⋊C8—44no—
385C4×C8—28no—
386C4×C8—24no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—22no—
402C24C22×D412yes—
403#41—832no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—81no—
414C2×Q8—81no—
415C2×Q82−1+412yes—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4C2×SD1614yes—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—42no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4C2×SD1618yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2511yescharacteristic
470SD16—48no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8C2×#4114yes—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4C2×SD1614yes—
519C4⋊C4AGL(1,ℤ/8)14yes—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4C2×SD1618yes—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—21no—
540C42—21no—
541C42—41no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—42no—
547C42—44no—
548C42—81no—
549C42C2×SD1614yes—
550C42C2×#4112yes—
551C42H(5,𝔽2)12yes—
552C42H(5,𝔽2)12yes—
553C422−1+412yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23C22×SD1614yes—
568C23C23×D411yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—84no—
574Q8—88no—
575D4—28no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—84no—
581D4—816no—
582D4—84no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—216no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—48no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—816no—
630C2×C4—88no—
631C2×C4#53512yes—
632C2×C4#53512yes—
633C2×C4#54712yes—
634C2×C4#54712yes—
635C2×C4C22×SD1612yes—
636C2×C4C22×SD1614yes—
637C2×C4C22×SD1612yes—
638C2×C4C22×SD1612yes—
639C2×C4C2×AGL(1,ℤ/8)14yes—
640C2×C4C2×AGL(1,ℤ/8)12yes—
641C2×C4#57712yes—
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×2−1+411yescharacteristic
646C2×C4C2×2−1+411yescharacteristic
647C2×C4#58511yescharacteristic
648C8—24no—
649C8—44no—
650C8—42no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22C2×#53511yescharacteristic
666C22C2×#54711yescharacteristic
667C22#348911yescharacteristic
668C22C23×SD1611yescharacteristic
669C22C22×AGL(1,ℤ/8)11yescharacteristic
670C22C2×#57711yescharacteristic
671C22#358611yescharacteristic
672C4—24no—
673C4—22no—
674C4—24no—
675C4—22no—
676C4—22no—
677C4—42no—
678C4—42no—
679C4—41no—
680C4—81no—
681C4—82no—
682C4—84no—
683C4—81no—
684C4—82no—
685C4—81no—
686C4—82no—
687C4#320814yes—
688C4#333514yes—
689C4#351711yescharacteristic
690C4#352011yescharacteristic
691C4#354711yescharacteristic
692C4#355211yescharacteristic
693C2—44no—
694C2—162no—
695C2#3892011yescharacteristic
696C2C2×#320811yescharacteristic
697C2#6083611yescharacteristic
698C2C2×#333511yescharacteristic
699C2#6125211yescharacteristic
700C2#6209911yescharacteristic
701C2#6215011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34941C2semidirect
3. #33196C2semidirect
4. #32282C2semidirect
5. #31932C2semidirect
6. #29921C2semidirect
7. #28687C2semidirect
8. #28530C2semidirect
9. #28280C2semidirect
10. #27727C2semidirect
11. #27680C2semidirect
12. #25699C2semidirect
13. #24653C2semidirect
14. #24120C2semidirect
15. #22795C2semidirect
16. #21981C2semidirect
17. C4×#460C2semidirect
295. C4×D4SD16semidirect
375. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#361792
#349412
#331962
#322822
#319322
#299212
#286872
#285302
#282802
#277272
#276802
#256992
#246532
#241202
#227952
#219812
C4×#4602