Finite Group Databasebeta

← #6615078

Subgroups — #6615078

Order 512.

1917 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36127C211yesmaximal, characteristic
2C4×#554C211yesmaximal, characteristic, complemented
3#33196C211yesmaximal, characteristic, complemented
4#33147C212yesmaximal, complemented
5#29219C211yesmaximal, characteristic, complemented
6#28280C211yesmaximal, characteristic, complemented
7#27162C214yesmaximal, complemented
8#26660C214yesmaximal, complemented
9#24967C212yesmaximal, complemented
10#24703C212yesmaximal, complemented
11#24154C212yesmaximal, complemented
12#24118C212yesmaximal, complemented
13#22963C212yesmaximal, complemented
14#22795C211yesmaximal, characteristic, complemented
15#22770C212yesmaximal, complemented
16C4×#460C211yesmaximal, characteristic, complemented
17#21386C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C2×#547—21no—
20C2×#532C2212yes—
21C2×#475C2212yes—
22C2×#476C2212yes—
23C2×#460C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#431C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2547—21no—
29#2537—21no—
30#2537C2212yes—
31#2432—22no—
32#2422—22no—
33#2422C2214yes—
34#2370—21no—
35#2369C2211yescharacteristic
36#2368—22no—
37#2360C2212yes—
38#2355C2211yescharacteristic
39#2349—21no—
40D4×C4⋊C4—21no—
41D4×C4⋊C4C2212yes—
42D4×C4⋊C4C2211yescharacteristic
43#2345C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#125—21no—
47C4×C22⋊Q8—22no—
48C4×C22⋊Q8C2212yes—
49C42×Q8C2211yescharacteristic
50C42×D4—21no—
51C42×D4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53#2099C2214yes—
54#2077C2214yes—
55#2065C2214yes—
56#2035C2214yes—
57#2034C2214yes—
58#2000C2214yes—
59#1990C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1988C2212yes—
63#1982C2212yes—
64#1982C2211yescharacteristic
65#1954C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1941C2214yes—
71#1933C2214yes—
72#1926C2212yes—
73#1923C2214yes—
74#1919C2214yes—
75#1919C2214yes—
76#1880C2212yes—
77#1877C2214yes—
78#1874C2214yes—
79#1874C2214yes—
80#1864C2214yes—
81#1853C2214yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1824C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2211yescharacteristic
88#1802C2212yes—
89#1797C2212yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2214yes—
93C4×#102C2214yes—
94C4×#102C2212yes—
95C22×#41—22no—
96C22×SD16C2312yes—
97#554—24no—
98#554—24no—
99#547—48no—
100#532—28no—
101#532—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#125—21no—
108C2×#124—22no—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2312yes—
113C2×#121—22no—
114C2×#121—22no—
115C2×#121C2314yes—
116C2×C4×Q8—21no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2312yes—
128C2×C4×D4C2314yes—
129C2×C4×D4C2311yescharacteristic
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C42—22no—
134C22×C42C2312yes—
135#476—28no—
136#475—44no—
137#460—24no—
138C4×SD16—28no—
139#431—44no—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2314yes—
159#397—24no—
160#394—24no—
161#394—24no—
162#391—21no—
163#391—22no—
164#391—22no—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#385—24no—
186#385—24no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#382—21no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2314yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C2×#95—22no—
225C43—21no—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#336C2314yes—
237#336C2312yes—
238C2×#41—48no—
239C2×#41—48no—
240C2×#41—84no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8C2411yescharacteristic
244C22×D4—24no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2412yes—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2412yes—
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—42no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C4⋊Q8—22no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—44no—
266C4⋊Q8—82no—
267C4⋊Q8#4114yescomplemented
268#127—22no—
269#127#4114yes—
270#125—28no—
271#124—48no—
272#123—416no—
273C22⋊Q8—216no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—48no—
277#121—28no—
278#121—44no—
279#121—48no—
280#121—48no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—41no—
286C4×Q8—82no—
287C4×Q8#4114yescomplemented
288C4×D4—28no—
289C4×D4—28no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—42no—
295C4×D4—44no—
296C4×D4—48no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4SD1618yescomplemented
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—21no—
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—22no—
356C2×C42—22no—
357C2×C42—84no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#106—24no—
369#106—28no—
370#106—44no—
371C4⋊C8—22no—
372C4⋊C8—24no—
373C4⋊C8—22no—
374C4⋊C8—24no—
375C4⋊C8—42no—
376#103—24no—
377#103—44no—
378#102—24no—
379#102—24no—
380#102—28no—
381#102—24no—
382#102—44no—
383C22⋊C8—44no—
384C22⋊C8—44no—
385C4×C8—24no—
386C4×C8—44no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403C24—22no—
404C24C22×D412yes—
405#41—832no—
406C2×Q8—24no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×Q8C2×#4112yes—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—84no—
434C2×D4C2×SD1614yes—
435C2×D4C2×#4114yes—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—84no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4C2×SD1618yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C2511yescharacteristic
478SD16—48no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—82no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4C2×SD1614yes—
521C4⋊C4C2×#4112yes—
522C4⋊C4C2×#4114yes—
523C4⋊C42−1+412yes—
524C22⋊C4—28no—
525C22⋊C4—24no—
526C22⋊C4—42no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4C2×SD1618yes—
535C42—24no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—21no—
543C42—21no—
544C42—41no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—44no—
550C42—81no—
551C42—82no—
552C42—82no—
553C42C2×SD1614yes—
554C42C2×#4112yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C42H(5,𝔽2)12yes—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—24no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23C22×SD1614yes—
572C23C23×D411yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—41no—
576Q8—41no—
577Q8—84no—
578Q8—88no—
579D4—28no—
580D4—24no—
581D4—24no—
582D4—48no—
583D4—42no—
584D4—42no—
585D4—48no—
586D4—816no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—216no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—84no—
634C2×C4—816no—
635C2×C4—88no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4#52512yes—
639C2×C4#52512yes—
640C2×C4#54612yes—
641C2×C4#54612yes—
642C2×C4C22×SD1612yes—
643C2×C4C22×SD1614yes—
644C2×C4C22×SD1612yes—
645C2×C4C22×SD1612yes—
646C2×C4#57512yes—
647C2×C4#57612yes—
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×2−1+411yescharacteristic
653C2×C4#58511yescharacteristic
654C8—42no—
655C8—42no—
656C8—44no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22C2×#52511yescharacteristic
672C22C2×#54611yescharacteristic
673C22#348911yescharacteristic
674C22C23×SD1611yescharacteristic
675C22C2×#57511yescharacteristic
676C22C2×#57611yescharacteristic
677C22#358611yescharacteristic
678C4—22no—
679C4—22no—
680C4—24no—
681C4—22no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—41no—
687C4—42no—
688C4—42no—
689C4—82no—
690C4—84no—
691C4—82no—
692C4—81no—
693C4#310314yes—
694C4#329514yes—
695C4#347511yescharacteristic
696C4#348211yescharacteristic
697C4#351111yescharacteristic
698C4#351711yescharacteristic
699C2—44no—
700C2—84no—
701C2#3640411yescharacteristic
702C2C2×#310311yescharacteristic
703C2#6059611yescharacteristic
704C2C2×#329511yescharacteristic
705C2#6115911yescharacteristic
706C2#6209911yescharacteristic
707C2#6214311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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. C4×#554C2semidirect
3. #33196C2semidirect
4. #33147C2semidirect
5. #29219C2semidirect
6. #28280C2semidirect
7. #27162C2semidirect
8. #26660C2semidirect
9. #24967C2semidirect
10. #24703C2semidirect
11. #24154C2semidirect
12. #24118C2semidirect
13. #22963C2semidirect
14. #22795C2semidirect
15. #22770C2semidirect
16. C4×#460C2semidirect
17. #21386C2semidirect
267. C4⋊Q8#41semidirect
287. C4×Q8#41semidirect
300. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#361272
C4×#5542
#331962
#331472
#292192
#282802
#271622
#266602
#249672
#247032
#241542
#241182
#229632
#227952
#227702
C4×#4602
#213862