Finite Group Databasebeta

← #6816553

Subgroups — #6816553

Order 512.

1954 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3384C211yesmaximal, characteristic, complemented
2C2×#3354C211yesmaximal, characteristic
3#39489C211yesmaximal, characteristic, complemented
4#39360C211yesmaximal, characteristic, complemented
5#33310C211yesmaximal, characteristic, complemented
6#31092C214yesmaximal, complemented
7#31086C214yesmaximal, complemented
8#29868C211yesmaximal, characteristic, complemented
9#29864C211yesmaximal, characteristic, complemented
10#28713C211yesmaximal, characteristic, complemented
11#28684C211yesmaximal, characteristic, complemented
12#26078C214yesmaximal, complemented
13#25835C211yesmaximal, characteristic, complemented
14#25834C211yesmaximal, characteristic, complemented
15#25700C211yesmaximal, characteristic, complemented
16#25699C211yesmaximal, characteristic, complemented
17#25046C214yesmaximal, complemented
18#21631C211yesmaximal, characteristic, complemented
19#21275C211yesmaximal, characteristic, complemented
20C2×#555C2211yescharacteristic
21C2×#554C2211yescharacteristic
22C2×D4×Q8C2211yescharacteristic
23C2×#547C2211yescharacteristic
24C2×#536—21no—
25#3384—28no—
26#3354—28no—
27C2×#501C2212yes—
28C2×#498C2211yescharacteristic
29C2×#498C2211yescharacteristic
30C2×#498C2211yescharacteristic
31C2×#498C2211yescharacteristic
32C2×#481C2212yes—
33C2×#482C2212yes—
34C2×#478C2212yes—
35C2×#476C2211yescharacteristic
36C2×#476C2211yescharacteristic
37C2×#476C2212yes—
38C2×#474C2211yescharacteristic
39C2×#474C2211yescharacteristic
40C2×#469C2212yes—
41C2×#446C2211yescharacteristic
42C2×#442C2212yes—
43C2×C4×D8C2211yescharacteristic
44C2×#436C2211yescharacteristic
45C2×#428C2212yes—
46#2868C2211yescharacteristic
47#2868C2211yescharacteristic
48#2680—24no—
49#2674—24no—
50#2621—21no—
51#2620—21no—
52#2567C2211yescharacteristic
53#2563C2211yescharacteristic
54#2547C2211yescharacteristic
55#2546C2211yescharacteristic
56#2501—21no—
57#2499—21no—
58#2441C2214yes—
59#2440C2214yes—
60#2434—21no—
61#2432—21no—
62#2422C2214yes—
63#2422C2214yes—
64#2317—21no—
65C4×#124C2211yescharacteristic
66#2293C2211yescharacteristic
67#2293C2211yescharacteristic
68#2088C2214yes—
69#2084C2214yes—
70#2083C2214yes—
71#2077C2214yes—
72#2071C2214yes—
73#2071C2214yes—
74#2059C2214yes—
75#2058C2214yes—
76#2034C2214yes—
77#2034C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1987C2211yescharacteristic
82#1984C2211yescharacteristic
83#1970C2212yes—
84#1954C2211yescharacteristic
85#1943C2214yes—
86#1942C2212yes—
87#1934C2212yes—
88#1934C2212yes—
89#1923C2214yes—
90#1864C2214yes—
91#1864C2214yes—
92#1854C2214yes—
93#1853C2214yes—
94#1844C2214yes—
95#1844C2214yes—
96#1831C2211yescharacteristic
97#1809C2214yes—
98C4×#107C2211yescharacteristic
99C4×#102C2211yescharacteristic
100C4×#102C2211yescharacteristic
101#1753C2214yes—
102C42×C8C2211yescharacteristic
103C22×#41—21no—
104C23×Q8—21no—
105C22×D8C2311yescharacteristic
106C22×M16C2312yes—
107#555—24no—
108#555—42no—
109#554—24no—
110#554—42no—
111D4×Q8—24no—
112D4×Q8—42no—
113#547—24no—
114#547—42no—
115#536—48no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—22no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2312yes—
125C2×#125C2312yes—
126C2×#124—21no—
127C2×#124—22no—
128C2×#124D411yescharacteristic
129C2×#123—24no—
130C2×#123—22no—
131C2×#123—24no—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—21no—
134C2×C22⋊Q8—21no—
135C2×C22⋊Q8—22no—
136C2×C22⋊Q8—22no—
137C2×C22⋊Q8—24no—
138C2×#121—21no—
139C2×#121C2312yes—
140C2×#121C2312yes—
141C2×C4×Q8C2311yescharacteristic
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C2×C4×D4C2311yescharacteristic
145C2×C4×D4C2311yescharacteristic
146C2×C4×D4C2311yescharacteristic
147C2×#117—21no—
148C2×#117—21no—
149C2×#117—21no—
150C2×#117C2312yes—
151C22×C4⋊C4—21no—
152C22×C4⋊C4C2312yes—
153#501—28no—
154#498—24no—
155#498—24no—
156#498—42no—
157#498—42no—
158#482—28no—
159#481—28no—
160#478—28no—
161#476—24no—
162#476—24no—
163#476—28no—
164#474—24no—
165#474—24no—
166#469—28no—
167#446—24no—
168#442—28no—
169C4×D8D418yes—
170#436—24no—
171#428—28no—
172C2×#107C2312yes—
173C2×#107C2312yes—
174C2×#107C2312yes—
175C2×#107C2311yescharacteristic
176C2×#106C2312yes—
177C2×#106C2312yes—
178C2×#106C2312yes—
179C2×#106C2312yes—
180C2×#106C2312yes—
181C2×#106C2312yes—
182C2×#106C2314yes—
183C2×#106C2314yes—
184C2×C4⋊C8C2312yes—
185C2×C4⋊C8C2311yescharacteristic
186C2×C4⋊C8C2311yescharacteristic
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2312yes—
191C2×#102C2312yes—
192C2×#102C2312yes—
193C2×#102C2311yescharacteristic
194C2×#102C2311yescharacteristic
195C2×C22⋊C8C2312yes—
196C2×C22⋊C8C2314yes—
197C2×#97C2312yes—
198C2×C4×C8C2312yes—
199C2×C4×C8C2312yes—
200C2×C4×C8C2311yescharacteristic
201C2×C4×C8C2311yescharacteristic
202#399—24no—
203#398—22no—
204#398—24no—
205#397—22no—
206#397—24no—
207#397—24no—
208#396—24no—
209#395—22no—
210#395—24no—
211#394—24no—
212#394—24no—
213#393—22no—
214#391—21no—
215#391—21no—
216#391—21no—
217#391D411yescharacteristic
218#390—21no—
219#390D411yescharacteristic
220#390C2312yes—
221#390C2312yes—
222#389—24no—
223#389C2314yes—
224#389C2314yes—
225#389C2312yes—
226#389C2312yes—
227#388—22no—
228#388—22no—
229#388D414yes—
230#387—24no—
231#385—24no—
232#385—24no—
233#385—24no—
234#384—21no—
235#384—21no—
236#384—22no—
237#384—21no—
238#384C2314yes—
239#384C2314yes—
240#382—21no—
241#382D411yescharacteristic
242#380C2314yes—
243#380C2314yes—
244C4×C4⋊C4C2311yescharacteristic
245C4×C4⋊C4C2311yescharacteristic
246C4×C22⋊C4—21no—
247C4×C22⋊C4—21no—
248C4×C22⋊C4C2314yes—
249C43C2311yescharacteristic
250#340C2312yes—
251#340C2314yes—
252#340C2312yes—
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#340C2312yes—
257#336C2314yes—
258#336C2314yes—
259C2×#41—44no—
260C2×#41—44no—
261C2×#41—82no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—22no—
265C22×Q8—21no—
266C22×Q8—21no—
267C22×Q8—44no—
268C22×Q8—44no—
269C22×Q8—82no—
270C22×Q8C2×D411yescharacteristic
271C22×D4—22no—
272C22×D4—22no—
273C22×D4C2×D411yescharacteristic
274C22×D4C2411yescharacteristic
275C22×D4C2411yescharacteristic
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—22no—
279C23×C4—22no—
280C23×C4—22no—
281C23×C4C2412yes—
282C2×D8—24no—
283C2×D8C2×D414yes—
284C2×M16—24no—
285C2×M16—28no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2411yescharacteristic
289C22×C8C2412yes—
290C22×C8C2411yescharacteristic
291C4⋊Q8—24no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—42no—
301C4⋊Q8—44no—
302C4⋊Q8—44no—
303C4⋊Q8—84no—
304C4⋊Q8—82no—
305#125—24no—
306#125—42no—
307#124—24no—
308#124—48no—
309#124—44no—
310#123—48no—
311#123—416no—
312#123—88no—
313C22⋊Q8—48no—
314C22⋊Q8—48no—
315C22⋊Q8—416no—
316C22⋊Q8—44no—
317C22⋊Q8—44no—
318C22⋊Q8—44no—
319#121—24no—
320#121—24no—
321#121—42no—
322#121—42no—
323#121—44no—
324C4×Q8—22no—
325C4×Q8—22no—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—42no—
333C4×D4—42no—
334C4×D4C2×D414yes—
335C4×D4C2×D414yes—
336#117—24no—
337#117—42no—
338#117—44no—
339#117—44no—
340#117—44no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—42no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2411yescharacteristic
370C2×C4⋊C4C2411yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D414yes—
386C2×C22⋊C4C2×D414yes—
387C2×C42—22no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D412yes—
392C2×C42C2×D411yescharacteristic
393C2×C42C2412yes—
394C2×C42C2412yes—
395C2×C42C2411yescharacteristic
396#107—24no—
397#107—24no—
398#107—24no—
399#107C2×D414yes—
400#106—24no—
401#106—24no—
402#106—24no—
403#106—24no—
404#106—24no—
405#106—24no—
406#106—44no—
407#106—44no—
408C4⋊C8—24no—
409C4⋊C8—22no—
410C4⋊C8—22no—
411#102—24no—
412#102—24no—
413#102—24no—
414#102—24no—
415#102—24no—
416#102—24no—
417#102C2×D414yes—
418#102C2×D414yes—
419C22⋊C8—24no—
420C22⋊C8—28no—
421#97—24no—
422C4×C8—24no—
423C4×C8—24no—
424C4×C8—22no—
425C4×C8C2×D414yes—
426#95—22no—
427#95—22no—
428#95—21no—
429#95—21no—
430#95—24no—
431#95—24no—
432#95—24no—
433#95—24no—
434#95C2×D414yes—
435#95C2×D414yes—
436C24—21no—
437C24—21no—
438C24C22×D412yes—
439#41—816no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—82no—
454C2×Q8—82no—
455C2×Q8—82no—
456C2×Q8—816no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—82no—
470C2×D4—84no—
471C2×D4C22×D412yes—
472C2×D4C22×D412yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—44no—
496C22×C4—82no—
497C22×C4—84no—
498C22×C4—82no—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C2511yescharacteristic
507D8—28no—
508M16—48no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8C22×D412yes—
517C2×C8C22×D412yes—
518C2×C82−1+414yes—
519C2×C82−1+414yes—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—48no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—48no—
549C4⋊C4—48no—
550C4⋊C4—84no—
551C4⋊C4—84no—
552C4⋊C4—84no—
553C4⋊C4C22×D412yes—
554C4⋊C4C22×D412yes—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—28no—
559C22⋊C4—28no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—48no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—84no—
571C42—22no—
572C42—24no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—44no—
578C42—42no—
579C42—42no—
580C42AGL(1,ℤ/8)14yes—
581C42C22×D412yes—
582C42H(5,𝔽2)12yes—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—24no—
587C23—22no—
588C23—24no—
589C23—22no—
590C23—42no—
591C23—42no—
592C23—41no—
593C23—41no—
594C23—42no—
595C23—42no—
596C23—41no—
597C23—41no—
598C23C23×D411yescharacteristic
599Q8—44no—
600Q8—88no—
601Q8—84no—
602Q8—84no—
603Q8—162no—
604Q8—162no—
605D4—22no—
606D4—22no—
607D4—22no—
608D4—22no—
609D4—44no—
610D4—42no—
611D4—42no—
612D4—88no—
613D4—164no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—48no—
635C2×C4—48no—
636C2×C4—44no—
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—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—88no—
654C2×C4—88no—
655C2×C4—164no—
656C2×C4D4212yes—
657C2×C4D4212yes—
658C2×C4C2×AGL(1,ℤ/8)14yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4#57714yes—
661C2×C4#57712yes—
662C2×C4C23×D411yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×2−1+412yes—
665C2×C4C2×2−1+412yes—
666C8—22no—
667C8—24no—
668C8—42no—
669C8—42no—
670C8#53614yescomplemented
671C22—44no—
672C22—44no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22C2×D4211yescharacteristic
687C22#349912yes—
688C22C22×AGL(1,ℤ/8)11yescharacteristic
689C22C2×#57712yes—
690C22C2×#57711yescharacteristic
691C4—24no—
692C4—24no—
693C4—22no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4#327214yes—
706C4C2×#53611yescharacteristic
707C4#349911yescharacteristic
708C4#355212yes—
709C2—82no—
710C2—82no—
711C2—82no—
712C2#3936011yescharacteristic
713C2C2×#327211yescharacteristic
714C2#6108711yescharacteristic
715C2#6247712yes—
716C2#6248612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3354C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3384C2semidirect
3. #39489C2semidirect
4. #39360C2semidirect
5. #33310C2semidirect
6. #31092C2semidirect
7. #31086C2semidirect
8. #29868C2semidirect
9. #29864C2semidirect
10. #28713C2semidirect
11. #28684C2semidirect
12. #26078C2semidirect
13. #25835C2semidirect
14. #25834C2semidirect
15. #25700C2semidirect
16. #25699C2semidirect
17. #25046C2semidirect
18. #21631C2semidirect
19. #21275C2semidirect
670. C8#536semidirect

Maximal subgroups

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

subgroupindex
C2×#33842
C2×#33542
#394892
#393602
#333102
#310922
#310862
#298682
#298642
#287132
#286842
#260782
#258352
#258342
#257002
#256992
#250462
#216312
#212752