Finite Group Databasebeta

← #6574556

Subgroups — #6574556

Order 512.

2261 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C212yesmaximal, complemented
2D4×C4⋊Q8C211yesmaximal, characteristic, complemented
3C4×#547C211yesmaximal, characteristic
4#33195C211yesmaximal, characteristic, complemented
5#32955C212yesmaximal, complemented
6#28512C211yesmaximal, characteristic, complemented
7#28283C211yesmaximal, characteristic, complemented
8#26466C212yesmaximal, complemented
9#26355C214yesmaximal, complemented
10#26044C212yesmaximal, complemented
11#25978C214yesmaximal, complemented
12#24845C212yesmaximal, complemented
13#24301C212yesmaximal, complemented
14#22663C212yesmaximal, complemented
15#22650C211yesmaximal, characteristic, complemented
16#22431C212yesmaximal
17C4×#474C211yesmaximal, characteristic, complemented
18C2×D4×Q8—21no—
19C2×#547C2211yescharacteristic
20C22×C4⋊Q8C2212yes—
21C2×C4×#41C2212yes—
22#3079—216no—
23C2×#477C2214yes—
24C2×#475C2212yes—
25C2×#475C2212yes—
26C2×#475C2212yes—
27C2×#476C2214yes—
28C2×#474C2212yes—
29C2×#474C2211yescharacteristic
30C2×#436C2212yes—
31C2×#428C2214yes—
32C2×#423C2212yes—
33C2×#417C2214yes—
34#2868C2211yescharacteristic
35#2857C2212yes—
36#2551—21no—
37#2551C2212yes—
38#2546—21no—
39#2440—22no—
40#2440C2214yes—
41#2434—22no—
42#2365C2211yescharacteristic
43#2362C2212yes—
44#2361—21no—
45#2359—22no—
46#2354C2211yescharacteristic
47D4×C4⋊C4—21no—
48D4×C4⋊C4C2211yescharacteristic
49D4×C4⋊C4C2212yes—
50#2344—21no—
51#2339C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C22⋊Q8C2212yes—
55C4×#123—22no—
56C4×#121—21no—
57C42×D4C2211yescharacteristic
58C42×D4C2211yescharacteristic
59C2×C4×C4⋊C4—21no—
60#2096C2214yes—
61#2088C2214yes—
62#2068C2214yes—
63#2066C2214yes—
64#2058C2214yes—
65#2024C2214yes—
66#1988C2212yes—
67#1988C2211yescharacteristic
68#1981C2212yes—
69#1956C2212yes—
70#1948C2212yes—
71#1947C2211yescharacteristic
72#1947C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1918C2212yes—
76#1894C2214yes—
77#1872C2214yes—
78#1864C2214yes—
79#1863C2214yes—
80#1862C2212yes—
81#1854C2214yes—
82#1853C2214yes—
83#1851C2214yes—
84#1844C2212yes—
85#1842C2211yescharacteristic
86#1809C2214yes—
87#1803C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92#1788C2214yes—
93C4×#107C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#102C2212yes—
96C22×#41C2312yes—
97C23×Q8—22no—
98C22×M16C2314yes—
99D4×Q8—48no—
100#547—24no—
101#547Q818yescomplemented
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—28no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—44no—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×#123—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8C2312yes—
117C2×#121—21no—
118C4×#41—28no—
119C4×#41—44no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C2×#117—22no—
134C2×#117C2312yes—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2314yes—
139C22×C42—22no—
140C22×C42—21no—
141C22×C42C2312yes—
142#477—216no—
143#476—216no—
144#475—28no—
145#475—28no—
146#475—44no—
147#474—28no—
148#474D418yescomplemented
149#436—28no—
150#428—216no—
151C2×#107C2314yes—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#106C2314yes—
155C2×#106C2314yes—
156C2×#106C2314yes—
157#423—28no—
158C2×C4⋊C8C2311yescharacteristic
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2314yes—
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2311yescharacteristic
164#417—216no—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#102C2314yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×C22⋊C8C2314yes—
173C2×C4×C8C2312yes—
174#398—24no—
175#395—24no—
176#395—24no—
177#391—21no—
178#391—22no—
179#391C2312yes—
180#391C2312yes—
181#390—21no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2314yes—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#386—22no—
199#386—22no—
200#385—22no—
201#385—24no—
202#385—24no—
203#385—22no—
204#384—21no—
205#384—21no—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2312yes—
209#384C2312yes—
210#381—22no—
211#380—22no—
212#380—21no—
213#380C2312yes—
214#380C2314yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C22⋊C4—22no—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—21no—
232C4×C22⋊C4—21no—
233C4×C22⋊C4D412yes—
234C4×C22⋊C4D412yes—
235C4×C22⋊C4C2312yes—
236C2×#95—22no—
237C43C2311yescharacteristic
238#340C2314yes—
239#340C2314yes—
240#340C2312yes—
241#340C2312yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#336C2314yes—
246#336C2312yes—
247C2×#41—24no—
248C2×#41—24no—
249C2×#41—24no—
250C2×#41—42no—
251C2×#41—42no—
252C2×#41C2×Q818yescomplemented
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—24no—
256C22×Q8—48no—
257C22×Q8—48no—
258C22×Q8—84no—
259C22×Q8C2412yes—
260C22×D4—22no—
261C22×D4C2×D411yescharacteristic
262C22×D4C2411yescharacteristic
263C23×C4—21no—
264C23×C4—22no—
265C23×C4—21no—
266C23×C4—24no—
267C23×C4—22no—
268C23×C4—21no—
269C23×C4C2×D412yes—
270C23×C4C2×D412yes—
271C23×C4C2412yes—
272C2×M16—28no—
273C2×M16—28no—
274C2×M16—44no—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C4⋊Q8—28no—
279C4⋊Q8—24no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—24no—
283C4⋊Q8—44no—
284C4⋊Q8—48no—
285C4⋊Q8—48no—
286C4⋊Q8—48no—
287C4⋊Q8—44no—
288C4⋊Q8—82no—
289C4⋊Q8C2×D414yes—
290C4⋊Q8C2×Q814yescomplemented
291#123—216no—
292C22⋊Q8—24no—
293C22⋊Q8—416no—
294C22⋊Q8—48no—
295C22⋊Q8C2×Q818yescomplemented
296#121—28no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—42no—
301C4×Q8—44no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—22no—
305C4×D4—24no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—41no—
309C4×D4—42no—
310C4×D4—48no—
311C4×D4—48no—
312C4×D4—48no—
313C4×D4C2×D414yes—
314C4×D4C2×Q814yes—
315C4×D4C2×Q814yescomplemented
316C4×D4#4114yes—
317#117—24no—
318#117—48no—
319#117—42no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—44no—
343C2×C4⋊C4—48no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2414yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—48no—
376C2×C42—44no—
377C2×C42C2×D412yes—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D412yes—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—28no—
388#107—42no—
389#107C2×D418yes—
390#106—28no—
391#106—28no—
392#106—44no—
393C4⋊C8—28no—
394C4⋊C8—24no—
395C4⋊C8—24no—
396C4⋊C8—22no—
397C4⋊C8—42no—
398C4⋊C8C2×D414yes—
399#103—28no—
400#103—28no—
401#103—44no—
402#103—44no—
403#102—28no—
404#102—24no—
405#102C2×D418yes—
406C22⋊C8—28no—
407C4×C8—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—24no—
420#95—24no—
421#95C2×D412yes—
422#95C2×D412yes—
423#95C2×D412yes—
424C24—21no—
425C24C22×D412yes—
426#41—216no—
427#41—48no—
428#41—48no—
429#41—48no—
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—48no—
437C2×Q8—48no—
438C2×Q8—44no—
439C2×Q8—42no—
440C2×Q8—44no—
441C2×Q8—42no—
442C2×Q8—44no—
443C2×Q8—84no—
444C2×Q8—832no—
445C2×Q8C22×Q814yes—
446C2×D4—28no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4C22⋊Q814yes—
455C2×D4C22×D412yes—
456C2×D4C22×Q812yes—
457C2×D42−1+412yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—28no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—48no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—48no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—81no—
492C22×C4—82no—
493C22×C4—84no—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×Q814yes—
504C22×C4C2×#4114yes—
505C22×C4C2511yescharacteristic
506M16—416no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—24no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—24no—
515C2×C8C22×D414yes—
516C4⋊C4—24no—
517C4⋊C4—28no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—48no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—81no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4C22⋊Q814yes—
549C4⋊C4#13714yes—
550C4⋊C4C22×D414yes—
551C4⋊C4C22×D412yes—
552C4⋊C4C22×Q812yes—
553C22⋊C4—28no—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—42no—
561C22⋊C4—48no—
562C22⋊C4—82no—
563C22⋊C4C22⋊Q818yes—
564C42—24no—
565C42—24no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—21no—
571C42—21no—
572C42—41no—
573C42—44no—
574C42—44no—
575C42—44no—
576C42—42no—
577C42—42no—
578C42—42no—
579C42—44no—
580C42AGL(1,ℤ/8)14yes—
581C42C22×D412yes—
582C42C22×Q812yes—
583C42C2×#4112yes—
584C42H(5,𝔽2)12yes—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23C2×C22⋊Q814yes—
599C23C23×D411yescharacteristic
600Q8—24no—
601Q8—24no—
602Q8—42no—
603Q8—42no—
604Q8—48no—
605Q8—816no—
606Q8—88no—
607D4—28no—
608D4—22no—
609D4—22no—
610D4—22no—
611D4—416no—
612D4D4×Q814yescomplemented
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—48no—
638C2×C4—416no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—48no—
648C2×C4—48no—
649C2×C4—48no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—88no—
663C2×C4—84no—
664C2×C4—816no—
665C2×C4C2×C22⋊Q812yes—
666C2×C4C2×C22⋊Q814yes—
667C2×C4C2×C22⋊Q812yes—
668C2×C4D4212yes—
669C2×C4D4212yes—
670C2×C4C2×AGL(1,ℤ/8)12yes—
671C2×C4C2×AGL(1,ℤ/8)12yes—
672C2×C4C2×AGL(1,ℤ/8)12yes—
673C2×C4C2×#13714yes—
674C2×C4C2×#13712yes—
675C2×C4#57712yes—
676C2×C4C23×D411yescharacteristic
677C2×C4C23×Q811yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C2×C4C2×H(5,𝔽2)11yescharacteristic
680C2×C4C2×2−1+411yescharacteristic
681C2×C4#58511yescharacteristic
682C8—24no—
683C8—44no—
684C8—42no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—24no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22C22×C22⋊Q811yescharacteristic
700C22C2×D4211yescharacteristic
701C22#348911yescharacteristic
702C22C22×AGL(1,ℤ/8)11yescharacteristic
703C22C22×#13711yescharacteristic
704C22C2×#57711yescharacteristic
705C22#358611yescharacteristic
706C4—22no—
707C4—22no—
708C4—24no—
709C4—22no—
710C4—24no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—44no—
715C4—41no—
716C4—44no—
717C4—84no—
718C4—82no—
719C4—82no—
720C4—81no—
721C4#307914yescomplemented
722C4#327214yes—
723C4C2×D4×Q811yescharacteristic
724C4D4×#4111yescharacteristic
725C4#350111yescharacteristic
726C4#352011yescharacteristic
727C2—44no—
728C2—44no—
729C2D4×C22⋊Q811yescharacteristic
730C2C2×#307911yescharacteristic
731C2#6049711yescharacteristic
732C2C2×#327211yescharacteristic
733C2D4×#13711yescharacteristic
734C2#6210711yescharacteristic
735C2#6214711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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
1. C2×#3079C2semidirect
2. D4×C4⋊Q8C2semidirect
4. #33195C2semidirect
5. #32955C2semidirect
6. #28512C2semidirect
7. #28283C2semidirect
8. #26466C2semidirect
9. #26355C2semidirect
10. #26044C2semidirect
11. #25978C2semidirect
12. #24845C2semidirect
13. #24301C2semidirect
14. #22663C2semidirect
15. #22650C2semidirect
17. C4×#474C2semidirect
101. #547Q8semidirect
148. #474D4semidirect
252. C2×#41C2×Q8semidirect
290. C4⋊Q8C2×Q8semidirect
295. C22⋊Q8C2×Q8semidirect
315. C4×D4C2×Q8semidirect
612. D4D4×Q8semidirect
721. C4#3079semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
D4×C4⋊Q82
C4×#5472
#331952
#329552
#285122
#282832
#264662
#263552
#260442
#259782
#248452
#243012
#226632
#226502
#224312
C4×#4742