Finite Group Databasebeta

← #6574553

Subgroups — #6574553

Order 512.

2299 conjugacy classes of subgroups.

The subgroup lattice has 746 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×#550C211yesmaximal, characteristic
4#33196C211yesmaximal, characteristic, complemented
5#32955C212yesmaximal, complemented
6#28508C211yesmaximal, characteristic, complemented
7#28279C211yesmaximal, characteristic, complemented
8#26464C212yesmaximal, complemented
9#26359C214yesmaximal, complemented
10#26046C212yesmaximal, complemented
11#25975C214yesmaximal, complemented
12#24844C212yesmaximal, complemented
13#24298C212yesmaximal, complemented
14#22665C212yesmaximal, complemented
15#22650C211yesmaximal, characteristic, complemented
16#22432C212yesmaximal
17C4×#475C211yesmaximal, characteristic, complemented
18C2×#550C2211yescharacteristic
19C2×D4×Q8—21no—
20C22×C4⋊Q8C2212yes—
21C2×C4×#41C2212yes—
22#3079—216no—
23C2×#477C2214yes—
24C2×#475C2212yes—
25C2×#475C2211yescharacteristic
26C2×#476C2214yes—
27C2×#474C2212yes—
28C2×#474C2212yes—
29C2×#474C2212yes—
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#2366—21no—
43#2366C2211yescharacteristic
44#2363—22no—
45#2363C2212yes—
46#2354—21no—
47#2354C2211yescharacteristic
48Q8×C4⋊C4C2211yescharacteristic
49D4×C4⋊C4—21no—
50D4×C4⋊C4C2212yes—
51#2341C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×#127—21no—
54C4×#127C2211yescharacteristic
55C4×#121—22no—
56C4×#121C2212yes—
57C42×Q8C2211yescharacteristic
58C42×D4—21no—
59C42×D4C2211yescharacteristic
60#2093C2214yes—
61#2087C2214yes—
62#2067C2214yes—
63#2065C2214yes—
64#2057C2214yes—
65#2023C2214yes—
66#1988C2212yes—
67#1988C2211yescharacteristic
68#1982C2212yes—
69#1957C2212yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1919C2212yes—
76#1895C2214yes—
77#1874C2214yes—
78#1865C2212yes—
79#1864C2214yes—
80#1863C2214yes—
81#1854C2214yes—
82#1852C2214yes—
83#1851C2214yes—
84#1845C2212yes—
85#1842C2211yescharacteristic
86#1809C2214yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92#1788C2214yes—
93C4×#106C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96C22×#41C2312yes—
97C23×Q8—22no—
98C22×M16C2314yes—
99#550—24no—
100#550Q818yescomplemented
101D4×Q8—48no—
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⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2311yescharacteristic
112C2×#127—21no—
113C2×#127C2311yescharacteristic
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—24no—
116C2×#121—22no—
117C2×#121C2312yes—
118C4×#41—28no—
119C4×#41—44no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—22no—
126C2×C4×D4—22no—
127C2×C4×D4—22no—
128C2×C4×D4—22no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4D411yescharacteristic
134C2×C4×D4D411yescharacteristic
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2311yescharacteristic
138C2×#117—22no—
139C2×#117C2312yes—
140C22×C4⋊C4—22no—
141C22×C4⋊C4C2314yes—
142C22×C42—22no—
143C22×C42C2312yes—
144#477—216no—
145#476—216no—
146#475—28no—
147#475D418yescomplemented
148#474—28no—
149#474—28no—
150#474—44no—
151#436—28no—
152#428—216no—
153C2×#107C2314yes—
154C2×#107C2314yes—
155C2×#107C2314yes—
156C2×#106C2314yes—
157C2×#106C2312yes—
158C2×#106C2312yes—
159#423—28no—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2314yes—
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2311yescharacteristic
166#417—216no—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×#102C2314yes—
173C2×#102C2314yes—
174C2×C22⋊C8C2314yes—
175C2×C4×C8C2312yes—
176#398—24no—
177#395—24no—
178#395—24no—
179#391—21no—
180#391—22no—
181#390—21no—
182#390—21no—
183#390—22no—
184#390—22no—
185#390—22no—
186#390—21no—
187#390—21no—
188#390C2312yes—
189#390C2312yes—
190#389—22no—
191#389—22no—
192#389—21no—
193#389—22no—
194#389—22no—
195#389C2312yes—
196#389C2312yes—
197#389C2314yes—
198#388—22no—
199#388—22no—
200#388—22no—
201#386—22no—
202#386—22no—
203#386—22no—
204#385—24no—
205#385—24no—
206#384—21no—
207#384—21no—
208#384—21no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2312yes—
212#384C2312yes—
213#380—22no—
214#380—22no—
215#380C2312yes—
216#380C2314yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2311yescharacteristic
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4D412yes—
234C4×C22⋊C4D412yes—
235C4×C22⋊C4C2312yes—
236C43—21no—
237C43C2311yescharacteristic
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#340C2312yes—
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×Q8C2411yescharacteristic
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4C2×D411yescharacteristic
265C22×D4C2412yes—
266C23×C4—22no—
267C23×C4—24no—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4C2×D412yes—
271C23×C4C2×D412yes—
272C23×C4C2412yes—
273C2×M16—28no—
274C2×M16—28no—
275C2×M16—44no—
276C22×C8C2414yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C4⋊Q8—28no—
280C4⋊Q8—24no—
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—
290#127—28no—
291#127—22no—
292#127C2×Q814yescomplemented
293C22⋊Q8—416no—
294C22⋊Q8—48no—
295#121—24no—
296#121—216no—
297#121C2×Q818yescomplemented
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—24no—
301C4×Q8—22no—
302C4×Q8—41no—
303C4×Q8—42no—
304C4×Q8—44no—
305C4×Q8C2×D414yes—
306C4×Q8C2×Q814yescomplemented
307C4×D4—28no—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—42no—
314C4×D4—48no—
315C4×D4—48no—
316C4×D4—48no—
317C4×D4—44no—
318C4×D4—44no—
319C4×D4—82no—
320C4×D4—84no—
321C4×D4C2×Q814yes—
322C4×D4#4114yes—
323#117—24no—
324#117—48no—
325#117—42no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—48no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2414yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4C2×D412yes—
369C2×C22⋊C4C2×D412yes—
370C2×C22⋊C4C2×D412yes—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—24no—
374C2×C42—24no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—48no—
383C2×C42C2×D412yes—
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D412yes—
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2412yes—
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393#107—28no—
394#107—28no—
395#107—44no—
396#106—28no—
397#106—42no—
398#106C2×D418yes—
399C4⋊C8—28no—
400C4⋊C8—24no—
401C4⋊C8—24no—
402C4⋊C8—22no—
403C4⋊C8—42no—
404C4⋊C8C2×D414yes—
405#103—28no—
406#103—24no—
407#103C2×D418yes—
408#102—28no—
409#102—28no—
410#102—44no—
411#102—44no—
412C22⋊C8—28no—
413C4×C8—24no—
414#95—22no—
415#95—22no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—22no—
420#95—22no—
421#95—24no—
422#95—24no—
423#95C2×D412yes—
424#95C2×D412yes—
425#95C2×D412yes—
426C24—22no—
427C24C22×D412yes—
428#41—216no—
429#41—48no—
430#41—48no—
431#41—48no—
432C2×Q8—24no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
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×D412yes—
446C2×Q8C22×Q812yes—
447C2×Q82−1+412yes—
448C2×D4—28no—
449C2×D4—28no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4—82no—
466C2×D4—82no—
467C2×D4C22⋊Q814yes—
468C2×D4C22×Q814yes—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—28no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—42no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—82no—
498C22×C4—84no—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×Q814yes—
509C22×C4C2×#4114yes—
510C22×C4C2511yescharacteristic
511M16—416no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—24no—
515C2×C8—22no—
516C2×C8—24no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—24no—
520C2×C8C22×D414yes—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—48no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—81no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4C22⋊Q814yes—
553C4⋊C4#13714yes—
554C4⋊C4C22×D414yes—
555C4⋊C4C22×D412yes—
556C4⋊C4C22×Q812yes—
557C22⋊C4—28no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—48no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4C22⋊Q818yes—
570C42—24no—
571C42—24no—
572C42—24no—
573C42—24no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—21no—
579C42—21no—
580C42—41no—
581C42—44no—
582C42—44no—
583C42—42no—
584C42—44no—
585C42—42no—
586C42—42no—
587C42—81no—
588C42AGL(1,ℤ/8)14yes—
589C42C22×D412yes—
590C42C22×Q812yes—
591C42C2×#4112yes—
592C42H(5,𝔽2)12yes—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—24no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23C2×C22⋊Q814yes—
607C23C23×D411yescharacteristic
608Q8—22no—
609Q8—22no—
610Q8—22no—
611Q8—48no—
612Q8—816no—
613Q8—88no—
614Q8D4×Q814yescomplemented
615D4—28no—
616D4—24no—
617D4—24no—
618D4—416no—
619D4—48no—
620D4—42no—
621D4—42no—
622D4—84no—
623D4—88no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—24no—
645C2×C4—21no—
646C2×C4—22no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—22no—
650C2×C4—48no—
651C2×C4—416no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—48no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—88no—
672C2×C4—816no—
673C2×C4—88no—
674C2×C4C2×C22⋊Q812yes—
675C2×C4C2×C22⋊Q814yes—
676C2×C4C2×C22⋊Q812yes—
677C2×C4D4212yes—
678C2×C4D4212yes—
679C2×C4C2×AGL(1,ℤ/8)12yes—
680C2×C4C2×AGL(1,ℤ/8)12yes—
681C2×C4C2×AGL(1,ℤ/8)12yes—
682C2×C4C2×#13714yes—
683C2×C4C2×#13712yes—
684C2×C4#57612yes—
685C2×C4C23×D411yescharacteristic
686C2×C4C23×Q811yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4C2×2−1+411yescharacteristic
690C2×C4#58511yescharacteristic
691C8—24no—
692C8—44no—
693C8—42no—
694C22—24no—
695C22—24no—
696C22—24no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22C22×C22⋊Q811yescharacteristic
709C22C2×D4211yescharacteristic
710C22#348911yescharacteristic
711C22C22×AGL(1,ℤ/8)11yescharacteristic
712C22C22×#13711yescharacteristic
713C22C2×#57611yescharacteristic
714C22#358611yescharacteristic
715C4—22no—
716C4—22no—
717C4—24no—
718C4—22no—
719C4—24no—
720C4—24no—
721C4—42no—
722C4—42no—
723C4—44no—
724C4—42no—
725C4—41no—
726C4—82no—
727C4—84no—
728C4—82no—
729C4—81no—
730C4#307914yescomplemented
731C4#327414yes—
732C4C2×D4×Q811yescharacteristic
733C4D4×#4111yescharacteristic
734C4#350111yescharacteristic
735C4#352011yescharacteristic
736C2—44no—
737C2—84no—
738C2D4×C22⋊Q811yescharacteristic
739C2C2×#307911yescharacteristic
740C2#6049411yescharacteristic
741C2C2×#327411yescharacteristic
742C2D4×#13711yescharacteristic
743C2#6211111yescharacteristic
744C2#6214911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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. #33196C2semidirect
5. #32955C2semidirect
6. #28508C2semidirect
7. #28279C2semidirect
8. #26464C2semidirect
9. #26359C2semidirect
10. #26046C2semidirect
11. #25975C2semidirect
12. #24844C2semidirect
13. #24298C2semidirect
14. #22665C2semidirect
15. #22650C2semidirect
17. C4×#475C2semidirect
100. #550Q8semidirect
147. #475D4semidirect
252. C2×#41C2×Q8semidirect
292. #127C2×Q8semidirect
297. #121C2×Q8semidirect
306. C4×Q8C2×Q8semidirect
614. Q8D4×Q8semidirect
730. 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×#5502
#331962
#329552
#285082
#282792
#264642
#263592
#260462
#259752
#248442
#242982
#226652
#226502
#224322
C4×#4752