Finite Group Databasebeta

← #6500556

Subgroups — #6500556

Order 512.

1974 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic
3#29241C212yesmaximal, complemented
4#29241C212yesmaximal, complemented
5#27681C212yesmaximal, complemented
6#26414C214yesmaximal, complemented
7#26342C214yesmaximal, complemented
8#25994C214yesmaximal, complemented
9#25963C214yesmaximal, complemented
10#24302C212yesmaximal, complemented
11#21940C211yesmaximal, characteristic
12#21930C211yesmaximal, characteristic
13#21759C211yesmaximal, characteristic
14C2×#1872C211yesmaximal, characteristic
15C2×#1852C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—21no—
18C22×C22⋊Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#477C2214yescomplemented
21C22×#107C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2440—22no—
26#2440—22no—
27#2440C2214yes—
28#2440C2214yes—
29C2×#398—21no—
30C2×#395—21no—
31#2368—22no—
32#2360C2212yes—
33#2360C2212yes—
34#2345—22no—
35#2345C2212yes—
36Q8×C22⋊C4C2212yes—
37#2329—21no—
38#2329—21no—
39#2315—21no—
40C4×C22⋊Q8C2212yes—
41C4×C22⋊Q8C2212yes—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#385—21no—
45C2×#384C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48#2273—21no—
49C2×C4×C22⋊C4C2211yescharacteristic
50#2100C2214yescomplemented
51#2095C2214yescomplemented
52#2087C2214yescomplemented
53#2087C2214yescomplemented
54#2068C2214yescomplemented
55#2068C2214yescomplemented
56#2057C2214yescomplemented
57#2024C2214yescomplemented
58#1987C2214yescomplemented
59#1947C2214yescomplemented
60#1921C2214yescomplemented
61#1921C2214yescomplemented
62#1892C2214yes—
63#1872C2212yes—
64#1872C2212yes—
65#1872C2214yes—
66#1872C2212yes—
67#1872C2212yes—
68#1863C2214yescomplemented
69#1863C2214yescomplemented
70#1852C2212yes—
71#1852C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1852C2214yes—
76#1852C2212yes—
77#1850C2214yes—
78#1850C2214yes—
79#1824C2212yes—
80#1808C2214yes—
81#1803C2214yescomplemented
82#1797C2212yes—
83#1797C2212yes—
84#1787C2212yes—
85C2×#340C2212yes—
86C2×#340C2212yes—
87C23×Q8—21no—
88C23×Q8C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C23×C8C2312yes—
92C23×C8C2312yes—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8C2314yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—48no—
102C2×C22⋊Q8Q818yescomplemented
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2312yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—44no—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4D411yescharacteristic
130C22×C22⋊C4D411yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#477—48no—
134C2×#107—44no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141C2×#103—44no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×C22⋊C8—24no—
147C2×C22⋊C8—24no—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152#398—24no—
153#398—24no—
154#395—24no—
155#395—24no—
156#391—22no—
157#391—22no—
158#391C2312yes—
159#391C2312yes—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389C2312yes—
165#389C2314yes—
166#389C2314yes—
167#389C2312yes—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#385—24no—
179#385—24no—
180#384—22no—
181#384C2314yes—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—22no—
187#380—22no—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
191#380C2312yes—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4C2312yes—
205C4×C22⋊C4C2312yes—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25C2×D411yescharacteristic
223C22×Q8—21no—
224C22×Q8—24no—
225C22×Q8—22no—
226C22×Q8—44no—
227C22×Q8—44no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8—82no—
231C22×Q8C2×Q814yescomplemented
232C22×Q8C2412yes—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×Q814yes—
256C23×C4#4114yes—
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C22×C8—24no—
261C22×C8—24no—
262C22×C8—42no—
263C22×C8—42no—
264C22×C8—42no—
265C22×C8—42no—
266C22×C8C2414yes—
267C22×C8C2414yes—
268C4⋊Q8—28no—
269C4⋊Q8—44no—
270C4⋊Q8—48no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—416no—
276C22⋊Q8—416no—
277C22⋊Q8C2×Q818yes—
278C22⋊Q8C2×Q818yescomplemented
279C4×Q8—44no—
280C4×Q8—44no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4C2×Q814yescomplemented
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4Q1618yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—42no—
349C2×C42—42no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354#107—28no—
355#107—44no—
356#107—44no—
357#107—44no—
358C4⋊C8—44no—
359C4⋊C8—44no—
360#103—28no—
361#103—44no—
362#103—44no—
363#103—44no—
364C22⋊C8—24no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—41no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24C22⋊Q814yes—
391C24C2×Q1614yes—
392C24AGL(1,ℤ/8)14yes—
393C24C22×D411yescharacteristic
394C24C22×D411yescharacteristic
395C24C22×D411yescharacteristic
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—48no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×Q8—84no—
406C2×Q8—84no—
407C2×Q8—816no—
408C2×Q8C22×Q814yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—816no—
472C22×C4C22⋊Q814yes—
473C22×C4C22⋊Q814yes—
474C22×C4C22⋊Q814yes—
475C22×C4C2×Q1614yes—
476C22×C4C2×Q1614yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×Q812yes—
485C22×C4C22×Q812yes—
486C22×C4C22×Q812yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2511yescharacteristic
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—48no—
498C2×C8—84no—
499C4⋊C4—24no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—42no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—81no—
523C4⋊C4—81no—
524C4⋊C4—84no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4C22×Q814yes—
528C4⋊C4H(5,𝔽2)14yes—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4C2×Q1618yes—
538C22⋊C4C2×Q1618yes—
539C42—24no—
540C42—24no—
541C42—42no—
542C42—42no—
543C42—82no—
544C42—82no—
545C42—82no—
546C42—82no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23#477116yescomplemented
566C23C2×C22⋊Q812yes—
567C23C2×C22⋊Q812yes—
568C23C2×C22⋊Q812yes—
569C23C22×Q1612yes—
570C23C22×Q1612yes—
571C23C22×Q1612yes—
572C23C2×AGL(1,ℤ/8)12yes—
573C23C2×AGL(1,ℤ/8)12yes—
574C23C2×AGL(1,ℤ/8)12yes—
575C23C23×D411yescharacteristic
576Q8—28no—
577Q8—44no—
578Q8—84no—
579Q8—816no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—162no—
642C2×C4—162no—
643C2×C4C2×C22⋊Q812yes—
644C2×C4C2×C22⋊Q812yes—
645C2×C4C2×C22⋊Q812yes—
646C2×C4C22×Q1614yes—
647C2×C4C22×Q1614yes—
648C2×C4C23×Q811yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×H(5,𝔽2)12yes—
651C2×C4#58512yes—
652C8—44no—
653C8—44no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—416no—
673C22C2×#47714yes—
674C22C2×#47714yes—
675C22C2×#47714yes—
676C22C22×C22⋊Q811yescharacteristic
677C22#348411yescharacteristic
678C22#348511yescharacteristic
679C22C23×Q1611yescharacteristic
680C22C22×AGL(1,ℤ/8)11yescharacteristic
681C22#358511yescharacteristic
682C22#358611yescharacteristic
683C4—24no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—24no—
688C4—42no—
689C4—44no—
690C4—44no—
691C4—42no—
692C4—44no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4#350112yes—
699C4#350512yes—
700C2—24no—
701C2—24no—
702C2—24no—
703C2#3534911yescharacteristic
704C2C22×#47711yescharacteristic
705C2#6049311yescharacteristic
706C2#6156311yescharacteristic
707C2#6157111yescharacteristic
708C2#6169811yescharacteristic
709C2#6170611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29241C2semidirect
4. #29241C2semidirect
5. #27681C2semidirect
6. #26414C2semidirect
7. #26342C2semidirect
8. #25994C2semidirect
9. #25963C2semidirect
10. #24302C2semidirect
20. C2×#477C22semidirect
50. #2100C22semidirect
51. #2095C22semidirect
52. #2087C22semidirect
53. #2087C22semidirect
54. #2068C22semidirect
55. #2068C22semidirect
56. #2057C22semidirect
57. #2024C22semidirect
58. #1987C22semidirect
59. #1947C22semidirect
60. #1921C22semidirect
61. #1921C22semidirect
68. #1863C22semidirect
69. #1863C22semidirect
81. #1803C22semidirect
102. C2×C22⋊Q8Q8semidirect
231. C22×Q8C2×Q8semidirect
278. C22⋊Q8C2×Q8semidirect
321. C2×C4⋊C4C2×Q8semidirect
565. C23#477semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
#342122
#292412
#276812
#264142
#263422
#259942
#259632
#243022
#219402
#219302
#217592
C2×#18722
C2×#18522