Finite Group Databasebeta

← #6591948

Subgroups — #6591948

Order 512.

1931 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2C4×D4×Q8C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#32957C212yesmaximal, complemented
5#29141C211yesmaximal, characteristic, complemented
6#28284C211yesmaximal, characteristic, complemented
7#27681C212yesmaximal, complemented
8#26532C212yesmaximal, complemented
9#26414C214yesmaximal, complemented
10#26112C212yesmaximal, complemented
11#25963C214yesmaximal, complemented
12#24837C212yesmaximal, complemented
13#22705C212yesmaximal
14D4×C4⋊C8C211yesmaximal, characteristic
15D4×#103C212yesmaximal
16C4×#477C211yesmaximal, characteristic, complemented
17C2×#1803C212yesmaximal, complemented
18C2×#554—21no—
19C2×D4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C2×#477C2211yescharacteristic
22C2×#477C2212yes—
23C2×#477C2212yes—
24C2×C8×D4C2212yes—
25C22×C4⋊C8C2212yes—
26C22×#103C2214yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2567—21no—
30#2552—21no—
31#2552C2212yes—
32#2502—22no—
33#2440—22no—
34#2440C2214yes—
35#2365—21no—
36#2365C2211yescharacteristic
37#2360—22no—
38#2360C2212yes—
39Q8×C4⋊C4C2211yescharacteristic
40D4×C4⋊C4—21no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2211yescharacteristic
44Q8×C22⋊C4C2212yes—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C22⋊Q8—22no—
49C4×C22⋊Q8C2212yes—
50C2×#384C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4—21no—
53C42×D4C2211yescharacteristic
54#2100C2214yes—
55#2095C2214yes—
56#2087C2214yes—
57#2068C2214yes—
58#2068C2214yes—
59#2024C2214yes—
60#1987C2211yescharacteristic
61#1987C2212yes—
62#1981C2212yes—
63#1956C2212yes—
64#1947C2212yes—
65#1947C2212yes—
66#1947C2211yescharacteristic
67#1947C2212yes—
68#1939C2212yes—
69#1921C2212yes—
70#1892C2214yes—
71#1872C2214yes—
72#1863C2214yes—
73#1863C2214yes—
74#1863C2212yes—
75#1852C2214yes—
76#1852C2214yes—
77#1850C2214yes—
78#1842C2211yescharacteristic
79#1834C2212yes—
80#1808C2214yes—
81#1803C2214yes—
82#1803C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2214yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1797C2212yes—
90#1787C2214yes—
91C4×#107C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94C2×#340C2214yes—
95C23×Q8C2312yes—
96C23×C8C2314yes—
97#554—48no—
98D4×Q8—24no—
99D4×Q8Q818yescomplemented
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#125—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8C2312yes—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—28no—
115C2×C4×Q8—44no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2314yes—
133C22×C42—22no—
134C22×C42C2312yes—
135#477—24no—
136#477—44no—
137#477—44no—
138C8×D4—28no—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×C4⋊C8—44no—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#103—48no—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C4×C8C2312yes—
160#398—24no—
161#398—24no—
162#395—24no—
163#391—21no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391C2312yes—
168#391C2312yes—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389C2312yes—
180#389C2314yes—
181#389C2312yes—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#385—24no—
188#385—22no—
189#385—24no—
190#384—21no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384—22no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2314yes—
199#384C2312yes—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#380C2314yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226C43—21no—
227C43C2311yescharacteristic
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#340C2312yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—24no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8C2×Q818yes—
248C22×Q8C2412yes—
249C22×Q8C2411yescharacteristic
250C22×D4C2×D411yescharacteristic
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—24no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C22×C8—28no—
258C22×C8—44no—
259C22×C8—44no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—28no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8—44no—
270C4⋊Q8—82no—
271C4⋊Q8C2×Q814yes—
272#125—48no—
273C22⋊Q8—24no—
274C22⋊Q8—216no—
275C22⋊Q8—416no—
276C22⋊Q8—48no—
277C22⋊Q8C2×Q818yes—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—24no—
281C4×Q8—22no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—48no—
286C4×Q8—42no—
287C4×Q8—42no—
288C4×Q8—41no—
289C4×Q8—48no—
290C4×Q8—44no—
291C4×Q8C2×Q814yes—
292C4×Q8#4114yes—
293C4×D4—28no—
294C4×D4—24no—
295C4×D4—44no—
296C4×D4—48no—
297C4×D4Q1618yes—
298C4×D4C2×Q814yes—
299C4×D4#4114yes—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—48no—
331C2×C4⋊C4—48no—
332C2×C4⋊C4—84no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—21no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—48no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#107—24no—
373#107—42no—
374#107—44no—
375#107—44no—
376C4⋊C8—24no—
377C4⋊C8—22no—
378C4⋊C8—22no—
379C4⋊C8—24no—
380C4⋊C8—44no—
381C4⋊C8—42no—
382#103—24no—
383#103—24no—
384#103—28no—
385#103—44no—
386#103—44no—
387#103—44no—
388#103—44no—
389C22⋊C8—28no—
390C4×C8—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407C24C22×D412yes—
408C2×Q8—28no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—216no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—48no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—44no—
426C2×Q8—42no—
427C2×Q8—82no—
428C2×Q8—82no—
429C2×Q8C22×Q814yes—
430C2×Q8C22×Q812yes—
431C2×Q8C2×#4112yes—
432C2×Q8H(5,𝔽2)12yes—
433C2×D4C22⋊Q814yes—
434C2×D4C2×Q1614yes—
435C2×D4AGL(1,ℤ/8)14yes—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—82no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4C2×Q1618yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×Q814yes—
474C22×C4C2×#4114yes—
475C22×C4C2511yescharacteristic
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—88no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4C22⋊Q814yes—
526C4⋊C4C2×Q1614yes—
527C4⋊C4C22×Q812yes—
528C4⋊C4C2×#4112yes—
529C22⋊C4—28no—
530C22⋊C4—24no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4C22⋊Q818yes—
534C22⋊C4C2×Q1618yes—
535C42—24no—
536C42—24no—
537C42—24no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—21no—
544C42—21no—
545C42—41no—
546C42—44no—
547C42—44no—
548C42—42no—
549C42—44no—
550C42—42no—
551C42—44no—
552C42—42no—
553C42—81no—
554C42C2×Q1614yes—
555C42C22×Q812yes—
556C42C2×#4112yes—
557C42C2×#4112yes—
558C42H(5,𝔽2)12yes—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23C2×C22⋊Q814yes—
564C23C22×Q1614yes—
565C23C2×AGL(1,ℤ/8)14yes—
566C23C23×D411yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—24no—
571Q8—24no—
572Q8—416no—
573Q8—42no—
574Q8—42no—
575Q8—84no—
576Q8—164no—
577Q8#55414yescomplemented
578D4#477116yescomplemented
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—216no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—48no—
607C2×C4—416no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—88no—
630C2×C4—168no—
631C2×C4C2×C22⋊Q812yes—
632C2×C4C2×C22⋊Q814yes—
633C2×C4C2×C22⋊Q812yes—
634C2×C4#54612yes—
635C2×C4#54612yes—
636C2×C4C22×Q1612yes—
637C2×C4C22×Q1614yes—
638C2×C4C22×Q1612yes—
639C2×C4C22×Q1612yes—
640C2×C4C2×AGL(1,ℤ/8)12yes—
641C2×C4#57712yes—
642C2×C4C23×Q811yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4#58511yescharacteristic
648C8—42no—
649C8—44no—
650C8—42no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22C2×#47718yes—
658C22C22×C22⋊Q811yescharacteristic
659C22C2×#54611yescharacteristic
660C22#348811yescharacteristic
661C22C23×Q1611yescharacteristic
662C22C22×AGL(1,ℤ/8)11yescharacteristic
663C22C2×#57711yescharacteristic
664C22#358611yescharacteristic
665C4—22no—
666C4—24no—
667C4—24no—
668C4—22no—
669C4—24no—
670C4—22no—
671C4—41no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—81no—
677C4—82no—
678C4—84no—
679C4—82no—
680C4—82no—
681C4C2×#47714yes—
682C4#329914yes—
683C4C2×#55411yescharacteristic
684C4#347511yescharacteristic
685C4#350111yescharacteristic
686C4#351911yescharacteristic
687C2—28no—
688C2#3615411yescharacteristic
689C2C22×#47711yescharacteristic
690C2#6049711yescharacteristic
691C2C2×#329911yescharacteristic
692C2#6116111yescharacteristic
693C2#6195211yescharacteristic
694C2#6198311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #33198C2semidirect
4. #32957C2semidirect
5. #29141C2semidirect
6. #28284C2semidirect
7. #27681C2semidirect
8. #26532C2semidirect
9. #26414C2semidirect
10. #26112C2semidirect
11. #25963C2semidirect
12. #24837C2semidirect
16. C4×#477C2semidirect
17. C2×#1803C2semidirect
99. D4×Q8Q8semidirect
577. Q8#554semidirect
578. D4#477semidirect

Maximal subgroups

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

subgroupindex
#361802
C4×D4×Q82
#331982
#329572
#291412
#282842
#276812
#265322
#264142
#261122
#259632
#248372
#227052
D4×C4⋊C82
D4×#1032
C4×#4772
C2×#18032