Finite Group Databasebeta

← #6491860

Subgroups — #6491860

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic
3#33198C212yesmaximal, complemented
4#33069C212yesmaximal, complemented
5#29999C212yesmaximal, complemented
6#29984C212yesmaximal, complemented
7#26112C212yesmaximal, complemented
8#26092C212yesmaximal, complemented
9#26046C212yesmaximal, complemented
10#26026C212yesmaximal, complemented
11#25994C214yesmaximal
12#25963C214yesmaximal
13#21982C211yesmaximal, characteristic
14#21972C211yesmaximal, characteristic
15#21961C211yesmaximal, characteristic
16#20734C211yesmaximal, characteristic
17C2×#1803C211yesmaximal, characteristic
18C2×#544—21no—
19C22×C4×Q8C2211yescharacteristic
20C2×#477C2214yes—
21C2×#431C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#103C2212yes—
24#2868C2212yes—
25#2859C2212yes—
26#2660—22no—
27#2552—21no—
28#2552C2212yes—
29#2551—21no—
30#2551C2212yes—
31#2440—22no—
32#2440C2214yes—
33#2355—21no—
34#2355—21no—
35#2355C2211yescharacteristic
36#2355C2211yescharacteristic
37Q8×C4⋊C4C2211yescharacteristic
38Q8×C4⋊C4C2211yescharacteristic
39#2345—22no—
40#2345C2212yes—
41Q8×C22⋊C4C2212yes—
42#2317—21no—
43#2295—22no—
44#2295C2212yes—
45#2285—21no—
46#2285C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C42×Q8C2212yes—
50C4×#117—21no—
51C4×#117C2211yescharacteristic
52#2100C2214yes—
53#2095C2214yes—
54#2087C2214yes—
55#2087C2214yes—
56#2068C2214yes—
57#2068C2214yes—
58#2057C2214yes—
59#2024C2214yes—
60#1987C2214yes—
61#1947C2212yes—
62#1947C2212yes—
63#1947C2212yes—
64#1947C2212yes—
65#1947C2212yes—
66#1947C2214yes—
67#1947C2212yes—
68#1939C2212yes—
69#1939C2212yes—
70#1921C2214yes—
71#1921C2214yes—
72#1863C2214yes—
73#1863C2214yes—
74#1852C2214yes—
75#1850C2214yes—
76#1845C2212yes—
77#1842C2212yes—
78#1834C2212yes—
79#1831C2212yes—
80#1824C2212yes—
81#1808C2214yes—
82#1803C2214yes—
83#1803C2212yes—
84#1803C2211yescharacteristic
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2211yescharacteristic
91#1787C2212yes—
92C4×#103C2212yes—
93C4×#103C2212yes—
94C2×#340C2212yes—
95C23×Q8C2311yescharacteristic
96C23×C8C2312yes—
97#544—48no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8C2314yes—
102C2×#125—22no—
103C2×#125—22no—
104C2×C22⋊Q8—24no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—24no—
108C2×C4×Q8Q818yescomplemented
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×#117—21no—
116C2×#117—21no—
117C2×#117D411yescharacteristic
118C2×#117D411yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126#477—48no—
127#431—24no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×C4⋊C8—24no—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2314yes—
139C2×#103—44no—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C4×C8C2312yes—
148#398—24no—
149#398—24no—
150#395—24no—
151#395—24no—
152#391—22no—
153#391—22no—
154#391C2312yes—
155#391C2312yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2314yes—
164#389C2314yes—
165#389C2312yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#385—24no—
173#384—21no—
174#384—21no—
175#384—22no—
176#384—22no—
177#384—21no—
178#384—21no—
179#384C2312yes—
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2311yescharacteristic
183#384C2314yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#382—21no—
187#382—21no—
188#382—22no—
189#382—21no—
190#382—21no—
191#380—21no—
192#380—21no—
193#380—22no—
194#380—22no—
195#380C2312yes—
196#380C2312yes—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376—22no—
220#376D412yes—
221C43—22no—
222C43C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#336C2314yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—41no—
241C22×Q8—41no—
242C22×Q8C2×Q814yes—
243C22×Q8C2411yescharacteristic
244C22×Q8C2411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C22×C8—24no—
252C22×C8—42no—
253C22×C8—42no—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260C4⋊Q8—48no—
261C4⋊Q8—84no—
262#125—48no—
263#125—48no—
264C22⋊Q8—416no—
265C4×Q8—28no—
266C4×Q8—28no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—42no—
274C4×Q8—42no—
275C4×Q8C2×Q814yes—
276C4×Q8C2×Q814yes—
277C4×Q8C2×Q814yes—
278C4×Q8C2×Q814yes—
279#117—24no—
280#117—44no—
281#117—44no—
282#117Q1618yes—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×Q814yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—28no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×Q814yes—
350C2×C42#4114yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—44no—
356#107—44no—
357#107—44no—
358#107—44no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361C4⋊C8—22no—
362C4⋊C8—22no—
363C4⋊C8—44no—
364C4⋊C8—44no—
365#103—28no—
366#103—24no—
367#103—24no—
368#103—44no—
369#103—42no—
370#103—42no—
371C22⋊C8—24no—
372C4×C8—24no—
373#95—22no—
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#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24C22×D411yescharacteristic
392C2×Q8—24no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—28no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—48no—
408C2×Q8—84no—
409C2×Q8—84no—
410C2×Q8C22×Q812yes—
411C2×Q8C22×Q812yes—
412C2×Q8H(5,𝔽2)12yes—
413C2×Q82−1+412yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4C22⋊Q814yes—
439C22×C4C22⋊Q814yes—
440C22×C4C2×Q1614yes—
441C22×C4AGL(1,ℤ/8)14yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×Q812yes—
452C22×C4C2×#4112yes—
453C22×C4C2511yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—84no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4C2×Q1614yes—
506C4⋊C4C2×Q1614yes—
507C4⋊C4C22×Q812yes—
508C4⋊C4C22×Q812yes—
509C22⋊C4—24no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4C2×Q1618yes—
513C42—24no—
514C42—24no—
515C42—24no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—44no—
526C42—44no—
527C42—44no—
528C42—82no—
529C42—82no—
530C42C22⋊Q814yes—
531C42C22⋊Q814yes—
532C42C2×Q1618yes—
533C42C22×Q812yes—
534C42C22×Q812yes—
535C42C2×#4112yes—
536C42C2×#4112yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23C2×C22⋊Q812yes—
542C23C22×Q1612yes—
543C23C2×AGL(1,ℤ/8)12yes—
544C23C23×D411yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—22no—
549Q8—22no—
550Q8—48no—
551Q8—41no—
552Q8—41no—
553Q8—168no—
554Q8#54414yescomplemented
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—84no—
605C2×C4—164no—
606C2×C4#477116yescomplemented
607C2×C4C2×C22⋊Q812yes—
608C2×C4C2×C22⋊Q812yes—
609C2×C4C2×C22⋊Q812yes—
610C2×C4C2×C22⋊Q812yes—
611C2×C4C2×C22⋊Q812yes—
612C2×C4C22×Q1614yes—
613C2×C4C22×Q1614yes—
614C2×C4C22×Q1612yes—
615C2×C4C22×Q1612yes—
616C2×C4C2×AGL(1,ℤ/8)12yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C23×Q811yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×2−1+411yescharacteristic
622C2×C4#58511yescharacteristic
623C2×C4#58511yescharacteristic
624C8—42no—
625C8—44no—
626C8—42no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22C2×#47714yes—
634C22C22×C22⋊Q811yescharacteristic
635C22#348811yescharacteristic
636C22#348911yescharacteristic
637C22C23×Q1611yescharacteristic
638C22C22×AGL(1,ℤ/8)11yescharacteristic
639C22#358511yescharacteristic
640C22#358611yescharacteristic
641C4—24no—
642C4—22no—
643C4—22no—
644C4—24no—
645C4—22no—
646C4—24no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—84no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4C2×#47714yes—
658C4C2×#47714yes—
659C4C2×#54411yescharacteristic
660C4#345811yescharacteristic
661C4#349411yescharacteristic
662C4#349511yescharacteristic
663C2—24no—
664C2#3525311yescharacteristic
665C2C22×#47711yescharacteristic
666C2#6049311yescharacteristic
667C2#6195211yescharacteristic
668C2#6196311yescharacteristic
669C2#6210011yescharacteristic
670C2#6211111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33198C2semidirect
4. #33069C2semidirect
5. #29999C2semidirect
6. #29984C2semidirect
7. #26112C2semidirect
8. #26092C2semidirect
9. #26046C2semidirect
10. #26026C2semidirect
108. C2×C4×Q8Q8semidirect
554. Q8#544semidirect
606. C2×C4#477semidirect

Maximal subgroups

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

subgroupindex
#352752
Q8×#1172
#331982
#330692
#299992
#299842
#261122
#260922
#260462
#260262
#259942
#259632
#219822
#219722
#219612
#207342
C2×#18032