Finite Group Databasebeta

← #6570673

Subgroups — #6570673

Order 512.

1759 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36275C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#29939C211yesmaximal, characteristic, complemented
4#29909C212yesmaximal, complemented
5#27961C211yesmaximal, characteristic, complemented
6#27858C212yesmaximal, complemented
7#27681C211yesmaximal, characteristic, complemented
8#27614C212yesmaximal, complemented
9#26282C212yesmaximal, complemented
10#26204C214yesmaximal, complemented
11#24996C212yesmaximal, complemented
12#24939C214yesmaximal, complemented
13#23757C211yesmaximal, characteristic, complemented
14C2×#2100C212yesmaximal, complemented
15#22705C212yesmaximal, complemented
16#22487C212yesmaximal, complemented
17#22469C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×C8×D4C2212yes—
22#2857C2211yescharacteristic
23#2855C2212yes—
24#2582C2211yescharacteristic
25#2567C2211yescharacteristic
26#2487C2212yes—
27#2453—21no—
28#2449—22no—
29#2441—21no—
30#2438—22no—
31#2417C2212yes—
32Q8×C4⋊C4—21no—
33Q8×C4⋊C4C2212yes—
34D4×C4⋊C4—21no—
35D4×C4⋊C4—22no—
36D4×C4⋊C4—21no—
37D4×C4⋊C4C2214yes—
38D4×C4⋊C4C2212yes—
39D4×C4⋊C4C2211yescharacteristic
40#2339—21no—
41#2317—22no—
42#2317—21no—
43C4×C22⋊Q8C2211yescharacteristic
44C2×#389C2212yes—
45#2285C2211yescharacteristic
46#2285C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49#2100C2212yes—
50#2100C2212yes—
51#2100C2212yes—
52#2100C2214yes—
53#2100C2214yes—
54#2100C2211yescharacteristic
55#2100C2212yes—
56#2095C2212yes—
57#2095C2211yescharacteristic
58#2083C2214yes—
59#2083C2211yescharacteristic
60#2083C2212yes—
61#2076C2214yes—
62#2076C2212yes—
63#2072C2214yes—
64#2072C2211yescharacteristic
65#2036C2212yes—
66#1990C2212yes—
67#1987C2212yes—
68#1984C2212yes—
69#1947C2212yes—
70#1947C2212yes—
71#1928C2214yes—
72#1921C2214yes—
73#1921C2214yes—
74#1921C2214yes—
75#1892C2214yes—
76#1872C2212yes—
77#1872C2214yes—
78#1869C2214yes—
79#1852C2212yes—
80#1852C2214yes—
81#1852C2214yes—
82#1834C2212yes—
83#1824C2214yes—
84#1824C2214yes—
85#1817C2212yes—
86#1808C2214yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89C4×#107C2212yes—
90C4×#103C2212yes—
91C4×C22⋊C8C2211yescharacteristic
92C2×#340C2214yes—
93C2×#340C2214yes—
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×C8C2314yes—
97#554—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2311yescharacteristic
104#520—48no—
105C4×#41—28no—
106C2×C4×Q8—22no—
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—24no—
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D412yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×#117—22no—
119C2×#117—21no—
120C2×#117D412yes—
121C2×#117D411yescharacteristic
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C8×D4—28no—
134C2×#107C2312yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C4×C8C2312yes—
146#400—22no—
147#399—22no—
148#398—22no—
149#398—22no—
150#396—22no—
151#395—22no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391C2312yes—
156#391C2311yescharacteristic
157#389—22no—
158#389—24no—
159#389—24no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2311yescharacteristic
171#387—22no—
172#387—22no—
173#387—21no—
174#386—22no—
175#386—22no—
176#385—22no—
177#385—21no—
178#385—24no—
179#385—24no—
180#385—24no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—22no—
185#384—21no—
186#384—21no—
187#384—22no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2314yes—
192#384C2312yes—
193#384C2311yescharacteristic
194#382—22no—
195#382—21no—
196#381—22no—
197#380—21no—
198#380—22no—
199#380—21no—
200#380—22no—
201#380C2312yes—
202#380C2314yes—
203#380C2312yes—
204#380C2312yes—
205#380C2311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—21no—
213C2×#95—22no—
214#340C2314yes—
215#340C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#336C2312yes—
228#336C2314yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232C2×#41—24no—
233C2×#41—24no—
234C2×#41Q1618yes—
235C22×Q8—21no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×D4C2×D412yes—
239C22×D4C2×D411yescharacteristic
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—24no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C22×C8—28no—
251C22×C8—44no—
252C22×C8—44no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257#125—44no—
258C22⋊Q8—28no—
259C22⋊Q8—22no—
260C22⋊Q8—41no—
261C22⋊Q8—48no—
262C22⋊Q8—82no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—48no—
267C4×D4—24no—
268C4×D4—28no—
269C4×D4—28no—
270C4×D4—416no—
271C4×D4—44no—
272C4×D4—44no—
273C4×D4—48no—
274C4×D4#4118yes—
275#117—24no—
276#117—28no—
277#117—48no—
278#117—44no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4—84no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—28no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—24no—
351#107—44no—
352#107—42no—
353#107—44no—
354C4⋊C8—24no—
355#103—42no—
356#103—42no—
357#103—44no—
358C22⋊C8—28no—
359C22⋊C8—42no—
360C22⋊C8—42no—
361C4×C8—24no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—24no—
369#95—21no—
370#95—21no—
371#95—24no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379C24C22×D411yescharacteristic
380C24C22×D412yes—
381#41—232no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—42no—
389C2×Q8—81no—
390C2×Q8—81no—
391C2×Q8C2×Q1614yes—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4#12314yes—
395C2×D4C2×Q1618yes—
396C2×D4C2×Q1614yes—
397C2×D4AGL(1,ℤ/8)14yes—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—82no—
430C22×C4—84no—
431C22×C4—82no—
432C22×C4—84no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4C2×Q1618yes—
437C22×C4C2×Q1614yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4118yes—
447C22×C4C2511yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—88no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—84no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—81no—
498C4⋊C4#12314yes—
499C4⋊C42−1+412yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—81no—
508C22⋊C4—82no—
509C22⋊C4#12318yes—
510C42—24no—
511C42—22no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—48no—
517C42—42no—
518C42—44no—
519C42—42no—
520C42C2×#4114yes—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23C2×#12314yes—
532C23C22×Q1612yes—
533C23C22×Q1614yes—
534C23C2×AGL(1,ℤ/8)14yes—
535C23C23×D411yescharacteristic
536Q8—28no—
537Q8—42no—
538Q8—42no—
539Q8—162no—
540D4—216no—
541D4#484116yes—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—216no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—88no—
594C2×C4—164no—
595C2×C4—168no—
596C2×C4—164no—
597C2×C4C2×#12314yes—
598C2×C4C2×#12312yes—
599C2×C4C2×#12312yes—
600C2×C4C22×Q1612yes—
601C2×C4C22×Q1614yes—
602C2×C4C22×Q1612yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4#57712yes—
605C2×C4C22×#4112yes—
606C2×C4C2×2−1+411yescharacteristic
607C2×C4#58512yes—
608C2×C4#58511yescharacteristic
609C8—42no—
610C8—44no—
611C8—42no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22C2×#48418yes—
626C22C22×#12311yescharacteristic
627C22#348911yescharacteristic
628C22#350011yescharacteristic
629C22C23×Q1611yescharacteristic
630C22C22×AGL(1,ℤ/8)11yescharacteristic
631C22C2×#57711yescharacteristic
632C22#358511yescharacteristic
633C4—22no—
634C4—24no—
635C4—22no—
636C4—22no—
637C4—24no—
638C4—42no—
639C4—42no—
640C4—41no—
641C4—81no—
642C4—82no—
643C4—81no—
644C4—82no—
645C4—84no—
646C4—82no—
647C4—82no—
648C4—81no—
649C4C2×#48414yes—
650C4#351512yes—
651C4#352112yes—
652C2—28no—
653C2—42no—
654C2#3606711yescharacteristic
655C2C22×#48411yescharacteristic
656C2#6052511yescharacteristic
657C2#6210011yescharacteristic
658C2#6212811yescharacteristic
659C2#6253011yescharacteristic
660C2#6257611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29939C2semidirect
4. #29909C2semidirect
5. #27961C2semidirect
6. #27858C2semidirect
7. #27681C2semidirect
8. #27614C2semidirect
9. #26282C2semidirect
10. #26204C2semidirect
11. #24996C2semidirect
12. #24939C2semidirect
13. #23757C2semidirect
14. C2×#2100C2semidirect
15. #22705C2semidirect
16. #22487C2semidirect

Maximal subgroups

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

subgroupindex
#362752
#341832
#299392
#299092
#279612
#278582
#276812
#276142
#262822
#262042
#249962
#249392
#237572
C2×#21002
#227052
#224872
#224692