Finite Group Databasebeta

← #6644654

Subgroups — #6644654

Order 512.

1772 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#35006C211yesmaximal, characteristic, complemented
2C2×#2566C211yesmaximal, characteristic, complemented
3#33004C212yesmaximal, complemented
4#32431C214yesmaximal, complemented
5#32349C214yesmaximal, complemented
6#31641C214yesmaximal, complemented
7#31523C214yesmaximal, complemented
8#29640C212yesmaximal, complemented
9#29103C211yesmaximal, characteristic, complemented
10#28301C211yesmaximal, characteristic
11#28117C212yesmaximal, complemented
12#27961C211yesmaximal, characteristic, complemented
13#27906C211yesmaximal, characteristic, complemented
14#27651C212yesmaximal, complemented
15#21912C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×#125C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#481C2212yes—
20C2×#481C2212yes—
21C2×#480C2212yes—
22C2×#447C2212yes—
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2854C2212yes—
26#2639—21no—
27#2618—21no—
28#2566—22no—
29#2566—22no—
30#2566C2214yes—
31#2566C2214yes—
32#2560—22no—
33#2558—22no—
34#2512C2212yescomplemented
35#2506C2212yescomplemented
36#2452C2212yescomplemented
37#2445C2212yescomplemented
38C2×#396—21no—
39C2×#396—21no—
40#2325—21no—
41C4×#121C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#385—21no—
45C2×#385—21no—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2100C2214yescomplemented
50#2094C2214yescomplemented
51#2093C2212yes—
52#2085C2214yescomplemented
53#2073C2214yescomplemented
54#2071C2212yes—
55#2065C2214yes—
56#2059C2214yescomplemented
57#2059C2214yescomplemented
58#2054C2214yes—
59#2025C2214yescomplemented
60#2025C2214yescomplemented
61#2023C2212yes—
62#2011C2212yes—
63#1990C2212yescomplemented
64#1970C2214yes—
65#1957C2212yescomplemented
66#1949C2214yescomplemented
67#1934C2214yes—
68#1926C2214yes—
69#1916C2214yescomplemented
70#1916C2214yescomplemented
71#1879C2214yes—
72#1879C2214yes—
73#1879C2214yes—
74#1864C2214yes—
75#1864C2214yes—
76#1844C2212yescomplemented
77#1834C2212yescomplemented
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1804C2214yescomplemented
83#1797C2212yes—
84#1786C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C22×D8C2314yes—
90#539—48no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#126—22no—
94C2×#125—22no—
95C2×#125—22no—
96C2×#125—44no—
97C2×#125C2314yes—
98C2×#123—22no—
99C2×#123—22no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—44no—
108C2×C4×D4C2314yes—
109C2×C4×D4C2312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#481—44no—
128#481—44no—
129#480—44no—
130#447—44no—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2312yes—
145#400—22no—
146#398—22no—
147#397—22no—
148#396—24no—
149#396—24no—
150#396—24no—
151#396—24no—
152#396—22no—
153#396—22no—
154#394—22no—
155#394—22no—
156#393—22no—
157#391—21no—
158#390—21no—
159#390—22no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390C2312yes—
164#390C2314yes—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#385—24no—
184#385—24no—
185#385—24no—
186#385—24no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384C2312yes—
191#384C2312yes—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#382—22no—
195#382C2314yes—
196#380—21no—
197#380C2312yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C25—21no—
220C22×Q8—21no—
221C22×D4—24no—
222C22×D4—22no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—24no—
227C22×D4—42no—
228C22×D4C2412yes—
229C22×D4C2412yes—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—44no—
236C23×C4—82no—
237C23×C4—82no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×D8—28no—
246C2×D8—44no—
247C2×D8—44no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—42no—
251#127—42no—
252#126—48no—
253#125—48no—
254#125—48no—
255#125—48no—
256#123—48no—
257#123—48no—
258C22⋊Q8—48no—
259C22⋊Q8—44no—
260#121—24no—
261#121—24no—
262#121—44no—
263#121—44no—
264C4×D4—48no—
265C4×D4—48no—
266C4×D4—44no—
267C4×D4—48no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4—82no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—44no—
341#106—44no—
342#106—44no—
343C4⋊C8—44no—
344#102—28no—
345#102—44no—
346#102—44no—
347#102—44no—
348#102—44no—
349#102—44no—
350C22⋊C8—24no—
351C22⋊C8—24no—
352C4×C8—42no—
353C4×C8—42no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—24no—
367C24—22no—
368C24—22no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24C22×D411yescharacteristic
376C2×Q8—42no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—416no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—81no—
396C2×D4—81no—
397C2×D4—81no—
398C2×D4—81no—
399C2×D42−1+414yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—816no—
440C22×C4#13514yes—
441C22×C4AGL(1,ℤ/8)14yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4H(5,𝔽2)14yes—
452C22×C4C2511yescharacteristic
453D8—88no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—81no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—82no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—24no—
512C42—41no—
513C42—41no—
514C42—42no—
515C42—44no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—24no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23#53512yes—
542C23C2×#13512yes—
543C23C2×AGL(1,ℤ/8)12yes—
544C23C23×D411yescharacteristic
545Q8—162no—
546D4—28no—
547D4—48no—
548D4—44no—
549D4—44no—
550D4—44no—
551D4—88no—
552D4—162no—
553D4—162no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
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—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—816no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—164no—
606C2×C4#53512yes—
607C2×C4C2×#13512yes—
608C2×C4C2×#13512yes—
609C2×C4C2×AGL(1,ℤ/8)12yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4C2×2−1+412yes—
613C2×C4#58512yes—
614C8—44no—
615C8—44no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—816no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22#320714yes—
638C22C2×#53511yescharacteristic
639C22#348511yescharacteristic
640C22#348611yescharacteristic
641C22C22×#13511yescharacteristic
642C22C22×AGL(1,ℤ/8)11yescharacteristic
643C22#358511yescharacteristic
644C22#358511yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4#353514yes—
661C2—24no—
662C2—84no—
663C2—162no—
664C2#3711811yescharacteristic
665C2C2×#320711yescharacteristic
666C2#6080611yescharacteristic
667C2#6169911yescharacteristic
668C2#6170711yescharacteristic
669C2#6181911yescharacteristic
670C2#6182511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28301C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35006C2semidirect
2. C2×#2566C2semidirect
3. #33004C2semidirect
4. #32431C2semidirect
5. #32349C2semidirect
6. #31641C2semidirect
7. #31523C2semidirect
8. #29640C2semidirect
9. #29103C2semidirect
11. #28117C2semidirect
12. #27961C2semidirect
13. #27906C2semidirect
14. #27651C2semidirect
15. #21912C2semidirect
34. #2512C22semidirect
35. #2506C22semidirect
36. #2452C22semidirect
37. #2445C22semidirect
49. #2100C22semidirect
50. #2094C22semidirect
52. #2085C22semidirect
53. #2073C22semidirect
56. #2059C22semidirect
57. #2059C22semidirect
59. #2025C22semidirect
60. #2025C22semidirect
63. #1990C22semidirect
65. #1957C22semidirect
66. #1949C22semidirect
69. #1916C22semidirect
70. #1916C22semidirect
76. #1844C22semidirect
77. #1834C22semidirect
82. #1804C22semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#25662
#330042
#324312
#323492
#316412
#315232
#296402
#291032
#283012
#281172
#279612
#279062
#276512
#219122