Finite Group Databasebeta

← #6793065

Subgroups — #6793065

Order 512.

1501 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#34945C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#31619C214yesmaximal, complemented
5#31562C214yesmaximal, complemented
6#29942C212yesmaximal, complemented
7#29218C211yesmaximal, characteristic, complemented
8#28283C211yesmaximal, characteristic, complemented
9#28151C212yesmaximal
10#26109C212yesmaximal, complemented
11#25348C212yesmaximal
12#24846C212yesmaximal, complemented
13#24818C212yesmaximal, complemented
14#24155C212yesmaximal, complemented
15#22650C211yesmaximal, characteristic
16#21892C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#477C2212yes—
20C2×#475C2212yes—
21C2×#462C2211yescharacteristic
22C2×C4×Q16C2212yes—
23C2×#436C2212yes—
24C2×#423C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2573—22no—
28#2571C2211yescharacteristic
29#2568C2211yescharacteristic
30#2552—21no—
31#2547C2211yescharacteristic
32#2542—21no—
33#2538C2212yes—
34#2538C2211yescharacteristic
35#2514—21no—
36#2494C2212yes—
37#2447—21no—
38#2441—22no—
39#2429—22no—
40#2423C2214yes—
41#2421C2212yes—
42#2401—22no—
43#2365—21no—
44#2354—21no—
45Q8×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#124—21no—
49C4×C22⋊Q8C2212yes—
50#2295—21no—
51#2283C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2101C2214yes—
55#2100C2214yes—
56#2086C2214yes—
57#2085C2214yes—
58#2077C2214yes—
59#2075C2214yes—
60#2066C2214yes—
61#2066C2214yes—
62#2036C2214yes—
63#2001C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1981C2212yes—
69#1981C2211yescharacteristic
70#1948C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1947C2211yescharacteristic
74#1939C2212yes—
75#1928C2212yes—
76#1894C2214yes—
77#1894C2214yes—
78#1894C2214yes—
79#1878C2212yes—
80#1878C2214yes—
81#1863C2214yes—
82#1851C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1809C2214yes—
86#1805C2214yes—
87#1805C2214yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1803C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C22×M16C2314yes—
97#533—28no—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125—22no—
105C2×#124—21no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2314yes—
109C2×C22⋊Q8C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2314yes—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C22×C42C2312yes—
122#477—44no—
123#475—44no—
124#462—24no—
125C4×Q16—44no—
126#436—44no—
127C2×#107C2314yes—
128C2×#107C2312yes—
129C2×#106C2314yes—
130#423—44no—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C4×C8C2312yes—
146#401—22no—
147#400—22no—
148#400—22no—
149#400—22no—
150#399—24no—
151#398—24no—
152#397—22no—
153#397—22no—
154#397—22no—
155#396—22no—
156#396—24no—
157#391—21no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—22no—
162#391C2312yes—
163#391C2312yes—
164#390—22no—
165#390—21no—
166#389—21no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—24no—
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#388—22no—
178#388—22no—
179#388—24no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—24no—
184#385—22no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#383—22no—
193#382—21no—
194#382C2312yes—
195#382C2312yes—
196#381—22no—
197#380—22no—
198#380C2312yes—
199#380C2314yes—
200#380C2312yes—
201#380C2312yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222#376—22no—
223C43—21no—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C22×D4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2412yes—
244C2×Q16—24no—
245C2×Q16—24no—
246C2×Q16—42no—
247C2×M16—44no—
248C2×M16—44no—
249C2×M16—44no—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8#4114yes—
261#125—48no—
262#124—44no—
263C22⋊Q8—28no—
264C22⋊Q8—24no—
265C22⋊Q8—42no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268C22⋊Q8—84no—
269C4×Q8—28no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—24no—
274C4×Q8—22no—
275C4×Q8—24no—
276C4×Q8—22no—
277C4×Q8—41no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—42no—
285#117—48no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—42no—
349#107—44no—
350#106—44no—
351C4⋊C8—22no—
352C4⋊C8—22no—
353C4⋊C8—24no—
354C4⋊C8—24no—
355C4⋊C8—42no—
356C4⋊C8—44no—
357#103—24no—
358#103—42no—
359#103—44no—
360#103—42no—
361#103—42no—
362#103—44no—
363#103—44no—
364C22⋊C8—44no—
365C4×C8—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24C22×D412yes—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8C2×#4114yes—
406C2×Q82−1+412yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2511yescharacteristic
441Q16—48no—
442M16—88no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—42no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—82no—
499C22⋊C4—84no—
500C42—24no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—21no—
507C42—21no—
508C42—41no—
509C42—48no—
510C42—42no—
511C42—42no—
512C42—44no—
513C42—42no—
514C42—44no—
515C42—42no—
516C42—42no—
517C42—81no—
518C42#13714yes—
519C42C2×#4112yes—
520C42H(5,𝔽2)12yes—
521C42H(5,𝔽2)12yes—
522C422−1+412yes—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23C23×D411yescharacteristic
531Q8—24no—
532Q8—22no—
533Q8—24no—
534Q8—22no—
535Q8—41no—
536Q8—42no—
537Q8—41no—
538Q8—42no—
539Q8—88no—
540Q8—84no—
541D4—44no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—84no—
593C2×C4#53712yes—
594C2×C4#53712yes—
595C2×C4#54612yes—
596C2×C4#54612yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13712yes—
599C2×C4C2×#13712yes—
600C2×C4#57512yes—
601C2×C4#57812yes—
602C2×C4C22×#4111yescharacteristic
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C2×C4C2×2−1+411yescharacteristic
607C2×C4#58511yescharacteristic
608C8—42no—
609C8—44no—
610C8—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22C2×#53711yescharacteristic
619C22C2×#54611yescharacteristic
620C22#348911yescharacteristic
621C22C22×#13711yescharacteristic
622C22C2×#57511yescharacteristic
623C22C2×#57811yescharacteristic
624C22#358511yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—41no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—81no—
635C4—82no—
636C4—82no—
637C4—81no—
638C4—82no—
639C4—81no—
640C4—84no—
641C4#325214yes—
642C4#332814yes—
643C4#351511yescharacteristic
644C4#352011yescharacteristic
645C4#354611yescharacteristic
646C4#355211yescharacteristic
647C2—82no—
648C2#3890811yescharacteristic
649C2C2×#325211yescharacteristic
650C2#6093011yescharacteristic
651C2C2×#332811yescharacteristic
652C2#6116011yescharacteristic
653C2#6211211yescharacteristic
654C2#6214611yescharacteristic
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. #31619C2semidirect
5. #31562C2semidirect
6. #29942C2semidirect
7. #29218C2semidirect
8. #28283C2semidirect
10. #26109C2semidirect
12. #24846C2semidirect
13. #24818C2semidirect
14. #24155C2semidirect
16. #21892C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#349452
#331982
#316192
#315622
#299422
#292182
#282832
#281512
#261092
#253482
#248462
#248182
#241552
#226502
#218922
C4×#4622