Finite Group Databasebeta

← #6434875

Subgroups — #6434875

Order 512.

1564 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#30163C214yesmaximal, complemented
4#30161C214yesmaximal, complemented
5#30150C212yesmaximal, complemented
6#30149C212yesmaximal, complemented
7#30146C212yesmaximal, complemented
8#30145C212yesmaximal, complemented
9#23671C212yesmaximal, complemented
10#23658C211yesmaximal, characteristic, complemented
11C2×#2071C211yesmaximal, characteristic
12#23612C212yesmaximal
13#23551C211yesmaximal, characteristic, complemented
14#23454C211yesmaximal, characteristic
15#21063C212yesmaximal
16#21039C212yesmaximal, complemented
17#20754C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#428C2212yes—
20C2×#427C2212yes—
21C2×#424C2212yes—
22C2×#418C2212yes—
23C22×#102C2212yes—
24#2860C2212yes—
25#2860C2212yes—
26#2644C2212yes—
27#2644C2212yes—
28#2513—21no—
29#2513—21no—
30#2512—21no—
31#2512—21no—
32#2480—22no—
33#2480—22no—
34#2360—22no—
35#2360—22no—
36#2355—21no—
37#2355—21no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4—21no—
40#2330C2214yescomplemented
41#2317C2211yescharacteristic
42#2317C2212yes—
43#2316C2211yescharacteristic
44#2309C2211yescharacteristic
45#2309C2211yescharacteristic
46C4×C4⋊Q8C2212yescomplemented
47C4×C4⋊Q8C2212yescomplemented
48#2296C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2274C2212yes—
52C4×#117C2211yescharacteristic
53#2088C2214yescomplemented
54#2088C2214yescomplemented
55#2087C2214yescomplemented
56#2086C2214yescomplemented
57#2084C2212yescomplemented
58#2084C2212yescomplemented
59#2083C2212yescomplemented
60#2083C2212yescomplemented
61#2071C2212yes—
62#2071C2212yes—
63#2071C2214yes—
64#2067C2214yes—
65#2067C2214yes—
66#2065C2214yes—
67#2065C2214yes—
68#2059C2212yescomplemented
69#2059C2212yescomplemented
70#2058C2211yescharacteristic, complemented
71#2058C2211yescharacteristic, complemented
72#2057C2211yescharacteristic, complemented
73#2057C2211yescharacteristic, complemented
74#2025C2214yes—
75#2023C2212yes—
76#2023C2212yes—
77#1926C2214yes—
78#1926C2214yes—
79#1925C2214yes—
80#1925C2214yes—
81#1921C2214yescomplemented
82#1920C2214yescomplemented
83#1919C2214yescomplemented
84#1918C2214yescomplemented
85#1832C2214yescomplemented
86#1831C2212yescomplemented
87#1831C2212yescomplemented
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#533—28no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#126—22no—
102C2×#126—22no—
103C2×#125C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2312yes—
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C42C2311yescharacteristic
122#428—44no—
123#427—28no—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×#107C2314yes—
127C2×#106C2312yes—
128C2×#106C2312yes—
129C2×#106C2314yes—
130#424—44no—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133#418—28no—
134C2×#102—44no—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139#397—22no—
140#397—22no—
141#397—22no—
142#397—22no—
143#397—22no—
144#397—22no—
145#396—22no—
146#396—22no—
147#391—22no—
148#391—22no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#390C2312yes—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—24no—
166#389—24no—
167#389C2314yes—
168#389C2314yes—
169#389C2311yescharacteristic
170#389C2311yescharacteristic
171#389C2311yescharacteristic
172#389C2311yescharacteristic
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—24no—
184#387—24no—
185#385—21no—
186#385—21no—
187#385—24no—
188#385—22no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196#382—21no—
197#380—22no—
198#379—22no—
199#379—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C22⋊C4—24no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215#376C2312yes—
216C43C2312yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234C25—21no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×D4—21no—
238C22×D4—24no—
239C22×D4—22no—
240C22×D4—24no—
241C22×D4—22no—
242C22×D4C2412yes—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×M16—24no—
249C2×M16—24no—
250C2×M16—24no—
251C22×C8—24no—
252C22×C8—24no—
253C22×C8—24no—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259#126—84no—
260#126—84no—
261#125—24no—
262#125—24no—
263C22⋊Q8—28no—
264C22⋊Q8—28no—
265C4×Q8—28no—
266C4×Q8—28no—
267C4×Q8—84no—
268C4×Q8—84no—
269#117—24no—
270#117—22no—
271#117—24no—
272#117—22no—
273#117—24no—
274#117—22no—
275#117—22no—
276#117#4118yes—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2412yes—
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—24no—
350#107—24no—
351#107—44no—
352#106—24no—
353#106—24no—
354#106—44no—
355C4⋊C8—44no—
356C4⋊C8—44no—
357#102—28no—
358#102—28no—
359#102—44no—
360#102—44no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—22no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—81no—
382C24—81no—
383C24—81no—
384C24—81no—
385C24C22×D411yescharacteristic
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—48no—
400C2×D4—48no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4114yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+414yes—
433C22×C42−1+412yes—
434C22×C4C2511yescharacteristic
435M16—48no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—48no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4C2×#4114yes—
482C4⋊C4C2×#4114yes—
483C22⋊C4—28no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C42—22no—
501C42—22no—
502C42—28no—
503C42—24no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42—84no—
515C42—84no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—162no—
531C23—162no—
532C23—162no—
533C23—162no—
534C23#52512yes—
535C23#53612yes—
536C23#53712yes—
537C23C23×D411yescharacteristic
538Q8—48no—
539Q8—48no—
540D4—44no—
541D4—44no—
542D4—84no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—162no—
592C2×C4—164no—
593C2×C4#52512yes—
594C2×C4#53612yes—
595C2×C4#53712yes—
596C2×C4#54118yes—
597C2×C4#54218yes—
598C2×C4#57612yes—
599C2×C4#57712yes—
600C2×C4C22×#4112yes—
601C2×C4C2×H(5,𝔽2)11yescharacteristic
602C2×C4C2×2−1+411yescharacteristic
603C2×C4C2×2−1+412yes—
604C8—42no—
605C8—42no—
606C8—44no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—168no—
621C22#263714yes—
622C22C2×#52511yescharacteristic
623C22C2×#53611yescharacteristic
624C22C2×#53711yescharacteristic
625C22C2×#57611yescharacteristic
626C22C2×#57711yescharacteristic
627C22#358511yescharacteristic
628C22#358611yescharacteristic
629C4—22no—
630C4—22no—
631C4—41no—
632C4—44no—
633C4—41no—
634C4—44no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—82no—
642C4—81no—
643C4—84no—
644C4—81no—
645C4#263714yes—
646C4C2×#54112yes—
647C4C2×#54212yes—
648C2—24no—
649C2—162no—
650C2C2×#263711yescharacteristic
651C2#6058411yescharacteristic
652C2#6058811yescharacteristic
653C2#6087711yescharacteristic
654C2#6088511yescharacteristic
655C2#6092211yescharacteristic
656C2#6092311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
2. #34214C2semidirect
3. #30163C2semidirect
4. #30161C2semidirect
5. #30150C2semidirect
6. #30149C2semidirect
7. #30146C2semidirect
8. #30145C2semidirect
9. #23671C2semidirect
10. #23658C2semidirect
13. #23551C2semidirect
16. #21039C2semidirect
17. #20754C2semidirect
40. #2330C22semidirect
46. C4×C4⋊Q8C22semidirect
47. C4×C4⋊Q8C22semidirect
53. #2088C22semidirect
54. #2088C22semidirect
55. #2087C22semidirect
56. #2086C22semidirect
57. #2084C22semidirect
58. #2084C22semidirect
59. #2083C22semidirect
60. #2083C22semidirect
68. #2059C22semidirect
69. #2059C22semidirect
70. #2058C22semidirect
71. #2058C22semidirect
72. #2057C22semidirect
73. #2057C22semidirect
81. #1921C22semidirect
82. #1920C22semidirect
83. #1919C22semidirect
84. #1918C22semidirect
85. #1832C22semidirect
86. #1831C22semidirect
87. #1831C22semidirect

Maximal subgroups

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

subgroupindex
#352662
#342142
#301632
#301612
#301502
#301492
#301462
#301452
#236712
#236582
C2×#20712
#236122
#235512
#234542
#210632
#210392
#207542