Finite Group Databasebeta

← #6434865

Subgroups — #6434865

Order 512.

1592 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#30158C214yesmaximal, complemented
4#30157C214yesmaximal, complemented
5#30150C212yesmaximal, complemented
6#30149C212yesmaximal, complemented
7#30146C212yesmaximal, complemented
8#30145C212yesmaximal, complemented
9#23669C212yesmaximal, complemented
10#23656C211yesmaximal, characteristic, complemented
11#23628C211yesmaximal, characteristic
12#23614C212yesmaximal
13#23552C211yesmaximal, characteristic, complemented
14C2×#2023C211yesmaximal, characteristic
15#21065C212yesmaximal
16#21035C212yesmaximal, complemented
17#20754C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
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#2369—21no—
35#2369—21no—
36#2365—21no—
37#2365—21no—
38#2345—22no—
39Q8×C22⋊C4—22no—
40#2331C2214yescomplemented
41#2317C2211yescharacteristic
42#2316C2212yes—
43#2316C2211yescharacteristic
44#2309C2211yescharacteristic
45#2309C2211yescharacteristic
46C4×C4⋊Q8C2212yescomplemented
47C4×C4⋊Q8C2212yescomplemented
48#2296C2211yescharacteristic
49C2×#390C2211yescharacteristic
50#2285C2211yescharacteristic
51#2274C2212yes—
52C4×#117C2211yescharacteristic
53#2088C2214yescomplemented
54#2088C2214yescomplemented
55#2087C2214yescomplemented
56#2086C2214yescomplemented
57#2085C2212yescomplemented
58#2085C2212yescomplemented
59#2084C2211yescharacteristic, complemented
60#2084C2211yescharacteristic, complemented
61#2083C2211yescharacteristic, complemented
62#2083C2211yescharacteristic, complemented
63#2073C2214yes—
64#2071C2212yes—
65#2071C2212yes—
66#2067C2214yes—
67#2067C2214yes—
68#2065C2214yes—
69#2065C2214yes—
70#2058C2212yescomplemented
71#2058C2212yescomplemented
72#2057C2212yescomplemented
73#2057C2212yescomplemented
74#2023C2212yes—
75#2023C2212yes—
76#2023C2214yes—
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×Q8—21no—
96C23×D4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#126—22no—
105C2×#126—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42C2311yescharacteristic
119#428—28no—
120#427—44no—
121C2×#107C2314yes—
122C2×#107C2312yes—
123C2×#107C2312yes—
124C2×#106C2314yes—
125C2×#106C2312yes—
126C2×#106C2312yes—
127#424—44no—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2314yes—
130#418—44no—
131C2×#102—28no—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136#397—22no—
137#397—22no—
138#397—22no—
139#397—22no—
140#397—22no—
141#397—22no—
142#396—22no—
143#396—22no—
144#391—22no—
145#391—22no—
146#391—24no—
147#391—24no—
148#390—21no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390C2312yes—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389C2314yes—
167#389C2314yes—
168#389C2311yescharacteristic
169#389C2311yescharacteristic
170#389C2311yescharacteristic
171#389C2311yescharacteristic
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—24no—
183#387—24no—
184#385—21no—
185#385—21no—
186#385—22no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#382—21no—
191#382C2312yes—
192#380—21no—
193#380—21no—
194#380—24no—
195#380—22no—
196#379—22no—
197#379—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
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#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234C25—21no—
235C22×Q8—28no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—82no—
239C22×Q8—82no—
240C22×D4—21no—
241C22×D4—24no—
242C22×D4—22no—
243C22×D4—24no—
244C22×D4—22no—
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×M16—24no—
252C2×M16—24no—
253C2×M16—24no—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264#126—84no—
265#126—84no—
266C4×Q8—84no—
267C4×Q8—84no—
268#117—22no—
269#117—24no—
270#117—22no—
271#117—24no—
272#117—24no—
273#117—22no—
274#117—22no—
275#117—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4#4118yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2414yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
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—82no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—28no—
346#107—42no—
347#107—42no—
348#106—28no—
349#106—42no—
350#106—42no—
351C4⋊C8—44no—
352C4⋊C8—44no—
353#102—28no—
354#102—28no—
355#102—44no—
356#102—44no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—22no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—81no—
380C24—81no—
381C24—81no—
382C24—81no—
383C24C22×D411yescharacteristic
384C2×Q8—28no—
385C2×Q8—28no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—82no—
389C2×Q8—82no—
390C2×Q8—816no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—24no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—48no—
399C2×D4—48no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
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—82no—
417C22×C4—82no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4114yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+414yes—
432C22×C42−1+412yes—
433C22×C4C2511yescharacteristic
434M16—48no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—48no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4C2×#4118yes—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—22no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—44no—
510C42—42no—
511C42—84no—
512C42—84no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—84no—
525C23—84no—
526C23—84no—
527C23—162no—
528C23—162no—
529C23—162no—
530C23—162no—
531C23#52512yes—
532C23#53612yes—
533C23#53712yes—
534C23C23×D411yescharacteristic
535Q8—416no—
536Q8—88no—
537D4—44no—
538D4—44no—
539D4—84no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—88no—
589C2×C4—162no—
590C2×C4#52512yes—
591C2×C4#53612yes—
592C2×C4#53712yes—
593C2×C4#54118yes—
594C2×C4#54218yes—
595C2×C4#57612yes—
596C2×C4#57712yes—
597C2×C4C22×#4112yes—
598C2×C4C2×H(5,𝔽2)11yescharacteristic
599C2×C4C2×2−1+411yescharacteristic
600C2×C4C2×2−1+412yes—
601C8—42no—
602C8—42no—
603C8—44no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—168no—
618C22#263714yes—
619C22C2×#52511yescharacteristic
620C22C2×#53611yescharacteristic
621C22C2×#53711yescharacteristic
622C22C2×#57611yescharacteristic
623C22C2×#57711yescharacteristic
624C22#358511yescharacteristic
625C22#358611yescharacteristic
626C4—22no—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—41no—
631C4—48no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—81no—
636C4—82no—
637C4—82no—
638C4—81no—
639C4—81no—
640C4—81no—
641C4—82no—
642C4#263714yes—
643C4C2×#54112yes—
644C4C2×#54212yes—
645C2—24no—
646C2—162no—
647C2C2×#263711yescharacteristic
648C2#6057811yescharacteristic
649C2#6058211yescharacteristic
650C2#6087711yescharacteristic
651C2#6088511yescharacteristic
652C2#6092811yescharacteristic
653C2#6092911yescharacteristic
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. #34216C2semidirect
3. #30158C2semidirect
4. #30157C2semidirect
5. #30150C2semidirect
6. #30149C2semidirect
7. #30146C2semidirect
8. #30145C2semidirect
9. #23669C2semidirect
10. #23656C2semidirect
13. #23552C2semidirect
16. #21035C2semidirect
17. #20754C2semidirect
40. #2331C22semidirect
46. C4×C4⋊Q8C22semidirect
47. C4×C4⋊Q8C22semidirect
53. #2088C22semidirect
54. #2088C22semidirect
55. #2087C22semidirect
56. #2086C22semidirect
57. #2085C22semidirect
58. #2085C22semidirect
59. #2084C22semidirect
60. #2084C22semidirect
61. #2083C22semidirect
62. #2083C22semidirect
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
#342162
#301582
#301572
#301502
#301492
#301462
#301452
#236692
#236562
#236282
#236142
#235522
C2×#20232
#210652
#210352
#207542