Finite Group Databasebeta

← #6461673

Subgroups — #6461673

Order 512.

1808 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#35266C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic, complemented
3#29453C214yesmaximal, complemented
4#29452C214yesmaximal, complemented
5#29445C214yesmaximal, complemented
6#29444C214yesmaximal, complemented
7#23668C211yesmaximal, characteristic, complemented
8#23612C211yesmaximal, characteristic, complemented
9#23612C211yesmaximal, characteristic, complemented
10#23540C211yesmaximal, characteristic
11C2×#1972C211yesmaximal, characteristic
12#21154C211yesmaximal, characteristic, complemented
13#21095C211yesmaximal, characteristic, complemented
14#21095C211yesmaximal, characteristic, complemented
15#21065C211yesmaximal, characteristic, complemented
16#21038C211yesmaximal, characteristic, complemented
17#20934C211yesmaximal, characteristic
18#20879C211yesmaximal, characteristic, complemented
19#20427C211yesmaximal, characteristic, complemented
20C23×D8C2211yescharacteristic
21C2×#518C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C2×#424C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#418C2211yescharacteristic
28C2×#418C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C2×C4×M16C2211yescharacteristic
34C22×#97C2211yescharacteristic
35#2860C2214yescomplemented
36#2644C2214yescomplemented
37#2513—22no—
38#2512—22no—
39#2480—24no—
40#2357—22no—
41#2354—22no—
42#2342—24no—
43#2319C2214yescomplemented
44#2317C2211yescharacteristic
45#2317C2211yescharacteristic
46#2316C2211yescharacteristic
47#2309C2211yescharacteristic
48#2309C2211yescharacteristic
49#2309C2211yescharacteristic
50#2296C2211yescharacteristic
51#2296C2211yescharacteristic
52#2293C2214yescomplemented
53C2×#390C2211yescharacteristic
54C2×#390C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2088C2214yescomplemented
58#2088C2214yescomplemented
59#2067C2214yescomplemented
60#2067C2214yescomplemented
61#2065C2214yescomplemented
62#2065C2214yescomplemented
63#2054C2214yes—
64#2054C2214yes—
65#1972C2212yes—
66#1972C2212yes—
67#1972C2212yes—
68#1972C2212yes—
69#1944C2212yescomplemented
70#1944C2212yescomplemented
71#1943C2212yescomplemented
72#1943C2212yescomplemented
73#1936C2214yescomplemented
74#1936C2214yescomplemented
75#1934C2214yescomplemented
76#1934C2214yescomplemented
77#1926C2214yescomplemented
78#1925C2214yescomplemented
79#1920C2214yescomplemented
80#1919C2214yescomplemented
81#1879C2214yes—
82#1879C2214yes—
83#1864C2214yescomplemented
84#1862C2214yescomplemented
85#1768C2212yescomplemented
86#1768C2212yescomplemented
87#1767C2212yescomplemented
88#1767C2212yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C23×D4C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×D8—24no—
100C22×D8—24no—
101C22×D8—42no—
102C22×D8C2312yes—
103C22×D8C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C2×#126—24no—
109#518—24no—
110#518—24no—
111C2×C4×D4—22no—
112C2×C4×D4C2314yes—
113C2×#117—24no—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C22⋊C4—22no—
125C22×C42C2311yescharacteristic
126#427—24no—
127#427—42no—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#106—24no—
131C2×#106C2312yes—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135#424—24no—
136#424—24no—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141#418—24no—
142#418—24no—
143#418—42no—
144#418—42no—
145C2×#102—24no—
146C2×#102—24no—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C4×M16—42no—
156C2×#97—42no—
157C2×#97C2311yescharacteristic
158C2×#97C2311yescharacteristic
159C2×#97C2311yescharacteristic
160C2×#97C2311yescharacteristic
161C2×#97C2312yes—
162C2×C4×C8C2312yes—
163#397—24no—
164#397—24no—
165#397—24no—
166#396—24no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—22no—
171#390—21no—
172#390—22no—
173#390—22no—
174#390C2314yes—
175#390C2314yes—
176#390C2314yes—
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#390C2312yes—
181#389—22no—
182#389C2314yes—
183#389C2314yes—
184#389C2314yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#388—24no—
190#388—24no—
191#388—24no—
192#388—24no—
193#388—24no—
194#388—24no—
195#388—24no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384—21no—
201#384—21no—
202#382—21no—
203#382—21no—
204#381—24no—
205#380—22no—
206#380—24no—
207#379—22no—
208#379—22no—
209#379—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2314yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—24no—
218C4×C22⋊C4D414yes—
219#376C2312yes—
220C43C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C25—21no—
234C25—21no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4#4114yes—
249C22×D4C2412yes—
250C22×D4C2412yes—
251C23×C4—24no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2411yescharacteristic
256C2×D8—24no—
257C2×D8—24no—
258C2×D8—24no—
259C2×D8—24no—
260C2×D8—42no—
261C2×D8—42no—
262C2×D8—416no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8#4114yes—
274C22×C8#4114yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279C22×C8C2411yescharacteristic
280C22×C8C2411yescharacteristic
281#126—88no—
282C4×D4—216no—
283C4×D4—28no—
284C4×D4—28no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—88no—
297#117#4114yes—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—216no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—84no—
337C2×C22⋊C4—84no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42C2×D412yes—
348C2×C42C2×D414yes—
349C2×C42C2×D414yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—24no—
355#107—42no—
356#106—24no—
357#106—24no—
358#106—24no—
359#106—42no—
360C4⋊C8—24no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364#102—28no—
365#102—24no—
366#102—24no—
367#102—24no—
368#102—24no—
369#102—28no—
370#102—44no—
371#102—44no—
372#97—41no—
373#97—41no—
374#97—41no—
375#97—41no—
376#97—42no—
377C4×C8—42no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95C2×D414yes—
387#95C2×D414yes—
388C24—22no—
389C24—24no—
390C24—22no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—82no—
397C24—82no—
398C24—82no—
399C24—82no—
400C24C22×D411yescharacteristic
401C2×D4—24no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4C2×#4114yes—
427C22×C4—22no—
428C22×C4—216no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4C22×D412yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C4C2511yescharacteristic
456D8—48no—
457D8—48no—
458M16—44no—
459M16—44no—
460C2×C8—28no—
461C2×C8—28no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8C2×#4112yes—
477C2×C8C2×#4112yes—
478C2×C8C2×#4112yes—
479C2×C8C2×#4112yes—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4C2×#4114yes—
504C22⋊C4—216no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—44no—
532C42—44no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—84no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—44no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—48no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—88no—
551C23—88no—
552C23—84no—
553C23—164no—
554C23—164no—
555C23—162no—
556C23—162no—
557C23—162no—
558C23—162no—
559C23#52512yes—
560C23#53212yes—
561C23#53712yes—
562C23C23×D411yescharacteristic
563D4—24no—
564D4—24no—
565D4—48no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—84no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—84no—
603C2×C4—816no—
604C2×C4—162no—
605C2×C4—162no—
606C2×C4#52512yes—
607C2×C4#53212yes—
608C2×C4#53712yes—
609C2×C4#55114yes—
610C2×C4#55214yes—
611C2×C4#55314yes—
612C2×C4#55614yes—
613C2×C4C22×#4111yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C8—22no—
620C8—22no—
621C8—22no—
622C8—22no—
623C8—42no—
624C8—42no—
625C22—24no—
626C22—22no—
627C22—22no—
628C22—24no—
629C22—44no—
630C22—48no—
631C22—84no—
632C22—82no—
633C22—82no—
634C22—84no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22—168no—
640C22—168no—
641C22#260114yes—
642C22C2×#52511yescharacteristic
643C22C2×#53211yescharacteristic
644C22C2×#53711yescharacteristic
645C22#358512yes—
646C22#358512yes—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—44no—
652C4—44no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4#260114yes—
659C4C2×#55111yescharacteristic
660C4C2×#55211yescharacteristic
661C4C2×#55311yescharacteristic
662C4C2×#55611yescharacteristic
663C2—24no—
664C2—84no—
665C2—162no—
666C2C2×#260111yescharacteristic
667C2#6057412yes—
668C2#6071911yescharacteristic
669C2#6071911yescharacteristic
670C2#6093712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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
1. #35266C2semidirect
2. #34255C2semidirect
3. #29453C2semidirect
4. #29452C2semidirect
5. #29445C2semidirect
6. #29444C2semidirect
7. #23668C2semidirect
8. #23612C2semidirect
9. #23612C2semidirect
12. #21154C2semidirect
13. #21095C2semidirect
14. #21095C2semidirect
15. #21065C2semidirect
16. #21038C2semidirect
18. #20879C2semidirect
19. #20427C2semidirect
35. #2860C22semidirect
36. #2644C22semidirect
43. #2319C22semidirect
52. #2293C22semidirect
57. #2088C22semidirect
58. #2088C22semidirect
59. #2067C22semidirect
60. #2067C22semidirect
61. #2065C22semidirect
62. #2065C22semidirect
69. #1944C22semidirect
70. #1944C22semidirect
71. #1943C22semidirect
72. #1943C22semidirect
73. #1936C22semidirect
74. #1936C22semidirect
75. #1934C22semidirect
76. #1934C22semidirect
77. #1926C22semidirect
78. #1925C22semidirect
79. #1920C22semidirect
80. #1919C22semidirect
83. #1864C22semidirect
84. #1862C22semidirect
85. #1768C22semidirect
86. #1768C22semidirect
87. #1767C22semidirect
88. #1767C22semidirect

Maximal subgroups

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

subgroupindex
#352662
#342552
#294532
#294522
#294452
#294442
#236682
#236122
#235402
C2×#19722
#211542
#210952
#210652
#210382
#209342
#208792
#204272