Finite Group Databasebeta

← #7505179

Subgroups — #7505179

Order 512.

1649 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35541C211yesmaximal, characteristic
2#35339C211yesmaximal, characteristic, complemented
3#31594C214yesmaximal, complemented
4#31593C214yesmaximal, complemented
5#31578C214yesmaximal, complemented
6#31577C214yesmaximal, complemented
7#23738C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic, complemented
9#23660C211yesmaximal, characteristic, complemented
10#23631C211yesmaximal, characteristic, complemented
11#23611C211yesmaximal, characteristic, complemented
12#23511C211yesmaximal, characteristic
13#21206C211yesmaximal, characteristic
14#21157C211yesmaximal, characteristic, complemented
15#21060C211yesmaximal, characteristic, complemented
16#21058C211yesmaximal, characteristic, complemented
17#21026C211yesmaximal, characteristic, complemented
18#20879C211yesmaximal, characteristic, complemented
19#20650C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C2×#518C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×#405C2211yescharacteristic
36#2864—22no—
37#2647—22no—
38#2570C2214yescomplemented
39#2570C2214yescomplemented
40#2569—24no—
41#2447C2214yes—
42#2445C2214yes—
43#2422—22no—
44#2420—22no—
45#2412—24no—
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2316C2211yescharacteristic
50#2316C2211yescharacteristic
51#2311C2211yescharacteristic
52#2297C2211yescharacteristic
53C2×#390C2211yescharacteristic
54#2285C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yescomplemented
58#2096C2212yescomplemented
59#2094C2212yescomplemented
60#2092C2214yes—
61#2092C2214yes—
62#2092C2212yescomplemented
63#2085C2214yescomplemented
64#2083C2214yescomplemented
65#2074C2214yescomplemented
66#2073C2214yescomplemented
67#2068C2214yescomplemented
68#2067C2214yescomplemented
69#2042C2214yes—
70#2041C2214yes—
71#1961C2212yes—
72#1959C2212yes—
73#1958C2212yes—
74#1957C2212yes—
75#1944C2214yescomplemented
76#1941C2214yescomplemented
77#1928C2212yescomplemented
78#1927C2212yescomplemented
79#1926C2214yes—
80#1926C2212yescomplemented
81#1925C2214yes—
82#1925C2212yescomplemented
83#1917C2214yescomplemented
84#1916C2214yescomplemented
85#1864C2214yes—
86#1862C2214yes—
87#1804C2214yes—
88#1802C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#535—24no—
104#535—42no—
105C2×#127C2312yes—
106C2×#124C2312yes—
107C2×#123—24no—
108C2×#121—22no—
109C2×#120—22no—
110#518—24no—
111#518—42no—
112C2×C4×D4—22no—
113C2×C4×D4C2314yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42C2311yescharacteristic
125#428—42no—
126#427—42no—
127C2×#107—42no—
128C2×#107C2314yes—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#106—24no—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136#424—42no—
137C2×C4⋊C8C2314yes—
138#418—24no—
139#418—42no—
140#417—42no—
141#417—42no—
142#416—24no—
143#416—42no—
144C2×#103C2311yescharacteristic
145C2×#103C2311yescharacteristic
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#102—24no—
150C2×#102—42no—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2311yescharacteristic
154C2×#102C2311yescharacteristic
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162#405—42no—
163C2×#97C2312yes—
164C2×C4×C8C2312yes—
165#400—24no—
166#400—24no—
167#398—24no—
168#398—24no—
169#397—24no—
170#397—24no—
171#394—24no—
172#394—24no—
173#391C2312yes—
174#391C2312yes—
175#390—21no—
176#390—21no—
177#390C2312yes—
178#390C2312yes—
179#390C2314yes—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389C2314yes—
186#389C2314yes—
187#389C2314yes—
188#387—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#386—22no—
193#386—22no—
194#386—24no—
195#385—24no—
196#385—22no—
197#385—22no—
198#384D412yes—
199#384C2312yes—
200#384C2312yes—
201#382—21no—
202#382—21no—
203#382D412yes—
204#382C2312yes—
205#382C2312yes—
206#382C2314yes—
207#381—24no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380—22no—
212#380D414yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2314yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219#376—21no—
220C2×#95—22no—
221C43—21no—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C25—21no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C22×Q8C2411yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—24no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4#4114yes—
250C22×D4C2411yescharacteristic
251C22×D4C2412yes—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—24no—
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8#4114yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2411yescharacteristic
275C22×C8C2411yescharacteristic
276#127—24no—
277#127—42no—
278#124—24no—
279#124—42no—
280#123—88no—
281#121—48no—
282#120—48no—
283C4×D4—28no—
284C4×D4—44no—
285C4×D4—48no—
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—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
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—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—48no—
332C2×C22⋊C4—84no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2412yes—
346C2×C42C2412yes—
347#107—44no—
348#107—42no—
349#107—42no—
350#106—24no—
351#106—24no—
352#106—42no—
353#106—42no—
354C4⋊C8—44no—
355#103—22no—
356#103—22no—
357#103—42no—
358#103—42no—
359#103—42no—
360#102—28no—
361#102—24no—
362#102—24no—
363#102—22no—
364#102—22no—
365#102—44no—
366#102—42no—
367#102—42no—
368#102—42no—
369#102—42no—
370#102—42no—
371#97—42no—
372C4×C8—42no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95C2×D414yes—
386#95C2×D414yes—
387C24—22no—
388C24—24no—
389C24—21no—
390C24—42no—
391C24—42no—
392C24—82no—
393C24—82no—
394C24C22×D411yescharacteristic
395#41—44no—
396#41—44no—
397#41—44no—
398#41—44no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×D4—28no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—48no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—44no—
427C2×D4—84no—
428C2×D4—84no—
429C2×D4C2×#4114yes—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—48no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C22×C4C2511yescharacteristic
465M16—44no—
466M16—44no—
467C2×C8—28no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—44no—
482C2×C8C2×#4112yes—
483C2×C8C2×#4112yes—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—82no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—82no—
534C42—82no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—48no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—88no—
549C23—164no—
550C23—162no—
551C23—162no—
552C23#53612yes—
553C23#54612yes—
554C23#55012yes—
555C23C23×D411yescharacteristic
556Q8—41no—
557Q8—41no—
558Q8—41no—
559Q8—41no—
560D4—24no—
561D4—24no—
562D4—42no—
563D4—41no—
564D4—41no—
565D4—48no—
566D4—41no—
567D4—42no—
568D4—41no—
569D4—84no—
570D4—168no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—168no—
605C2×C4—162no—
606C2×C4—162no—
607C2×C4—162no—
608C2×C4#53612yes—
609C2×C4#53914yes—
610C2×C4#53914yes—
611C2×C4#54612yes—
612C2×C4#55012yes—
613C2×C4#55214yes—
614C2×C4#56214yes—
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C2×C4C2×2−1+411yescharacteristic
621C8—22no—
622C8—22no—
623C8—42no—
624C8—42no—
625C8—42no—
626C22—24no—
627C22—24no—
628C22—22no—
629C22—22no—
630C22—48no—
631C22—44no—
632C22—84no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—81no—
637C22—82no—
638C22—81no—
639C22—81no—
640C22—168no—
641C22#269414yes—
642C22C2×#53611yescharacteristic
643C22C2×#54611yescharacteristic
644C22C2×#55011yescharacteristic
645C22#358412yes—
646C22#358512yes—
647C4—44no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—82no—
652C4—82no—
653C4—81no—
654C4—81no—
655C4—82no—
656C4—81no—
657C4—82no—
658C4—82no—
659C4#269414yes—
660C4C2×#53911yescharacteristic
661C4C2×#53911yescharacteristic
662C4C2×#55211yescharacteristic
663C4C2×#56211yescharacteristic
664C2—24no—
665C2—84no—
666C2—161no—
667C2C2×#269411yescharacteristic
668C2#6087512yes—
669C2#6120112yes—
670C2#6141211yescharacteristic
671C2#6141711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35339C2semidirect
3. #31594C2semidirect
4. #31593C2semidirect
5. #31578C2semidirect
6. #31577C2semidirect
7. #23738C2semidirect
8. C2×#2092C2semidirect
9. #23660C2semidirect
10. #23631C2semidirect
11. #23611C2semidirect
14. #21157C2semidirect
15. #21060C2semidirect
16. #21058C2semidirect
17. #21026C2semidirect
18. #20879C2semidirect
19. #20650C2semidirect
38. #2570C22semidirect
39. #2570C22semidirect
57. #2097C22semidirect
58. #2096C22semidirect
59. #2094C22semidirect
62. #2092C22semidirect
63. #2085C22semidirect
64. #2083C22semidirect
65. #2074C22semidirect
66. #2073C22semidirect
67. #2068C22semidirect
68. #2067C22semidirect
75. #1944C22semidirect
76. #1941C22semidirect
77. #1928C22semidirect
78. #1927C22semidirect
80. #1926C22semidirect
82. #1925C22semidirect
83. #1917C22semidirect
84. #1916C22semidirect

Maximal subgroups

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

subgroupindex
#355412
#353392
#315942
#315932
#315782
#315772
#237382
C2×#20922
#236602
#236312
#236112
#235112
#212062
#211572
#210602
#210582
#210262
#208792
#206502