Finite Group Databasebeta

← #6477207

Subgroups — #6477207

Order 512.

1713 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35261C211yesmaximal, characteristic
2C2×#2629C211yesmaximal, characteristic, complemented
3#33216C212yesmaximal, complemented
4#33055C212yesmaximal, complemented
5#30369C214yesmaximal, complemented
6#29169C214yesmaximal, complemented
7#27966C211yesmaximal, characteristic, complemented
8#27909C211yesmaximal, characteristic, complemented
9#27885C212yesmaximal, complemented
10#25913C211yesmaximal, characteristic, complemented
11#25815C211yesmaximal, characteristic, complemented
12#25682C212yesmaximal, complemented
13#24558C212yesmaximal, complemented
14#24550C212yesmaximal, complemented
15#22081C212yesmaximal, complemented
16#22075C212yesmaximal, complemented
17#20379C211yesmaximal, characteristic, complemented
18C2×#540C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C2×#499C2212yescomplemented
22C2×#490C2212yescomplemented
23C2×#480C2212yes—
24C2×#480C2212yes—
25C2×#407C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C2×#405C2212yes—
28#2856C2211yescharacteristic
29#2851C2211yescharacteristic
30#2847C2212yes—
31#2629—22no—
32#2629—22no—
33#2629C2212yes—
34#2629C2212yes—
35#2629C2212yes—
36#2629C2212yes—
37#2629C2211yescharacteristic
38#2624C2212yes—
39#2619C2211yescharacteristic
40#2476—22no—
41#2471—22no—
42#2443—22no—
43#2424—22no—
44C2×#398—22no—
45C2×#396—22no—
46#2324C2212yes—
47#2314C2211yescharacteristic
48C4×#121C2212yes—
49C2×#389C2212yes—
50C2×#384C2211yescharacteristic
51C2×#380C2212yes—
52#2100C2214yescomplemented
53#2093C2214yescomplemented
54#2076C2214yes—
55#2075C2214yes—
56#2059C2214yescomplemented
57#2034C2214yes—
58#2033C2214yes—
59#2025C2214yescomplemented
60#1992C2212yescomplemented
61#1992C2212yescomplemented
62#1991C2212yescomplemented
63#1957C2212yescomplemented
64#1943C2214yescomplemented
65#1942C2214yescomplemented
66#1936C2214yescomplemented
67#1934C2214yescomplemented
68#1921C2214yescomplemented
69#1918C2214yescomplemented
70#1895C2214yes—
71#1893C2214yes—
72#1874C2212yes—
73#1874C2212yes—
74#1864C2214yescomplemented
75#1862C2214yescomplemented
76#1853C2212yes—
77#1852C2212yes—
78#1845C2212yescomplemented
79#1836C2212yescomplemented
80#1823C2212yes—
81#1823C2212yes—
82#1809C2214yes—
83#1808C2214yes—
84#1750C2212yes—
85#1750C2212yes—
86#1749C2211yescharacteristic
87#1749C2211yescharacteristic
88C4×C22⋊C8C2211yescharacteristic
89C4×C22⋊C8C2211yescharacteristic
90#1729C2212yes—
91C24×C4C2311yescharacteristic
92C22×M16C2312yes—
93C23×C8C2312yes—
94#540—44no—
95C2×#127C2311yescharacteristic
96C2×#125—22no—
97C2×#125—44no—
98C2×#125C2311yescharacteristic
99C2×#125C2312yes—
100C2×#125C2312yes—
101C2×#123—22no—
102C2×#121—22no—
103C2×#121—22no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#121C2312yes—
107C2×#120—22no—
108C2×C4×D4—22no—
109C2×C4×D4C2312yes—
110C2×#117—44no—
111C2×#117D412yes—
112C2×#117D412yes—
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4D412yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4—22no—
124C22×C22⋊C4D412yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#499—44no—
128#490—44no—
129#480—28no—
130#480—28no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139#407—24no—
140C2×C22⋊C8—24no—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2311yescharacteristic
144C2×C22⋊C8C2311yescharacteristic
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147#405—44no—
148C2×#97C2312yes—
149C2×#97C2312yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2312yes—
152#399—22no—
153#398—24no—
154#398—24no—
155#398—24no—
156#398—24no—
157#398—22no—
158#397—22no—
159#396—24no—
160#396—24no—
161#396—24no—
162#396—24no—
163#396—22no—
164#394—22no—
165#394—22no—
166#394—22no—
167#392—22no—
168#390—22no—
169#390—22no—
170#390C2312yes—
171#390C2312yes—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#389C2314yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—22no—
182#387—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#385—22no—
189#385—22no—
190#384—22no—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382C2311yescharacteristic
195#380—22no—
196#380—22no—
197#380C2312yes—
198#380C2312yes—
199#380C2314yes—
200#380C2314yes—
201#380C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204C2×#95—22no—
205C2×#95—22no—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218C25C2×D411yescharacteristic
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4C2412yes—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—42no—
231C23×C4C2×D412yes—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×M16—24no—
240C2×M16—24no—
241C2×M16—42no—
242C22×C8—24no—
243C22×C8—24no—
244C22×C8—42no—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249#127—24no—
250#125—24no—
251#125—24no—
252#125—24no—
253#125—44no—
254#125—48no—
255#123—48no—
256#121—24no—
257#121—24no—
258#121—28no—
259#121—48no—
260#121—44no—
261#121—84no—
262#120—84no—
263C4×D4—44no—
264C4×D4—84no—
265#117—24no—
266#117—24no—
267#117—42no—
268#117—42no—
269#117—44no—
270#117—44no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—84no—
289C2×C4⋊C4—84no—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—84no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—42no—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327#107—28no—
328#107—28no—
329#107—44no—
330#106—44no—
331#102—28no—
332#102—28no—
333#102—44no—
334#102—44no—
335C22⋊C8—22no—
336C22⋊C8—22no—
337C22⋊C8—24no—
338C22⋊C8—24no—
339C22⋊C8#4114yes—
340C22⋊C8#4114yes—
341#97—24no—
342#97—42no—
343C4×C8—24no—
344C4×C8—42no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—41no—
361C24—42no—
362C24C22×D411yescharacteristic
363C24C22×D411yescharacteristic
364C24C22×D411yescharacteristic
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4—82no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C42−1+414yes—
431C22×C4C2511yescharacteristic
432M16—48no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—48no—
444C2×C8C2×#4114yes—
445C2×C8C2×#4114yes—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—82no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—42no—
492C42—82no—
493C42H(5,𝔽2)14yes—
494C23—22no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—44no—
507C23—48no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—82no—
517C23—84no—
518C23#53212yes—
519C23#53612yes—
520C23#57614yes—
521C23C23×D411yescharacteristic
522D4—44no—
523D4—44no—
524D4—88no—
525D4—164no—
526D4—164no—
527D4—164no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—84no—
572C2×C4—88no—
573C2×C4—88no—
574C2×C4—164no—
575C2×C4#53212yes—
576C2×C4#53414yes—
577C2×C4#53612yes—
578C2×C4C22×#4111yescharacteristic
579C2×C4C22×#4111yescharacteristic
580C2×C4C2×H(5,𝔽2)12yes—
581C2×C4C2×2−1+412yes—
582C8—24no—
583C8—24no—
584C8—42no—
585C8—42no—
586C22—22no—
587C22—22no—
588C22—24no—
589C22—24no—
590C22—22no—
591C22—24no—
592C22—41no—
593C22—41no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—42no—
599C22—42no—
600C22—88no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22#317614yes—
609C22C2×#53211yescharacteristic
610C22C2×#53411yescharacteristic
611C22C2×#53611yescharacteristic
612C22C2×#57612yes—
613C22#358612yes—
614C4—22no—
615C4—22no—
616C4—24no—
617C4—41no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—84no—
622C4—82no—
623C4—84no—
624C4—82no—
625C4—82no—
626C4#352012yes—
627C4#352012yes—
628C2—24no—
629C2—42no—
630C2—81no—
631C2—162no—
632C2#3508511yescharacteristic
633C2C2×#317611yescharacteristic
634C2#6070511yescharacteristic
635C2#6076212yes—
636C2#6087112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35261C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2629C2semidirect
3. #33216C2semidirect
4. #33055C2semidirect
5. #30369C2semidirect
6. #29169C2semidirect
7. #27966C2semidirect
8. #27909C2semidirect
9. #27885C2semidirect
10. #25913C2semidirect
11. #25815C2semidirect
12. #25682C2semidirect
13. #24558C2semidirect
14. #24550C2semidirect
15. #22081C2semidirect
16. #22075C2semidirect
17. #20379C2semidirect
21. C2×#499C22semidirect
22. C2×#490C22semidirect
52. #2100C22semidirect
53. #2093C22semidirect
56. #2059C22semidirect
59. #2025C22semidirect
60. #1992C22semidirect
61. #1992C22semidirect
62. #1991C22semidirect
63. #1957C22semidirect
64. #1943C22semidirect
65. #1942C22semidirect
66. #1936C22semidirect
67. #1934C22semidirect
68. #1921C22semidirect
69. #1918C22semidirect
74. #1864C22semidirect
75. #1862C22semidirect
78. #1845C22semidirect
79. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#352612
C2×#26292
#332162
#330552
#303692
#291692
#279662
#279092
#278852
#259132
#258152
#256822
#245582
#245502
#220812
#220752
#203792