Finite Group Databasebeta

← #6434847

Subgroups — #6434847

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2644C211yesmaximal, characteristic
2#34074C211yesmaximal, characteristic, complemented
3#30163C214yesmaximal, complemented
4#30161C214yesmaximal, complemented
5#30158C214yesmaximal, complemented
6#30157C214yesmaximal, complemented
7#23670C212yesmaximal, complemented
8#23657C211yesmaximal, characteristic, complemented
9#23628C211yesmaximal, characteristic
10#23618C212yesmaximal
11#23554C211yesmaximal, characteristic, complemented
12#23454C211yesmaximal, characteristic
13#21060C212yesmaximal
14#21034C212yesmaximal, complemented
15#20753C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#428C2212yes—
18C2×#428C2212yes—
19C2×#423C2212yes—
20C2×#418C2212yes—
21C2×#418C2212yes—
22#2644—22no—
23#2644—22no—
24#2644C2214yes—
25#2644C2214yes—
26C2×#397—21no—
27C2×#397—21no—
28C2×#397—21no—
29C2×#396—21no—
30#2331C2212yescomplemented
31#2331C2212yescomplemented
32#2330C2212yescomplemented
33#2330C2212yescomplemented
34#2329—21no—
35#2329—21no—
36#2329—21no—
37#2329—21no—
38#2296C2212yes—
39#2295—22no—
40#2295—22no—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47#2285C2211yescharacteristic
48C4×#117C2212yes—
49C2×#376C2211yescharacteristic
50#2088C2214yescomplemented
51#2088C2214yescomplemented
52#2087C2214yescomplemented
53#2086C2214yescomplemented
54#2085C2214yescomplemented
55#2084C2212yescomplemented
56#2083C2212yescomplemented
57#2073C2214yes—
58#2071C2212yes—
59#2071C2212yes—
60#2067C2214yes—
61#2067C2214yes—
62#2065C2214yes—
63#2065C2214yes—
64#2059C2214yescomplemented
65#2058C2212yescomplemented
66#2057C2212yescomplemented
67#2025C2214yes—
68#2023C2212yes—
69#2023C2212yes—
70#1926C2214yes—
71#1926C2214yes—
72#1925C2214yes—
73#1925C2214yes—
74#1921C2214yescomplemented
75#1920C2214yescomplemented
76#1919C2214yescomplemented
77#1918C2214yescomplemented
78#1832C2214yescomplemented
79#1832C2214yescomplemented
80#1736C2212yes—
81#1736C2212yes—
82#1736C2212yes—
83#1736C2212yes—
84#1728C2211yescharacteristic
85#1728C2212yes—
86#1728C2211yescharacteristic
87C23×D4C2311yescharacteristic
88C22×M16C2312yes—
89C22×M16C2312yes—
90#533—28no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#125C2312yes—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C4×Q8—21no—
97C2×C4×Q8—21no—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#428—28no—
116#428—44no—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121#423—44no—
122C2×C4⋊C8C2314yes—
123C2×C4⋊C8C2314yes—
124#418—28no—
125#418—44no—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130#397—24no—
131#397—24no—
132#397—24no—
133#397—24no—
134#397—24no—
135#397—24no—
136#396—24no—
137#396—24no—
138#391—22no—
139#391—22no—
140#391—22no—
141#391—22no—
142#390—21no—
143#390—21no—
144#390—22no—
145#390—22no—
146#390C2314yes—
147#390C2312yes—
148#390C2312yes—
149#390C2314yes—
150#389—21no—
151#389—21no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2314yes—
165#389C2314yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#385—22no—
175#385—22no—
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#382C2312yes—
179#380—21no—
180#380—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191#376—22no—
192#376D412yes—
193#376D412yes—
194#376C2312yes—
195#376C2312yes—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95D411yescharacteristic
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202C2×#95D411yescharacteristic
203C43—22no—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2312yes—
215#336C2312yes—
216#336C2314yes—
217#336C2314yes—
218C25—21no—
219C22×Q8—21no—
220C22×Q8—21no—
221C22×D4—21no—
222C22×D4—24no—
223C22×D4—22no—
224C22×D4—24no—
225C22×D4—22no—
226C22×D4C2412yes—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2411yescharacteristic
237C2×M16—24no—
238C2×M16—24no—
239C2×M16—24no—
240C2×M16—24no—
241C2×M16—24no—
242C2×M16—24no—
243C22×C8C2414yes—
244C22×C8C2414yes—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247#125—24no—
248#125—24no—
249C22⋊Q8—28no—
250C22⋊Q8—28no—
251C4×Q8—28no—
252C4×Q8—28no—
253#117—24no—
254#117—24no—
255#117—24no—
256#117#4118yes—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—82no—
301C2×C22⋊C4—82no—
302C2×C22⋊C4—82no—
303C2×C22⋊C4—82no—
304C2×C22⋊C4—82no—
305C2×C22⋊C4—82no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2412yes—
323C2×C42C2412yes—
324#107—28no—
325#107—44no—
326#106—28no—
327#106—44no—
328C4⋊C8—44no—
329C4⋊C8—44no—
330#102—28no—
331#102—28no—
332#102—44no—
333#102—44no—
334#95—24no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95C2×D412yes—
349#95C2×D412yes—
350#95C2×D412yes—
351#95C2×D412yes—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D412yes—
355#95C2×D412yes—
356C24—22no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24C22×D411yescharacteristic
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—81no—
368C2×Q8—81no—
369C2×Q8—81no—
370C2×Q8—81no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—24no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—48no—
379C2×D4—48no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4114yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C42−1+414yes—
426C22×C42−1+412yes—
427C22×C4C2511yescharacteristic
428M16—48no—
429M16—48no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—82no—
458C4⋊C4—81no—
459C4⋊C4—82no—
460C4⋊C4—81no—
461C4⋊C4—84no—
462C4⋊C4—82no—
463C4⋊C4—84no—
464C4⋊C4—82no—
465C4⋊C4C2×#4114yes—
466C4⋊C4C2×#4114yes—
467C22⋊C4—28no—
468C22⋊C4—44no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C22⋊C4—82no—
472C22⋊C4—84no—
473C22⋊C4—82no—
474C22⋊C4—84no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C42—22no—
478C42—22no—
479C42—28no—
480C42—24no—
481C42—24no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—24no—
486C42—22no—
487C42—44no—
488C42—82no—
489C42—82no—
490C42—82no—
491C42—82no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—84no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23#52512yes—
511C23#53612yes—
512C23#53712yes—
513C23C23×D411yescharacteristic
514Q8—84no—
515Q8—84no—
516D4—44no—
517D4—44no—
518D4—84no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4#52512yes—
574C2×C4#53612yes—
575C2×C4#53712yes—
576C2×C4#54118yes—
577C2×C4#54218yes—
578C2×C4C22×#4112yes—
579C2×C4C2×H(5,𝔽2)11yescharacteristic
580C2×C4C2×2−1+411yescharacteristic
581C2×C4C2×2−1+412yes—
582C8—44no—
583C8—44no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—168no—
598C22#263714yes—
599C22C2×#52511yescharacteristic
600C22C2×#53611yescharacteristic
601C22C2×#53711yescharacteristic
602C22#358511yescharacteristic
603C22#358511yescharacteristic
604C22#358511yescharacteristic
605C22#358611yescharacteristic
606C4—41no—
607C4—41no—
608C4—44no—
609C4—41no—
610C4—41no—
611C4—44no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4#263714yes—
622C4C2×#54112yes—
623C4C2×#54212yes—
624C2—24no—
625C2—162no—
626C2C2×#263711yescharacteristic
627C2#6056711yescharacteristic
628C2#6056811yescharacteristic
629C2#6086811yescharacteristic
630C2#6087011yescharacteristic
631C2#6091911yescharacteristic
632C2#6092011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34074C2semidirect
3. #30163C2semidirect
4. #30161C2semidirect
5. #30158C2semidirect
6. #30157C2semidirect
7. #23670C2semidirect
8. #23657C2semidirect
11. #23554C2semidirect
14. #21034C2semidirect
15. #20753C2semidirect
30. #2331C22semidirect
31. #2331C22semidirect
32. #2330C22semidirect
33. #2330C22semidirect
50. #2088C22semidirect
51. #2088C22semidirect
52. #2087C22semidirect
53. #2086C22semidirect
54. #2085C22semidirect
55. #2084C22semidirect
56. #2083C22semidirect
64. #2059C22semidirect
65. #2058C22semidirect
66. #2057C22semidirect
74. #1921C22semidirect
75. #1920C22semidirect
76. #1919C22semidirect
77. #1918C22semidirect
78. #1832C22semidirect
79. #1832C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26442
#340742
#301632
#301612
#301582
#301572
#236702
#236572
#236282
#236182
#235542
#234542
#210602
#210342
#207532