Finite Group Databasebeta

← #6644703

Subgroups — #6644703

Order 512.

1640 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
1#35006C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3#33005C212yesmaximal, complemented
4#32453C214yesmaximal, complemented
5#32367C214yesmaximal, complemented
6#31661C214yesmaximal, complemented
7#31543C214yesmaximal, complemented
8#29638C212yesmaximal, complemented
9#29111C211yesmaximal, characteristic, complemented
10#28314C211yesmaximal, characteristic
11#28118C212yesmaximal, complemented
12#27969C211yesmaximal, characteristic, complemented
13#27918C211yesmaximal, characteristic
14#27651C212yesmaximal, complemented
15#21916C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#481C2212yes—
20C2×#480C2212yes—
21C2×#480C2212yes—
22C2×#448C2212yes—
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2854C2212yes—
26#2639—21no—
27#2618—21no—
28#2568—22no—
29#2568—22no—
30#2568C2214yes—
31#2568C2214yes—
32#2560—22no—
33#2558—22no—
34#2512C2212yes—
35#2506C2212yes—
36#2452C2212yes—
37#2445C2212yes—
38C2×#400—21no—
39C2×#400—21no—
40#2325—21no—
41C4×#121C2212yes—
42C2×#391—21no—
43C2×#391—21no—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2101C2214yes—
50#2095C2212yes—
51#2092C2214yes—
52#2085C2214yes—
53#2073C2214yes—
54#2072C2212yes—
55#2068C2214yes—
56#2057C2214yes—
57#2057C2214yes—
58#2055C2214yes—
59#2024C2212yes—
60#2023C2214yes—
61#2023C2214yes—
62#2014C2212yes—
63#1993C2212yes—
64#1973C2214yes—
65#1961C2212yes—
66#1949C2214yes—
67#1935C2214yes—
68#1928C2214yes—
69#1917C2214yes—
70#1917C2214yes—
71#1880C2214yes—
72#1880C2214yes—
73#1877C2214yes—
74#1865C2214yes—
75#1865C2214yes—
76#1847C2212yes—
77#1836C2212yes—
78#1824C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1804C2214yes—
83#1797C2212yes—
84#1787C2212yes—
85#1749C2211yescharacteristic
86#1749C2211yescharacteristic
87C23×Q8—21no—
88C23×Q8C2311yescharacteristic
89C22×SD16C2314yes—
90#539—48no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2314yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#126—22no—
97C2×#125—22no—
98C2×#123—22no—
99C2×#123—22no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×Q8—22no—
106C2×C4×Q8—44no—
107C2×C4×Q8C2314yes—
108C2×C4×D4—22no—
109C2×C4×D4C2312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#481—44no—
122#480—44no—
123#480—44no—
124#448—44no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2312yes—
138C2×#97C2312yes—
139#400—24no—
140#400—24no—
141#400—24no—
142#400—24no—
143#400—22no—
144#398—22no—
145#397—22no—
146#396—22no—
147#396—22no—
148#394—22no—
149#394—22no—
150#393—22no—
151#391—24no—
152#391—24no—
153#391—24no—
154#391—24no—
155#391—21no—
156#391—22no—
157#391—21no—
158#391—21no—
159#391C2314yes—
160#391C2312yes—
161#391C2312yes—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#389—21no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#387—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#385—22no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#382—22no—
185#382—21no—
186#382—21no—
187#382C2314yes—
188#382C2312yes—
189#382C2312yes—
190#380—21no—
191#380C2312yes—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4D412yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C22⋊C4—22no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C22×Q8—24no—
220C22×Q8—21no—
221C22×Q8—24no—
222C22×Q8—21no—
223C22×Q8—24no—
224C22×Q8—42no—
225C22×Q8—44no—
226C22×Q8—44no—
227C22×Q8—44no—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×SD16—28no—
246C2×SD16—44no—
247C2×SD16—44no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—48no—
251C4⋊Q8—42no—
252C4⋊Q8—48no—
253#127—42no—
254#126—48no—
255#125—48no—
256#123—48no—
257#123—48no—
258C22⋊Q8—48no—
259C22⋊Q8—44no—
260#121—24no—
261#121—24no—
262#121—44no—
263#121—44no—
264C4×Q8—48no—
265C4×Q8—48no—
266C4×D4—44no—
267C4×D4—48no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
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—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—44no—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2412yes—
321C2×C42C2412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324#107—44no—
325#107—44no—
326#106—44no—
327C4⋊C8—44no—
328#103—28no—
329#103—44no—
330#103—44no—
331#102—44no—
332#102—44no—
333#102—44no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336#97—42no—
337#97—42no—
338#95—24no—
339#95—22no—
340#95—24no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—22no—
346#95—24no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95C2×D412yes—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360C24—22no—
361C24C22×D411yescharacteristic
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—416no—
367C2×Q8—42no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—48no—
373C2×Q8—81no—
374C2×Q8—81no—
375C2×Q8—816no—
376C2×Q82−1+414yes—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—81no—
387C2×D4—81no—
388C2×D4—81no—
389C2×D4—81no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4AGL(1,ℤ/8)14yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4H(5,𝔽2)14yes—
442C22×C4C2511yescharacteristic
443SD16—88no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C42—24no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—42no—
492C42—41no—
493C42—41no—
494C42—48no—
495C42—44no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23#53512yes—
508C23C2×AGL(1,ℤ/8)12yes—
509C23#57512yes—
510C23C23×D411yescharacteristic
511Q8—28no—
512Q8—48no—
513Q8—44no—
514Q8—816no—
515Q8—162no—
516D4—44no—
517D4—44no—
518D4—88no—
519D4—162no—
520D4—162no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—44no—
550C2×C4—48no—
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—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—164no—
579C2×C4#53512yes—
580C2×C4C2×AGL(1,ℤ/8)14yes—
581C2×C4C2×H(5,𝔽2)12yes—
582C2×C4C2×2−1+412yes—
583C2×C4#58512yes—
584C8—44no—
585C8—44no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22#321314yes—
600C22C2×#53511yescharacteristic
601C22#348511yescharacteristic
602C22#348611yescharacteristic
603C22C22×AGL(1,ℤ/8)11yescharacteristic
604C22C2×#57511yescharacteristic
605C22#358611yescharacteristic
606C22#358611yescharacteristic
607C4—24no—
608C4—22no—
609C4—42no—
610C4—41no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—44no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4#353514yes—
624C2—24no—
625C2—162no—
626C2#3711811yescharacteristic
627C2C2×#321311yescharacteristic
628C2#6080711yescharacteristic
629C2#6171211yescharacteristic
630C2#6172211yescharacteristic
631C2#6182811yescharacteristic
632C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2568C2semidirect
3. #33005C2semidirect
4. #32453C2semidirect
5. #32367C2semidirect
6. #31661C2semidirect
7. #31543C2semidirect
8. #29638C2semidirect
9. #29111C2semidirect
11. #28118C2semidirect
12. #27969C2semidirect
14. #27651C2semidirect
15. #21916C2semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#25682
#330052
#324532
#323672
#316612
#315432
#296382
#291112
#283142
#281182
#279692
#279182
#276512
#219162