Finite Group Databasebeta

← #7174902

Subgroups — #7174902

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#59815C211yesmaximal, characteristic
3#32016C214yesmaximal, complemented
4#31278C214yesmaximal, complemented
5#29255C211yesmaximal, characteristic, complemented
6#28201C211yesmaximal, characteristic, complemented
7#25882C212yesmaximal, complemented
8#25347C212yesmaximal, complemented
9#25012C212yesmaximal, complemented
10#24881C214yesmaximal, complemented
11#24045C214yesmaximal, complemented
12#23815C211yesmaximal, characteristic, complemented
13#22431C212yesmaximal, complemented
14#22264C211yesmaximal, characteristic, complemented
15#22218C211yesmaximal, characteristic, complemented
16C2×#559—21no—
17C2×#550C2212yes—
18C2×#547C2212yes—
19C2×#487C2211yescharacteristic
20C2×#481C2214yes—
21C2×#474C2212yes—
22C2×#436C2212yes—
23C2×#417C2214yes—
24#2724—22no—
25#2682—22no—
26#2582—21no—
27#2547C2212yes—
28#2547C2212yes—
29#2541—21no—
30#2538C2212yes—
31#2537C2212yes—
32#2486C2214yes—
33#2447—22no—
34#2447—22no—
35#2445—22no—
36#2416—22no—
37#2407C2214yes—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2099C2212yes—
47#2096C2212yes—
48#2083C2212yes—
49#2077C2214yes—
50#2076C2214yes—
51#2071C2212yes—
52#2041C2212yes—
53#2041C2212yes—
54#2036C2214yes—
55#2036C2214yes—
56#2034C2214yes—
57#2034C2214yes—
58#2024C2212yes—
59#2023C2212yes—
60#1990C2211yescharacteristic
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1954C2212yes—
64#1948C2212yes—
65#1927C2214yes—
66#1925C2214yes—
67#1895C2214yes—
68#1894C2214yes—
69#1872C2214yes—
70#1870C2214yes—
71#1870C2214yes—
72#1865C2214yes—
73#1865C2214yes—
74#1853C2214yes—
75#1844C2211yescharacteristic
76#1831C2211yescharacteristic
77#1809C2214yes—
78#1802C2212yes—
79#1788C2214yes—
80#1788C2214yes—
81#1788C2214yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84#1744C2211yescharacteristic
85C22×#41C2314yes—
86C22×M16C2314yes—
87#559—48no—
88#550—48no—
89#547—28no—
90#547—44no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127—22no—
95C2×#127D411yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#124—22no—
98C2×#124—22no—
99C2×#123—24no—
100C2×#123—24no—
101C2×C22⋊Q8C2314yes—
102C2×#121—22no—
103C2×#121—22no—
104C2×#121—24no—
105C2×#121C2314yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C22×C4⋊C4—22no—
119C22×C4⋊C4C2314yes—
120#487—24no—
121#481—48no—
122#474—44no—
123#436—44no—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128#417—48no—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#102C2314yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144#398—24no—
145#397—24no—
146#397—24no—
147#397—24no—
148#396—24no—
149#394—24no—
150#394—24no—
151#392—24no—
152#391—22no—
153#391—22no—
154#391—21no—
155#391—21no—
156#391C2312yes—
157#390—22no—
158#390—21no—
159#390—21no—
160#390D412yes—
161#390D412yes—
162#390D412yes—
163#390C2312yes—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389C2312yes—
172#389C2312yes—
173#386—24no—
174#385—24no—
175#385—24no—
176#385—24no—
177#384D411yescharacteristic
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#381—24no—
183#380—22no—
184#380C2314yes—
185#380C2314yes—
186#380C2314yes—
187#380C2314yes—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C43C2311yescharacteristic
195#340C2314yes—
196#340C2312yes—
197#340C2314yes—
198#340C2312yes—
199#340C2312yes—
200#340C2314yes—
201#340C2312yes—
202#336C2312yes—
203#336C2312yes—
204#336C2312yes—
205#336C2312yes—
206C2×#41—28no—
207C2×#41—28no—
208C2×#41—44no—
209C2×#41—44no—
210C2×#41—44no—
211C2×#41—44no—
212C22×Q8—21no—
213C22×Q8—21no—
214C22×Q8C2412yes—
215C22×D4—24no—
216C22×D4—24no—
217C22×D4—21no—
218C22×D4—21no—
219C22×D4C2×D412yes—
220C22×D4C2×D412yes—
221C22×D4C2×D412yes—
222C22×D4C2412yes—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4C2414yes—
227C2×M16—44no—
228C2×M16—44no—
229C2×M16—44no—
230C22×C8C2412yes—
231C22×C8C2414yes—
232C22×C8C2412yes—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—42no—
237C4⋊Q8#4114yes—
238#127—24no—
239#127—24no—
240#127—48no—
241#124—48no—
242#124—48no—
243#123—416no—
244#123—416no—
245C22⋊Q8—28no—
246C22⋊Q8—44no—
247#121—28no—
248#121—44no—
249#121—48no—
250#121—416no—
251#121—84no—
252C4×Q8—44no—
253C4×Q8—44no—
254C4×Q8—44no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—44no—
260C4×D4—44no—
261C4×D4—48no—
262C4×D4—48no—
263C4×D4—48no—
264C4×D4—44no—
265C4×D4—82no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—48no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4C2×D414yes—
305C2×C22⋊C4C2×D414yes—
306C2×C22⋊C4C2×D414yes—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317#107—44no—
318#106—44no—
319#106—44no—
320C4⋊C8—24no—
321#103—28no—
322#103—44no—
323#103—44no—
324#103—44no—
325#102—24no—
326#102—28no—
327#102—24no—
328#102—42no—
329#102—42no—
330#102—44no—
331C22⋊C8—44no—
332C22⋊C8—44no—
333C22⋊C8—44no—
334C4×C8—22no—
335C4×C8—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346C24—22no—
347C24C22×D414yes—
348#41—416no—
349#41—88no—
350#41—88no—
351#41—88no—
352C2×Q8—22no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8C2×#4114yes—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—42no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4H(5,𝔽2)14yes—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D414yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2511yescharacteristic
419M16—88no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—84no—
458C4⋊C4—81no—
459C22⋊C4—28no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—82no—
470C22⋊C4—84no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—41no—
475C42—41no—
476C42—44no—
477C42—44no—
478C42—44no—
479C42AGL(1,ℤ/8)14yes—
480C42C2×#4112yes—
481C42H(5,𝔽2)12yes—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23C23×D411yescharacteristic
497Q8—28no—
498Q8—44no—
499Q8—84no—
500Q8—84no—
501D4—24no—
502D4—24no—
503D4—48no—
504D4—48no—
505D4—48no—
506D4—42no—
507D4—42no—
508D4—84no—
509D4—84no—
510D4—816no—
511D4—168no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—816no—
553C2×C4—88no—
554C2×C4—168no—
555C2×C4#54612yes—
556C2×C4#54612yes—
557C2×C4C2×AGL(1,ℤ/8)12yes—
558C2×C4C2×AGL(1,ℤ/8)12yes—
559C2×C4C2×AGL(1,ℤ/8)12yes—
560C2×C4#57712yes—
561C2×C4C22×#4111yescharacteristic
562C2×C4C2×H(5,𝔽2)12yes—
563C2×C4C2×H(5,𝔽2)11yescharacteristic
564C2×C4#58512yes—
565C8—42no—
566C8—44no—
567C8—42no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22C2×#54611yescharacteristic
583C22#348411yescharacteristic
584C22#348511yescharacteristic
585C22C22×AGL(1,ℤ/8)11yescharacteristic
586C22C2×#57711yescharacteristic
587C22#358611yescharacteristic
588C22#358611yescharacteristic
589C4—22no—
590C4—22no—
591C4—22no—
592C4—44no—
593C4—44no—
594C4—41no—
595C4—41no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4#333214yes—
603C4#351312yes—
604C4#355812yes—
605C2—84no—
606C2—84no—
607C2#4514011yescharacteristic
608C2C2×#333211yescharacteristic
609C2#6122111yescharacteristic
610C2#6158511yescharacteristic
611C2#6168011yescharacteristic
612C2#6171711yescharacteristic
613C2#6180911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59815C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59827C2semidirect
3. #32016C2semidirect
4. #31278C2semidirect
5. #29255C2semidirect
6. #28201C2semidirect
7. #25882C2semidirect
8. #25347C2semidirect
9. #25012C2semidirect
10. #24881C2semidirect
11. #24045C2semidirect
12. #23815C2semidirect
13. #22431C2semidirect
14. #22264C2semidirect
15. #22218C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#598152
#320162
#312782
#292552
#282012
#258822
#253472
#250122
#248812
#240452
#238152
#224312
#222642
#222182