Finite Group Databasebeta

← #7605694

Subgroups — #7605694

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic, complemented
2#59823C211yesmaximal, characteristic, complemented
3#32550C214yesmaximal, complemented
4#29287C211yesmaximal, characteristic, complemented
5#28251C211yesmaximal, characteristic
6#28197C214yesmaximal, complemented
7#27736C212yesmaximal, complemented
8#27735C212yesmaximal, complemented
9#26110C212yesmaximal, complemented
10#26107C212yesmaximal, complemented
11#25321C212yesmaximal, complemented
12#25318C212yesmaximal, complemented
13#23001C212yesmaximal, complemented
14#23001C212yesmaximal, complemented
15#21712C211yesmaximal, characteristic, complemented
16C4×#486C211yesmaximal, characteristic, complemented
17C4×#486C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×#486C2211yescharacteristic
20C2×#486C2211yescharacteristic
21C2×#459C2212yes—
22C2×#459C2212yes—
23C2×#446C2212yes—
24C2×#442C2214yes—
25#2804—22no—
26#2732—22no—
27#2580—22no—
28#2580C2214yes—
29#2571C2212yes—
30#2563—21no—
31#2563C2212yes—
32#2552C2212yes—
33#2542C2212yes—
34#2538—21no—
35#2538C2214yes—
36#2487—22no—
37#2454—22no—
38#2417—22no—
39#2366C2212yes—
40#2366C2212yes—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#127C2211yescharacteristic
49#2101C2214yes—
50#2100C2214yes—
51#2088C2214yes—
52#2088C2214yes—
53#2073C2214yes—
54#2071C2214yes—
55#2054C2214yes—
56#2054C2214yes—
57#1990C2211yescharacteristic
58#1990C2211yescharacteristic
59#1988C2212yes—
60#1987C2212yes—
61#1984C2212yes—
62#1957C2211yescharacteristic
63#1957C2211yescharacteristic
64#1948C2212yes—
65#1948C2212yes—
66#1939C2212yes—
67#1939C2212yes—
68#1936C2214yes—
69#1936C2214yes—
70#1934C2212yes—
71#1934C2212yes—
72#1934C2212yes—
73#1934C2212yes—
74#1879C2214yes—
75#1879C2214yes—
76#1864C2214yes—
77#1864C2214yes—
78#1842C2211yescharacteristic
79#1836C2214yes—
80#1834C2211yescharacteristic
81#1834C2211yescharacteristic
82#1804C2214yes—
83#1802C2214yes—
84#1802C2212yes—
85#1802C2212yes—
86#1770C2212yes—
87#1770C2212yes—
88C4×#102C2212yes—
89C4×#102C2212yes—
90C4×#102C2212yes—
91C4×#102C2212yes—
92C42×C8C2211yescharacteristic
93C22×D8C2314yes—
94#554—48no—
95C2×C4⋊Q8D411yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127C2311yescharacteristic
99C2×#127C2311yescharacteristic
100C2×#125—22no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—22no—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2314yes—
116C2×C4×D4C2314yes—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119#486—24no—
120#486—24no—
121#459—28no—
122#459—28no—
123#446—44no—
124#442—48no—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#102C2312yes—
131C2×#102C2312yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#97C2314yes—
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147#400—24no—
148#399—24no—
149#399—24no—
150#398—24no—
151#396—24no—
152#396—24no—
153#393—24no—
154#393—24no—
155#391—21no—
156#391—21no—
157#391D412yes—
158#390—22no—
159#390—22no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2312yes—
166#390C2314yes—
167#390C2314yes—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389D412yes—
173#389D412yes—
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#389C2314yes—
178#386—24no—
179#386—24no—
180#385—24no—
181#385—24no—
182#385—24no—
183#385—24no—
184#384D411yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2314yes—
190#382—22no—
191#382C2314yes—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—24no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C43C2311yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2312yes—
207#340C2314yes—
208#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#336C2314yes—
217#336C2314yes—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8C2×D412yes—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2412yes—
226C22×D4C2412yes—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—24no—
231C2×D8—28no—
232C2×D8—28no—
233C2×D8—44no—
234C22×C8C2414yes—
235C22×C8C2411yescharacteristic
236C22×C8C2411yescharacteristic
237C22×C8C2411yescharacteristic
238C22×C8C2411yescharacteristic
239C4⋊Q8—24no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244#127—22no—
245#127—22no—
246#127#4114yes—
247#127#4114yes—
248#125—48no—
249C22⋊Q8—48no—
250C22⋊Q8—416no—
251C22⋊Q8—84no—
252C4×Q8—42no—
253C4×Q8—42no—
254C4×Q8—44no—
255C4×Q8—82no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—44no—
261C4×D4—44no—
262C4×D4—44no—
263C4×D4—44no—
264C4×D4—48no—
265C4×D4—44no—
266C4×D4—44no—
267C4×D4—82no—
268C4×D4—82no—
269C4×D4—84no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—84no—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2414yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322#106—44no—
323C4⋊C8—24no—
324C4⋊C8—24no—
325C4⋊C8—42no—
326C4⋊C8—44no—
327#102—28no—
328#102—28no—
329#102—24no—
330#102—24no—
331#102—24no—
332#102—24no—
333#102—24no—
334#102—24no—
335#102—42no—
336#102—42no—
337#97—28no—
338C4×C8—22no—
339C4×C8—22no—
340C4×C8—22no—
341C4×C8—22no—
342C4×C8—22no—
343C4×C8—41no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95C2×D414yes—
353#95C2×D414yes—
354C24—22no—
355C24—22no—
356C2×Q8—24no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—81no—
364C2×Q8—81no—
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—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4—81no—
387C2×D4C2×#4114yes—
388C2×D4C2×#4114yes—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—48no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4C22×D414yes—
408C22×C4C22×D414yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2511yescharacteristic
416D8—416no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—21no—
426C2×C8—24no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8H(5,𝔽2)12yes—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—84no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—82no—
465C4⋊C4—84no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C22⋊C4—84no—
474C22⋊C4—82no—
475C22⋊C4—84no—
476C22⋊C4—82no—
477C22⋊C4—84no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—24no—
482C42—21no—
483C42—24no—
484C42—41no—
485C42—44no—
486C42—44no—
487C42AGL(1,ℤ/8)14yes—
488C42C2×#4112yes—
489C42C2×#4112yes—
490C422−1+412yes—
491C422−1+412yes—
492C23—24no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23C23×D411yescharacteristic
507Q8—84no—
508Q8—84no—
509Q8—84no—
510D4—24no—
511D4—24no—
512D4—24no—
513D4—24no—
514D4—48no—
515D4—48no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—84no—
521D4—84no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—21no—
532C2×C4—24no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—21no—
536C2×C4—24no—
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—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—88no—
556C2×C4—88no—
557C2×C4—168no—
558C2×C4—168no—
559C2×C4#54812yes—
560C2×C4#54812yes—
561C2×C4#54812yes—
562C2×C4#54812yes—
563C2×C4C2×AGL(1,ℤ/8)12yes—
564C2×C4C2×AGL(1,ℤ/8)12yes—
565C2×C4C2×AGL(1,ℤ/8)12yes—
566C2×C4#57612yes—
567C2×C4#57712yes—
568C2×C4C22×#4111yescharacteristic
569C2×C4C22×#4111yescharacteristic
570C2×C4C2×H(5,𝔽2)11yescharacteristic
571C2×C4C2×2−1+411yescharacteristic
572C2×C4C2×2−1+411yescharacteristic
573C2×C4#58511yescharacteristic
574C8—22no—
575C8—41no—
576C8—44no—
577C8—41no—
578C8—41no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22C2×#54811yescharacteristic
594C22C2×#54811yescharacteristic
595C22#349611yescharacteristic
596C22C22×AGL(1,ℤ/8)11yescharacteristic
597C22C2×#57611yescharacteristic
598C22C2×#57711yescharacteristic
599C22#358411yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—44no—
604C4—41no—
605C4—44no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4#336614yes—
613C4#337314yes—
614C4#348111yescharacteristic
615C4#351511yescharacteristic
616C4#351511yescharacteristic
617C4#354411yescharacteristic
618C2—84no—
619C2—84no—
620C2#5966011yescharacteristic
621C2C2×#336611yescharacteristic
622C2C2×#337311yescharacteristic
623C2#6132511yescharacteristic
624C2#6132611yescharacteristic
625C2#6225911yescharacteristic
626C2#6234211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28251C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59864C2semidirect
2. #59823C2semidirect
3. #32550C2semidirect
4. #29287C2semidirect
6. #28197C2semidirect
7. #27736C2semidirect
8. #27735C2semidirect
9. #26110C2semidirect
10. #26107C2semidirect
11. #25321C2semidirect
12. #25318C2semidirect
13. #23001C2semidirect
14. #23001C2semidirect
15. #21712C2semidirect
16. C4×#486C2semidirect
17. C4×#486C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#598232
#325502
#292872
#282512
#281972
#277362
#277352
#261102
#261072
#253212
#253182
#230012
#217122
C4×#4862