Finite Group Databasebeta

← #6478011

Subgroups — #6478011

Order 512.

1768 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33146C211yesmaximal, characteristic, complemented
4#33106C211yesmaximal, characteristic
5#33094C212yesmaximal, complemented
6#29966C211yesmaximal, characteristic, complemented
7#29754C211yesmaximal, characteristic
8#29638C212yesmaximal, complemented
9#27126C214yesmaximal, complemented
10#26998C214yesmaximal, complemented
11#26856C214yesmaximal, complemented
12#26688C214yesmaximal, complemented
13#22075C212yesmaximal, complemented
14#21912C211yesmaximal, characteristic, complemented
15C4×#480C212yesmaximal, complemented
16C2×#539—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8—21no—
19C2×#480C2212yes—
20C2×#480C2212yes—
21#2857C2211yescharacteristic
22#2856C2211yescharacteristic
23#2854C2212yes—
24#2639—21no—
25#2618—21no—
26#2560—22no—
27#2558—22no—
28#2512C2212yes—
29#2506C2212yes—
30#2452C2212yes—
31#2445C2212yes—
32#2365—22no—
33#2365—22no—
34#2365C2214yes—
35#2365C2214yes—
36#2325—21no—
37C4×#121C2212yes—
38C2×#391—21no—
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#387—21no—
42C2×#387—21no—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47#2099C2212yes—
48#2086C2214yes—
49#2083C2212yes—
50#2067C2214yes—
51#2057C2212yes—
52#2057C2214yes—
53#2057C2214yes—
54#2041C2212yes—
55#2023C2214yes—
56#2023C2214yes—
57#1988C2212yes—
58#1982C2212yes—
59#1947C2212yes—
60#1947C2212yes—
61#1942C2214yes—
62#1934C2214yes—
63#1925C2214yes—
64#1925C2214yes—
65#1925C2214yes—
66#1925C2214yes—
67#1918C2214yes—
68#1918C2214yes—
69#1918C2214yes—
70#1918C2214yes—
71#1918C2214yes—
72#1916C2214yes—
73#1916C2214yes—
74#1874C2212yes—
75#1874C2212yes—
76#1862C2214yes—
77#1852C2212yes—
78#1852C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84C4×#107C2214yes—
85C4×#102C2214yes—
86C4×C22⋊C8C2212yes—
87C23×Q8—21no—
88#539—48no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8C2314yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#126—22no—
96C2×#125—22no—
97C2×#123—22no—
98C2×#123—22no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—22no—
101C2×#121—21no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8—48no—
107C2×C4×D4—22no—
108C2×C4×D4C2312yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C42—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#480—28no—
125#480—28no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C4×C8C2314yes—
139#400—22no—
140#398—22no—
141#397—22no—
142#396—22no—
143#396—22no—
144#394—22no—
145#394—22no—
146#393—22no—
147#391—24no—
148#391—24no—
149#391—21no—
150#390—21no—
151#390—21no—
152#390C2312yes—
153#390C2312yes—
154#390C2312yes—
155#390C2312yes—
156#389—21no—
157#389—22no—
158#389—21no—
159#389—21no—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2314yes—
166#389C2312yes—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#387—22no—
172#387—24no—
173#387—24no—
174#387—24no—
175#387—24no—
176#386—22no—
177#386—22no—
178#386—22no—
179#385—22no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#380—21no—
183#380C2312yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4D412yes—
190C4×C4⋊C4C2314yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2312yes—
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×#95D411yescharacteristic
204C2×#95D411yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2312yes—
214#336C2312yes—
215#336C2314yes—
216#336C2314yes—
217#336C2314yes—
218C22×Q8—28no—
219C22×Q8—24no—
220C22×Q8—21no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C2412yes—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—216no—
242C4⋊Q8—28no—
243C4⋊Q8—28no—
244C4⋊Q8—42no—
245#127—42no—
246#126—48no—
247#125—48no—
248#123—48no—
249#123—48no—
250C22⋊Q8—44no—
251C22⋊Q8—48no—
252#121—24no—
253#121—24no—
254#121—44no—
255#121—44no—
256C4×Q8—416no—
257C4×Q8—416no—
258C4×D4—44no—
259C4×D4—48no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4D818yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4#4114yes—
291C2×C4⋊C4#4114yes—
292C2×C4⋊C4C2412yes—
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—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330#107—28no—
331#107—28no—
332#107—28no—
333#107—44no—
334C4⋊C8—42no—
335C4⋊C8—42no—
336#102—28no—
337#102—28no—
338#102—28no—
339#102—44no—
340C22⋊C8—24no—
341C22⋊C8—24no—
342C4×C8—44no—
343#95—24no—
344#95—24no—
345#95—22no—
346#95—22no—
347#95—24no—
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—216no—
363C2×Q8—432no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—42no—
367C2×Q8—81no—
368C2×Q8—81no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—42no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4—81no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4—81no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4C2×D814yes—
422C22×C4C2×D814yes—
423C22×C4C2×D814yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2511yescharacteristic
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C4⋊C4—24no—
443C4⋊C4—216no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4C2×D818yes—
470C4⋊C4C2×D818yes—
471C4⋊C4C2×#4114yes—
472C4⋊C4C2×#4114yes—
473C4⋊C42−1+414yes—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C42—24no—
486C42—22no—
487C42—22no—
488C42—24no—
489C42—22no—
490C42—22no—
491C42—42no—
492C42—41no—
493C42—41no—
494C42—48no—
495C42—44no—
496C42—44no—
497C42—48no—
498C42—44no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23#52512yes—
511C23C22×D812yes—
512C23#57512yes—
513C23C23×D411yescharacteristic
514Q8—432no—
515Q8—162no—
516D4—44no—
517D4—44no—
518D4—88no—
519D4—162no—
520D4—162no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—48no—
555C2×C4—48no—
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—42no—
572C2×C4—42no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—84no—
576C2×C4—164no—
577C2×C4#52512yes—
578C2×C4C22×D814yes—
579C2×C4C22×D812yes—
580C2×C4C22×D814yes—
581C2×C4C22×D812yes—
582C2×C4C22×#4111yescharacteristic
583C2×C4C22×#4111yescharacteristic
584C2×C4C2×2−1+412yes—
585C2×C4#58512yes—
586C8—44no—
587C8—44no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22#310014yes—
602C22C2×#52511yescharacteristic
603C22#348511yescharacteristic
604C22#348611yescharacteristic
605C22C23×D811yescharacteristic
606C22C2×#57511yescharacteristic
607C22#358611yescharacteristic
608C22#358611yescharacteristic
609C4—22no—
610C4—24no—
611C4—24no—
612C4—22no—
613C4—22no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—42no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4#346112yes—
626C4#350712yes—
627C2—24no—
628C2—162no—
629C2#3509211yescharacteristic
630C2C2×#310011yescharacteristic
631C2#6056611yescharacteristic
632C2#6169411yescharacteristic
633C2#6171211yescharacteristic
634C2#6181711yescharacteristic
635C2#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×#2365C2semidirect
3. #33146C2semidirect
5. #33094C2semidirect
6. #29966C2semidirect
8. #29638C2semidirect
9. #27126C2semidirect
10. #26998C2semidirect
11. #26856C2semidirect
12. #26688C2semidirect
13. #22075C2semidirect
14. #21912C2semidirect
15. C4×#480C2semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#23652
#331462
#331062
#330942
#299662
#297542
#296382
#271262
#269982
#268562
#266882
#220752
#219122
C4×#4802