Finite Group Databasebeta

← #6626460

Subgroups — #6626460

Order 512.

1669 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36564C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#33149C211yesmaximal, characteristic
4#33147C212yesmaximal
5#29302C212yesmaximal
6#29244C212yesmaximal
7#27017C214yesmaximal, complemented
8#26875C214yesmaximal, complemented
9#26440C212yesmaximal, complemented
10#25811C211yesmaximal, characteristic
11#25453C212yesmaximal, complemented
12#24301C211yesmaximal, characteristic
13#23163C212yesmaximal, complemented
14#22970C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic
16#21581C212yesmaximal
17C4×#484C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8—21no—
20C2×#498C2212yes—
21C2×#484C2211yescharacteristic
22C2×#475C2212yes—
23#2857C2212yes—
24#2857C2211yescharacteristic
25#2586—22no—
26#2567C2211yescharacteristic
27#2551C2211yescharacteristic
28#2549—22no—
29#2507—21no—
30#2507C2212yes—
31#2451C2212yes—
32#2440—21no—
33#2369—21no—
34#2369C2211yescharacteristic
35#2365—22no—
36#2365—21no—
37#2365—21no—
38#2365C2212yes—
39#2365C2212yes—
40#2365C2214yes—
41Q8×C4⋊C4C2211yescharacteristic
42#2345—21no—
43C4×C22⋊Q8C2212yes—
44C4×C22⋊Q8C2211yescharacteristic
45C2×#391—21no—
46C2×#389C2212yes—
47C2×#387—22no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2086C2214yes—
52#2084C2214yes—
53#2066C2214yes—
54#2058C2214yes—
55#2058C2212yes—
56#2057C2214yes—
57#2024C2212yes—
58#1990C2212yes—
59#1988C2212yes—
60#1988C2214yes—
61#1987C2212yes—
62#1948C2214yes—
63#1947C2212yes—
64#1944C2214yes—
65#1944C2214yes—
66#1939C2212yes—
67#1928C2212yes—
68#1927C2214yes—
69#1921C2212yes—
70#1921C2214yes—
71#1920C2214yes—
72#1919C2214yes—
73#1919C2214yes—
74#1918C2214yes—
75#1917C2214yes—
76#1872C2211yescharacteristic
77#1872C2212yes—
78#1863C2214yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1852C2212yes—
83#1852C2211yescharacteristic
84#1824C2211yescharacteristic
85#1824C2212yes—
86#1797C2212yes—
87#1797C2211yescharacteristic
88#1776C2214yes—
89C4×#107C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2311yescharacteristic
107C2×C4×Q8—22no—
108C2×C4×Q8—24no—
109C2×C4×Q8—22no—
110C2×C4×Q8—48no—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C42—22no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#498—28no—
125#484Q818yescomplemented
126#475—28no—
127C2×#107C2312yes—
128C2×#107C2314yes—
129C2×#107C2312yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2311yescharacteristic
140C2×C22⋊C8C2311yescharacteristic
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143#398—22no—
144#398—22no—
145#398—22no—
146#395—22no—
147#395—22no—
148#395—22no—
149#391—22no—
150#391—24no—
151#391—22no—
152#391—21no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391C2312yes—
157#391C2311yescharacteristic
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2314yes—
169#389C2312yes—
170#389C2311yescharacteristic
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2314yes—
175#387—22no—
176#387—22no—
177#387—21no—
178#387—22no—
179#387—24no—
180#387—24no—
181#387—24no—
182#387—24no—
183#385—22no—
184#385—21no—
185#384—21no—
186#384C2312yes—
187#384C2311yescharacteristic
188#380—21no—
189#380—22no—
190#380C2312yes—
191#380C2311yescharacteristic
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C2×#95—21no—
215C2×#95—22no—
216C2×#95—22no—
217C2×#95D412yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—28no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×Q8C2411yescharacteristic
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2411yescharacteristic
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—28no—
253C4⋊Q8—28no—
254C4⋊Q8—28no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8#4118yes—
260C22⋊Q8—22no—
261C22⋊Q8—84no—
262C22⋊Q8—82no—
263C22⋊Q8C2×Q814yes—
264C4×Q8—24no—
265C4×Q8—44no—
266C4×Q8—48no—
267C4×Q8—416no—
268C4×Q8—48no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—48no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×Q814yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2414yes—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42—48no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335#107—42no—
336#107—44no—
337#107C2×Q818yescomplemented
338#106—28no—
339#106—28no—
340#106—28no—
341C4⋊C8—24no—
342C4⋊C8—44no—
343#103—28no—
344#103—42no—
345#103C2×Q818yescomplemented
346C22⋊C8—24no—
347C22⋊C8—22no—
348C22⋊C8C2×Q814yes—
349#97—44no—
350C4×C8—24no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—21no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24C22×D411yescharacteristic
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—44no—
381C2×Q8—432no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—81no—
388C2×Q8—82no—
389C2×Q8—82no—
390C2×Q8—81no—
391C2×Q8C22×Q812yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4#13714yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×Q812yes—
434C22×C42−1+412yes—
435C22×C4C2511yescharacteristic
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8C22×Q814yes—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—42no—
465C4⋊C4—48no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—81no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4AGL(1,ℤ/8)14yes—
481C4⋊C4C22×Q814yes—
482C4⋊C4C22×Q812yes—
483C4⋊C4C2×#4118yes—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—44no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—24no—
492C42—42no—
493C42—42no—
494C42—44no—
495C42—48no—
496C42—42no—
497C42—48no—
498C42—44no—
499C42—42no—
500C42—82no—
501C42—82no—
502C42C2×#4114yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23D4×Q812yes—
508C23C2×#13712yes—
509C23#57712yes—
510C23C23×D411yescharacteristic
511Q8—24no—
512Q8—416no—
513Q8—416no—
514Q8—42no—
515Q8—164no—
516Q8—162no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—48no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—416no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4#52512yes—
574C2×C4#52512yes—
575C2×C4D4×Q812yes—
576C2×C4C2×AGL(1,ℤ/8)14yes—
577C2×C4C2×AGL(1,ℤ/8)12yes—
578C2×C4C2×#13712yes—
579C2×C4C2×#13712yes—
580C2×C4C23×Q811yescharacteristic
581C2×C4C22×#4112yes—
582C2×C4C2×2−1+411yescharacteristic
583C2×C4#58512yes—
584C8—24no—
585C8—42no—
586C8—44no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22#338714yes—
594C22C2×#52511yescharacteristic
595C22C2×D4×Q811yescharacteristic
596C22#348511yescharacteristic
597C22C22×AGL(1,ℤ/8)11yescharacteristic
598C22C22×#13711yescharacteristic
599C22C2×#57711yescharacteristic
600C22#358611yescharacteristic
601C4—24no—
602C4—22no—
603C4—22no—
604C4—22no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—48no—
609C4—41no—
610C4—42no—
611C4—44no—
612C4—42no—
613C4—82no—
614C4—82no—
615C4—81no—
616C4—81no—
617C4—82no—
618C4#311014yes—
619C4Q8×#4112yes—
620C4#351612yes—
621C2—24no—
622C2#3654711yescharacteristic
623C2C2×#311011yescharacteristic
624C2#6058211yescharacteristic
625C2C2×#338711yescharacteristic
626C2Q8×AGL(1,ℤ/8)11yescharacteristic
627C2#6172311yescharacteristic
628C2#6172811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
7. #27017C2semidirect
8. #26875C2semidirect
9. #26440C2semidirect
11. #25453C2semidirect
13. #23163C2semidirect
14. #22970C2semidirect
125. #484Q8semidirect
337. #107C2×Q8semidirect
345. #103C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365642
C2×#23652
#331492
#331472
#293022
#292442
#270172
#268752
#264402
#258112
#254532
#243012
#231632
#229702
#226292
#215812
C4×#4842