Finite Group Databasebeta

← #6502459

Subgroups — #6502459

Order 512.

1884 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2453C211yesmaximal, characteristic, complemented
2#34206C211yesmaximal, characteristic
3#28118C212yesmaximal, complemented
4#27613C211yesmaximal, characteristic, complemented
5#27586C211yesmaximal, characteristic, complemented
6#27429C214yesmaximal, complemented
7#27001C214yesmaximal, complemented
8#26896C214yesmaximal, complemented
9#26374C214yesmaximal, complemented
10#24305C212yesmaximal, complemented
11#24060C211yesmaximal, characteristic
12#23931C211yesmaximal, characteristic
13#22075C212yesmaximal, complemented
14#21912C211yesmaximal, characteristic, complemented
15C4×#448C212yesmaximal, complemented
16C2×#532C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C2×#520—21no—
19C22×C4×Q8C2211yescharacteristic
20C2×#480C2212yes—
21C2×#478C2214yes—
22C2×#448C2212yes—
23C2×C4×SD16C2214yes—
24#2453—22no—
25#2453—22no—
26#2453C2214yes—
27#2453C2214yes—
28C2×#400—21no—
29C2×#398—21no—
30#2367—22no—
31#2366C2212yes—
32#2365C2212yes—
33#2354C2212yes—
34#2352—22no—
35D4×C4⋊C4C2212yes—
36#2325—21no—
37#2325—21no—
38#2311—21no—
39C4×#121C2212yes—
40C4×#121C2212yes—
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#389—21no—
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#387—21no—
47#2285C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2100C2212yes—
50#2095C2212yes—
51#2084C2212yes—
52#2072C2212yes—
53#2068C2214yes—
54#2059C2214yes—
55#2057C2212yes—
56#2055C2214yes—
57#2041C2212yes—
58#2025C2214yes—
59#2024C2212yes—
60#2014C2212yes—
61#1989C2214yes—
62#1942C2214yes—
63#1934C2214yes—
64#1928C2214yes—
65#1928C2214yes—
66#1919C2214yes—
67#1918C2214yes—
68#1916C2214yes—
69#1916C2214yes—
70#1880C2214yes—
71#1877C2214yes—
72#1877C2214yes—
73#1874C2212yes—
74#1874C2212yes—
75#1865C2214yes—
76#1862C2214yes—
77#1862C2214yes—
78#1853C2212yes—
79#1852C2212yes—
80#1823C2212yes—
81#1817C2211yescharacteristic
82#1817C2211yescharacteristic
83#1797C2212yes—
84#1787C2212yes—
85C4×#103C2214yes—
86C4×C22⋊C8C2212yes—
87C22×#41—21no—
88C23×Q8C2311yescharacteristic
89C22×SD16C2314yes—
90#532—28no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#125—22no—
94C2×#125—44no—
95C2×#125C2314yes—
96C2×#124—21no—
97C2×#121—21no—
98C2×#121C2312yes—
99#520—48no—
100C2×C4×Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—44no—
103C2×C4×Q8C2314yes—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×#117—21no—
112C2×#117—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#480—28no—
129#478—48no—
130#448—28no—
131C4×SD16—48no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C4×C8C2314yes—
145#400—24no—
146#400—24no—
147#398—24no—
148#398—24no—
149#391—22no—
150#391—22no—
151#391—21no—
152#391—21no—
153#391C2314yes—
154#391C2312yes—
155#391C2312yes—
156#390—22no—
157#390C2312yes—
158#390C2312yes—
159#389—24no—
160#389—24no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—22no—
175#387—24no—
176#387—24no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#385—22no—
182#385—22no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#382—21no—
186#380—21no—
187#380—22no—
188#380—22no—
189#380C2312yes—
190C4×C4⋊C4—21no—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
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#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C2×#41—28no—
221C2×#41—44no—
222C2×#41—44no—
223C22×Q8—24no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—24no—
227C22×Q8—42no—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×SD16—28no—
248C2×SD16—28no—
249C2×SD16—44no—
250C22×C8C2414yes—
251C22×C8C2414yes—
252C4⋊Q8—24no—
253#127—24no—
254#125—28no—
255#125—44no—
256#125—48no—
257#124—28no—
258#121—28no—
259#121—24no—
260#121#4118yes—
261C4×Q8—28no—
262C4×Q8—44no—
263C4×Q8—48no—
264C4×Q8—48no—
265C4×D4—28no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—44no—
269C4×D4—44no—
270C4×D4—48no—
271C4×D4—48no—
272C4×D4—48no—
273#117—48no—
274#117—44no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—48no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—44no—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2412yes—
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337#107—28no—
338#107—44no—
339#106—28no—
340C4⋊C8—44no—
341#103—28no—
342#103—28no—
343#103—44no—
344#102—28no—
345#102—28no—
346#102—44no—
347C22⋊C8—24no—
348C22⋊C8—24no—
349C4×C8—44no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24—22no—
370C24C22×D411yescharacteristic
371#41—432no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—416no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q82−1+414yes—
383C2×D4—24no—
384C2×D4—28no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—42no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—42no—
393C2×D4—81no—
394C2×D4—81no—
395C2×D4C2×#4114yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4#13514yes—
443C22×C4AGL(1,ℤ/8)14yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C2×#4114yes—
454C22×C4C2511yescharacteristic
455SD16—416no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4C2×#4114yes—
489C4⋊C4H(5,𝔽2)14yes—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—24no—
504C42—24no—
505C42—42no—
506C42—42no—
507C42—48no—
508C42—44no—
509C42—44no—
510C42—44no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—24no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23#52512yes—
523C23C2×#13512yes—
524C23C2×AGL(1,ℤ/8)12yes—
525C23C23×D411yescharacteristic
526Q8—28no—
527Q8—48no—
528Q8—48no—
529Q8—44no—
530D4—28no—
531D4—48no—
532D4—416no—
533D4—44no—
534D4—84no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—44no—
564C2×C4—416no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—164no—
592C2×C4—164no—
593C2×C4#52512yes—
594C2×C4C2×#13512yes—
595C2×C4C2×#13512yes—
596C2×C4C2×AGL(1,ℤ/8)14yes—
597C2×C4C22×#4112yes—
598C2×C4C2×H(5,𝔽2)12yes—
599C2×C4C2×2−1+412yes—
600C8—44no—
601C8—44no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22#310514yes—
616C22C2×#52511yescharacteristic
617C22#348411yescharacteristic
618C22#348611yescharacteristic
619C22C22×#13511yescharacteristic
620C22C22×AGL(1,ℤ/8)11yescharacteristic
621C22#358611yescharacteristic
622C22#358611yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—24no—
627C4—44no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4#352014yes—
640C2—24no—
641C2—84no—
642C2#3535711yescharacteristic
643C2C2×#310511yescharacteristic
644C2#6056911yescharacteristic
645C2#6156511yescharacteristic
646C2#6157211yescharacteristic
647C2#6182211yescharacteristic
648C2#6182811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
1. C2×#2453C2semidirect
3. #28118C2semidirect
4. #27613C2semidirect
5. #27586C2semidirect
6. #27429C2semidirect
7. #27001C2semidirect
8. #26896C2semidirect
9. #26374C2semidirect
10. #24305C2semidirect
13. #22075C2semidirect
14. #21912C2semidirect
15. C4×#448C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24532
#342062
#281182
#276132
#275862
#274292
#270012
#268962
#263742
#243052
#240602
#239312
#220752
#219122
C4×#4482