Finite Group Databasebeta

← #6502272

Subgroups — #6502272

Order 512.

1924 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2446C211yesmaximal, characteristic, complemented
2#34206C211yesmaximal, characteristic
3#29235C212yesmaximal, complemented
4#27583C211yesmaximal, characteristic, complemented
5#27583C211yesmaximal, characteristic, complemented
6#27192C214yesmaximal, complemented
7#26690C214yesmaximal, complemented
8#26586C214yesmaximal, complemented
9#26166C214yesmaximal, complemented
10#24209C212yesmaximal, complemented
11#23931C211yesmaximal, characteristic
12#23931C211yesmaximal, characteristic
13#22041C212yesmaximal, complemented
14#21912C211yesmaximal, characteristic, complemented
15C4×#448C212yesmaximal, complemented
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#520—21no—
19C22×C4×Q8C2211yescharacteristic
20C2×#461C2214yes—
21C2×#448C2212yes—
22C2×#448C2212yes—
23C2×C4×SD16C2214yes—
24#2446—22no—
25#2446—22no—
26#2446C2214yes—
27#2446C2214yes—
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×#391C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#387—21no—
46C2×#387—21no—
47#2285C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2096C2212yes—
50#2096C2212yes—
51#2088C2214yes—
52#2072C2212yes—
53#2072C2212yes—
54#2065C2214yes—
55#2043C2214yes—
56#2024C2212yes—
57#2024C2212yes—
58#2014C2212yes—
59#2014C2212yes—
60#2013C2214yes—
61#1981C2214yes—
62#1968C2214yes—
63#1933C2214yes—
64#1927C2214yes—
65#1927C2214yes—
66#1927C2214yes—
67#1920C2214yes—
68#1916C2214yes—
69#1916C2214yes—
70#1880C2214yes—
71#1877C2214yes—
72#1877C2214yes—
73#1877C2214yes—
74#1874C2212yes—
75#1874C2212yes—
76#1865C2214yes—
77#1864C2214yes—
78#1853C2212yes—
79#1853C2212yes—
80#1817C2211yescharacteristic
81#1817C2211yescharacteristic
82#1797C2212yes—
83#1787C2212yes—
84#1787C2212yes—
85C4×#103C2214yes—
86C4×C22⋊C8C2212yes—
87C22×#41—21no—
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C22×SD16C2314yes—
91C22×SD16C2314yes—
92#532—28no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8C2314yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127C2311yescharacteristic
98C2×#124—21no—
99C2×#121—21no—
100C2×#121C2312yes—
101#520—48no—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—44no—
105C2×C4×Q8C2314yes—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117—21no—
114C2×#117—22no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#461—48no—
129#448—28no—
130#448—28no—
131C4×SD16—48no—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C4×C8C2314yes—
146#400—24no—
147#400—24no—
148#398—24no—
149#398—24no—
150#391—22no—
151#391—22no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391C2312yes—
157#391C2312yes—
158#391C2314yes—
159#391C2312yes—
160#391C2312yes—
161#390—22no—
162#390C2312yes—
163#390C2312yes—
164#389—22no—
165#389C2314yes—
166#389C2312yes—
167#389C2312yes—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#387—24no—
173#387—24no—
174#387—22no—
175#387—22no—
176#387—24no—
177#387—24no—
178#386—22no—
179#386—22no—
180#386—22no—
181#386—22no—
182#385—22no—
183#385—22no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#382—21no—
187#380—21no—
188#380—22no—
189#380—22no—
190#380C2312yes—
191C4×C4⋊C4—21no—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4D412yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
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—21no—
227C22×Q8—24no—
228C22×Q8—24no—
229C22×Q8—42no—
230C22×Q8—42no—
231C22×Q8—42no—
232C22×Q8C2412yes—
233C22×Q8C2412yes—
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×SD16—28no—
253C2×SD16—28no—
254C2×SD16—28no—
255C2×SD16—44no—
256C2×SD16—44no—
257C2×SD16—44no—
258C22×C8C2414yes—
259C22×C8C2414yes—
260C4⋊Q8—24no—
261C4⋊Q8—28no—
262C4⋊Q8—44no—
263C4⋊Q8—48no—
264#127—24no—
265#124—28no—
266#121—28no—
267#121—24no—
268#121#4118yes—
269C4×Q8—28no—
270C4×Q8—44no—
271C4×Q8—48no—
272C4×Q8—48no—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—44no—
277C4×D4—44no—
278C4×D4—48no—
279C4×D4—48no—
280C4×D4—48no—
281#117—44no—
282#117—48no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4SD1618yescomplemented
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
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×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—44no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339#106—44no—
340#106—44no—
341C4⋊C8—44no—
342#103—28no—
343#103—28no—
344#103—28no—
345#103—44no—
346#103—44no—
347#103—44no—
348#102—44no—
349#102—44no—
350C22⋊C8—24no—
351C22⋊C8—24no—
352C4×C8—44no—
353#95—24no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—22no—
375C24C22×D411yescharacteristic
376#41—432no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—416no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—88no—
395C2×Q8H(5,𝔽2)14yes—
396C2×D4—24no—
397C2×D4—28no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4C2×#4114yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4C2×SD1614yes—
455C22×C4C2×SD1614yes—
456C22×C4C2×SD1614yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4114yes—
467C22×C4C2511yescharacteristic
468SD16—416no—
469SD16—88no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4C2×SD1618yes—
497C4⋊C4C2×SD1618yes—
498C4⋊C4C2×#4114yes—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—24no—
514C42—42no—
515C42—42no—
516C42—48no—
517C42—44no—
518C42—44no—
519C42—44no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23#52512yes—
532C23C22×SD1612yes—
533C23#57512yes—
534C23C23×D411yescharacteristic
535Q8—28no—
536Q8—48no—
537Q8—48no—
538Q8—44no—
539Q8—44no—
540Q8—44no—
541Q8—84no—
542D4—28no—
543D4—48no—
544D4—416no—
545D4—44no—
546D4—84no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—416no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—164no—
603C2×C4—164no—
604C2×C4#52512yes—
605C2×C4C22×SD1612yes—
606C2×C4C22×SD1614yes—
607C2×C4C22×SD1614yes—
608C2×C4C22×SD1612yes—
609C2×C4C22×#4112yes—
610C2×C4C2×H(5,𝔽2)12yes—
611C2×C4#58512yes—
612C8—44no—
613C8—44no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22#310114yes—
628C22C2×#52511yescharacteristic
629C22#348411yescharacteristic
630C22#348711yescharacteristic
631C22C23×SD1611yescharacteristic
632C22C2×#57511yescharacteristic
633C22#358611yescharacteristic
634C22#358611yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—24no—
639C4—24no—
640C4—44no—
641C4—44no—
642C4—42no—
643C4—42no—
644C4—41no—
645C4—41no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4#351314yes—
652C2—24no—
653C2—84no—
654C2#3535611yescharacteristic
655C2C2×#310111yescharacteristic
656C2#6056911yescharacteristic
657C2#6156211yescharacteristic
658C2#6157811yescharacteristic
659C2#6188311yescharacteristic
660C2#6189411yescharacteristic
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×#2446C2semidirect
3. #29235C2semidirect
4. #27583C2semidirect
5. #27583C2semidirect
6. #27192C2semidirect
7. #26690C2semidirect
8. #26586C2semidirect
9. #26166C2semidirect
10. #24209C2semidirect
13. #22041C2semidirect
14. #21912C2semidirect
15. C4×#448C2semidirect
304. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#24462
#342062
#292352
#275832
#271922
#266902
#265862
#261662
#242092
#239312
#220412
#219122
C4×#4482