Finite Group Databasebeta

← #6571134

Subgroups — #6571134

Order 512.

1652 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36473C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#30129C212yesmaximal, complemented
4#29937C211yesmaximal, characteristic, complemented
5#29494C212yesmaximal, complemented
6#28205C212yesmaximal, complemented
7#27522C214yesmaximal, complemented
8#27226C214yesmaximal
9#26462C212yesmaximal, complemented
10#25856C211yesmaximal, characteristic, complemented
11#25320C212yesmaximal, complemented
12#24209C211yesmaximal, characteristic
13#23816C212yesmaximal, complemented
14#23755C211yesmaximal, characteristic
15#23214C212yesmaximal, complemented
16#23094C212yesmaximal, complemented
17#22632C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#487C2212yes—
21C2×#475C2214yes—
22C2×#461C2212yes—
23C2×#448C2212yes—
24#2860C2212yes—
25#2857C2211yescharacteristic
26#2604—22no—
27#2551C2211yescharacteristic
28#2542C2211yescharacteristic
29#2521—22no—
30#2496C2212yes—
31#2492—21no—
32#2491C2212yes—
33#2415—21no—
34#2369—21no—
35#2369—21no—
36#2365C2211yescharacteristic
37#2365C2212yes—
38#2365C2212yes—
39#2362—21no—
40#2354C2211yescharacteristic
41#2345—22no—
42Q8×C22⋊C4C2214yes—
43#2331—22no—
44#2316—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×#121C2211yescharacteristic
49#2285C2211yescharacteristic
50#2274—21no—
51C4×#117C2211yescharacteristic
52#2100C2211yescharacteristic
53#2100C2212yes—
54#2094C2211yescharacteristic
55#2084C2211yescharacteristic
56#2084C2212yes—
57#2077C2214yes—
58#2077C2212yes—
59#2072C2214yes—
60#2071C2211yescharacteristic
61#2066C2214yes—
62#2055C2214yes—
63#2043C2212yes—
64#2043C2212yes—
65#2043C2212yes—
66#2034C2212yes—
67#2024C2214yes—
68#2013C2212yes—
69#1988C2212yes—
70#1981C2212yes—
71#1968C2214yes—
72#1956C2212yes—
73#1948C2212yes—
74#1939C2212yes—
75#1933C2214yes—
76#1927C2214yes—
77#1927C2214yes—
78#1923C2214yes—
79#1880C2214yes—
80#1880C2214yes—
81#1865C2214yes—
82#1865C2214yes—
83#1831C2212yes—
84#1803C2214yes—
85#1797C2211yescharacteristic
86#1787C2214yes—
87#1787C2212yes—
88C4×#106C2212yes—
89C8×C4⋊C4C2212yes—
90C4×#103C2214yes—
91C4×#103C2214yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C23×Q8C2312yes—
96C22×SD16C2314yes—
97#556—48no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#126—22no—
107C2×#126—22no—
108C2×#124—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×#121C2311yescharacteristic
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2314yes—
117C2×C4×D4—21no—
118C2×C4×D4C2311yescharacteristic
119C2×#117—21no—
120C2×#117—21no—
121C2×#117D411yescharacteristic
122C2×#117D411yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C42C2311yescharacteristic
126#487—28no—
127#475—48no—
128#461—28no—
129#448—44no—
130C2×#106C2312yes—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2314yes—
148#400—22no—
149#399—22no—
150#397—22no—
151#395—22no—
152#394—22no—
153#393—22no—
154#391—21no—
155#391—22no—
156#391—21no—
157#391—24no—
158#391—24no—
159#391C2314yes—
160#391C2314yes—
161#390C2312yes—
162#390C2311yescharacteristic
163#390C2312yes—
164#389—22no—
165#389—22no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—21no—
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—21no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—24no—
184#387—24no—
185#387—22no—
186#387—22no—
187#386—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2311yescharacteristic
192#382—21no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—24no—
197#380C2312yes—
198#380C2312yes—
199#380C2311yescharacteristic
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—24no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C22×Q8—28no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—44no—
239C22×Q8—42no—
240C22×Q8—42no—
241C22×Q8C2414yes—
242C22×D4—21no—
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×SD16—28no—
250C2×SD16—44no—
251C2×SD16—44no—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263C4⋊Q8#4118yes—
264#126—48no—
265#126—48no—
266#124—48no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270#121—22no—
271#121—41no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—48no—
278C4×D4—24no—
279C4×D4—44no—
280#117—42no—
281#117—44no—
282#117—44no—
283#117SD1618yescomplemented
284C2×C4⋊C4—28no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
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—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#106—24no—
348#106—42no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#103—28no—
353#103—28no—
354#103—28no—
355#103—28no—
356#103—44no—
357#103—44no—
358#103—44no—
359#102—42no—
360#102—42no—
361C22⋊C8—42no—
362C22⋊C8—42no—
363C4×C8—24no—
364C4×C8—44no—
365#95—21no—
366#95—24no—
367#95—24no—
368#95—21no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C24C22×D411yescharacteristic
386C2×Q8—28no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—88no—
401C2×Q8—88no—
402C2×Q8C2×#4118yes—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—81no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4C2×SD1614yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444SD16—88no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4C2×SD1614yes—
487C4⋊C4C2×SD1614yes—
488C4⋊C4H(5,𝔽2)12yes—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—42no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—82no—
501C22⋊C4C2×SD1618yes—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—22no—
509C42—42no—
510C42—42no—
511C42—44no—
512C42—42no—
513C42—44no—
514C42—44no—
515C42—84no—
516C42—84no—
517C42C2×SD1618yes—
518C42C2×#4114yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C22×SD1612yes—
533C23C23×D411yescharacteristic
534Q8—28no—
535Q8—28no—
536Q8—44no—
537Q8—44no—
538Q8—88no—
539Q8—88no—
540Q8—84no—
541D4—42no—
542D4—42no—
543D4—84no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—88no—
592C2×C4—84no—
593C2×C4#52512yes—
594C2×C4#52512yes—
595C2×C4C22×SD1614yes—
596C2×C4C22×SD1612yes—
597C2×C4C22×SD1614yes—
598C2×C4C22×SD1612yes—
599C2×C4#57512yes—
600C2×C4#57812yes—
601C2×C4C22×#4112yes—
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4#58512yes—
604C2×C4#58511yescharacteristic
605C8—42no—
606C8—44no—
607C8—42no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22C2×#52511yescharacteristic
623C22#348811yescharacteristic
624C22#350011yescharacteristic
625C22C23×SD1611yescharacteristic
626C22C2×#57511yescharacteristic
627C22C2×#57811yescharacteristic
628C22#358611yescharacteristic
629C4—24no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—24no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—48no—
638C4—42no—
639C4—82no—
640C4—81no—
641C4—81no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4#310314yes—
646C4#351412yes—
647C4#352112yes—
648C2—42no—
649C2—161no—
650C2#3607511yescharacteristic
651C2C2×#310311yescharacteristic
652C2#6058311yescharacteristic
653C2#6194811yescharacteristic
654C2#6199811yescharacteristic
655C2#6252611yescharacteristic
656C2#6258111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. #30129C2semidirect
4. #29937C2semidirect
5. #29494C2semidirect
6. #28205C2semidirect
7. #27522C2semidirect
9. #26462C2semidirect
10. #25856C2semidirect
11. #25320C2semidirect
13. #23816C2semidirect
15. #23214C2semidirect
16. #23094C2semidirect
17. #22632C2semidirect
283. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#364732
#342162
#301292
#299372
#294942
#282052
#275222
#272262
#264622
#258562
#253202
#242092
#238162
#237552
#232142
#230942
#226322