Finite Group Databasebeta

← #6436171

Subgroups — #6436171

Order 512.

1798 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×#3083C212yesmaximal
2#35275C211yesmaximal, characteristic, complemented
3#34253C211yesmaximal, characteristic
4#29875C214yesmaximal, complemented
5#29872C214yesmaximal, complemented
6#29871C212yesmaximal, complemented
7#29870C212yesmaximal, complemented
8#29869C212yesmaximal, complemented
9#29864C212yesmaximal, complemented
10#23660C212yesmaximal, complemented
11#23657C211yesmaximal, characteristic, complemented
12C2×#2083C211yesmaximal, characteristic, complemented
13#23634C212yesmaximal
14#23630C211yesmaximal, characteristic
15#23627C211yesmaximal, characteristic
16#21315C212yesmaximal, complemented
17#20754C211yesmaximal, characteristic, complemented
18C2×#544C2212yes—
19C2×#520C2211yescharacteristic
20#3083—216no—
21C2×#479C2214yes—
22C2×#477C2214yes—
23C2×#478C2214yes—
24C2×#474C2214yes—
25C2×#428C2212yes—
26C22×#107C2212yes—
27C2×#424C2212yes—
28C2×#417C2212yes—
29C2×#416C2212yes—
30#2868C2212yes—
31#2859C2212yes—
32#2660C2214yes—
33#2552—21no—
34#2552—21no—
35#2551—21no—
36#2551—21no—
37#2440—22no—
38#2440—22no—
39#2353—22no—
40#2352—22no—
41#2347—21no—
42#2346—21no—
43#2345—21no—
44#2344—21no—
45#2320C2212yes—
46#2319C2212yes—
47#2317C2212yes—
48#2316C2211yescharacteristic
49#2311C2211yescharacteristic
50#2297C2211yescharacteristic
51#2295C2212yes—
52#2293C2212yes—
53C2×#389C2211yescharacteristic
54#2285C2211yescharacteristic
55C2×#384C2211yescharacteristic
56#2274C2211yescharacteristic
57C4×#117C2211yescharacteristic
58#2085C2214yes—
59#2085C2214yes—
60#2085C2214yes—
61#2084C2212yes—
62#2083C2212yes—
63#2083C2212yes—
64#2083C2214yes—
65#2083C2212yes—
66#2083C2214yes—
67#2083C2214yes—
68#2074C2212yes—
69#2074C2214yes—
70#2073C2212yes—
71#2073C2214yes—
72#2072C2211yescharacteristic
73#2072C2211yescharacteristic
74#2072C2212yes—
75#2072C2212yes—
76#2072C2214yes—
77#2071C2211yescharacteristic
78#2071C2211yescharacteristic
79#2071C2212yes—
80#2071C2212yes—
81#2071C2214yes—
82#1989C2214yes—
83#1989C2214yes—
84#1988C2214yes—
85#1987C2214yes—
86#1832C2214yes—
87#1831C2212yes—
88#1831C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1729C2212yes—
94#1728C2211yescharacteristic
95C2×#336C2211yescharacteristic
96C22×#41C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—28no—
100#544—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2314yes—
105C2×#125—22no—
106C2×#125C2314yes—
107C2×C22⋊Q8—24no—
108#520—24no—
109#520D418yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4C2312yes—
113C2×#117—21no—
114C2×#117—22no—
115C2×#117C2311yescharacteristic
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#479—216no—
128#478—216no—
129#477—216no—
130#474—216no—
131#428—28no—
132C2×#107—28no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137#424—28no—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140#417—28no—
141#416—28no—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148#398—24no—
149#398—24no—
150#395—24no—
151#395—24no—
152#391C2312yes—
153#391C2311yescharacteristic
154#391C2311yescharacteristic
155#390C2312yes—
156#390C2311yescharacteristic
157#390C2311yescharacteristic
158#389—21no—
159#389—21no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#386—21no—
181#386—21no—
182#386—22no—
183#385—24no—
184#384—22no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384C2314yes—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2311yescharacteristic
193#383—22no—
194#382—22no—
195#382C2314yes—
196#382C2312yes—
197#381—22no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380—22no—
203#380—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221#376—22no—
222#376C2312yes—
223C2×#95—21no—
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#336C2311yescharacteristic
236#336C2311yescharacteristic
237#336C2312yes—
238#336C2312yes—
239#336C2314yes—
240#336C2314yes—
241#336C2312yes—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41C2×D414yes—
246C22×Q8—24no—
247C22×Q8C2411yescharacteristic
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×M16—24no—
259C2×M16—24no—
260C2×M16—24no—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8—24no—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C4⋊Q8—28no—
268C4⋊Q8—28no—
269C4⋊Q8—48no—
270C4⋊Q8—84no—
271C4⋊Q8—84no—
272#125—28no—
273#125—28no—
274#125—48no—
275C22⋊Q8—416no—
276C4×Q8—24no—
277C4×Q8C2×D418yes—
278C4×D4—216no—
279C4×D4—24no—
280C4×D4C2×D418yes—
281#117—24no—
282#117—28no—
283#117—24no—
284#117—24no—
285#117—24no—
286#117—84no—
287#117C2×D414yes—
288#117#4114yes—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357#107—28no—
358#107—28no—
359#107—28no—
360#106—28no—
361C4⋊C8—28no—
362C4⋊C8—28no—
363#103—24no—
364#103—24no—
365#103—28no—
366#102—24no—
367#102—24no—
368#102—28no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C24C22×D411yescharacteristic
391#41—216no—
392#41—48no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—48no—
397C2×Q8—84no—
398C2×Q8—84no—
399C2×Q8C22×D412yes—
400C2×D4—24no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4C22×D412yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—82no—
433C22×C4—81no—
434C22×C4—82no—
435C22×C4—81no—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C2×#4112yes—
447C22×C42−1+414yes—
448C22×C42−1+414yes—
449C22×C4C2511yescharacteristic
450M16—48no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—48no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4#12114yes—
495C4⋊C4#12114yes—
496C4⋊C4C22×D414yes—
497C22⋊C4—24no—
498C22⋊C4—28no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—82no—
505C22⋊C4—81no—
506C22⋊C4—82no—
507C22⋊C4#12118yes—
508C42—28no—
509C42—22no—
510C42—22no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—42no—
517C42—42no—
518C42—44no—
519C42—42no—
520C42—82no—
521C42—82no—
522C42C22×D414yes—
523C42C2×#4114yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23C2×#12112yes—
536C23#53612yes—
537C23#53612yes—
538C23C23×D411yescharacteristic
539Q8—24no—
540Q8—42no—
541Q8—168no—
542D4—24no—
543D4—48no—
544D4—42no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4—164no—
595C2×C4—162no—
596C2×C4C2×#12114yes—
597C2×C4C2×#12114yes—
598C2×C4C2×#12112yes—
599C2×C4#53612yes—
600C2×C4#53612yes—
601C2×C4#53618yes—
602C2×C4#54118yes—
603C2×C4#57612yes—
604C2×C4#57812yes—
605C2×C4C23×D411yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×2−1+412yes—
608C2×C4C2×2−1+412yes—
609C8—42no—
610C8—42no—
611C8—44no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22#262114yes—
626C22C22×#12111yescharacteristic
627C22C2×#53611yescharacteristic
628C22C2×#53611yescharacteristic
629C22C2×#57611yescharacteristic
630C22C2×#57811yescharacteristic
631C22#358511yescharacteristic
632C22#358611yescharacteristic
633C4—24no—
634C4—24no—
635C4—22no—
636C4—22no—
637C4—44no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—42no—
643C4—84no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—81no—
648C4—81no—
649C4#262114yes—
650C4C2×#53612yes—
651C4C2×#54112yes—
652C2—24no—
653C2—82no—
654C2C2×#262111yescharacteristic
655C2#6042811yescharacteristic
656C2#6043211yescharacteristic
657C2#6087111yescharacteristic
658C2#6087411yescharacteristic
659C2#6088011yescharacteristic
660C2#6088711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. #35275C2semidirect
4. #29875C2semidirect
5. #29872C2semidirect
6. #29871C2semidirect
7. #29870C2semidirect
8. #29869C2semidirect
9. #29864C2semidirect
10. #23660C2semidirect
11. #23657C2semidirect
12. C2×#2083C2semidirect
16. #21315C2semidirect
17. #20754C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30832
#352752
#342532
#298752
#298722
#298712
#298702
#298692
#298642
#236602
#236572
C2×#20832
#236342
#236302
#236272
#213152
#207542