Finite Group Databasebeta

← #6993541

Subgroups — #6993541

Order 512.

1603 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36272C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic, complemented
3#32820C212yesmaximal, complemented
4#32280C214yesmaximal, complemented
5#31934C214yesmaximal
6#30305C212yesmaximal, complemented
7#29981C212yesmaximal, complemented
8#29688C211yesmaximal, characteristic, complemented
9#29554C212yesmaximal, complemented
10#28530C212yesmaximal, complemented
11#27729C212yesmaximal, complemented
12#25811C211yesmaximal, characteristic, complemented
13#25453C212yesmaximal, complemented
14#24815C212yesmaximal, complemented
15#24299C211yesmaximal, characteristic
16#23568C211yesmaximal, characteristic
17#22632C211yesmaximal, characteristic, complemented
18C2×#552—21no—
19C2×#533C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#498C2212yes—
22C2×#475C2212yes—
23C2×#476C2212yes—
24C2×C4×SD16C2212yes—
25#2868C2212yes—
26#2862C2212yes—
27#2859C2212yes—
28#2857C2211yescharacteristic
29#2660—22no—
30#2611—22no—
31#2567C2212yes—
32#2567C2212yes—
33#2553—22no—
34#2552C2211yescharacteristic
35#2546—21no—
36#2546—21no—
37#2542C2211yescharacteristic
38#2502C2214yes—
39#2496C2212yes—
40#2489—21no—
41#2434—22no—
42#2422C2212yes—
43#2415—21no—
44#2365C2211yescharacteristic
45#2360—21no—
46D4×C4⋊C4C2211yescharacteristic
47#2317—21no—
48#2317—21no—
49C4×#121C2211yescharacteristic
50#2293C2212yes—
51C4×#117C2211yescharacteristic
52#2100C2214yes—
53#2096C2214yes—
54#2094C2214yes—
55#2084C2214yes—
56#2084C2214yes—
57#2072C2214yes—
58#2066C2214yes—
59#2066C2214yes—
60#2058C2211yescharacteristic
61#2058C2212yes—
62#2058C2212yes—
63#2058C2212yes—
64#2058C2212yes—
65#2043C2211yescharacteristic
66#2043C2212yes—
67#2035C2212yes—
68#2035C2214yes—
69#2023C2211yescharacteristic
70#2023C2212yes—
71#2013C2211yescharacteristic
72#2000C2212yes—
73#1990C2212yes—
74#1987C2212yes—
75#1968C2214yes—
76#1948C2214yes—
77#1939C2212yes—
78#1933C2214yes—
79#1927C2214yes—
80#1923C2214yes—
81#1877C2214yes—
82#1874C2212yes—
83#1853C2212yes—
84#1853C2214yes—
85#1844C2214yes—
86#1834C2214yes—
87#1831C2212yes—
88#1803C2212yes—
89#1802C2214yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C23×Q8—21no—
96C22×SD16C2312yes—
97#552—48no—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#126—22no—
110C2×#125—21no—
111C2×#125—22no—
112C2×#125—21no—
113C2×#123—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8C2312yes—
119C2×#121C2311yescharacteristic
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8C2312yes—
123C2×C4×D4—21no—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×#117—21no—
127C2×#117—22no—
128C2×#117D411yescharacteristic
129C2×#117D411yescharacteristic
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134#498—44no—
135#476—44no—
136#475—44no—
137C4×SD16—44no—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C4×C8C2314yes—
155C2×C4×C8C2312yes—
156#400—22no—
157#399—22no—
158#398—22no—
159#398—24no—
160#398—24no—
161#397—22no—
162#397—22no—
163#395—24no—
164#395—24no—
165#394—22no—
166#391—21no—
167#391—21no—
168#391—22no—
169#391—22no—
170#391C2314yes—
171#391C2314yes—
172#390C2312yes—
173#390C2311yescharacteristic
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389C2312yes—
183#389C2311yescharacteristic
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#388—21no—
188#388—22no—
189#388—22no—
190#387—22no—
191#387—22no—
192#386—21no—
193#385—24no—
194#385—24no—
195#385—22no—
196#385—22no—
197#384—22no—
198#384—22no—
199#384C2314yes—
200#384C2314yes—
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2312yes—
204#382—22no—
205#382—22no—
206#382—22no—
207#380—21no—
208#380—21no—
209#380C2312yes—
210#380C2311yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221#376D412yes—
222C43C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—44no—
240C22×Q8—82no—
241C22×Q8—82no—
242C22×Q8C2412yes—
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×SD16—24no—
252C2×SD16—24no—
253C2×SD16—42no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—84no—
266C4⋊Q8—84no—
267#126—48no—
268#125—44no—
269#125—44no—
270#125—48no—
271#123—48no—
272C22⋊Q8—42no—
273C22⋊Q8—42no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278#121—22no—
279#121—41no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—42no—
287C4×D4—44no—
288#117—24no—
289#117—44no—
290#117—48no—
291#117SD1618yescomplemented
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356#106—42no—
357#106—44no—
358#106—44no—
359#106—44no—
360#106—44no—
361C4⋊C8—24no—
362C4⋊C8—44no—
363#103—24no—
364#103—44no—
365#103—42no—
366#103—42no—
367#103—44no—
368#102—42no—
369#102—44no—
370C22⋊C8—24no—
371C22⋊C8—42no—
372C4×C8—28no—
373C4×C8—24no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—21no—
392C24C22×D411yescharacteristic
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—816no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—81no—
418C2×D42−1+412yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—41no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4C2×SD1614yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2511yescharacteristic
456SD16—48no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C82−1+414yes—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—82no—
497C4⋊C4—81no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4C2×SD1614yes—
503C4⋊C4C2×SD1614yes—
504C4⋊C4#13714yes—
505C22⋊C4—24no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4C2×SD1618yes—
517C42—22no—
518C42—22no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—42no—
526C42—44no—
527C42—44no—
528C42—42no—
529C42C2×SD1618yes—
530C42H(5,𝔽2)14yes—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C22×SD1612yes—
545C23C23×D411yescharacteristic
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—84no—
551Q8—88no—
552Q8—88no—
553Q8—84no—
554D4—24no—
555D4—42no—
556D4—84no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4—84no—
606C2×C4#53512yes—
607C2×C4#53512yes—
608C2×C4C22×SD1614yes—
609C2×C4C22×SD1612yes—
610C2×C4C22×SD1614yes—
611C2×C4C22×SD1612yes—
612C2×C4C2×#13714yes—
613C2×C4C2×#13712yes—
614C2×C4#57712yes—
615C2×C4C2×H(5,𝔽2)12yes—
616C2×C4C2×2−1+411yescharacteristic
617C2×C4C2×2−1+412yes—
618C2×C4#58511yescharacteristic
619C8—24no—
620C8—42no—
621C8—44no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22C2×#53511yescharacteristic
637C22#348911yescharacteristic
638C22#349911yescharacteristic
639C22C23×SD1611yescharacteristic
640C22C22×#13711yescharacteristic
641C22C2×#57711yescharacteristic
642C22#358511yescharacteristic
643C4—24no—
644C4—24no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—24no—
649C4—42no—
650C4—41no—
651C4—81no—
652C4—81no—
653C4—82no—
654C4—84no—
655C4—82no—
656C4—82no—
657C4—81no—
658C4—82no—
659C4#320614yes—
660C4#353812yes—
661C4#354712yes—
662C2—42no—
663C2—82no—
664C2#4299411yescharacteristic
665C2C2×#320611yescharacteristic
666C2#6081211yescharacteristic
667C2#6209611yescharacteristic
668C2#6214711yescharacteristic
669C2#6246911yescharacteristic
670C2#6250411yescharacteristic
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. #35281C2semidirect
3. #32820C2semidirect
4. #32280C2semidirect
6. #30305C2semidirect
7. #29981C2semidirect
8. #29688C2semidirect
9. #29554C2semidirect
10. #28530C2semidirect
11. #27729C2semidirect
12. #25811C2semidirect
13. #25453C2semidirect
14. #24815C2semidirect
17. #22632C2semidirect
291. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#362722
#352812
#328202
#322802
#319342
#303052
#299812
#296882
#295542
#285302
#277292
#258112
#254532
#248152
#242992
#235682
#226322