Finite Group Databasebeta

← #6856263

Subgroups — #6856263

Order 512.

1624 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#32906C211yesmaximal, characteristic, complemented
4#32444C214yesmaximal, complemented
5#31652C214yesmaximal, complemented
6#29719C212yesmaximal
7#28419C212yesmaximal, complemented
8#27978C212yesmaximal
9#27738C212yesmaximal, complemented
10#27680C212yesmaximal, complemented
11#25902C211yesmaximal, characteristic, complemented
12#25324C212yesmaximal, complemented
13#24680C212yesmaximal, complemented
14#24299C211yesmaximal, characteristic
15#22629C211yesmaximal, characteristic, complemented
16#22050C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#487C2212yes—
21C2×#476C2212yes—
22C2×#462C2212yes—
23C2×#451C2212yes—
24C2×#448C2211yescharacteristic
25C2×C4×SD16C2212yes—
26#2857C2211yescharacteristic
27#2850C2212yes—
28#2596—22no—
29#2571C2211yescharacteristic
30#2567—22no—
31#2567—21no—
32#2567—21no—
33#2567C2212yes—
34#2567C2212yes—
35#2567C2214yes—
36#2547C2211yescharacteristic
37#2514C2212yes—
38#2510—22no—
39#2494—21no—
40#2487—21no—
41#2448C2212yes—
42C2×#395—22no—
43#2365C2211yescharacteristic
44#2354C2211yescharacteristic
45#2352—21no—
46#2347—21no—
47C4×C22⋊Q8C2212yes—
48C4×#121C2211yescharacteristic
49C2×#391C2212yes—
50C2×#389—21no—
51C2×#389—21no—
52C2×#384C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2099C2214yes—
55#2096C2214yes—
56#2074C2214yes—
57#2072C2214yes—
58#2071C2214yes—
59#2058C2212yes—
60#2056C2214yes—
61#2055C2214yes—
62#2024C2214yes—
63#2023C2212yes—
64#2015C2214yes—
65#1990C2212yes—
66#1987C2212yes—
67#1971C2214yes—
68#1959C2214yes—
69#1948C2214yes—
70#1939C2212yes—
71#1935C2214yes—
72#1927C2214yes—
73#1917C2214yes—
74#1880C2212yes—
75#1878C2214yes—
76#1878C2214yes—
77#1874C2212yes—
78#1874C2211yescharacteristic
79#1872C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1865C2212yes—
83#1853C2211yescharacteristic
84#1853C2212yes—
85#1836C2214yes—
86#1803C2214yes—
87#1803C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1787C2211yescharacteristic
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×Q16C2314yes—
98C22×SD16C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#124C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×#121C2311yescharacteristic
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—44no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—21no—
115C2×C4×D4C2311yescharacteristic
116C2×#117—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#487—44no—
129#476—44no—
130#462—44no—
131#451—28no—
132#448—24no—
133C4×SD16—44no—
134C2×#106C2314yes—
135C2×#106C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#401—22no—
153#400—22no—
154#400—22no—
155#399—22no—
156#397—22no—
157#397—22no—
158#395—24no—
159#395—24no—
160#395—24no—
161#395—24no—
162#391—22no—
163#391—22no—
164#391C2314yes—
165#391C2314yes—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#390C2312yes—
170#390C2311yescharacteristic
171#389—22no—
172#389—22no—
173#389—22no—
174#389—24no—
175#389—24no—
176#389—22no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389C2311yescharacteristic
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—21no—
187#388—22no—
188#388—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#386—22no—
193#386—22no—
194#386—21no—
195#385—22no—
196#384—22no—
197#384—22no—
198#384C2314yes—
199#384C2314yes—
200#384C2311yescharacteristic
201#383—22no—
202#382—21no—
203#382C2312yes—
204#380—21no—
205#380—21no—
206#380—21no—
207#380—21no—
208#380—21no—
209#380C2311yescharacteristic
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—82no—
244C22×Q8—82no—
245C22×Q8#4114yes—
246C22×Q8C2412yes—
247C22×Q8C2412yes—
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259C2×Q16—28no—
260C2×Q16—44no—
261C2×Q16—44no—
262C2×SD16—24no—
263C2×SD16—42no—
264C2×SD16—42no—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274#124—44no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277#121—22no—
278#121—22no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×D4—24no—
287C4×D4—44no—
288#117—44no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—84no—
314C2×C4⋊C4—84no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—44no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353#106—42no—
354#106—44no—
355C4⋊C8—24no—
356C4⋊C8—44no—
357#103—24no—
358#103—28no—
359#103—42no—
360#103—42no—
361#103—44no—
362#103—44no—
363#103—44no—
364#102—42no—
365#102—44no—
366C22⋊C8—22no—
367C22⋊C8—24no—
368C22⋊C8—22no—
369#97—44no—
370C4×C8—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—21no—
393C24C22×D411yescharacteristic
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—48no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—48no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—816no—
415C2×Q8C2×#4114yes—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4#13714yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4112yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469Q16—88no—
470SD16—48no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—82no—
513C4⋊C4#13714yes—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—42no—
520C22⋊C4—82no—
521C22⋊C4—81no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—24no—
528C42—24no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—44no—
533C42—44no—
534C42—42no—
535C42—82no—
536C42H(5,𝔽2)14yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23#54812yes—
549C23C2×#13712yes—
550C23#57712yes—
551C23C23×D411yescharacteristic
552Q8—24no—
553Q8—24no—
554Q8—48no—
555Q8—42no—
556Q8—42no—
557Q8—44no—
558Q8—44no—
559Q8—88no—
560Q8—88no—
561D4—42no—
562D4—42no—
563D4—84no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4—84no—
618C2×C4#53512yes—
619C2×C4#53512yes—
620C2×C4#54812yes—
621C2×C4C2×#13714yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4C2×#13712yes—
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×H(5,𝔽2)12yes—
627C2×C4C2×2−1+411yescharacteristic
628C2×C4#58512yes—
629C8—42no—
630C8—44no—
631C8—42no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22#336914yes—
646C22C2×#53511yescharacteristic
647C22C2×#54811yescharacteristic
648C22#348511yescharacteristic
649C22C22×#13711yescharacteristic
650C22C22×#13711yescharacteristic
651C22C2×#57711yescharacteristic
652C22#358511yescharacteristic
653C4—24no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—41no—
662C4—81no—
663C4—82no—
664C4—81no—
665C4—81no—
666C4—82no—
667C4—82no—
668C4—84no—
669C4#321814yes—
670C4#350512yes—
671C4#354112yes—
672C2—24no—
673C2—161no—
674C2#3972211yescharacteristic
675C2C2×#321811yescharacteristic
676C2#6081211yescharacteristic
677C2C2×#336911yescharacteristic
678C2#6131911yescharacteristic
679C2#6171911yescharacteristic
680C2#6173511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2567C2semidirect
3. #32906C2semidirect
4. #32444C2semidirect
5. #31652C2semidirect
7. #28419C2semidirect
9. #27738C2semidirect
10. #27680C2semidirect
11. #25902C2semidirect
12. #25324C2semidirect
13. #24680C2semidirect
15. #22629C2semidirect
16. #22050C2semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#25672
#329062
#324442
#316522
#297192
#284192
#279782
#277382
#276802
#259022
#253242
#246802
#242992
#226292
#220502
C4×#4482