Finite Group Databasebeta

← #6469760

Subgroups — #6469760

Order 512.

1663 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic
2#34225C211yesmaximal, characteristic
3#29565C214yesmaximal, complemented
4#29207C211yesmaximal, characteristic, complemented
5#29144C211yesmaximal, characteristic, complemented
6#28748C211yesmaximal, characteristic, complemented
7#28440C211yesmaximal, characteristic, complemented
8#27892C211yesmaximal, characteristic
9#27889C211yesmaximal, characteristic
10#25600C211yesmaximal, characteristic
11#25600C211yesmaximal, characteristic
12#24664C214yesmaximal
13#23586C214yesmaximal
14#21691C211yesmaximal, characteristic, complemented
15C4×#494C211yesmaximal, characteristic, complemented
16#21647C211yesmaximal, characteristic, complemented
17C4×#487C211yesmaximal, characteristic, complemented
18#21516C214yesmaximal, complemented
19C4×#433C211yesmaximal, characteristic
20C2×#541C2212yes—
21C2×#533C2211yescharacteristic
22C2×#494C2211yescharacteristic
23C2×#494C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#487C2211yescharacteristic
26C2×#462C2214yes—
27C2×#441C2214yes—
28C2×C4×Q16C2214yes—
29C2×#433C2212yes—
30#2868C2211yescharacteristic
31#2859C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2642C2212yes—
35#2621C2212yes—
36#2506—21no—
37#2506—21no—
38#2506—21no—
39#2506—21no—
40#2496—21no—
41#2496—21no—
42#2496—21no—
43#2496—21no—
44#2370—21no—
45#2370—21no—
46#2362—21no—
47#2362—21no—
48#2356—21no—
49#2351—21no—
50#2347—21no—
51#2346—21no—
52#2327C2212yes—
53#2323C2212yes—
54#2323C2212yes—
55#2320C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62#2285C2211yescharacteristic
63#2276C2212yes—
64C4×#117C2211yescharacteristic
65#2059C2214yes—
66#2043C2214yes—
67#2036C2214yes—
68#2035C2214yes—
69#2026C2214yes—
70#2016C2214yes—
71#2001C2214yes—
72#2001C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1981C2214yes—
76#1971C2214yes—
77#1969C2214yes—
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1949C2214yes—
85#1935C2214yes—
86#1933C2214yes—
87#1928C2214yes—
88#1927C2214yes—
89#1878C2214yes—
90#1878C2214yes—
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1805C2214yes—
96#1768C2212yes—
97C4×C4⋊C8C2211yescharacteristic
98C4×C4⋊C8C2211yescharacteristic
99#1763C2214yes—
100C4×#103C2214yes—
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×Q16C2314yes—
108C22×Q16C2314yes—
109#541—48no—
110#533—24no—
111#533—24no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2311yescharacteristic
118C2×#126—22no—
119C2×#126—22no—
120C2×#125—22no—
121C2×#125—22no—
122C2×#125—21no—
123C2×#125C2312yes—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2312yes—
127C2×C22⋊Q8C2312yes—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×#117—21no—
137C2×#117—21no—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#494—24no—
142#494—24no—
143#487—24no—
144#487—24no—
145#462—216no—
146#441—48no—
147C4×Q16—48no—
148#433—28no—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×#97C2312yes—
164C2×#97C2312yes—
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2312yes—
170C2×C4×C8C2312yes—
171#400—22no—
172#400—22no—
173#400—22no—
174#400—22no—
175#398—22no—
176#398—22no—
177#397—22no—
178#397—22no—
179#391—21no—
180#391—21no—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#389—22no—
188#389—22no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389C2312yes—
208#389C2312yes—
209#388—22no—
210#388—22no—
211#388—22no—
212#388—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#384—21no—
222#384—21no—
223#384—21no—
224#384—21no—
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#383—22no—
230#382—21no—
231#382—22no—
232#382C2312yes—
233#380—21no—
234#380—21no—
235#380—21no—
236#380—21no—
237#380—21no—
238#380C2312yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—22no—
251C4×C4⋊C4—22no—
252C4×C4⋊C4—22no—
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4C2312yes—
258C4×C4⋊C4C2312yes—
259C4×C4⋊C4C2312yes—
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4D411yescharacteristic
262C4×C22⋊C4D411yescharacteristic
263C4×C22⋊C4D411yescharacteristic
264#376—22no—
265#376C2311yescharacteristic
266#376C2311yescharacteristic
267C43C2311yescharacteristic
268C43C2311yescharacteristic
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#340C2314yes—
273#336C2312yes—
274#336C2312yes—
275#336C2312yes—
276#336C2312yes—
277C22×Q8—21no—
278C22×Q8C2412yes—
279C22×Q8C2412yes—
280C22×Q8C2412yes—
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2411yescharacteristic
284C2×Q16—28no—
285C2×Q16—28no—
286C2×Q16—28no—
287C2×Q16—28no—
288C2×Q16—44no—
289C2×Q16—44no—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—28no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—44no—
302C4⋊Q8#4114yes—
303C4⋊Q8#4114yes—
304#126—48no—
305#126—48no—
306#125—28no—
307#125—44no—
308#125—48no—
309#125—48no—
310C22⋊Q8—28no—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C4×Q8—28no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—28no—
319C4×Q8—28no—
320C4×Q8—28no—
321C4×Q8—28no—
322C4×Q8—82no—
323C4×Q8—82no—
324#117—28no—
325#117—24no—
326#117—24no—
327#117—82no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402#107—44no—
403#106—44no—
404C4⋊C8—24no—
405C4⋊C8#4118yes—
406#103—28no—
407#103—28no—
408#103—28no—
409#103—28no—
410#103—44no—
411#103—44no—
412#103—44no—
413#103—44no—
414C22⋊C8—24no—
415C22⋊C8—24no—
416#97—24no—
417#97—24no—
418C4×C8—24no—
419C4×C8—22no—
420C4×C8—22no—
421C4×C8—22no—
422C4×C8—22no—
423C4×C8—24no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452C24C22×D411yescharacteristic
453C2×Q8—24no—
454C2×Q8—22no—
455C2×Q8—24no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—24no—
459C2×Q8—24no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—24no—
463C2×Q8—24no—
464C2×Q8—24no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8C2×#4114yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C2511yescharacteristic
500Q16—416no—
501Q16—416no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8C2×#4114yes—
513C2×C8C2×#4114yes—
514C4⋊C4—28no—
515C4⋊C4—28no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4C2×#4114yes—
555C22⋊C4—28no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—82no—
563C42—28no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—21no—
576C42—21no—
577C42—22no—
578C42—44no—
579C42—44no—
580C42—44no—
581C42—44no—
582C42—82no—
583C42—82no—
584C42—82no—
585C42C2×#4112yes—
586C42C2×#4112yes—
587C42C2×#4114yes—
588C42H(5,𝔽2)14yes—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23C23×D411yescharacteristic
597Q8—28no—
598Q8—48no—
599Q8—44no—
600Q8—44no—
601Q8—44no—
602Q8—44no—
603Q8—44no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4#52512yes—
662C2×C4#52512yes—
663C2×C4#53212yes—
664C2×C4#53212yes—
665C2×C4#53512yes—
666C2×C4#53512yes—
667C2×C4#57712yes—
668C2×C4#57812yes—
669C2×C4#57812yes—
670C2×C4#57812yes—
671C2×C4C22×#4111yescharacteristic
672C2×C4C22×#4112yes—
673C2×C4C22×#4111yescharacteristic
674C2×C4C2×H(5,𝔽2)12yes—
675C8—24no—
676C8—24no—
677C8—42no—
678C8—42no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22C2×#52511yescharacteristic
687C22C2×#53211yescharacteristic
688C22C2×#53511yescharacteristic
689C22C2×#57711yescharacteristic
690C22C2×#57811yescharacteristic
691C22C2×#57811yescharacteristic
692C22C2×#57811yescharacteristic
693C4—22no—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—44no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—44no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4#312014yes—
713C4#312714yes—
714C4#348112yes—
715C4#348312yes—
716C2—81no—
717C2#3497711yescharacteristic
718C2C2×#312011yescharacteristic
719C2C2×#312711yescharacteristic
720C2#6071511yescharacteristic
721C2#6072311yescharacteristic
722C2#6082111yescharacteristic
723C2#6082411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #29565C2semidirect
4. #29207C2semidirect
5. #29144C2semidirect
6. #28748C2semidirect
7. #28440C2semidirect
14. #21691C2semidirect
15. C4×#494C2semidirect
16. #21647C2semidirect
17. C4×#487C2semidirect
18. #21516C2semidirect

Maximal subgroups

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

subgroupindex
#353192
#342252
#295652
#292072
#291442
#287482
#284402
#278922
#278892
#256002
#246642
#235862
#216912
C4×#4942
#216472
C4×#4872
#215162
C4×#4332