Finite Group Databasebeta

← #6467826

Subgroups — #6467826

Order 512.

1858 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3185C211yesmaximal, characteristic
2C2×#3170C211yesmaximal, characteristic
3#35322C211yesmaximal, characteristic
4#34225C211yesmaximal, characteristic
5#28748C211yesmaximal, characteristic, complemented
6#28719C211yesmaximal, characteristic, complemented
7#28688C211yesmaximal, characteristic, complemented
8#28656C211yesmaximal, characteristic, complemented
9#28653C211yesmaximal, characteristic, complemented
10#28616C211yesmaximal, characteristic, complemented
11#28616C211yesmaximal, characteristic, complemented
12#28588C214yesmaximal
13#28440C211yesmaximal, characteristic, complemented
14#24764C214yesmaximal
15#23691C214yesmaximal, complemented
16#21516C214yesmaximal, complemented
17#21409C211yesmaximal, characteristic
18#21400C211yesmaximal, characteristic
19C4×#424C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#533C2211yescharacteristic
23#3185—28no—
24#3170—28no—
25C2×#500C2211yescharacteristic
26C2×#498C2211yescharacteristic
27C2×#494C2211yescharacteristic
28C2×#494C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#487C2211yescharacteristic
32C2×#487C2211yescharacteristic
33C2×#470C2214yes—
34C2×#467C2214yes—
35C2×#462C2214yes—
36C2×#441C2214yes—
37C2×C4×Q16C2214yes—
38C2×#424C2212yes—
39C2×C4×M16C2211yescharacteristic
40C22×C4×C8C2211yescharacteristic
41#2868C2211yescharacteristic
42#2862C2211yescharacteristic
43#2862C2211yescharacteristic
44#2862C2211yescharacteristic
45#2621C2212yes—
46#2621C2212yes—
47#2565—21no—
48#2565—21no—
49#2565—21no—
50#2565—21no—
51#2499—21no—
52#2499—21no—
53#2499—21no—
54#2499—21no—
55#2370—21no—
56#2370—21no—
57#2362—21no—
58#2362—21no—
59#2356—21no—
60#2351—21no—
61#2347—21no—
62#2346—21no—
63#2327C2212yes—
64#2327C2212yes—
65#2323C2212yes—
66#2323C2212yes—
67#2320C2212yes—
68C4×C4⋊Q8C2211yescharacteristic
69C4×C4⋊Q8C2211yescharacteristic
70C4×C4⋊Q8C2211yescharacteristic
71C4×C4⋊Q8C2211yescharacteristic
72#2285C2211yescharacteristic
73#2276C2212yes—
74C4×#117C2211yescharacteristic
75#2085C2214yes—
76#2074C2214yes—
77#2068C2214yes—
78#2066C2214yes—
79#2056C2214yes—
80#2056C2214yes—
81#2043C2214yes—
82#2016C2214yes—
83#1990C2211yescharacteristic
84#1990C2211yescharacteristic
85#1990C2211yescharacteristic
86#1990C2211yescharacteristic
87#1981C2214yes—
88#1956C2211yescharacteristic
89#1956C2211yescharacteristic
90#1956C2211yescharacteristic
91#1956C2211yescharacteristic
92#1949C2214yes—
93#1928C2214yes—
94#1927C2214yes—
95#1894C2214yes—
96#1892C2214yes—
97#1878C2214yes—
98#1878C2214yes—
99#1805C2214yes—
100#1768C2212yes—
101#1768C2212yes—
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104C4×C4⋊C8C2211yescharacteristic
105C4×C4⋊C8C2211yescharacteristic
106#1763C2214yes—
107C4×#103C2214yes—
108#1729C2212yes—
109C22×Q16C2314yes—
110C22×Q16C2314yes—
111C22×M16C2312yes—
112C23×C8C2312yes—
113#533—24no—
114#533—24no—
115#533—24no—
116#533—24no—
117#533—24no—
118#533D418yes—
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2312yes—
124C2×C4⋊Q8C2311yescharacteristic
125C2×C4⋊Q8C2311yescharacteristic
126C2×C4⋊Q8C2311yescharacteristic
127C2×#125—21no—
128C2×#125—21no—
129C2×#125—21no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8C2312yes—
134C2×C22⋊Q8C2312yes—
135C2×C22⋊Q8C2312yes—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8—21no—
139C2×C4×Q8—21no—
140C2×C4×Q8—21no—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2312yes—
144C2×C4×Q8C2312yes—
145C2×C4×Q8C2312yes—
146C2×#117—21no—
147C2×#117—21no—
148C2×#117—21no—
149C2×#117—21no—
150C2×#117C2311yescharacteristic
151C2×#117C2311yescharacteristic
152C22×C42C2311yescharacteristic
153#500—24no—
154#498—24no—
155#494—24no—
156#494—24no—
157#487—24no—
158#487—24no—
159#487—24no—
160#487—24no—
161#470—216no—
162#467—216no—
163#462—216no—
164#441—216no—
165C4×Q16—216no—
166C2×#107C2314yes—
167C2×#106C2314yes—
168#424—28no—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173C2×#103C2314yes—
174C2×#103C2314yes—
175C2×#103C2314yes—
176C2×#103C2314yes—
177C2×#103C2314yes—
178C2×#103C2314yes—
179C2×#103C2314yes—
180C2×#103C2314yes—
181C4×M16—24no—
182C2×#97C2312yes—
183C2×C4×C8—24no—
184C2×C4×C8C2311yescharacteristic
185C2×C4×C8C2311yescharacteristic
186C2×C4×C8C2312yes—
187C2×C4×C8C2311yescharacteristic
188C2×C4×C8C2311yescharacteristic
189#398—22no—
190#398—22no—
191#398—22no—
192#398—22no—
193#397—22no—
194#397—22no—
195#397—22no—
196#397—22no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391C2312yes—
208#391C2312yes—
209#391C2312yes—
210#391C2312yes—
211#391C2312yes—
212#391C2312yes—
213#389—22no—
214#389—22no—
215#389—21no—
216#389—21no—
217#389—21no—
218#389—21no—
219#389—21no—
220#389—21no—
221#389C2312yes—
222#389C2312yes—
223#388—22no—
224#388—22no—
225#388—22no—
226#388—22no—
227#387—22no—
228#387—22no—
229#387—22no—
230#387—22no—
231#387—22no—
232#387—22no—
233#387—22no—
234#387—22no—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—22no—
239#384—21no—
240#384—21no—
241#384—21no—
242#384—21no—
243#384C2311yescharacteristic
244#384C2311yescharacteristic
245#383—22no—
246#382—21no—
247#382—21no—
248#382—21no—
249#382—21no—
250#382—21no—
251#382—22no—
252#382C2312yes—
253#382C2312yes—
254#380—21no—
255#380C2312yes—
256C4×C4⋊C4—21no—
257C4×C4⋊C4—21no—
258C4×C4⋊C4—21no—
259C4×C4⋊C4—21no—
260C4×C4⋊C4—21no—
261C4×C4⋊C4—21no—
262C4×C4⋊C4—21no—
263C4×C4⋊C4—22no—
264C4×C4⋊C4—22no—
265C4×C4⋊C4—22no—
266C4×C4⋊C4D411yescharacteristic
267C4×C4⋊C4D411yescharacteristic
268C4×C4⋊C4D411yescharacteristic
269C4×C4⋊C4D411yescharacteristic
270C4×C4⋊C4C2312yes—
271C4×C4⋊C4C2312yes—
272C4×C4⋊C4C2312yes—
273C4×C22⋊C4D411yescharacteristic
274C4×C22⋊C4D411yescharacteristic
275C4×C22⋊C4D411yescharacteristic
276C4×C22⋊C4D411yescharacteristic
277#376—22no—
278#376C2311yescharacteristic
279#376C2311yescharacteristic
280C43C2311yescharacteristic
281C43C2311yescharacteristic
282#340C2314yes—
283#340C2314yes—
284#340C2314yes—
285#340C2314yes—
286#336C2312yes—
287#336C2312yes—
288#336C2312yes—
289#336C2312yes—
290C22×Q8—21no—
291C22×Q8—21no—
292C22×Q8—21no—
293C22×Q8C2412yes—
294C22×Q8C2412yes—
295C22×Q8C2412yes—
296C23×C4C2×D411yescharacteristic
297C23×C4C2×D411yescharacteristic
298C23×C4C2411yescharacteristic
299C2×Q16—28no—
300C2×Q16—28no—
301C2×Q16—28no—
302C2×Q16—28no—
303C2×Q16—44no—
304C2×Q16—44no—
305C2×M16—24no—
306C2×M16—24no—
307C2×M16—42no—
308C22×C8—24no—
309C22×C8—24no—
310C22×C8—42no—
311C22×C8C2412yes—
312C22×C8C2412yes—
313C22×C8C2412yes—
314C22×C8C2412yes—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C4⋊Q8—22no—
318C4⋊Q8—22no—
319C4⋊Q8—22no—
320C4⋊Q8—22no—
321C4⋊Q8—22no—
322C4⋊Q8—22no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325C4⋊Q8—28no—
326C4⋊Q8—44no—
327C4⋊Q8C2×D414yes—
328C4⋊Q8C2×D414yes—
329#125—28no—
330#125—44no—
331#125—44no—
332C22⋊Q8—28no—
333C22⋊Q8—24no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337C22⋊Q8—24no—
338C22⋊Q8—44no—
339C22⋊Q8—44no—
340C22⋊Q8C2×D418yes—
341C4×Q8—28no—
342C4×Q8—24no—
343C4×Q8—24no—
344C4×Q8—24no—
345C4×Q8—24no—
346C4×Q8—24no—
347C4×Q8—28no—
348C4×Q8—44no—
349C4×Q8—44no—
350C4×Q8—44no—
351C4×Q8—82no—
352C4×Q8—82no—
353C4×Q8C2×D418yes—
354#117—28no—
355#117—24no—
356#117—42no—
357#117—44no—
358#117—44no—
359#117—82no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4C2×D411yescharacteristic
388C2×C4⋊C4C2×D411yescharacteristic
389C2×C4⋊C4C2×D411yescharacteristic
390C2×C4⋊C4C2×D411yescharacteristic
391C2×C4⋊C4C2412yes—
392C2×C4⋊C4C2412yes—
393C2×C4⋊C4C2412yes—
394C2×C4⋊C4C2412yes—
395C2×C4⋊C4C2412yes—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4C2×D411yescharacteristic
401C2×C22⋊C4C2×D411yescharacteristic
402C2×C22⋊C4C2×D411yescharacteristic
403C2×C22⋊C4C2×D411yescharacteristic
404C2×C42—22no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—22no—
409C2×C42—22no—
410C2×C42—22no—
411C2×C42—22no—
412C2×C42C2×D411yescharacteristic
413C2×C42C2×D411yescharacteristic
414C2×C42C2×D411yescharacteristic
415C2×C42C2×D411yescharacteristic
416C2×C42C2×D411yescharacteristic
417C2×C42C2×D411yescharacteristic
418C2×C42C2×D411yescharacteristic
419C2×C42C2×D411yescharacteristic
420C2×C42C2×D411yescharacteristic
421C2×C42C2×D411yescharacteristic
422C2×C42C2×D414yes—
423C2×C42#4114yes—
424C2×C42C2411yescharacteristic
425C2×C42C2411yescharacteristic
426C2×C42C2411yescharacteristic
427C2×C42C2411yescharacteristic
428C2×C42C2411yescharacteristic
429C2×C42C2411yescharacteristic
430#107—28no—
431#106—28no—
432C4⋊C8—24no—
433C4⋊C8—24no—
434C4⋊C8—24no—
435C4⋊C8#4118yes—
436#103—28no—
437#103—28no—
438#103—28no—
439#103—28no—
440#103—28no—
441#103—28no—
442#103—44no—
443#103—44no—
444#97—24no—
445C4×C8—24no—
446C4×C8—22no—
447C4×C8—22no—
448C4×C8—22no—
449C4×C8—22no—
450#95—22no—
451#95—22no—
452#95—22no—
453#95—22no—
454#95—22no—
455#95—22no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—22no—
461#95—22no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—22no—
469#95—22no—
470#95—22no—
471#95—22no—
472#95C2×D411yescharacteristic
473#95C2×D411yescharacteristic
474#95C2×D411yescharacteristic
475#95C2×D411yescharacteristic
476#95C2×D411yescharacteristic
477#95C2×D411yescharacteristic
478#95C2×D411yescharacteristic
479#95C2×D411yescharacteristic
480C24C22×D411yescharacteristic
481C2×Q8—24no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—24no—
486C2×Q8—24no—
487C2×Q8—24no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—24no—
491C2×Q8—24no—
492C2×Q8—24no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—42no—
496C2×Q8—42no—
497C2×Q8—42no—
498C2×Q8—42no—
499C2×Q8—42no—
500C2×Q8—42no—
501C2×Q8C22×D414yes—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4#12114yes—
517C22×C4#12114yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D412yes—
528C22×C4C22×D411yescharacteristic
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532C22×C4C2×#4112yes—
533C22×C4C2511yescharacteristic
534Q16—416no—
535Q16—416no—
536M16—48no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—48no—
548C2×C8C2×#4114yes—
549C2×C8C2×#4114yes—
550C4⋊C4—24no—
551C4⋊C4—24no—
552C4⋊C4—24no—
553C4⋊C4—28no—
554C4⋊C4—28no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—24no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—24no—
564C4⋊C4—41no—
565C4⋊C4—41no—
566C4⋊C4—44no—
567C4⋊C4—44no—
568C4⋊C4—44no—
569C4⋊C4—42no—
570C4⋊C4—42no—
571C4⋊C4—42no—
572C4⋊C4—42no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—44no—
576C4⋊C4—44no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—44no—
580C4⋊C4—44no—
581C4⋊C4—44no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—42no—
585C4⋊C4—42no—
586C4⋊C4—82no—
587C4⋊C4—82no—
588C4⋊C4C22×D414yes—
589C22⋊C4—28no—
590C22⋊C4—22no—
591C22⋊C4—22no—
592C22⋊C4—41no—
593C22⋊C4—41no—
594C22⋊C4—44no—
595C22⋊C4—44no—
596C22⋊C4—82no—
597C42—28no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—22no—
606C42—22no—
607C42—22no—
608C42—22no—
609C42—44no—
610C42—44no—
611C42—44no—
612C42—44no—
613C42—82no—
614C42—82no—
615C42—82no—
616C42#12114yes—
617C42#12114yes—
618C42C22×D412yes—
619C42C22×D412yes—
620C42C2×#4112yes—
621C42C2×#4112yes—
622C42C2×#4114yes—
623C23—21no—
624C23—21no—
625C23—21no—
626C23—21no—
627C23—21no—
628C23—21no—
629C23C2×#12112yes—
630C23C23×D411yescharacteristic
631Q8—28no—
632Q8—44no—
633Q8—48no—
634Q8—44no—
635Q8—44no—
636Q8—44no—
637Q8—44no—
638Q8—84no—
639Q8—84no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—28no—
653C2×C4—28no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—21no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4C2×#12112yes—
697C2×C4C2×#12112yes—
698C2×C4C2×#12112yes—
699C2×C4C2×#12112yes—
700C2×C4C2×#12112yes—
701C2×C4#53212yes—
702C2×C4#53212yes—
703C2×C4#57712yes—
704C2×C4#57812yes—
705C2×C4#57812yes—
706C2×C4#57812yes—
707C2×C4C23×D411yescharacteristic
708C2×C4C22×#4112yes—
709C2×C4C22×#4111yescharacteristic
710C2×C4#58512yes—
711C8—24no—
712C8—24no—
713C8—42no—
714C8—42no—
715C22—22no—
716C22—22no—
717C22—22no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22C22×#12111yescharacteristic
723C22C2×#53211yescharacteristic
724C22#344911yescharacteristic
725C22C2×#57711yescharacteristic
726C22C2×#57811yescharacteristic
727C22C2×#57811yescharacteristic
728C22C2×#57811yescharacteristic
729C4—22no—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—44no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—44no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4#304414yes—
749C4#304614yes—
750C4D4×#4112yes—
751C4#348012yes—
752C2—42no—
753C2#3493711yescharacteristic
754C2C2×#304411yescharacteristic
755C2C2×#304611yescharacteristic
756C2#6071511yescharacteristic
757C2#6072311yescharacteristic
758C2#6100811yescharacteristic
759C2#6101311yescharacteristic
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
5. #28748C2semidirect
6. #28719C2semidirect
7. #28688C2semidirect
8. #28656C2semidirect
9. #28653C2semidirect
10. #28616C2semidirect
11. #28616C2semidirect
13. #28440C2semidirect
15. #23691C2semidirect
16. #21516C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31852
C2×#31702
#353222
#342252
#287482
#287192
#286882
#286562
#286532
#286162
#285882
#284402
#247642
#236912
#215162
#214092
#214002
C4×#4242