Finite Group Databasebeta

← #6529799

Subgroups — #6529799

Order 512.

1891 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3108C211yesmaximal, characteristic
2C2×#2982C211yesmaximal, characteristic, complemented
3#35111C211yesmaximal, characteristic
4C2×#2581C211yesmaximal, characteristic, complemented
5#28268C212yesmaximal, complemented
6#28266C212yesmaximal, complemented
7#28190C212yesmaximal, complemented
8#28189C212yesmaximal, complemented
9#25474C214yesmaximal, complemented
10#25297C214yesmaximal, complemented
11#24601C212yesmaximal, complemented
12#24600C212yesmaximal
13#24503C212yesmaximal
14#24502C212yesmaximal, complemented
15#21983C211yesmaximal, characteristic, complemented
16#21861C211yesmaximal, characteristic
17#20735C211yesmaximal, characteristic, complemented
18C2×#529C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3108—28no—
22C2×#483C2211yescharacteristic
23C2×#480C2211yescharacteristic
24C2×#479C2212yes—
25C2×#464C2212yes—
26C2×#451C2211yescharacteristic
27C2×#448C2211yescharacteristic
28#2982—28no—
29C2×C8×Q8C2212yes—
30C2×C8×Q8C2212yes—
31C2×C8×Q8C2214yes—
32C2×C4×Q16C2212yes—
33C2×C4×SD16C2212yes—
34C2×#431C2212yes—
35C2×#431C2212yes—
36C2×#431C2211yescharacteristic
37C2×#431C2211yescharacteristic
38#2855C2212yes—
39#2854C2212yes—
40#2623—21no—
41#2614—21no—
42#2604—21no—
43#2604—21no—
44#2599—22no—
45#2592C2212yes—
46#2584C2212yes—
47#2581—22no—
48#2581—21no—
49#2581—21no—
50#2581C2212yes—
51#2581C2211yescharacteristic
52#2581C2212yes—
53#2581C2211yescharacteristic
54#2581C2214yes—
55#2580C2211yescharacteristic
56#2580C2211yescharacteristic
57C2×#400—21no—
58C2×#395—21no—
59#2322—22no—
60C2×#391C2212yes—
61C2×#389C2212yes—
62C2×#387—22no—
63#2283C2211yescharacteristic
64C2×#382C2211yescharacteristic
65#2088C2214yes—
66#2086C2214yes—
67#2068C2214yes—
68#2067C2214yes—
69#2066C2214yes—
70#2066C2214yes—
71#2056C2214yes—
72#2055C2214yes—
73#1991C2212yes—
74#1969C2214yes—
75#1968C2214yes—
76#1958C2212yes—
77#1949C2214yes—
78#1942C2214yes—
79#1941C2214yes—
80#1934C2214yes—
81#1933C2214yes—
82#1933C2214yes—
83#1933C2214yes—
84#1874C2212yes—
85#1872C2212yes—
86#1853C2212yes—
87#1852C2212yes—
88#1846C2212yes—
89#1835C2212yes—
90#1823C2212yes—
91#1805C2212yes—
92#1805C2212yes—
93#1805C2212yes—
94#1804C2212yes—
95#1787C2212yes—
96C4×#103C2214yes—
97C23×Q8D411yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×Q16C2312yes—
100C22×SD16C2312yes—
101#529—24no—
102#529—24no—
103C2×#126—21no—
104C2×#125C2311yescharacteristic
105C2×#124C2311yescharacteristic
106C2×#123—22no—
107C2×#123—21no—
108C2×C22⋊Q8C2311yescharacteristic
109C2×#121C2311yescharacteristic
110C2×C4×Q8—24no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—24no—
114C2×C4×Q8—24no—
115C2×C4×Q8—24no—
116C2×C4×Q8D412yes—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2312yes—
125C2×C4×D4—21no—
126C2×C4×D4C2311yescharacteristic
127C2×#117—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4D412yes—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C42—21no—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#483—24no—
138#480—24no—
139#479—28no—
140#464—28no—
141#451—24no—
142#448—24no—
143C8×Q8—216no—
144C8×Q8—28no—
145C8×Q8—28no—
146C4×Q16—28no—
147C4×SD16—28no—
148#431—24no—
149#431—28no—
150#431—28no—
151#431D418yes—
152C2×#107C2312yes—
153C2×#107C2314yes—
154C2×#106C2312yes—
155C2×#106C2314yes—
156C2×C4⋊C8C2314yes—
157C2×C4⋊C8C2314yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#102C2312yes—
167C2×#102C2314yes—
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2311yescharacteristic
170C2×C22⋊C8C2311yescharacteristic
171C2×C4×C8C2314yes—
172C2×C4×C8C2314yes—
173C2×C4×C8C2312yes—
174C2×C4×C8C2312yes—
175#400—24no—
176#400—22no—
177#400—22no—
178#400—21no—
179#400—21no—
180#399—21no—
181#399—21no—
182#398—22no—
183#397—22no—
184#395—22no—
185#395—22no—
186#395—24no—
187#395—21no—
188#395—21no—
189#395—22no—
190#393—21no—
191#393—21no—
192#393—22no—
193#391—22no—
194#391—22no—
195#391C2314yes—
196#391C2312yes—
197#391C2314yes—
198#391C2312yes—
199#390C2312yes—
200#390C2312yes—
201#389—22no—
202#389—22no—
203#389—22no—
204#389—22no—
205#389C2312yes—
206#389C2314yes—
207#389C2314yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#388—22no—
212#388—22no—
213#388—22no—
214#388—22no—
215#387—22no—
216#387—22no—
217#387—24no—
218#387—24no—
219#387—24no—
220#387—24no—
221#387—22no—
222#387—22no—
223#386—22no—
224#385—22no—
225#385—22no—
226#385—22no—
227#383—21no—
228#382—21no—
229#382—21no—
230#382C2312yes—
231#382C2311yescharacteristic
232#382C2311yescharacteristic
233#382C2312yes—
234#381—22no—
235#381—21no—
236#380C2312yes—
237C4×C4⋊C4C2312yes—
238C4×C22⋊C4—21no—
239#376—21no—
240C2×#95—21no—
241C2×#95—22no—
242C2×#95—22no—
243C2×#95—22no—
244C2×#95—21no—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251C22×Q8—24no—
252C22×Q8—24no—
253C22×Q8—28no—
254C22×Q8—21no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—42no—
258C22×Q8C2×D412yes—
259C22×Q8C2×D411yescharacteristic
260C22×Q8C2×D411yescharacteristic
261C22×Q8C2411yescharacteristic
262C22×Q8C2412yes—
263C22×D4—21no—
264C22×D4C2411yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—21no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D412yes—
271C23×C4C2×D412yes—
272C23×C4C2×D411yescharacteristic
273C23×C4C2411yescharacteristic
274C2×Q16—24no—
275C2×Q16—24no—
276C2×Q16—42no—
277C2×SD16—24no—
278C2×SD16—24no—
279C2×SD16—42no—
280C22×C8C2414yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283#126—44no—
284#125—22no—
285#125—22no—
286#124—22no—
287#124—22no—
288#123—44no—
289#123—48no—
290C22⋊Q8—22no—
291C22⋊Q8—22no—
292#121—22no—
293#121—22no—
294C4×Q8—28no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—22no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—22no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—22no—
309C4×Q8—44no—
310C4×Q8—44no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—44no—
314#117—44no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—48no—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—44no—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D414yes—
371C2×C42#4114yes—
372C2×C42C2412yes—
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—24no—
377#107—44no—
378#106—24no—
379#106—44no—
380C4⋊C8—28no—
381C4⋊C8—28no—
382C4⋊C8—24no—
383C4⋊C8C2×D418yes—
384#103—28no—
385#103—24no—
386#103—24no—
387#103—24no—
388#103—44no—
389#103—44no—
390#102—24no—
391#102—44no—
392C22⋊C8—24no—
393C22⋊C8C2×D414yes—
394C22⋊C8C2×D414yes—
395C4×C8—28no—
396C4×C8—28no—
397C4×C8—24no—
398C4×C8C2×D418yes—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—22no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423C24—21no—
424C24C22×D411yescharacteristic
425C2×Q8—232no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—48no—
444C2×Q8—48no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q82−1+414yes—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—42no—
453C2×D4—44no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—82no—
481C22×C4#12114yes—
482C22×C4#12114yes—
483C22×C4#13518yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C2×#4112yes—
495C22×C4C2511yescharacteristic
496Q16—48no—
497SD16—48no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8C22×D414yes—
506C2×C8C22×D414yes—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C22⋊C4—42no—
539C22⋊C4—44no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C42—24no—
549C42—24no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—82no—
558C42#12114yes—
559C42#12114yes—
560C42C22×D412yes—
561C42C22×D412yes—
562C42C2×#4112yes—
563C42C2×#4112yes—
564C23—22no—
565C23—21no—
566C23—21no—
567C23—42no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23C2×#12112yes—
573C23C2×#13514yes—
574C23C23×D411yescharacteristic
575Q8—216no—
576Q8—28no—
577Q8—28no—
578Q8—24no—
579Q8—24no—
580Q8—48no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585D4—42no—
586D4—42no—
587D4—84no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—28no—
595C2×C4—216no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—88no—
634C2×C4—84no—
635C2×C4C2×#12112yes—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4C2×#12112yes—
639C2×C4C2×#12112yes—
640C2×C4C2×#13514yes—
641C2×C4C2×#13514yes—
642C2×C4C23×D411yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×2−1+412yes—
645C2×C4#58512yes—
646C8—24no—
647C8—24no—
648C8—44no—
649C22—22no—
650C22—22no—
651C22—24no—
652C22—24no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—82no—
657C22—82no—
658C22#305014yes—
659C22C22×#12111yescharacteristic
660C22#348611yescharacteristic
661C22#348711yescharacteristic
662C22C22×#13512yes—
663C22#358612yes—
664C4—22no—
665C4—22no—
666C4—24no—
667C4—24no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—82no—
675C4—81no—
676C4—81no—
677C4—81no—
678C4—82no—
679C4—82no—
680C4—84no—
681C4#348912yes—
682C4#349512yes—
683C2—24no—
684C2—161no—
685C2#3566111yescharacteristic
686C2C2×#305011yescharacteristic
687C2#6045511yescharacteristic
688C2#6182212yes—
689C2#6188212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2982C2semidirect
4. C2×#2581C2semidirect
5. #28268C2semidirect
6. #28266C2semidirect
7. #28190C2semidirect
8. #28189C2semidirect
9. #25474C2semidirect
10. #25297C2semidirect
11. #24601C2semidirect
14. #24502C2semidirect
15. #21983C2semidirect
17. #20735C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31082
C2×#29822
#351112
C2×#25812
#282682
#282662
#281902
#281892
#254742
#252972
#246012
#246002
#245032
#245022
#219832
#218612
#207352