Finite Group Databasebeta

← #6519707

Subgroups — #6519707

Order 512.

2090 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3218C211yesmaximal, characteristic
2C2×#3108C211yesmaximal, characteristic
3C2×#2546C211yesmaximal, characteristic, complemented
4#34243C211yesmaximal, characteristic
5#28124C212yesmaximal, complemented
6#28118C212yesmaximal
7#25382C214yesmaximal, complemented
8#25290C214yesmaximal, complemented
9#24732C212yesmaximal, complemented
10#24730C212yesmaximal, complemented
11#24632C212yesmaximal, complemented
12#24630C212yesmaximal, complemented
13#24212C212yesmaximal, complemented
14#24209C212yesmaximal
15#21953C211yesmaximal, characteristic, complemented
16#21804C211yesmaximal, characteristic, complemented
17#20740C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C2×#529C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#3218—28no—
23C2×#497C2212yes—
24C2×#491C2212yes—
25#3108—28no—
26C2×#483C2211yescharacteristic
27C2×#480C2211yescharacteristic
28C2×#479C2212yes—
29C2×#464C2212yes—
30C2×#462C2212yes—
31C2×#462C2214yes—
32C2×#461C2212yes—
33C2×#461C2214yes—
34C2×#451C2212yes—
35C2×#451C2211yescharacteristic
36C2×#448C2212yes—
37C2×#448C2211yescharacteristic
38C2×C4×Q16C2212yes—
39C2×C4×SD16C2212yes—
40C2×#432C2211yescharacteristic
41C2×#431C2211yescharacteristic
42#2546—21no—
43#2546—22no—
44#2546—21no—
45#2546C2214yes—
46#2546C2212yes—
47#2546C2212yes—
48C2×#398—22no—
49#2370C2212yes—
50#2367C2212yes—
51#2359—22no—
52#2355C2211yescharacteristic
53#2354C2211yescharacteristic
54#2352—21no—
55Q8×C4⋊C4C2211yescharacteristic
56D4×C4⋊C4—21no—
57D4×C4⋊C4C2211yescharacteristic
58#2334—22no—
59#2314—21no—
60C4×C22⋊Q8C2212yes—
61C4×#121C2212yes—
62C2×#391C2212yes—
63C2×#389—21no—
64C2×#389—21no—
65#2285C2211yescharacteristic
66#2274—21no—
67C2×C4×C4⋊C4C2211yescharacteristic
68#2068C2214yes—
69#2066C2214yes—
70#2056C2214yes—
71#2055C2214yes—
72#2043C2214yes—
73#2043C2214yes—
74#2016C2214yes—
75#2013C2214yes—
76#1981C2214yes—
77#1981C2214yes—
78#1981C2214yes—
79#1956C2212yes—
80#1956C2212yes—
81#1940C2214yes—
82#1939C2212yes—
83#1939C2212yes—
84#1928C2214yes—
85#1927C2214yes—
86#1878C2214yes—
87#1877C2214yes—
88#1823C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2212yes—
92#1797C2212yes—
93#1787C2212yes—
94#1787C2212yes—
95#1787C2212yes—
96#1763C2214yes—
97C23×Q8C2311yescharacteristic
98C23×Q8C2312yes—
99C22×Q16C2312yes—
100C22×Q16C2314yes—
101C22×SD16C2312yes—
102C22×SD16C2314yes—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—24no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8—48no—
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2314yes—
113C2×C4⋊Q8C2314yes—
114#529—24no—
115#529D418yes—
116C2×#126—21no—
117C2×#125C2311yescharacteristic
118C2×#124C2311yescharacteristic
119C2×#123—21no—
120C2×C22⋊Q8C2311yescharacteristic
121C2×#121C2311yescharacteristic
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—24no—
125C2×C4×Q8—24no—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2311yescharacteristic
129C2×C4×Q8C2312yes—
130C2×C4×D4—21no—
131C2×C4×D4—22no—
132C2×C4×D4C2311yescharacteristic
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—22no—
140C22×C4⋊C4—22no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4D411yescharacteristic
144C22×C4⋊C4D411yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C42—21no—
147C22×C42C2312yes—
148C22×C42C2311yescharacteristic
149#497—28no—
150#491—28no—
151#483—24no—
152#480—24no—
153#479—28no—
154#464—28no—
155#462—28no—
156#462—48no—
157#461—28no—
158#461—48no—
159#451—24no—
160#451—28no—
161#448—24no—
162#448—28no—
163C4×Q16—28no—
164C4×SD16—28no—
165#432—24no—
166#431—24no—
167C2×#107C2312yes—
168C2×#106C2312yes—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2314yes—
172C2×C4⋊C8C2314yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2314yes—
177C2×#103C2314yes—
178C2×#103C2314yes—
179C2×#103C2314yes—
180C2×#103C2314yes—
181C2×#103C2314yes—
182C2×#103C2314yes—
183C2×#103C2314yes—
184C2×#102C2312yes—
185C2×C22⋊C8C2311yescharacteristic
186C2×C22⋊C8C2311yescharacteristic
187C2×C22⋊C8C2312yes—
188C2×#97C2312yes—
189C2×C4×C8C2312yes—
190#398—24no—
191#398—24no—
192#398—24no—
193#398—24no—
194#391—22no—
195#391—22no—
196#391C2314yes—
197#391C2312yes—
198#391C2314yes—
199#390C2312yes—
200#389—22no—
201#389—22no—
202#389—22no—
203#389—24no—
204#389—24no—
205#389—22no—
206#389—22no—
207#389—22no—
208#389C2312yes—
209#389C2312yes—
210#388—22no—
211#388—22no—
212#388—22no—
213#388—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#386—22no—
219#385—22no—
220#385—22no—
221#385—22no—
222#384—21no—
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#382—21no—
226#382—21no—
227#380—21no—
228#380—21no—
229#380—21no—
230#380C2312yes—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4D411yescharacteristic
237C4×C4⋊C4D412yes—
238C4×C4⋊C4D411yescharacteristic
239C4×C4⋊C4C2312yes—
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—22no—
248C2×#95—22no—
249C2×#95—22no—
250C2×#95D412yes—
251#340C2314yes—
252#340C2314yes—
253#336C2314yes—
254#336C2314yes—
255C22×Q8—24no—
256C22×Q8—21no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—24no—
260C22×Q8—22no—
261C22×Q8—24no—
262C22×Q8—42no—
263C22×Q8—44no—
264C22×Q8—44no—
265C22×Q8C2414yes—
266C22×Q8C2411yescharacteristic
267C22×Q8C2412yes—
268C22×D4—21no—
269C22×D4C2411yescharacteristic
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—22no—
273C23×C4—21no—
274C23×C4—22no—
275C23×C4C2×D412yes—
276C23×C4C2×D412yes—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2411yescharacteristic
280C2×Q16—24no—
281C2×Q16—24no—
282C2×Q16—28no—
283C2×Q16—42no—
284C2×Q16—44no—
285C2×Q16—44no—
286C2×SD16—24no—
287C2×SD16—24no—
288C2×SD16—28no—
289C2×SD16—42no—
290C2×SD16—44no—
291C2×SD16—44no—
292C22×C8C2414yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C4⋊Q8—28no—
296C4⋊Q8—28no—
297C4⋊Q8—24no—
298C4⋊Q8—24no—
299C4⋊Q8—24no—
300C4⋊Q8—24no—
301C4⋊Q8—44no—
302C4⋊Q8—44no—
303C4⋊Q8—48no—
304C4⋊Q8—48no—
305C4⋊Q8—44no—
306C4⋊Q8—44no—
307#126—28no—
308#125—22no—
309#125C2×D414yes—
310#124—22no—
311#124C2×D414yes—
312#123—28no—
313C22⋊Q8—22no—
314C22⋊Q8C2×D414yes—
315#121—22no—
316#121C2×D414yes—
317C4×Q8—22no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8—44no—
325C4×Q8C2×D414yes—
326C4×D4—28no—
327C4×D4—22no—
328C4×D4—44no—
329C4×D4—84no—
330C4×D4C2×D414yes—
331#117—28no—
332#117—44no—
333#117—82no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—48no—
361C2×C4⋊C4—48no—
362C2×C4⋊C4—44no—
363C2×C4⋊C4—44no—
364C2×C4⋊C4—44no—
365C2×C4⋊C4—44no—
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D412yes—
369C2×C4⋊C4C2×D412yes—
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2×D411yescharacteristic
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2411yescharacteristic
376C2×C4⋊C4C2411yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C42—21no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—44no—
394C2×C42—44no—
395C2×C42C2×D412yes—
396C2×C42C2×D412yes—
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D414yes—
400C2×C42#4114yes—
401C2×C42C2412yes—
402C2×C42C2412yes—
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405#107—24no—
406#106—24no—
407C4⋊C8—24no—
408C4⋊C8—24no—
409C4⋊C8—44no—
410C4⋊C8—44no—
411#103—24no—
412#103—28no—
413#103—24no—
414#103—24no—
415#103—28no—
416#103—28no—
417#103—44no—
418#103—44no—
419#103—44no—
420#103—44no—
421#103—44no—
422#102—24no—
423C22⋊C8—22no—
424C22⋊C8—22no—
425C22⋊C8—24no—
426#97—24no—
427C4×C8—24no—
428#95—24no—
429#95—24no—
430#95—24no—
431#95—24no—
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#95C2×D412yes—
441#95C2×D412yes—
442#95C2×D412yes—
443#95C2×D412yes—
444C24—21no—
445C24C22×D411yescharacteristic
446C2×Q8—24no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—24no—
451C2×Q8—22no—
452C2×Q8—24no—
453C2×Q8—22no—
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—24no—
460C2×Q8—24no—
461C2×Q8—48no—
462C2×Q8—48no—
463C2×Q8—42no—
464C2×Q8—44no—
465C2×Q8—44no—
466C2×Q8—42no—
467C2×Q8—44no—
468C2×Q8—88no—
469C2×Q8—88no—
470C2×Q8C22×D412yes—
471C2×Q8H(5,𝔽2)14yes—
472C2×D4—24no—
473C2×D4—21no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4C22×D412yes—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—42no—
508C22×C4—44no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—44no—
512C22×C4—82no—
513C22×C4—82no—
514C22×C4#12114yes—
515C22×C4#12114yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D411yescharacteristic
526C22×C4C2×#4112yes—
527C22×C4C2511yescharacteristic
528Q16—48no—
529Q16—88no—
530SD16—48no—
531SD16—88no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—24no—
535C2×C8—24no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—24no—
554C4⋊C4—24no—
555C4⋊C4—48no—
556C4⋊C4—48no—
557C4⋊C4—48no—
558C4⋊C4—48no—
559C4⋊C4—42no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4—44no—
566C4⋊C4—44no—
567C4⋊C4—42no—
568C4⋊C4—42no—
569C4⋊C4—81no—
570C4⋊C4—81no—
571C4⋊C4—82no—
572C4⋊C4—82no—
573C4⋊C4#13714yes—
574C4⋊C4#13714yes—
575C4⋊C4C22×D412yes—
576C4⋊C4C22×D412yes—
577C22⋊C4—24no—
578C22⋊C4—24no—
579C22⋊C4—24no—
580C22⋊C4—24no—
581C22⋊C4—24no—
582C22⋊C4—44no—
583C22⋊C4—44no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C42—24no—
587C42—24no—
588C42—24no—
589C42—42no—
590C42—42no—
591C42—44no—
592C42—44no—
593C42—44no—
594C42—82no—
595C42—82no—
596C42—82no—
597C42#12114yes—
598C42#12114yes—
599C42C22×D412yes—
600C42C22×D412yes—
601C42C2×#4112yes—
602C42C2×#4112yes—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—22no—
608C23—41no—
609C23—41no—
610C23—41no—
611C23—41no—
612C23—41no—
613C23—41no—
614C23C2×#12112yes—
615C23#57712yes—
616C23#57812yes—
617C23C23×D411yescharacteristic
618Q8—24no—
619Q8—24no—
620Q8—24no—
621Q8—48no—
622Q8—42no—
623Q8—42no—
624Q8—42no—
625Q8—44no—
626Q8—44no—
627Q8—44no—
628Q8—44no—
629Q8—88no—
630D4—24no—
631D4—48no—
632D4—42no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—28no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—22no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—22no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—28no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—24no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—48no—
660C2×C4—48no—
661C2×C4—48no—
662C2×C4—42no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—48no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—88no—
684C2×C4—84no—
685C2×C4—164no—
686C2×C4C2×#12112yes—
687C2×C4C2×#12112yes—
688C2×C4C2×#12112yes—
689C2×C4C2×#12112yes—
690C2×C4C2×#12112yes—
691C2×C4C2×#13714yes—
692C2×C4C2×#13714yes—
693C2×C4C2×#13712yes—
694C2×C4C2×#13712yes—
695C2×C4C23×D411yescharacteristic
696C2×C4C22×#4111yescharacteristic
697C2×C4C2×H(5,𝔽2)12yes—
698C2×C4#58512yes—
699C8—42no—
700C8—44no—
701C8—42no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22—81no—
715C22#304614yes—
716C22C22×#12111yescharacteristic
717C22#348411yescharacteristic
718C22#348711yescharacteristic
719C22C22×#13711yescharacteristic
720C22C22×#13711yescharacteristic
721C22C2×#57711yescharacteristic
722C22C2×#57811yescharacteristic
723C4—22no—
724C4—22no—
725C4—24no—
726C4—24no—
727C4—22no—
728C4—22no—
729C4—42no—
730C4—42no—
731C4—42no—
732C4—44no—
733C4—42no—
734C4—42no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—84no—
739C4#303914yes—
740C4#348812yes—
741C4#349412yes—
742C2—24no—
743C2—82no—
744C2#3556311yescharacteristic
745C2C2×#303911yescharacteristic
746C2C2×#304611yescharacteristic
747C2#6159311yescharacteristic
748C2#6160211yescharacteristic
749C2#6190011yescharacteristic
750C2#6190611yescharacteristic
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
3. C2×#2546C2semidirect
5. #28124C2semidirect
7. #25382C2semidirect
8. #25290C2semidirect
9. #24732C2semidirect
10. #24730C2semidirect
11. #24632C2semidirect
12. #24630C2semidirect
13. #24212C2semidirect
15. #21953C2semidirect
16. #21804C2semidirect
17. #20740C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32182
C2×#31082
C2×#25462
#342432
#281242
#281182
#253822
#252902
#247322
#247302
#246322
#246302
#242122
#242092
#219532
#218042
#207402