Finite Group Databasebeta

← #6375879

Subgroups — #6375879

Order 512.

2154 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3044C211yesmaximal, characteristic
2#34276C211yesmaximal, characteristic
3#34254C211yesmaximal, characteristic, complemented
4#22164C214yesmaximal, complemented
5#22163C214yesmaximal, complemented
6#22150C214yesmaximal, complemented
7#22148C214yesmaximal, complemented
8#21262C211yesmaximal, characteristic
9#21237C211yesmaximal, characteristic
10C2×#1969C211yesmaximal, characteristic, complemented
11#21173C211yesmaximal, characteristic, complemented
12#21094C211yesmaximal, characteristic, complemented
13#21083C211yesmaximal, characteristic
14#20935C211yesmaximal, characteristic, complemented
15#20931C211yesmaximal, characteristic
16#20881C211yesmaximal, characteristic
17#20877C211yesmaximal, characteristic, complemented
18#20651C211yesmaximal, characteristic, complemented
19C4×#424C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C23×Q16C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#519C2211yescharacteristic
26#3044C4116yescomplemented
27C2×#464C2214yes—
28C2×#462C2214yes—
29C2×#428C2211yescharacteristic
30C22×#107C2211yescharacteristic
31C2×#424C2211yescharacteristic
32C2×#424C2211yescharacteristic
33C2×#417C2211yescharacteristic
34C2×#419C2211yescharacteristic
35C2×#419C2211yescharacteristic
36C2×#419C2211yescharacteristic
37C2×#416C2211yescharacteristic
38C2×#416C2211yescharacteristic
39C22×#103C2211yescharacteristic
40C22×#103C2211yescharacteristic
41C2×C4×M16C2211yescharacteristic
42C22×C4×C8C2211yescharacteristic
43#2362—22no—
44#2360—22no—
45#2359—24no—
46#2356—22no—
47#2355—22no—
48#2343—24no—
49#2327C2212yes—
50#2326C2212yes—
51#2320C2214yes—
52#2317C2211yescharacteristic
53#2317C2211yescharacteristic
54#2310C2211yescharacteristic
55C4×C4⋊Q8C2212yes—
56C4×#124C2212yes—
57#2297C2211yescharacteristic
58#2296C2211yescharacteristic
59#2295C2214yes—
60C2×#391C2211yescharacteristic
61C2×#391C2211yescharacteristic
62#2274C2211yescharacteristic
63C4×#117C2211yescharacteristic
64#1983C2214yes—
65#1981C2214yes—
66#1973C2212yes—
67#1971C2212yes—
68#1969C2212yes—
69#1969C2214yes—
70#1969C2214yes—
71#1968C2212yes—
72#1949C2214yes—
73#1947C2214yes—
74#1935C2214yes—
75#1934C2212yes—
76#1934C2212yes—
77#1933C2212yes—
78#1933C2212yes—
79#1933C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1877C2214yes—
84#1865C2214yes—
85#1863C2214yes—
86#1863C2214yes—
87#1862C2214yes—
88#1805C2214yes—
89#1803C2214yes—
90#1768C2214yes—
91C4×C4⋊C8C2214yes—
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1735C2211yescharacteristic
95#1735C2211yescharacteristic
96#1729C2211yescharacteristic
97#1729C2211yescharacteristic
98C22×#41C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C23×Q8C2311yescharacteristic
101C2×#137—42no—
102C2×#137—42no—
103C2×#137C2×C418yes—
104C22×Q16—42no—
105C22×Q16—42no—
106C22×Q16C2×C418yescomplemented
107C22×Q16C2314yes—
108C22×Q16C2312yes—
109C22×SD16C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#536—24no—
115#536C2×C418yes—
116C2×C4⋊Q8C2312yes—
117C2×#124C2312yes—
118C2×#123—22no—
119C2×C22⋊Q8—22no—
120#520—42no—
121#520—42no—
122#519—24no—
123#519—42no—
124#519—42no—
125#519C2×C418yescomplemented
126C2×C4×Q8—22no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8C2314yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4—22no—
134C2×C4×D4C2312yes—
135C2×#117—21no—
136C2×#117—22no—
137C2×#117—22no—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C22×C4⋊C4—21no—
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C42C2311yescharacteristic
147#464C2×C4132yescomplemented
148#462C2×C4132yescomplemented
149#428—42no—
150C2×#107—42no—
151C2×#107C2314yes—
152C2×#107C2312yes—
153C2×#106C2312yes—
154#424—24no—
155#424C2×C418yescomplemented
156C2×C4⋊C8C2314yes—
157C2×C4⋊C8C2314yes—
158#419—42no—
159#419—42no—
160#419C2×C418yescomplemented
161#417—42no—
162#416—42no—
163#416C2×C418yes—
164C2×#103—42no—
165C2×#103—42no—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2314yes—
171C2×#103C2312yes—
172C2×#103C2314yes—
173C2×#103C2312yes—
174C2×#103C2314yes—
175C2×#102C2312yes—
176C2×#102C2312yes—
177C2×#102C2312yes—
178C4×M16—42no—
179C2×#97C2312yes—
180C2×C4×C8—42no—
181C2×C4×C8C2312yes—
182C2×C4×C8C2312yes—
183C2×C4×C8C2312yes—
184#391—21no—
185#391—21no—
186#391—22no—
187#391—22no—
188#391C2311yescharacteristic
189#391C2312yes—
190#391C2314yes—
191#391C2312yes—
192#391C2312yes—
193#391C2311yescharacteristic
194#391C2312yes—
195#390C2311yescharacteristic
196#390C2311yescharacteristic
197#389—22no—
198#389—24no—
199#389C2314yes—
200#389C2312yes—
201#389C2312yes—
202#388—24no—
203#388—24no—
204#387—24no—
205#387—24no—
206#387—24no—
207#387—24no—
208#387—24no—
209#387—24no—
210#386—22no—
211#385—22no—
212#385—22no—
213#385—22no—
214#385—24no—
215#384—21no—
216#384C2312yes—
217#383—24no—
218#382—21no—
219#382—24no—
220#382C2312yes—
221#381—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—24no—
224C4×C4⋊C4—24no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4D412yes—
227C4×C4⋊C4D412yes—
228C4×C4⋊C4C2314yes—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—24no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4D414yes—
233#376—24no—
234#376C2312yes—
235C2×#95—22no—
236C2×#95—22no—
237C43C2312yes—
238#340C2312yes—
239#340C2314yes—
240#340C2312yes—
241#340C2314yes—
242#340C2312yes—
243#340C2312yes—
244#336C2314yes—
245#336C2314yes—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41C22×C414yes—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—22no—
264C22×Q8C22×C414yes—
265C22×Q8C22×C414yes—
266C22×Q8C2412yes—
267C22×Q8C2411yescharacteristic
268C22×Q8C2412yes—
269C22×D4—21no—
270C22×D4C2411yescharacteristic
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—22no—
277C23×C4C2×D412yes—
278C23×C4C2411yescharacteristic
279#137—416no—
280C2×Q16—416no—
281C2×Q16—44no—
282C2×Q16—44no—
283C2×Q16—42no—
284C2×Q16—42no—
285C2×Q16C22×C4116yes—
286C2×Q16C22×C418yes—
287C2×SD16—42no—
288C2×SD16—42no—
289C2×SD16C22×C418yes—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16C22×C414yes—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8C22×C414yes—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2412yes—
306C4⋊Q8—24no—
307C4⋊Q8C22×C418yes—
308#124—24no—
309#124C22×C418yes—
310#123—216no—
311C22⋊Q8—216no—
312C4×Q8—216no—
313C4×Q8—28no—
314C4×Q8—42no—
315C4×Q8—42no—
316C4×Q8—42no—
317C4×Q8—42no—
318C4×Q8—42no—
319C4×Q8—42no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×Q8C22×C4116yes—
323C4×D4—42no—
324C4×D4—42no—
325C4×D4—48no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—216no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—48no—
335#117—44no—
336#117C22×C414yes—
337#117C22×C414yes—
338#117#4114yes—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—44no—
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C42—24no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—24no—
383C2×C42C2×D414yes—
384C2×C42C2×D412yes—
385C2×C42C2×D414yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2412yes—
390#107—44no—
391#107—42no—
392#106—42no—
393C4⋊C8—28no—
394C4⋊C8C22×C4116yes—
395#103—44no—
396#103—44no—
397#103—42no—
398#103—42no—
399#103—44no—
400#103—42no—
401#103—42no—
402#103C22×C418yes—
403#103C22×C4116yes—
404#102—42no—
405#102—42no—
406#102C22×C418yes—
407#97—42no—
408C4×C8—42no—
409C4×C8—42no—
410C4×C8—42no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—22no—
416#95—24no—
417#95—24no—
418#95—22no—
419#95—24no—
420#95—24no—
421#95—22no—
422#95—24no—
423#95—22no—
424#95C2×D414yes—
425#95C2×D414yes—
426C24—21no—
427C24C22×D411yescharacteristic
428#41—28no—
429#41—28no—
430#41—44no—
431#41—44no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—28no—
438C2×Q8—28no—
439C2×Q8—28no—
440C2×Q8—28no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—21no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—21no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—44no—
458C2×Q8—44no—
459C2×Q8—44no—
460C2×Q8—44no—
461C2×Q8C23×C414yes—
462C2×Q8C23×C412yes—
463C2×Q8C23×C414yes—
464C2×D4—24no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4C23×C412yes—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—44no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—84no—
498C22×C4—82no—
499C22×C4—82no—
500C22×C4C23×C412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D414yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C2×#4112yes—
508C22×C4H(5,𝔽2)12yes—
509C22×C4H(5,𝔽2)12yes—
510C22×C4H(5,𝔽2)12yes—
511C22×C42−1+412yes—
512C22×C4C2511yescharacteristic
513Q16—416no—
514Q16—48no—
515SD16—48no—
516M16—28no—
517M16—44no—
518C2×C8—28no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—44no—
530C2×C8C23×C414yes—
531C2×C8C23×C414yes—
532C4⋊C4—216no—
533C4⋊C4—216no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—22no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—84no—
554C4⋊C4—84no—
555C4⋊C4—82no—
556C4⋊C4C4×D418yes—
557C4⋊C4C23×C414yes—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—216no—
561C22⋊C4—44no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—44no—
565C22⋊C4—82no—
566C22⋊C4C4×D418yes—
567C42—216no—
568C42—24no—
569C42—24no—
570C42—28no—
571C42—21no—
572C42—21no—
573C42—22no—
574C42—22no—
575C42—21no—
576C42—21no—
577C42—44no—
578C42—44no—
579C42—44no—
580C42—44no—
581C42—42no—
582C42—42no—
583C42—84no—
584C42C23×C414yes—
585C42C2×#4114yes—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—42no—
590C23—41no—
591C23—41no—
592C23—42no—
593C23C2×C4×D412yes—
594C23#53512yes—
595C23#53712yes—
596C23C23×D411yescharacteristic
597Q8—24no—
598Q8—24no—
599Q8—22no—
600Q8—24no—
601Q8—24no—
602Q8—22no—
603Q8—48no—
604Q8—48no—
605Q8—42no—
606Q8—42no—
607Q8—41no—
608Q8—42no—
609Q8—42no—
610Q8—41no—
611D4—22no—
612D4—22no—
613D4—48no—
614D4—41no—
615D4—41no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—48no—
634C2×C4—48no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—48no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—164no—
655C2×C4—162no—
656C2×C4—162no—
657C2×C4C2×C4×D418yes—
658C2×C4C2×C4×D412yes—
659C2×C4#51814yes—
660C2×C4#52014yes—
661C2×C4#53512yes—
662C2×C4#53712yes—
663C2×C4#54014yes—
664C2×C4#54214yes—
665C2×C4#57814yes—
666C2×C4C24×C411yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×H(5,𝔽2)11yescharacteristic
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C2×C4C2×2−1+411yescharacteristic
672C8—24no—
673C8—24no—
674C8—42no—
675C8—42no—
676C22—24no—
677C22—24no—
678C22—22no—
679C22—22no—
680C22—44no—
681C22—42no—
682C22—82no—
683C22—81no—
684C22—81no—
685C22#232414yes—
686C22C22×C4×D411yescharacteristic
687C22C2×#53511yescharacteristic
688C22C2×#53711yescharacteristic
689C22C2×#57812yes—
690C22#358412yes—
691C4—24no—
692C4—28no—
693C4—42no—
694C4—44no—
695C4—42no—
696C4—44no—
697C4—44no—
698C4—42no—
699C4—42no—
700C4—82no—
701C4—81no—
702C4—82no—
703C4—82no—
704C4—81no—
705C4#232414yes—
706C4C2×#51811yescharacteristic
707C4C2×#52011yescharacteristic
708C4C2×#54011yescharacteristic
709C4C2×#54211yescharacteristic
710C2—24no—
711C2—82no—
712C2C2×#232411yescharacteristic
713C2C4×#57812yes—
714C2#6081911yescharacteristic
715C2#6082111yescharacteristic
716C2#6093612yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C2×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. #34254C2semidirect
4. #22164C2semidirect
5. #22163C2semidirect
6. #22150C2semidirect
7. #22148C2semidirect
10. C2×#1969C2semidirect
11. #21173C2semidirect
12. #21094C2semidirect
14. #20935C2semidirect
17. #20877C2semidirect
18. #20651C2semidirect
19. C4×#424C2semidirect
26. #3044C4semidirect
106. C22×Q16C2×C4semidirect
125. #519C2×C4semidirect
147. #464C2×C4semidirect
148. #462C2×C4semidirect
155. #424C2×C4semidirect
160. #419C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30442
#342762
#342542
#221642
#221632
#221502
#221482
#212622
#212372
C2×#19692
#211732
#210942
#210832
#209352
#209312
#208812
#208772
#206512
C4×#4242