Finite Group Databasebeta

← #6811444

Subgroups — #6811444

Order 512.

2119 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C212yesmaximal, complemented
2D4×C4⋊Q8C211yesmaximal, characteristic
3#34929C211yesmaximal, characteristic
4#33163C211yesmaximal, characteristic, complemented
5#30530C214yesmaximal, complemented
6#30449C214yesmaximal, complemented
7#29664C212yesmaximal, complemented
8#28462C212yesmaximal, complemented
9#28440C211yesmaximal, characteristic, complemented
10#28255C211yesmaximal, characteristic, complemented
11#27769C212yesmaximal, complemented
12#25760C212yesmaximal
13#24652C212yesmaximal, complemented
14#24243C212yesmaximal
15#24212C212yesmaximal, complemented
16#22795C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic, complemented
18C2×D4×Q8C2212yes—
19C2×#533C2212yes—
20C22×C4⋊Q8C2212yes—
21#3209—216no—
22C2×#494C2214yes—
23C2×#494C2212yes—
24C2×#487C2214yes—
25C2×#462C2212yes—
26C2×#462C2212yes—
27C2×#462C2212yes—
28C2×#462C2212yes—
29C2×#462C2212yes—
30C2×#462C2212yes—
31C2×#462C2212yes—
32C2×#462C2211yescharacteristic
33C2×#451C2214yes—
34C2×#451C2214yes—
35C2×#451C2214yes—
36C2×C4×Q16C2212yes—
37C2×#431C2212yes—
38#2868C2211yescharacteristic
39#2857C2212yes—
40#2556—22no—
41#2551—21no—
42#2551—21no—
43#2551C2211yescharacteristic
44#2546C2212yes—
45#2546C2211yescharacteristic
46#2542C2211yescharacteristic
47#2538C2211yescharacteristic
48#2506—21no—
49#2492C2212yes—
50#2452—21no—
51#2440—22no—
52#2440—22no—
53#2434C2214yes—
54#2429C2212yes—
55#2396—22no—
56D4×C4⋊C4—21no—
57D4×C4⋊C4—21no—
58C4×C4⋊Q8—21no—
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×C22⋊Q8C2212yes—
62#2293—21no—
63C42×Q8C2211yescharacteristic
64C42×D4C2211yescharacteristic
65#2056C2214yes—
66#2043C2214yes—
67#2036C2214yes—
68#2024C2214yes—
69#2016C2214yes—
70#2016C2214yes—
71#2016C2214yes—
72#2001C2214yes—
73#2001C2214yes—
74#2001C2214yes—
75#1981C2212yes—
76#1981C2212yes—
77#1981C2212yes—
78#1981C2211yescharacteristic
79#1969C2214yes—
80#1956C2212yes—
81#1954C2212yes—
82#1947C2211yescharacteristic
83#1933C2214yes—
84#1928C2212yes—
85#1923C2214yes—
86#1878C2212yes—
87#1842C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1803C2211yescharacteristic
91#1797C2212yes—
92#1787C2214yes—
93C8×C4⋊C4C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#103C2212yes—
96C23×Q8C2314yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×Q16C2314yes—
100C22×Q16C2314yes—
101C22×Q16C2312yes—
102D4×Q8—44no—
103D4×Q8—44no—
104#533—28no—
105#533—44no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—28no—
109C2×C4⋊Q8—22no—
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8—44no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2314yes—
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2312yes—
120C2×#125—22no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8C2312yes—
124C2×C22⋊Q8C2312yes—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2314yes—
127C2×C4×Q8—22no—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2311yescharacteristic
131C2×C4×Q8C2311yescharacteristic
132C2×C4×Q8C2311yescharacteristic
133C2×C4×D4—22no—
134C2×C4×D4D411yescharacteristic
135C2×C4×D4D411yescharacteristic
136C2×C4×D4C2311yescharacteristic
137C2×#117—22no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4—22no—
140C22×C42C2312yes—
141#494—216no—
142#494—44no—
143#487—216no—
144#462—28no—
145#462—28no—
146#462—28no—
147#462—44no—
148#462—44no—
149#462—44no—
150#462—44no—
151#462D418yescomplemented
152#451—216no—
153#451—216no—
154#451—48no—
155C4×Q16—44no—
156#431—28no—
157C2×#107C2312yes—
158C2×C4⋊C8C2311yescharacteristic
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2311yescharacteristic
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2312yes—
174C2×C22⋊C8C2314yes—
175C2×C22⋊C8C2314yes—
176C2×C4×C8C2314yes—
177C2×C4×C8C2312yes—
178#400—22no—
179#399—22no—
180#398—22no—
181#398—22no—
182#398—24no—
183#396—22no—
184#396—22no—
185#395—22no—
186#395—24no—
187#395—24no—
188#393—22no—
189#391—22no—
190#391C2312yes—
191#391C2312yes—
192#391C2314yes—
193#391C2312yes—
194#390—21no—
195#390—21no—
196#389—21no—
197#389—21no—
198#389—22no—
199#389—22no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—22no—
207#389—22no—
208#389—22no—
209#389C2312yes—
210#388—22no—
211#388—22no—
212#387—22no—
213#385—22no—
214#385—24no—
215#385—24no—
216#384—21no—
217#384—21no—
218#384—21no—
219#384—21no—
220#384—21no—
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#380—22no—
224#380—22no—
225#380C2312yes—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4D411yescharacteristic
235C4×C4⋊C4D411yescharacteristic
236C4×C4⋊C4C2311yescharacteristic
237C4×C4⋊C4C2311yescharacteristic
238C4×C4⋊C4C2311yescharacteristic
239C4×C4⋊C4C2311yescharacteristic
240C4×C22⋊C4—22no—
241C4×C22⋊C4D412yes—
242C4×C22⋊C4D412yes—
243C4×C22⋊C4C2312yes—
244C43—21no—
245C43C2311yescharacteristic
246#340C2314yes—
247#340C2312yes—
248#340C2312yes—
249#336C2314yes—
250#336C2312yes—
251C22×Q8—22no—
252C22×Q8—24no—
253C22×Q8—28no—
254C22×Q8—44no—
255C22×Q8—44no—
256C22×Q8—44no—
257C22×Q8—44no—
258C22×Q8—44no—
259C22×Q8C2412yes—
260C22×Q8C2414yes—
261C22×Q8C2411yescharacteristic
262C22×Q8C2412yes—
263C22×D4C2×D411yescharacteristic
264C23×C4—24no—
265C23×C4C2×D412yes—
266C23×C4C2×D412yes—
267C23×C4C2412yes—
268C2×Q16—28no—
269C2×Q16—28no—
270C2×Q16—28no—
271C2×Q16—42no—
272C2×Q16—42no—
273C2×Q16—44no—
274C2×Q16—44no—
275C2×Q16—44no—
276C2×Q16—44no—
277C2×Q16—44no—
278C2×Q16—44no—
279C2×Q16—44no—
280C2×Q16—44no—
281C2×Q16—44no—
282C2×Q16C2×D418yes—
283C22×C8C2412yes—
284C22×C8C2414yes—
285C22×C8C2412yes—
286C4⋊Q8—24no—
287C4⋊Q8—28no—
288C4⋊Q8—24no—
289C4⋊Q8—24no—
290C4⋊Q8—22no—
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—42no—
297C4⋊Q8—48no—
298C4⋊Q8—48no—
299C4⋊Q8—44no—
300C4⋊Q8—42no—
301C4⋊Q8—42no—
302C4⋊Q8—42no—
303C4⋊Q8—48no—
304C4⋊Q8—44no—
305C4⋊Q8C2×D414yes—
306#125—48no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—416no—
311C22⋊Q8—42no—
312C22⋊Q8—42no—
313C22⋊Q8—42no—
314C22⋊Q8—44no—
315C22⋊Q8—44no—
316C22⋊Q8—48no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—24no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8—48no—
326C4×Q8—41no—
327C4×Q8C2×D414yes—
328C4×D4—24no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—48no—
332C4×D4Q1618yescomplemented
333#117—48no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
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—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—48no—
364C2×C4⋊C4—48no—
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2411yescharacteristic
371C2×C4⋊C4C2411yescharacteristic
372C2×C4⋊C4C2411yescharacteristic
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4C2×D412yes—
379C2×C22⋊C4C2×D412yes—
380C2×C22⋊C4C2×D412yes—
381C2×C42—21no—
382C2×C42—24no—
383C2×C42—24no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D412yes—
393C2×C42C2×D412yes—
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2412yes—
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401#107—42no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—22no—
405C4⋊C8—42no—
406C4⋊C8C2×D414yes—
407#103—28no—
408#103—28no—
409#103—28no—
410#103—24no—
411#103—28no—
412#103—42no—
413#103—42no—
414#103—44no—
415#103—44no—
416#103—44no—
417#103C2×D418yes—
418C22⋊C8—28no—
419C22⋊C8—44no—
420C4×C8—28no—
421C4×C8—24no—
422#95—22no—
423#95—22no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—24no—
434#95—24no—
435#95C2×D412yes—
436#95C2×D412yes—
437#95C2×D412yes—
438C24C22×D412yes—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—22no—
451C2×Q8—24no—
452C2×Q8—24no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—24no—
456C2×Q8—22no—
457C2×Q8—21no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—24no—
461C2×Q8—44no—
462C2×Q8—48no—
463C2×Q8—48no—
464C2×Q8—44no—
465C2×Q8—82no—
466C2×Q8—84no—
467C2×Q8—88no—
468C2×Q8—88no—
469C2×Q8—88no—
470C2×Q8—88no—
471C2×Q8C22×D414yes—
472C2×Q8C22×D412yes—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4C2×Q1614yes—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—42no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—84no—
499C22×C4C2×Q1618yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C2511yescharacteristic
510Q16—48no—
511Q16—88no—
512Q16—88no—
513Q16—88no—
514Q16—88no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—24no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—22no—
523C2×C8C22×D414yes—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—48no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—81no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4C2×Q1614yes—
563C4⋊C4#13714yes—
564C4⋊C4C22×D412yes—
565C4⋊C4H(5,𝔽2)12yes—
566C22⋊C4—24no—
567C22⋊C4—24no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—48no—
572C22⋊C4—82no—
573C22⋊C4C2×Q1618yes—
574C42—24no—
575C42—24no—
576C42—24no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—21no—
581C42—22no—
582C42—21no—
583C42—41no—
584C42—42no—
585C42—44no—
586C42—42no—
587C42—44no—
588C42—42no—
589C42—42no—
590C42—81no—
591C42C2×Q1614yes—
592C42C22×D412yes—
593C42H(5,𝔽2)12yes—
594C42H(5,𝔽2)12yes—
595C42H(5,𝔽2)12yes—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23C22×Q1614yes—
603C23C23×D411yescharacteristic
604Q8—22no—
605Q8—24no—
606Q8—22no—
607Q8—24no—
608Q8—41no—
609Q8—42no—
610Q8—41no—
611Q8—42no—
612Q8—42no—
613Q8—44no—
614Q8—42no—
615Q8—44no—
616Q8—42no—
617Q8—44no—
618Q8—42no—
619Q8—44no—
620Q8—88no—
621Q8—816no—
622D4—28no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—216no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—48no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—816no—
673C2×C4#53512yes—
674C2×C4#53512yes—
675C2×C4D4212yes—
676C2×C4D4212yes—
677C2×C4C22×Q1612yes—
678C2×C4C22×Q1614yes—
679C2×C4C22×Q1612yes—
680C2×C4C22×Q1612yes—
681C2×C4C2×#13714yes—
682C2×C4C2×#13712yes—
683C2×C4#57812yes—
684C2×C4C23×D411yescharacteristic
685C2×C4C2×H(5,𝔽2)11yescharacteristic
686C2×C4C2×H(5,𝔽2)11yescharacteristic
687C2×C4C2×H(5,𝔽2)11yescharacteristic
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4#58511yescharacteristic
690C8—24no—
691C8—44no—
692C8—42no—
693C22—24no—
694C22—24no—
695C22—24no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22C2×#53511yescharacteristic
701C22C2×D4211yescharacteristic
702C22#348811yescharacteristic
703C22C23×Q1611yescharacteristic
704C22C22×#13711yescharacteristic
705C22C2×#57811yescharacteristic
706C22#358611yescharacteristic
707C4—22no—
708C4—24no—
709C4—22no—
710C4—24no—
711C4—22no—
712C4—41no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—44no—
717C4—42no—
718C4—82no—
719C4—81no—
720C4—82no—
721C4—84no—
722C4—82no—
723C4#320914yescomplemented
724C4D4×Q1614yes—
725C4#349611yescharacteristic
726C4#349811yescharacteristic
727C4#354611yescharacteristic
728C4#354911yescharacteristic
729C2—44no—
730C2#3934211yescharacteristic
731C2C2×#320911yescharacteristic
732C2#6083711yescharacteristic
733C2C2×D4×Q1611yescharacteristic
734C2D4×#13711yescharacteristic
735C2#6195211yescharacteristic
736C2#6200011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3209C2semidirect
4. #33163C2semidirect
5. #30530C2semidirect
6. #30449C2semidirect
7. #29664C2semidirect
8. #28462C2semidirect
9. #28440C2semidirect
10. #28255C2semidirect
11. #27769C2semidirect
13. #24652C2semidirect
15. #24212C2semidirect
17. C4×#462C2semidirect
151. #462D4semidirect
332. C4×D4Q16semidirect
723. C4#3209semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
D4×C4⋊Q82
#349292
#331632
#305302
#304492
#296642
#284622
#284402
#282552
#277692
#257602
#246522
#242432
#242122
#227952
C4×#4622