Finite Group Databasebeta

← #216713

Subgroups — #216713

Order 512.

1788 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C2×C4⋊#95C211yesmaximal, characteristic, complemented
4#9884C212yesmaximal, complemented
5#9680C212yesmaximal, complemented
6#9147C214yesmaximal, complemented
7#8577C214yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525—21no—
10C23×C4⋊C4—21no—
11C23×C42C2211yescharacteristic
12C2×#431—21no—
13#2857C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2856C411yescharacteristic, complemented
16#2856C2211yescharacteristic
17#2854—21no—
18#2854C412yescomplemented
19#2850—21no—
20#2850—21no—
21#2847—21no—
22#2847C2212yes—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121C412yescomplemented
26C4×#121C2212yes—
27#2297—21no—
28#2285C411yescharacteristic, complemented
29C22×#95—21no—
30C22×#95C2212yes—
31#1817—21no—
32#1797—21no—
33#1797C2212yes—
34#1779C2212yes—
35C4×C22⋊C8—21no—
36C4×C22⋊C8C2212yes—
37C2×#327C2218yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42#1294C2218yes—
43C22×#41—21no—
44C24×C4—22no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48#532—44no—
49C2×C4⋊Q8C2×C411yescharacteristic, complemented
50C2×#127C2×C411yescharacteristic, complemented
51C2×#124—21no—
52#525—44no—
53#525—44no—
54C2×#123—22no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121C2×C412yes—
63C2×#121C2×C412yes—
64C2×#121C2312yes—
65C2×#120—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C412yescomplemented
70C2×#117—21no—
71C22×C4⋊C4—28no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—44no—
75C22×C4⋊C4—44no—
76C22×C42—44no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#431—44no—
90C2×C4⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8C2314yes—
94C2×C4×C8—22no—
95#391—21no—
96#391—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390C2×C412yescomplemented
102#390C2×C412yescomplemented
103#389—21no—
104#389—21no—
105#389C2×C412yescomplemented
106#389C2×C412yescomplemented
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#384C2×C411yescharacteristic, complemented
118#384C2×C411yescharacteristic, complemented
119#382—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380C2×C412yes—
124#380C2×C412yes—
125#380C2312yes—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2×C412yescomplemented
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—44no—
134C2×#95—44no—
135C2×#95C2314yes—
136#336—22no—
137#336—22no—
138#327—48no—
139C25C2×D411yescharacteristic
140C2×#41—44no—
141C2×#41—44no—
142C2×#41—44no—
143C22×Q8—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C22×C412yes—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—28no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4D814yes—
178C23×C4SD1614yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2411yescharacteristic
189C22×C8—24no—
190C4⋊Q8—42no—
191#127—42no—
192#124—44no—
193#123—48no—
194C22⋊Q8—44no—
195C22⋊Q8—42no—
196C22⋊Q8—42no—
197#121—24no—
198#121—24no—
199#121—44no—
200#121—44no—
201#121—44no—
202#121—44no—
203#121—44no—
204#121—42no—
205#121—42no—
206#121—42no—
207#121—42no—
208#120—44no—
209#120—44no—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4—48no—
213C4×D4—44no—
214C4×D4—44no—
215#117—44no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—432no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C4⋊C8—44no—
274C22⋊C8—24no—
275C22⋊C8—24no—
276C22⋊C8—44no—
277C22⋊C8—44no—
278C4×C8—44no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—28no—
286#95—44no—
287#95—44no—
288#95—44no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24C2×C22⋊C411yescharacteristic
299C24C2×C22⋊C411yescharacteristic
300C24C2×D812yes—
301C24C2×SD1612yes—
302C24C22×D411yescharacteristic
303#41—816no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×D4—24no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C22×C4—216no—
329C22×C4—216no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—416no—
387C22×C4—44no—
388C22×C4—416no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—416no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—88no—
397C22×C4—88no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4#9914yes—
401C22×C4#10214yes—
402C22×C4#10214yes—
403C22×C4#10214yes—
404C22×C4#10214yes—
405C22×C4#10414yes—
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×D812yes—
417C22×C4C2×D812yes—
418C22×C4C2×SD1612yes—
419C22×C4C2×SD1612yes—
420C22×C4AGL(1,ℤ/8)12yes—
421C22×C4#13712yes—
422C22×C4C23×C411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4H(5,𝔽2)12yes—
425C22×C42−1+412yes—
426C2×C8—44no—
427C2×C8—44no—
428C2×C8—44no—
429C4⋊C4—24no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—416no—
433C4⋊C4—416no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—44no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23C2×#9912yes—
506C23#41012yes—
507C23C2×#10212yes—
508C23C2×#10212yes—
509C23C2×#10412yes—
510C23#42112yes—
511C23C22×C22⋊C411yescharacteristic
512C23#53511yescharacteristic
513C23#53611yescharacteristic
514C23C22×D811yescharacteristic
515C23C22×SD1611yescharacteristic
516C23C2×AGL(1,ℤ/8)11yescharacteristic
517C23C2×#13711yescharacteristic
518Q8—84no—
519D4—84no—
520D4—84no—
521D4—82no—
522D4—82no—
523D4—82no—
524D4—82no—
525D4—88no—
526D4—88no—
527D4—84no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—88no—
589C2×C4—832no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4C2×#9912yes—
595C2×C4C2×#9912yes—
596C2×C4C2×#10212yes—
597C2×C4C2×#10212yes—
598C2×C4C2×#10212yes—
599C2×C4C2×#10212yes—
600C2×C4#41712yes—
601C2×C4#41712yes—
602C2×C4C2×#10412yes—
603C2×C4C2×#10412yes—
604C2×C4#52012yes—
605C2×C4#53712yes—
606C8—84no—
607C22—28no—
608C22—28no—
609C22—28no—
610C22—28no—
611C22—22no—
612C22—22no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—21no—
618C22—21no—
619C22—22no—
620C22—21no—
621C22—21no—
622C22—22no—
623C22—22no—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—22no—
628C22—21no—
629C22—21no—
630C22—22no—
631C22—21no—
632C22—21no—
633C22—22no—
634C22—22no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22#150214yes—
643C22#151914yes—
644C22#231111yescharacteristic
645C22C22×#9911yescharacteristic
646C22C2×#41011yescharacteristic
647C22C22×#10211yescharacteristic
648C22C2×#41711yescharacteristic
649C22C22×#10411yescharacteristic
650C22C2×#42111yescharacteristic
651C22#320511yescharacteristic
652C22#320611yescharacteristic
653C22#322611yescharacteristic
654C22#322711yescharacteristic
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—41no—
665C4—41no—
666C4—88no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#150414yes—
671C2—21no—
672C2—21no—
673C2—21no—
674C2—21no—
675C2—24no—
676C2—24no—
677C2—162no—
678C2C2×#150211yescharacteristic
679C2C2×#150411yescharacteristic
680C2C2×#151911yescharacteristic
681C2#1758811yescharacteristic
682C2#2185911yescharacteristic
683C2#2187411yescharacteristic
684C2#2189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. C2×C4⋊#95C2semidirect
4. #9884C2semidirect
5. #9680C2semidirect
6. #9147C2semidirect
7. #8577C2semidirect
8. C2×#532C4semidirect
13. #2857C4semidirect
15. #2856C4semidirect
18. #2854C4semidirect
25. C4×#121C4semidirect
28. #2285C4semidirect
49. C2×C4⋊Q8C2×C4semidirect
50. C2×#127C2×C4semidirect
69. C2×C4×D4C2×C4semidirect
101. #390C2×C4semidirect
102. #390C2×C4semidirect
105. #389C2×C4semidirect
106. #389C2×C4semidirect
117. #384C2×C4semidirect
118. #384C2×C4semidirect
127. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#366242
#207142
C2×C4⋊#952
#98842
#96802
#91472
#85772