Finite Group Databasebeta

← #373669

Subgroups — #373669

Order 512.

2518 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#417C211yesmaximal, characteristic
2C2×#2341C211yesmaximal, characteristic
3C2×#1865C212yesmaximal, complemented
4C2×#1853C211yesmaximal, characteristic, complemented
5C2×#1788C211yesmaximal, characteristic
6#14130C211yesmaximal, characteristic, complemented
7#11121C212yesmaximal, complemented
8#11066C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10722C211yesmaximal, characteristic, complemented
11#10715C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#106C2212yes—
18C2×#417—28no—
19C2×#417—24no—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25#2341—24no—
26#2341—24no—
27C2×#390—21no—
28C2×#390—21no—
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#389C2212yes—
32C2×#385—21no—
33C2×#380—21no—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37C2×C4×C22⋊C4C2211yescharacteristic
38#1865—28no—
39#1853—24no—
40#1788C418yescomplemented
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1444—21no—
47#1442—21no—
48C4⋊#95C2212yes—
49C22⋊#95—21no—
50C22×#41—22no—
51C22×#41—22no—
52C22×#41—22no—
53C22×#41—24no—
54C22×#41—24no—
55C22×#41—42no—
56C22×#41—42no—
57C22×#41C2×C414yes—
58C22×#41D414yes—
59C22×#41Q814yes—
60C23×Q8C2311yescharacteristic
61C23×D4—21no—
62C23×D4C2311yescharacteristic
63C24×C4—21no—
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C22×M16—22no—
67C22×M16—22no—
68C22×M16—22no—
69C22×M16—24no—
70C22×M16—42no—
71C23×C8C2312yes—
72C23×C8C2312yes—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—24no—
78C2×C4×Q8—24no—
79C2×C4×Q8—42no—
80C2×C4×Q8—42no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4D411yescharacteristic
100C22×C22⋊C4D411yescharacteristic
101C22×C42—21no—
102C22×C42—21no—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C2×#106—24no—
106C2×#106—24no—
107C2×#106—42no—
108#417—232no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103C2×C418yescomplemented
114C2×#103D418yes—
115C2×#102—24no—
116C2×#102—24no—
117C2×#102—24no—
118C2×#102—42no—
119C2×#102—42no—
120C2×#102C2×C418yescomplemented
121C2×C22⋊C8—22no—
122C2×C22⋊C8—22no—
123C2×C22⋊C8C2×C414yes—
124#390—28no—
125#390—28no—
126#390—22no—
127#390—22no—
128#390—22no—
129#390—22no—
130#389—24no—
131#389—42no—
132#385—44no—
133#380—28no—
134#380—22no—
135#380—22no—
136#380C2×C414yes—
137#380D414yes—
138C4×C22⋊C4—28no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95D411yescharacteristic
147C2×#95D411yescharacteristic
148#340—24no—
149#340—24no—
150#340—42no—
151#336—24no—
152#336—42no—
153C25—21no—
154C25C2×D411yescharacteristic
155C2×#41—28no—
156C2×#41—28no—
157C2×#41—28no—
158C2×#41—216no—
159C2×#41—28no—
160C2×#41—216no—
161C2×#41—28no—
162C2×#41—48no—
163C2×#41—44no—
164C2×#41—44no—
165C2×#41—48no—
166C2×#41—48no—
167C2×#41—48no—
168C2×#41—48no—
169C2×#41—48no—
170C2×#41—48no—
171C2×#41—48no—
172C2×#41C4⋊C4116yescomplemented
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8C22×C412yes—
179C22×Q8C2×D412yes—
180C22×Q8C2×D414yes—
181C22×Q8C2×Q812yes—
182C22×Q8#4114yes—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—21no—
188C22×D4—22no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4C22×C412yes—
196C22×D4C2×D412yes—
197C22×D4C2×Q812yes—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—82no—
217C23×C4C22×C412yes—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D412yes—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×Q812yes—
226C23×C4#4112yes—
227C23×C4#4112yes—
228C23×C4C2411yescharacteristic
229C2×M16—28no—
230C2×M16—48no—
231C2×M16—48no—
232C2×M16—44no—
233C2×M16—44no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8C22×C414yes—
247C22×C8C2×D414yes—
248C4×Q8—48no—
249C4×Q8—48no—
250C4×Q8—44no—
251C4×Q8—44no—
252C4×Q8—44no—
253C4×Q8—44no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4C22×C414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—82no—
305C2×C22⋊C4—82no—
306C2×C22⋊C4—82no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—44no—
318C2×C42—44no—
319#106—48no—
320#103—216no—
321#103—216no—
322#102—216no—
323#102—48no—
324C22⋊C8—28no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—44no—
330#95—44no—
331#95—82no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—42no—
344C24—42no—
345C24—41no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—42no—
350C24—42no—
351C24C4×D412yes—
352C24#12012yes—
353C24C22⋊Q812yes—
354C24C22×D411yescharacteristic
355#41—232no—
356#41—416no—
357#41—432no—
358#41—416no—
359#41—432no—
360#41—416no—
361C2×Q8—22no—
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8C2×C4⋊C414yes—
375C2×Q8C4×D418yes—
376C2×Q8#12014yes—
377C2×Q8C22⋊Q818yes—
378C2×Q8#12314yes—
379C2×D4—216no—
380C2×D4—24no—
381C2×D4—22no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—416no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—416no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4C2×C4⋊C414yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—216no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—416no—
450C22×C4—48no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—42no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4C2×C4⋊C414yes—
476C22×C4C4×D414yes—
477C22×C4C4×D412yes—
478C22×C4#12012yes—
479C22×C4C22⋊Q814yes—
480C22×C4C22⋊Q812yes—
481C22×C4#12314yes—
482C22×C4AGL(1,ℤ/8)12yes—
483C22×C4#13712yes—
484C22×C4C23×C411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×Q811yescharacteristic
489C22×C4C2×#4112yes—
490C22×C4C2×#4111yescharacteristic
491C22×C4C2×#4111yescharacteristic
492C22×C4H(5,𝔽2)12yes—
493C22×C4H(5,𝔽2)12yes—
494M16—416no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—42no—
502C2×C8—42no—
503C2×C8—42no—
504C2×C8—42no—
505C2×C8—42no—
506C2×C8—42no—
507C2×C8—42no—
508C2×C8—42no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—42no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—88no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C42—48no—
536C42—48no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—42no—
542C42—42no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—24no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—24no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—44no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—82no—
581C23—82no—
582C23—82no—
583C23—82no—
584C23—82no—
585C23—82no—
586C23—82no—
587C23—82no—
588C23#38514yes—
589C23C2×C4×D411yescharacteristic
590C23C2×#12011yescharacteristic
591C23C2×C22⋊Q811yescharacteristic
592C23D4212yes—
593C23#54612yes—
594C23C2×AGL(1,ℤ/8)11yescharacteristic
595C23C2×#13711yescharacteristic
596C23#57511yescharacteristic
597C23#57511yescharacteristic
598C23#57712yes—
599C23#57712yes—
600Q8—24no—
601Q8—28no—
602Q8—24no—
603Q8—28no—
604Q8—24no—
605Q8#38518yescomplemented
606D4—24no—
607D4—24no—
608D4—432no—
609D4—44no—
610D4—42no—
611D4—44no—
612D4—42no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—28no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—432no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—816no—
662C2×C4—82no—
663C2×C4—82no—
664C2×C4—82no—
665C2×C4—82no—
666C2×C4—82no—
667C2×C4—82no—
668C2×C4—82no—
669C2×C4—82no—
670C2×C4—82no—
671C2×C4—82no—
672C2×C4—82no—
673C2×C4—82no—
674C2×C4#38514yes—
675C2×C4C22×C4⋊C411yescharacteristic
676C2×C4C2×C4×D411yescharacteristic
677C2×C4C2×C4×D412yes—
678C2×C4#51812yes—
679C2×C4C2×#12011yescharacteristic
680C2×C4C2×C22⋊Q811yescharacteristic
681C2×C4C2×C22⋊Q812yes—
682C2×C4C2×#12311yescharacteristic
683C2×C4C2×#12311yescharacteristic
684C2×C4#53411yescharacteristic
685C2×C4#53411yescharacteristic
686C2×C4#54312yes—
687C2×C4D4212yes—
688C2×C4#54612yes—
689C2×C4#54612yes—
690C2×C4#54612yes—
691C2×C4#55112yes—
692C2×C4#55112yes—
693C8—44no—
694C8—44no—
695C22—24no—
696C22—24no—
697C22—24no—
698C22—42no—
699C22—42no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—42no—
705C22—42no—
706C22—44no—
707C22—44no—
708C22—44no—
709C22—82no—
710C22—82no—
711C22—84no—
712C22—84no—
713C22—84no—
714C22—84no—
715C22—82no—
716C22—82no—
717C22C2×#38511yescharacteristic
718C22#235812yes—
719C22#240311yescharacteristic
720C22#240511yescharacteristic
721C22#243712yes—
722C22C4×AGL(1,ℤ/8)11yescharacteristic
723C22C4×#13711yescharacteristic
724C22#295212yes—
725C22#300611yescharacteristic
726C22#300711yescharacteristic
727C22#300911yescharacteristic
728C22#301011yescharacteristic
729C22#307911yescharacteristic
730C22#308011yescharacteristic
731C22#308412yes—
732C22#326512yes—
733C22#327112yes—
734C22#327512yes—
735C22#327812yes—
736C22#331612yes—
737C22#331712yes—
738C22#332612yes—
739C22#333212yes—
740C4—24no—
741C4—28no—
742C4—42no—
743C4—42no—
744C4—84no—
745C4—81no—
746C4—82no—
747C4—84no—
748C4—81no—
749C4—82no—
750C4C2×#38511yescharacteristic
751C4#231511yescharacteristic
752C4#235812yes—
753C4#240511yescharacteristic
754C4#243712yes—
755C4#248311yescharacteristic
756C2—44no—
757C2—84no—
758C2#1412711yescharacteristic
759C2#2087511yescharacteristic
760C2#2088211yescharacteristic
761C2#2286712yes—
762C2#2288312yes—
763C2#2386611yescharacteristic
764C2#2387011yescharacteristic
765C2#2404511yescharacteristic
766C2#2405211yescharacteristic
767C2#2596712yes—
768C2#2597712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1865C2semidirect
4. C2×#1853C2semidirect
6. #14130C2semidirect
7. #11121C2semidirect
8. #11066C2semidirect
9. #10789C2semidirect
10. #10722C2semidirect
11. #10715C2semidirect
40. #1788C4semidirect
113. C2×#103C2×C4semidirect
120. C2×#102C2×C4semidirect
172. C2×#41C4⋊C4semidirect
605. Q8#385semidirect

Maximal subgroups

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

subgroupindex
C22×#4172
C2×#23412
C2×#18652
C2×#18532
C2×#17882
#141302
#111212
#110662
#107892
#107222
#107152