Finite Group Databasebeta

← C2×#29718 #6432964

Subgroups — C2×#29718 #6432964

Order 512.

4076 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#450C211yesmaximal, characteristic, complemented
2C22×#447C211yesmaximal, characteristic
3C2×#2850C211yesmaximal, characteristic, complemented
4C22×#390C211yesmaximal, characteristic, complemented
5#29718C2116yesmaximal, direct product factor
6C2×#2073C212yesmaximal, complemented
7C2×#2055C212yesmaximal, complemented
8C2×#2054C212yesmaximal, complemented
9C2×#2012C212yesmaximal, complemented
10C2×#1874C211yesmaximal, characteristic, complemented
11C2×#1872C211yesmaximal, characteristic, complemented
12C2×#1797C211yesmaximal, characteristic, complemented
13C24×D4C2211yescharacteristic
14C23×SD16C2212yes—
15C23×D8C2212yes—
16C22×#124C2211yescharacteristic, complemented
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4—21no—
21C23×C22⋊C4—21no—
22C23×C42C2211yescharacteristic
23C2×#450—28no—
24C2×#450C2218yescomplemented
25C2×#447—28no—
26C2×#447C2218yes—
27C22×C4⋊C8C2212yescomplemented
28C22×#103C2212yescomplemented
29C22×#102C2212yes—
30C22×#102C2212yes—
31C22×#102C2212yescomplemented
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2850—24no—
35#2850C2218yescomplemented
36C2×#391C2211yescharacteristic, complemented
37C2×#390—21no—
38C2×#390—21no—
39C2×#390—24no—
40C2×#390C2212yes—
41C2×#390C2212yes—
42C2×#390C2218yescomplemented
43C2×#390C2211yescharacteristic, complemented
44C2×#388—21no—
45C2×#388—21no—
46C2×#386—21no—
47C2×#386—21no—
48C2×#382C2211yescharacteristic, complemented
49#2073C22116yescomplemented
50#2055C22116yescomplemented
51#2054C22116yescomplemented
52#2012C22116yescomplemented
53#1874C2218yescomplemented
54#1872C2218yescomplemented
55#1797C2218yescomplemented
56C2×#340C2212yescomplemented
57C2×#340C2212yescomplemented
58C2×#336C2212yescomplemented
59C26—21no—
60C23×Q8C2311yescharacteristic
61C23×D4—21no—
62C23×D4—22no—
63C23×D4—21no—
64C23×D4—24no—
65C23×D4—44no—
66C23×D4—44no—
67C23×D4D418yes—
68C23×D4C2311yescharacteristic
69C23×D4C2312yes—
70C23×D4C2314yes—
71C24×C4—21no—
72C24×C4D411yescharacteristic
73C24×C4D411yescharacteristic
74C24×C4C2311yescharacteristic
75C22×SD16—28no—
76C22×SD16—44no—
77C22×SD16—44no—
78C22×SD16C2318yes—
79C22×D8—28no—
80C22×D8—44no—
81C22×D8—44no—
82C22×D8C2318yes—
83C23×C8C2312yes—
84C23×C8C2312yes—
85C2×#124—22no—
86C2×#124—44no—
87C2×#124C2314yescomplemented
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8C2314yes—
92C2×#121—22no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#121C2314yes—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—22no—
107C22×C22⋊C4—22no—
108C22×C22⋊C4—22no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—48no—
117C22×C22⋊C4—48no—
118C22×C42—44no—
119C22×C42D414yes—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2314yes—
125#450—232no—
126#447—232no—
127C2×C4⋊C8—44no—
128C2×C4⋊C8C2318yescomplemented
129C2×#103—44no—
130C2×#103C2318yescomplemented
131C2×#102—28no—
132C2×#102—28no—
133C2×#102—44no—
134C2×#102C2318yes—
135C2×#102C2318yes—
136C2×#102C2318yescomplemented
137C2×C22⋊C8—24no—
138C2×C22⋊C8—24no—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141#391—22no—
142#391C2314yescomplemented
143#390—22no—
144#390—24no—
145#390—24no—
146#390—28no—
147#390—28no—
148#390C2318yes—
149#390C2318yes—
150#390C2314yescomplemented
151#388—28no—
152#388—28no—
153#386—28no—
154#386—28no—
155#382—22no—
156#382C2314yescomplemented
157C2×#95—21no—
158C2×#95—21no—
159C2×#95—21no—
160C2×#95—21no—
161#340C2318yescomplemented
162#340C2318yescomplemented
163#336C2318yescomplemented
164C25—28no—
165C25—24no—
166C25—22no—
167C25—21no—
168C25—44no—
169C25—44no—
170C25—44no—
171C25—44no—
172C25—44no—
173C25—44no—
174C25C2×D411yescharacteristic
175C22×Q8—21no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8C2412yes—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—216no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—21no—
188C22×D4—28no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—432no—
198C22×D4—48no—
199C22×D4—48no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—48no—
203C22×D4—48no—
204C22×D4—432no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4C2×D418yes—
209C22×D4C2×D4116yes—
210C22×D4C2412yes—
211C22×D4C2414yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D414yes—
232C23×C4C2×D412yes—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D414yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D414yes—
237C23×C4#4114yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C2×SD16—216no—
241C2×SD16—48no—
242C2×SD16—48no—
243C2×SD16—432no—
244C2×D8—216no—
245C2×D8—48no—
246C2×D8—48no—
247C2×D8—432no—
248C22×C8—24no—
249C22×C8—24no—
250C22×C8—24no—
251C22×C8—24no—
252C22×C8—24no—
253C22×C8—24no—
254C22×C8C2414yes—
255C22×C8C2414yes—
256#124—48no—
257#124—48no—
258C22⋊Q8—28no—
259C22⋊Q8—28no—
260C22⋊Q8—48no—
261#121—28no—
262#121—28no—
263#121—48no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—432no—
297C2×C22⋊C4—432no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—48no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—48no—
307C2×C42—48no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D414yes—
313C2×C42C2×D414yes—
314C2×C42C2412yes—
315C2×C42C2412yes—
316C4⋊C8—48no—
317#103—48no—
318#102—216no—
319#102—216no—
320#102—48no—
321C22⋊C8—28no—
322C22⋊C8—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—216no—
332C24—28no—
333C24—24no—
334C24—22no—
335C24—48no—
336C24—432no—
337C24—48no—
338C24—48no—
339C24—432no—
340C24—48no—
341C24—44no—
342C24—44no—
343C24—48no—
344C24—44no—
345C24—48no—
346C24—44no—
347C24—44no—
348C24—44no—
349C24—432no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—816no—
358C24—816no—
359C24—816no—
360C24—816no—
361C24#12114yes—
362C24#13514yes—
363C24AGL(1,ℤ/8)14yes—
364C24C22×D412yes—
365C24C22×D411yescharacteristic
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×D4—216no—
376C2×D4—28no—
377C2×D4—216no—
378C2×D4—216no—
379C2×D4—22no—
380C2×D4—24no—
381C2×D4—22no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—432no—
389C2×D4—432no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—464no—
393C2×D4—432no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—42no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—816no—
407C2×D4C22×D418yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—216no—
414C22×C4—24no—
415C22×C4—216no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—816no—
455C22×C4#12114yes—
456C22×C4#12118yes—
457C22×C4AGL(1,ℤ/8)14yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4112yes—
468C22×C4H(5,𝔽2)14yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C4C2511yescharacteristic
471SD16—432no—
472D8—432no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—48no—
480C2×C8—48no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C42—24no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—44no—
507C42—44no—
508C23—28no—
509C23—28no—
510C23—22no—
511C23—22no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—22no—
517C23—22no—
518C23—28no—
519C23—432no—
520C23—432no—
521C23—416no—
522C23—44no—
523C23—44no—
524C23—432no—
525C23—416no—
526C23—416no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—8128no—
539C23—816no—
540C23—816no—
541C23—816no—
542C23—816no—
543C23—88no—
544C23—88no—
545C23—88no—
546C23—88no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23#463116yes—
555C23C2×#12114yes—
556C23C2×#12112yes—
557C23#53412yes—
558C23#53512yes—
559C23C2×#13514yes—
560C23C2×#13512yes—
561C23C2×AGL(1,ℤ/8)14yes—
562C23C2×AGL(1,ℤ/8)12yes—
563C23C2×AGL(1,ℤ/8)12yes—
564C23#57512yes—
565C23C23×D411yescharacteristic
566Q8—44no—
567Q8—44no—
568Q8—84no—
569D4—28no—
570D4—28no—
571D4—432no—
572D4—416no—
573D4—44no—
574D4—44no—
575D4—44no—
576D4—44no—
577D4—816no—
578D4—84no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—216no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—816no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4C2×#12114yes—
620C2×C4C2×#12112yes—
621C2×C4#53412yes—
622C2×C4#53418yes—
623C2×C4#53512yes—
624C2×C4#53818yes—
625C2×C4C2×AGL(1,ℤ/8)14yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4C2×H(5,𝔽2)12yes—
630C8—44no—
631C8—44no—
632C22—232no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—48no—
640C22—48no—
641C22—48no—
642C22—864no—
643C22—832no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—168no—
656C22#261714yes—
657C22C2×#46318yes—
658C22C2×#46314yes—
659C22#303614yes—
660C22#318814yes—
661C22#319414yes—
662C22#321114yes—
663C22#321714yes—
664C22C22×#12111yescharacteristic
665C22C2×#53411yescharacteristic
666C22C2×#53511yescharacteristic
667C22C22×#13511yescharacteristic
668C22C22×AGL(1,ℤ/8)11yescharacteristic
669C22C22×AGL(1,ℤ/8)11yescharacteristic
670C22C2×#57511yescharacteristic
671C4—24no—
672C4—24no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—84no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#261714yes—
681C4C2×#53412yes—
682C4C2×#53812yes—
683C2—28no—
684C2—88no—
685C2—162no—
686C2#2971818yesdirect product factor
687C2C2×#261711yescharacteristic
688C2C22×#46311yescharacteristic
689C2C2×#303611yescharacteristic
690C2C2×#318811yescharacteristic
691C2C2×#319411yescharacteristic
692C2C2×#321111yescharacteristic
693C2C2×#321711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#450C2semidirect
3. C2×#2850C2semidirect
4. C22×#390C2semidirect
5. #29718C2direct (N × H)
6. C2×#2073C2semidirect
7. C2×#2055C2semidirect
8. C2×#2054C2semidirect
9. C2×#2012C2semidirect
10. C2×#1874C2semidirect
11. C2×#1872C2semidirect
12. C2×#1797C2semidirect
16. C22×#124C22semidirect
24. C2×#450C22semidirect
27. C22×C4⋊C8C22semidirect
28. C22×#103C22semidirect
31. C22×#102C22semidirect
35. #2850C22semidirect
36. C2×#391C22semidirect
42. C2×#390C22semidirect
43. C2×#390C22semidirect
48. C2×#382C22semidirect
49. #2073C22semidirect
50. #2055C22semidirect
51. #2054C22semidirect
52. #2012C22semidirect
53. #1874C22semidirect
54. #1872C22semidirect
55. #1797C22semidirect
56. C2×#340C22semidirect
57. C2×#340C22semidirect
58. C2×#336C22semidirect
87. C2×#124C23semidirect
128. C2×C4⋊C8C23semidirect
130. C2×#103C23semidirect
136. C2×#102C23semidirect
142. #391C23semidirect
150. #390C23semidirect
156. #382C23semidirect
161. #340C23semidirect
162. #340C23semidirect
163. #336C23semidirect
686. C2#29718direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4502
C22×#4472
C2×#28502
C22×#3902
#297182
C2×#20732
C2×#20552
C2×#20542
C2×#20122
C2×#18742
C2×#18722
C2×#17972