Finite Group Databasebeta

← C2×#24067 #6452889

Subgroups — C2×#24067 #6452889

Order 512.

3830 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C22×#484C211yesmaximal, characteristic, complemented
3C22×#481C211yesmaximal, characteristic
4C22×#389C211yesmaximal, characteristic, complemented
5#24067C2116yesmaximal, direct product factor
6C2×#2059C212yesmaximal, complemented
7C2×#2042C212yesmaximal, complemented
8C2×#1874C211yesmaximal, characteristic
9C2×#1872C211yesmaximal, characteristic, complemented
10C2×#1865C212yesmaximal, complemented
11C2×#1862C212yesmaximal, complemented
12C2×#1817C211yesmaximal, characteristic, complemented
13C2×#529—24no—
14C2×#529—28no—
15C2×#529—28no—
16C2×#529C2218yes—
17C22×#126—21no—
18C22×#125C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×#123—21no—
21C22×C22⋊Q8C2211yescharacteristic
22C22×#121C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×D4—21no—
25C22×C4×D4C2211yescharacteristic
26C22×#117—21no—
27C23×C4⋊C4—21no—
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C23×C42C2211yescharacteristic
31C2×#484—28no—
32C2×#484C2218yescomplemented
33C2×#481—28no—
34C2×#481C2218yes—
35C22×#107C2212yes—
36C22×#106C2212yes—
37C22×#103C2212yes—
38C22×#103C2212yes—
39C22×#102C2212yes—
40C22×#102C2212yes—
41C22×C22⋊C8C2211yescharacteristic
42C22×C22⋊C8C2211yescharacteristic
43C2×#389—21no—
44C2×#389—21no—
45C2×#389—24no—
46C2×#389C2212yes—
47C2×#389C2212yes—
48C2×#389C2218yescomplemented
49C22×#95—21no—
50#2059C22116yescomplemented
51#2042C22116yescomplemented
52#1874C2218yes—
53#1872C2218yescomplemented
54#1865C22116yescomplemented
55#1862C22116yescomplemented
56#1817C2218yescomplemented
57C2×#340C2212yes—
58C2×#340C2212yes—
59C2×#336C2212yes—
60C2×#336C2212yes—
61C23×Q8C2311yescharacteristic
62C23×D4—21no—
63C23×D4C2311yescharacteristic
64C24×C4—21no—
65C24×C4—21no—
66C24×C4—21no—
67C24×C4D411yescharacteristic
68C24×C4D411yescharacteristic
69C24×C4C2311yescharacteristic
70C23×C8C2312yes—
71C23×C8C2312yes—
72#529—264no—
73#529—232no—
74#529—232no—
75C2×#126—216no—
76C2×#126—28no—
77C2×#125—24no—
78C2×#125—24no—
79C2×#125—22no—
80C2×#125C2314yes—
81C2×#124—24no—
82C2×#124—24no—
83C2×#124—22no—
84C2×#124C2314yes—
85C2×#123—216no—
86C2×#123—28no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8D418yes—
90C2×C22⋊Q8C2314yes—
91C2×#121—24no—
92C2×#121—22no—
93C2×#121D418yes—
94C2×#121C2314yes—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—22no—
98C2×C4×Q8C2314yes—
99C2×C4×D4—216no—
100C2×C4×D4—24no—
101C2×C4×D4—28no—
102C2×C4×D4—24no—
103C2×C4×D4—22no—
104C2×C4×D4C2314yes—
105C2×#117—216no—
106C2×#117—28no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—28no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—28no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—48no—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C42—24no—
134C22×C42—24no—
135C22×C42—21no—
136C22×C42D411yescharacteristic
137C22×C42D411yescharacteristic
138C22×C42D414yes—
139C22×C42C2314yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#484—232no—
143#481—232no—
144C2×#107—28no—
145C2×#107C2318yes—
146C2×#106—28no—
147C2×#106C2318yes—
148C2×#103—28no—
149C2×#103—28no—
150C2×#103C2318yes—
151C2×#103C2318yes—
152C2×#102—28no—
153C2×#102—28no—
154C2×#102C2318yes—
155C2×#102C2318yes—
156C2×C22⋊C8—24no—
157C2×C22⋊C8—24no—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160#389—24no—
161#389—24no—
162#389—28no—
163#389—28no—
164#389C2318yes—
165#389C2318yes—
166C2×#95—22no—
167C2×#95—22no—
168C2×#95—28no—
169#340C2318yes—
170#340C2318yes—
171#336C2318yes—
172#336C2318yes—
173C25—21no—
174C25C2×D411yescharacteristic
175C22×Q8—22no—
176C22×Q8—21no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8C2×D414yes—
180C22×Q8C2412yes—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—21no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4C2×D414yes—
190C22×D4C2412yes—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4C2×D414yes—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D414yes—
221C23×C4C2×D412yes—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C22×C8—24no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8—24no—
232C22×C8—24no—
233C22×C8—24no—
234C22×C8C2414yes—
235C22×C8C2414yes—
236#126—264no—
237#125—216no—
238#125—28no—
239#125—28no—
240#124—216no—
241#124—28no—
242#124—28no—
243#123—264no—
244C22⋊Q8—216no—
245C22⋊Q8—28no—
246C22⋊Q8C2×D4116yes—
247#121—216no—
248#121—28no—
249#121C2×D4116yes—
250C4×Q8—216no—
251C4×Q8—28no—
252C4×Q8—28no—
253C4×D4—264no—
254C4×D4—216no—
255C4×D4—28no—
256C4×D4—28no—
257#117—264no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—216no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—216no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—432no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—48no—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4#4118yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2414yes—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C42—28no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—216no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—28no—
316C2×C42—28no—
317C2×C42C2×D414yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D414yes—
321C2×C42C2412yes—
322C2×C42C2412yes—
323#107—216no—
324#106—216no—
325#103—216no—
326#103—216no—
327#102—216no—
328#102—216no—
329C22⋊C8—28no—
330C22⋊C8—28no—
331#95—28no—
332#95—28no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24#12014yes—
346C24#13514yes—
347C24#13714yes—
348C24C22×D412yes—
349C24C22×D411yescharacteristic
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—216no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—48no—
358C2×Q8C22×D414yes—
359C2×D4—216no—
360C2×D4—24no—
361C2×D4—216no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—432no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—48no—
371C2×D4C22×D414yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—44no—
408C22×C4—416no—
409C22×C4—432no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—432no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—416no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—816no—
431C22×C4#12014yes—
432C22×C4#12018yes—
433C22×C4AGL(1,ℤ/8)14yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C2×#4114yes—
445C22×C42−1+414yes—
446C22×C4C2511yescharacteristic
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—48no—
454C2×C8—48no—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—416no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—416no—
472C4⋊C4C22×D414yes—
473C4⋊C4C22×D414yes—
474C4⋊C4C2×#4118yes—
475C22⋊C4—216no—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—216no—
480C42—216no—
481C42—28no—
482C42—24no—
483C42—24no—
484C42—28no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—22no—
493C23—22no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—48no—
501C23—42no—
502C23—42no—
503C23—48no—
504C23—48no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23—84no—
513C23#452116yes—
514C23C2×#12014yes—
515C23C2×#12012yes—
516C23#52512yes—
517C23#53612yes—
518C23C2×#13514yes—
519C23C2×#13512yes—
520C23C2×AGL(1,ℤ/8)12yes—
521C23C2×#13714yes—
522C23C2×#13712yes—
523C23#57512yes—
524C23C23×D411yescharacteristic
525Q8—28no—
526Q8—48no—
527Q8—44no—
528D4—28no—
529D4—432no—
530D4—48no—
531D4—44no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—216no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—416no—
555C2×C4—432no—
556C2×C4—432no—
557C2×C4—416no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—416no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—816no—
577C2×C4—88no—
578C2×C4C2×#12014yes—
579C2×C4C2×#12012yes—
580C2×C4#52512yes—
581C2×C4#53612yes—
582C2×C4#54718yes—
583C2×C4#54718yes—
584C2×C4C2×AGL(1,ℤ/8)14yes—
585C2×C4C23×D411yescharacteristic
586C2×C4C23×D411yescharacteristic
587C2×C4C22×#4112yes—
588C2×C4C2×2−1+412yes—
589C8—44no—
590C8—44no—
591C22—232no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—816no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22#240614yes—
607C22C2×#45218yes—
608C22C2×#45214yes—
609C22#300614yes—
610C22#310514yes—
611C22#311714yes—
612C22#323214yes—
613C22#323914yes—
614C22C22×#12011yescharacteristic
615C22C2×#52511yescharacteristic
616C22C2×#53611yescharacteristic
617C22C22×#13511yescharacteristic
618C22C22×AGL(1,ℤ/8)11yescharacteristic
619C22C22×#13711yescharacteristic
620C22C2×#57511yescharacteristic
621C4—24no—
622C4—24no—
623C4—48no—
624C4—44no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—42no—
629C4—44no—
630C4—84no—
631C4#240614yes—
632C4C2×#54712yes—
633C4C2×#54712yes—
634C2—28no—
635C2—84no—
636C2#2406718yesdirect product factor
637C2C2×#240611yescharacteristic
638C2C22×#45211yescharacteristic
639C2C2×#300611yescharacteristic
640C2C2×#310511yescharacteristic
641C2C2×#311711yescharacteristic
642C2C2×#323211yescharacteristic
643C2C2×#323911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
2. C22×#484C2semidirect
4. C22×#389C2semidirect
5. #24067C2direct (N × H)
6. C2×#2059C2semidirect
7. C2×#2042C2semidirect
9. C2×#1872C2semidirect
10. C2×#1865C2semidirect
11. C2×#1862C2semidirect
12. C2×#1817C2semidirect
32. C2×#484C22semidirect
48. C2×#389C22semidirect
50. #2059C22semidirect
51. #2042C22semidirect
53. #1872C22semidirect
54. #1865C22semidirect
55. #1862C22semidirect
56. #1817C22semidirect
636. C2#24067direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5292
C22×#4842
C22×#4812
C22×#3892
#240672
C2×#20592
C2×#20422
C2×#18742
C2×#18722
C2×#18652
C2×#18622
C2×#18172