Finite Group Databasebeta

← C2×#18758 #6348672

Subgroups — C2×#18758 #6348672

Order 512.

4142 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#3227C211yesmaximal, characteristic, complemented
3C2×#3103C211yesmaximal, characteristic, complemented
4C2×#3036C211yesmaximal, characteristic
5#18758C2116yesmaximal, direct product factor
6C2×#1651C214yesmaximal, complemented
7C2×#1627C214yesmaximal, complemented
8C2×#1549C211yesmaximal, characteristic, complemented
9C2×#1487C211yesmaximal, characteristic, complemented
10C2×#1478C211yesmaximal, characteristic, complemented
11C24×Q8—21no—
12C22×AGL(1,ℤ/8)—21no—
13C2×D4×Q8—22no—
14C2×D4×Q8—24no—
15C2×D4×Q8—24no—
16C2×D4×Q8—22no—
17C2×D4×Q8—22no—
18C2×D4×Q8—24no—
19C2×D4×Q8C2214yes—
20C2×D4×Q8C2218yescomplemented
21C2×#532C2211yescharacteristic
22C22×C4⋊Q8—21no—
23C22×C4⋊Q8C2211yescharacteristic
24C22×C22⋊Q8—21no—
25C22×C22⋊Q8—22no—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4—21no—
28C22×C4×D4C2211yescharacteristic
29#3227—24no—
30#3227C2218yescomplemented
31C2×#498C2212yescomplemented
32C2×#486C2212yescomplemented
33#3103—24no—
34#3103C2218yescomplemented
35C2×#481—21no—
36C2×#481—22no—
37C2×#475C2212yescomplemented
38C2×#476—22no—
39#3036—24no—
40#3036C2218yes—
41C2×#460C2212yescomplemented
42C2×#463—22no—
43C2×#461C2212yes—
44C2×#459C2212yes—
45C2×#448—21no—
46C2×C4×SD16—22no—
47C2×#431C2211yescharacteristic
48C2×#431C2211yescharacteristic
49C2×#423C2211yescharacteristic
50C2×#417—21no—
51#1651C22132yescomplemented
52#1627C22132yescomplemented
53#1549C2218yescomplemented
54C2×#332C2214yescomplemented
55#1487C2218yescomplemented
56#1478C2218yescomplemented
57C2×Q8⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yes—
59C22×#41—21no—
60C23×Q8—22no—
61C23×Q8—28no—
62C23×Q8—24no—
63C23×Q8—24no—
64C23×Q8—28no—
65C23×Q8—21no—
66C23×Q8—416no—
67C23×Q8D411yescharacteristic
68C23×D4D411yescharacteristic
69C24×C4—22no—
70C24×C4—21no—
71C2×AGL(1,ℤ/8)—28no—
72C2×AGL(1,ℤ/8)—48no—
73C22×SD16—22no—
74C22×SD16—22no—
75C22×D8—22no—
76C22×M16—21no—
77D4×Q8—216no—
78D4×Q8—216no—
79D4×Q8—216no—
80D4×Q8—28no—
81D4×Q8—216no—
82D4×Q8—28no—
83D4×Q8—216no—
84D4×Q8—28no—
85D4×Q8—216no—
86D4×Q8C23116yes—
87#532—22no—
88#532—22no—
89#532—22no—
90#532C2314yes—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8D414yes—
100C2×C4⋊Q8D412yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2314yes—
104C2×#127C2311yescharacteristic
105C2×#124—21no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—216no—
109C2×C22⋊Q8—28no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—28no—
112C2×C22⋊Q8—28no—
113C2×C22⋊Q8—216no—
114C2×#121—21no—
115C2×#121—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8—22no—
120C2×C4×Q8D414yes—
121C2×C4×Q8D412yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2314yes—
124C2×C4×D4—28no—
125C2×C4×D4—24no—
126C2×C4×D4—24no—
127C2×C4×D4—28no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131C2×C4×D4—22no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4D412yes—
135C2×C4×D4D414yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2314yes—
138C22×C4⋊C4—22no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4D411yescharacteristic
142C22×C4⋊C4D412yes—
143C22×C4⋊C4D411yescharacteristic
144C22×C22⋊C4—22no—
145C22×C22⋊C4—21no—
146C22×C42—21no—
147C22×C42C2311yescharacteristic
148#498—24no—
149#498C2318yescomplemented
150#486—24no—
151#486C2318yescomplemented
152#481—28no—
153#481—216no—
154#476—216no—
155#475—24no—
156#475C2318yescomplemented
157#463—216no—
158#461—24no—
159#461C2318yes—
160#460—24no—
161#460C2318yescomplemented
162#459—24no—
163#459C2318yes—
164#448—28no—
165C4×SD16—216no—
166#431—22no—
167#431—22no—
168#431C2314yes—
169#431C2314yes—
170C2×#106—22no—
171C2×#106—24no—
172#423—22no—
173#423C2314yes—
174C2×C4⋊C8—21no—
175C2×C4⋊C8—21no—
176C2×C4⋊C8C2312yes—
177C2×C4⋊C8C2312yes—
178#417—28no—
179C2×#103—22no—
180C2×#103—22no—
181C2×#102—24no—
182C2×#102—22no—
183C2×#102—22no—
184C2×C22⋊C8—21no—
185C2×C22⋊C8—21no—
186C2×C4×C8—21no—
187C2×C4×C8C2312yes—
188#332C23116yescomplemented
189Q8⋊C8C23116yescomplemented
190D4⋊C8C23116yes—
191C25—21no—
192C2×#41—24no—
193C2×#41—42no—
194C2×#41—44no—
195C2×#41—44no—
196C22×Q8—28no—
197C22×Q8—216no—
198C22×Q8—28no—
199C22×Q8—216no—
200C22×Q8—232no—
201C22×Q8—216no—
202C22×Q8—22no—
203C22×Q8—21no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—21no—
209C22×Q8—416no—
210C22×Q8—48no—
211C22×Q8—464no—
212C22×Q8—432no—
213C22×Q8—464no—
214C22×Q8—416no—
215C22×Q8—416no—
216C22×Q8C2×D414yes—
217C22×Q8C2×D412yes—
218C22×D4—22no—
219C22×D4—21no—
220C22×D4—24no—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4—41no—
225C22×D4SD1618yescomplemented
226C22×D4C2×D412yes—
227C22×D4C2×D414yes—
228C23×C4—21no—
229C23×C4—216no—
230C23×C4—28no—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—21no—
234C23×C4—24no—
235C23×C4—22no—
236C23×C4—24no—
237C23×C4—22no—
238C23×C4—416no—
239C23×C4—48no—
240C23×C4—41no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244AGL(1,ℤ/8)—432no—
245C2×SD16—28no—
246C2×SD16—28no—
247C2×SD16—48no—
248C2×SD16—48no—
249C2×D8—28no—
250C2×D8—48no—
251C2×M16—24no—
252C2×M16—44no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C4⋊Q8—216no—
257C4⋊Q8—232no—
258C4⋊Q8—216no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—21no—
269C4⋊Q8C2×D418yes—
270C4⋊Q8C2×D414yes—
271C4⋊Q8C2412yes—
272C4⋊Q8C2414yes—
273#127—21no—
274#127—21no—
275#127—21no—
276#127C2412yes—
277#124—24no—
278#124—42no—
279C22⋊Q8—216no—
280C22⋊Q8—232no—
281C22⋊Q8—232no—
282C22⋊Q8—264no—
283C22⋊Q8—216no—
284C22⋊Q8—232no—
285#121—24no—
286#121—24no—
287#121—42no—
288#121—42no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—22no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—22no—
296C4×Q8C2×D418yes—
297C4×Q8C2×D414yes—
298C4×Q8C2414yes—
299C4×D4—216no—
300C4×D4—232no—
301C4×D4—216no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—22no—
307C4×D4—24no—
308C4×D4—22no—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—42no—
312C4×D4—42no—
313C4×D4C2×D414yes—
314C4×D4C2×D418yes—
315C4×D4C2414yes—
316C2×C4⋊C4—216no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D414yes—
338C2×C4⋊C4C2×D414yes—
339C2×C4⋊C4C2×D418yes—
340C2×C22⋊C4—216no—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—41no—
347C2×C22⋊C4—41no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360#106—28no—
361#106—216no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364C4⋊C8—22no—
365C4⋊C8—22no—
366C4⋊C8C2414yes—
367C4⋊C8C2414yes—
368#103—28no—
369#103—28no—
370#102—28no—
371#102—28no—
372#102—216no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375C4×C8—24no—
376C4×C8—22no—
377C4×C8C2414yes—
378C24—24no—
379C24—22no—
380C24—28no—
381C24—44no—
382C24—44no—
383C24—41no—
384C24#12011yescharacteristic
385#41—48no—
386#41—48no—
387#41—88no—
388C2×Q8—216no—
389C2×Q8—28no—
390C2×Q8—216no—
391C2×Q8—232no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—28no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—464no—
400C2×Q8—432no—
401C2×Q8—48no—
402C2×Q8—416no—
403C2×Q8—416no—
404C2×Q8—4128no—
405C2×Q8—464no—
406C2×Q8—432no—
407C2×Q8—432no—
408C2×Q8—41no—
409C2×Q8—48no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8AGL(1,ℤ/8)18yescomplemented
413C2×Q8C22×D414yes—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—28no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—28no—
422C2×D4—216no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—42no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—84no—
432C2×D4C2×SD1618yes—
433C2×D4C2×SD16116yescomplemented
434C2×D4C22×D414yes—
435C22×C4—28no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—232no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—464no—
451C22×C4—432no—
452C22×C4—48no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—41no—
460C22×C4—44no—
461C22×C4—84no—
462C22×C4#12012yes—
463C22×C4#12011yescharacteristic
464C22×C4#12011yescharacteristic
465C22×C4#12012yes—
466C22×C4#12012yes—
467C22×C4#12011yescharacteristic
468C22×C4C2×SD1614yes—
469C22×C4#13714yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476SD16—416no—
477SD16—416no—
478D8—416no—
479M16—48no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—44no—
484C2×C8—44no—
485C2×C8—44no—
486C4⋊C4—232no—
487C4⋊C4—28no—
488C4⋊C4—24no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—28no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4C22×D414yes—
504C4⋊C4C22×D414yes—
505C4⋊C4C22×D418yes—
506C22⋊C4—232no—
507C22⋊C4—28no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C42—28no—
513C42—24no—
514C42—24no—
515C42#12012yes—
516C42#12012yes—
517C42#12012yes—
518C42C22×D412yes—
519C42C22×D411yescharacteristic
520C42C22×D412yes—
521C42C22×D411yescharacteristic
522C42C22×D412yes—
523C42C22×D411yescharacteristic
524C42C2511yescharacteristic
525C23—22no—
526C23—22no—
527C23—24no—
528C23—28no—
529C23—216no—
530C23—48no—
531C23—432no—
532C23—48no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23#44814yes—
540C23C2×#12012yes—
541C23C2×#12011yescharacteristic
542Q8—216no—
543Q8—28no—
544Q8—24no—
545Q8—464no—
546Q8—416no—
547Q8—48no—
548Q8—48no—
549Q8—42no—
550Q8—42no—
551Q8—82no—
552Q8C2×AGL(1,ℤ/8)18yescomplemented
553D4—28no—
554D4—216no—
555D4—24no—
556D4—44no—
557D4—42no—
558D4—44no—
559D4—42no—
560D4—82no—
561D4—88no—
562D4—82no—
563D4—82no—
564D4C22×SD1618yes—
565C2×C4—216no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—464no—
579C2×C4—416no—
580C2×C4—48no—
581C2×C4—416no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—82no—
590C2×C4—88no—
591C2×C4#44814yes—
592C2×C4#44818yes—
593C2×C4C2×#12012yes—
594C2×C4C2×#12011yescharacteristic
595C2×C4C2×#12012yes—
596C2×C4C2×#12012yes—
597C2×C4C2×#12011yescharacteristic
598C2×C4C2×#12011yescharacteristic
599C2×C4C22×SD1612yes—
600C2×C4C22×SD1614yes—
601C2×C4C2×AGL(1,ℤ/8)14yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13714yes—
604C2×C4C23×D411yescharacteristic
605C2×C4C23×D411yescharacteristic
606C2×C4C23×D411yescharacteristic
607C8—44no—
608C8—44no—
609C8—44no—
610C22—28no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—432no—
615C22—416no—
616C22—44no—
617C22—44no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—164no—
622C22C2×#44814yes—
623C22C2×#44812yes—
624C22#300312yes—
625C22#302012yes—
626C22C22×#12011yescharacteristic
627C4—28no—
628C4—22no—
629C4—21no—
630C4—22no—
631C4—21no—
632C4—48no—
633C4—44no—
634C4—44no—
635C4—82no—
636C4C2×#44814yes—
637C4C2×#44812yes—
638C4#300312yes—
639C4C23×SD1611yescharacteristic
640C4C22×AGL(1,ℤ/8)12yes—
641C4C22×#13711yescharacteristic
642C2—24no—
643C2—48no—
644C2—162no—
645C2#1875814yesdirect product factor
646C2C22×#44811yescharacteristic
647C2C2×#300311yescharacteristic
648C2C2×#302011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC2×#3036C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×D4×Q8C2semidirect
2. C2×#3227C2semidirect
3. C2×#3103C2semidirect
5. #18758C2direct (N × H)
6. C2×#1651C2semidirect
7. C2×#1627C2semidirect
8. C2×#1549C2semidirect
9. C2×#1487C2semidirect
10. C2×#1478C2semidirect
20. C2×D4×Q8C22semidirect
30. #3227C22semidirect
31. C2×#498C22semidirect
32. C2×#486C22semidirect
34. #3103C22semidirect
37. C2×#475C22semidirect
41. C2×#460C22semidirect
51. #1651C22semidirect
52. #1627C22semidirect
53. #1549C22semidirect
54. C2×#332C22semidirect
55. #1487C22semidirect
56. #1478C22semidirect
57. C2×Q8⋊C8C22semidirect
149. #498C23semidirect
151. #486C23semidirect
156. #475C23semidirect
161. #460C23semidirect
188. #332C23semidirect
189. Q8⋊C8C23semidirect
225. C22×D4SD16semidirect
412. C2×Q8AGL(1,ℤ/8)semidirect
433. C2×D4C2×SD16semidirect
552. Q8C2×AGL(1,ℤ/8)semidirect
645. C2#18758direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×Q82
C2×#32272
C2×#31032
C2×#30362
#187582
C2×#16512
C2×#16272
C2×#15492
C2×#14872
C2×#14782