Finite Group Databasebeta

← C2×#18757 #6348671

Subgroups — C2×#18757 #6348671

Order 512.

4072 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#547C211yesmaximal, characteristic
2C2×#3229C211yesmaximal, characteristic, complemented
3C2×#3101C211yesmaximal, characteristic, complemented
4C2×#3077C211yesmaximal, characteristic, complemented
5#18757C2116yesmaximal, direct product factor
6C2×#1654C214yesmaximal, complemented
7C2×#1635C214yesmaximal, complemented
8C2×#1546C211yesmaximal, characteristic, complemented
9C2×#1486C211yesmaximal, characteristic, complemented
10C2×#1478C211yesmaximal, characteristic, complemented
11C23×#41—21no—
12C2×#547—22no—
13C2×#547—24no—
14C2×#547—22no—
15C2×#547—24no—
16C2×#547—24no—
17C2×#547—22no—
18C2×#547C2214yes—
19C2×#547C2218yes—
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#123—22no—
23C22×C22⋊Q8—21no—
24C22×#121—21no—
25C22×C4×D4C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C23×C4⋊C4—21no—
28#3229—24no—
29#3229C2218yescomplemented
30C2×#498C2212yescomplemented
31C2×#487C2212yescomplemented
32#3101—24no—
33#3101C2218yescomplemented
34C2×#483—21no—
35C2×#483—22no—
36#3077—24no—
37#3077C2218yescomplemented
38C2×#475—22no—
39C2×#478—22no—
40C2×#476C2212yescomplemented
41C2×#476C2212yescomplemented
42C2×#474C2212yescomplemented
43C2×#461C2212yescomplemented
44C2×#450—21no—
45C2×C4×SD16—22no—
46C2×#431C2211yescharacteristic
47C2×#431C2211yescharacteristic
48C2×#428—21no—
49C2×#423C2211yescharacteristic
50C2×#418—21no—
51#1654C22132yescomplemented
52#1635C22132yescomplemented
53#1546C2218yescomplemented
54C2×#332C2214yescomplemented
55#1486C2218yescomplemented
56#1478C2218yescomplemented
57C2×D4⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yescomplemented
59C22×#41—28no—
60C22×#41—24no—
61C22×#41—24no—
62C22×#41—28no—
63C22×#41—48no—
64C22×#41—48no—
65C23×Q8—21no—
66C23×D4—22no—
67C23×D4D411yescharacteristic
68C23×D4D411yescharacteristic
69C24×C4—22no—
70C24×C4—21no—
71C24×C4—21no—
72C22×SD16—22no—
73C22×M16—21no—
74#547—216no—
75#547—216no—
76#547—216no—
77#547—216no—
78#547—28no—
79#547—28no—
80#547—216no—
81#547—216no—
82#547—28no—
83#547C23116yes—
84#533—22no—
85#533—22no—
86#533—22no—
87#533C2314yes—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8D412yes—
93C2×C4⋊Q8D414yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2314yes—
98C2×#125—21no—
99C2×#123—216no—
100C2×#123—28no—
101C2×#123—28no—
102C2×#123—216no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×#121—28no—
110C2×#121—24no—
111C2×#121—24no—
112C2×#121—28no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4D412yes—
124C2×C4×D4D414yes—
125C2×C4×D4D414yes—
126C2×C4×D4D412yes—
127C2×C4×D4C2314yes—
128C2×C4×D4C2314yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×#117—21no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—28no—
134C22×C4⋊C4—24no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—24no—
137C22×C4⋊C4—28no—
138C22×C4⋊C4D411yescharacteristic
139C22×C4⋊C4D412yes—
140C22×C4⋊C4D411yescharacteristic
141C22×C22⋊C4—22no—
142C22×C22⋊C4—21no—
143C22×C22⋊C4—21no—
144C22×C42C2311yescharacteristic
145#498—24no—
146#498C2318yescomplemented
147#487—24no—
148#487C2318yescomplemented
149#483—28no—
150#483—216no—
151#478—216no—
152#476—24no—
153#476—24no—
154#476C2318yescomplemented
155#476C2318yescomplemented
156#475—216no—
157#474—24no—
158#474C2318yescomplemented
159#461—24no—
160#461C2318yescomplemented
161#450—28no—
162C4×SD16—216no—
163#431—22no—
164#431—22no—
165#431C2314yes—
166#431C2314yes—
167#428—28no—
168C2×#107—22no—
169C2×#106—22no—
170C2×#106—24no—
171C2×#106—22no—
172#423—22no—
173#423C2314yes—
174C2×C4⋊C8—21no—
175C2×C4⋊C8—21no—
176C2×C4⋊C8C2312yes—
177C2×C4⋊C8C2312yes—
178#418—28no—
179C2×#103—24no—
180C2×#103—22no—
181C2×#102—22no—
182C2×#102—22no—
183C2×#102—22no—
184C2×C22⋊C8—21no—
185C2×C22⋊C8—21no—
186C2×C4×C8—21no—
187C2×C4×C8C2312yes—
188#332C23116yescomplemented
189D4⋊C8C23116yescomplemented
190D4⋊C8C23116yescomplemented
191C25—21no—
192C25—21no—
193C2×#41—232no—
194C2×#41—216no—
195C2×#41—216no—
196C2×#41—416no—
197C2×#41—432no—
198C2×#41—432no—
199C2×#41—464no—
200C2×#41—416no—
201C2×#41—432no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8—44no—
209C22×Q8—44no—
210C22×D4—216no—
211C22×D4—28no—
212C22×D4—22no—
213C22×D4—21no—
214C22×D4—22no—
215C22×D4—21no—
216C22×D4—24no—
217C22×D4—44no—
218C22×D4—416no—
219C22×D4SD1618yescomplemented
220C22×D4C2×D412yes—
221C22×D4C2×D412yes—
222C22×D4C2×D414yes—
223C22×D4C2×D414yes—
224C23×C4—216no—
225C23×C4—28no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—24no—
234C23×C4—22no—
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—48no—
238C23×C4—416no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C2×SD16—28no—
245C2×SD16—48no—
246C2×M16—24no—
247C2×M16—44no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8C2×D418yes—
265C4⋊Q8C2×D414yes—
266C4⋊Q8C2412yes—
267C4⋊Q8C2412yes—
268C4⋊Q8C2414yes—
269#125—24no—
270#125—42no—
271#123—264no—
272#123—232no—
273#123—232no—
274C22⋊Q8—232no—
275C22⋊Q8—216no—
276C22⋊Q8—216no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—42no—
280C22⋊Q8—42no—
281#121—232no—
282#121—216no—
283#121—216no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—42no—
287C4×Q8—42no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4C2×D418yes—
303C4×D4C2×D414yes—
304C4×D4C2×D414yes—
305C4×D4C2×D418yes—
306C4×D4C2414yes—
307C4×D4C2414yes—
308#117—24no—
309#117—42no—
310C2×C4⋊C4—216no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—232no—
313C2×C4⋊C4—216no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—216no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—41no—
330C2×C4⋊C4—41no—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D414yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2×D418yes—
337C2×C22⋊C4—216no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—41no—
348C2×C42—41no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357#107—28no—
358#106—28no—
359#106—28no—
360#106—216no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363C4⋊C8—22no—
364C4⋊C8—22no—
365C4⋊C8C2414yes—
366C4⋊C8C2414yes—
367#103—28no—
368#103—216no—
369#102—28no—
370#102—28no—
371#102—28no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374C4×C8—24no—
375C4×C8—22no—
376C4×C8C2414yes—
377C24—24no—
378C24—22no—
379C24—28no—
380C24—24no—
381C24—22no—
382C24—44no—
383C24—44no—
384C24—44no—
385C24—44no—
386C24—44no—
387C24#12011yescharacteristic
388#41—464no—
389#41—4128no—
390#41—432no—
391#41—432no—
392C2×Q8—28no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—44no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8—44no—
401C2×Q8—48no—
402C2×Q8—44no—
403C2×Q8—48no—
404C2×Q8—416no—
405C2×Q8—48no—
406C2×Q8—44no—
407C2×D4—232no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—28no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—216no—
417C2×D4—28no—
418C2×D4—48no—
419C2×D4—44no—
420C2×D4—48no—
421C2×D4—432no—
422C2×D4—464no—
423C2×D4C2×SD1618yes—
424C2×D4C2×SD16116yescomplemented
425C2×D4C22×D414yes—
426C2×D4C22×D414yes—
427C22×C4—24no—
428C22×C4—232no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—48no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—44no—
446C22×C4—432no—
447C22×C4—464no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—48no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4#12012yes—
460C22×C4#12011yescharacteristic
461C22×C4#12011yescharacteristic
462C22×C4#12012yes—
463C22×C4#12012yes—
464C22×C4#12011yescharacteristic
465C22×C4C2×SD1614yes—
466C22×C4AGL(1,ℤ/8)14yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473SD16—416no—
474M16—48no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—44no—
479C2×C8—44no—
480C2×C8—44no—
481C4⋊C4—232no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—28no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—28no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4C22×D414yes—
502C4⋊C4C22×D418yes—
503C4⋊C4C22×D414yes—
504C22⋊C4—232no—
505C22⋊C4—28no—
506C22⋊C4—24no—
507C22⋊C4—28no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—44no—
512C42—44no—
513C42#12012yes—
514C42#12012yes—
515C42#12012yes—
516C42C22×D411yescharacteristic
517C42C22×D412yes—
518C42C22×D412yes—
519C42C22×D412yes—
520C42C22×D411yescharacteristic
521C42C22×D411yescharacteristic
522C42C2511yescharacteristic
523C23—22no—
524C23—22no—
525C23—24no—
526C23—28no—
527C23—216no—
528C23—24no—
529C23—48no—
530C23—432no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—44no—
535C23—44no—
536C23—48no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—816no—
541C23#45014yes—
542C23C2×#12012yes—
543C23C2×#12011yescharacteristic
544Q8—44no—
545Q8—416no—
546Q8—48no—
547Q8—44no—
548Q8—44no—
549Q8—44no—
550Q8—82no—
551Q8—82no—
552D4—28no—
553D4—216no—
554D4—24no—
555D4—24no—
556D4—24no—
557D4—48no—
558D4—464no—
559D4—44no—
560D4C22×SD1618yes—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—416no—
572C2×C4—48no—
573C2×C4—464no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—416no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—416no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4#45018yes—
590C2×C4#45014yes—
591C2×C4C2×#12012yes—
592C2×C4C2×#12011yescharacteristic
593C2×C4C2×#12012yes—
594C2×C4C2×#12012yes—
595C2×C4C2×#12011yescharacteristic
596C2×C4C2×#12011yescharacteristic
597C2×C4C22×SD1612yes—
598C2×C4C22×SD1614yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×AGL(1,ℤ/8)14yes—
601C2×C4#57514yes—
602C2×C4C23×D411yescharacteristic
603C2×C4C23×D411yescharacteristic
604C2×C4C23×D411yescharacteristic
605C8—44no—
606C8—44no—
607C8—44no—
608C22—28no—
609C22—22no—
610C22—22no—
611C22—28no—
612C22—432no—
613C22—416no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—816no—
620C22—88no—
621C22C2×#45014yes—
622C22C2×#45012yes—
623C22#300512yes—
624C22#302012yes—
625C22C22×#12011yescharacteristic
626C4—22no—
627C4—21no—
628C4—22no—
629C4—21no—
630C4—44no—
631C4—48no—
632C4—44no—
633C4—82no—
634C4—82no—
635C4C2×#45014yes—
636C4C2×#45012yes—
637C4#300512yes—
638C4C23×SD1611yescharacteristic
639C4C22×AGL(1,ℤ/8)11yescharacteristic
640C4C2×#57512yes—
641C2—24no—
642C2—48no—
643C2—84no—
644C2#1875714yesdirect product factor
645C2C22×#45011yescharacteristic
646C2C2×#300511yescharacteristic
647C2C2×#302011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC22×#547C2
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
2. C2×#3229C2semidirect
3. C2×#3101C2semidirect
4. C2×#3077C2semidirect
5. #18757C2direct (N × H)
6. C2×#1654C2semidirect
7. C2×#1635C2semidirect
8. C2×#1546C2semidirect
9. C2×#1486C2semidirect
10. C2×#1478C2semidirect
29. #3229C22semidirect
30. C2×#498C22semidirect
31. C2×#487C22semidirect
33. #3101C22semidirect
37. #3077C22semidirect
40. C2×#476C22semidirect
41. C2×#476C22semidirect
42. C2×#474C22semidirect
43. C2×#461C22semidirect
51. #1654C22semidirect
52. #1635C22semidirect
53. #1546C22semidirect
54. C2×#332C22semidirect
55. #1486C22semidirect
56. #1478C22semidirect
57. C2×D4⋊C8C22semidirect
58. C2×D4⋊C8C22semidirect
146. #498C23semidirect
148. #487C23semidirect
154. #476C23semidirect
155. #476C23semidirect
158. #474C23semidirect
160. #461C23semidirect
188. #332C23semidirect
189. D4⋊C8C23semidirect
190. D4⋊C8C23semidirect
219. C22×D4SD16semidirect
424. C2×D4C2×SD16semidirect
644. C2#18757direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5472
C2×#32292
C2×#31012
C2×#30772
#187572
C2×#16542
C2×#16352
C2×#15462
C2×#14862
C2×#14782