Finite Group Databasebeta

← C2×#18773 #6348688

Subgroups — C2×#18773 #6348688

Order 512.

3224 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#555C211yesmaximal, characteristic
2C2×#3209C211yesmaximal, characteristic, complemented
3C2×#3101C211yesmaximal, characteristic, complemented
4C2×#3077C211yesmaximal, characteristic, complemented
5#18773C2116yesmaximal, direct product factor
6C2×#1646C214yesmaximal, complemented
7C2×#1634C214yesmaximal, complemented
8C2×#1548C211yesmaximal, characteristic, complemented
9C2×#1486C211yesmaximal, characteristic, complemented
10C2×#1478C211yesmaximal, characteristic, complemented
11C2×#555—24no—
12C2×#555—22no—
13C2×#555—22no—
14C2×#555—24no—
15C2×#555—24no—
16C2×#555—22no—
17C2×#555C2214yes—
18C2×#555C2218yes—
19C2×#533C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#124—22no—
22C22×#121—21no—
23C22×#121—21no—
24C22×C4×D4C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C22×#117—21no—
27C22×#117—21no—
28#3209—24no—
29#3209C2218yescomplemented
30C2×#494C2212yescomplemented
31C2×#487C2212yescomplemented
32#3101—24no—
33#3101C2218yescomplemented
34C2×#483—21no—
35#3077—24no—
36#3077C2218yescomplemented
37C2×#478—22no—
38C2×#476C2212yescomplemented
39C2×#476C2212yescomplemented
40C2×#474C2212yescomplemented
41C2×#462—22no—
42C2×#461C2212yescomplemented
43C2×#451—22no—
44C2×#450—21no—
45C2×C4×SD16—22no—
46C2×#431C2211yescharacteristic
47C2×#431C2211yescharacteristic
48C2×#428—21no—
49C2×#423C2211yescharacteristic
50C2×#418—21no—
51#1646C22132yescomplemented
52#1634C22132yescomplemented
53#1548C2218yescomplemented
54C2×#333C2214yescomplemented
55#1486C2218yescomplemented
56#1478C2218yescomplemented
57C2×D4⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yescomplemented
59C23×Q8D412yes—
60C23×D4—22no—
61C23×D4D411yescharacteristic
62C23×D4D411yescharacteristic
63C24×C4—21no—
64C24×C4—21no—
65C22×Q16—24no—
66C22×SD16—22no—
67C22×M16—21no—
68#555—216no—
69#555—216no—
70#555—216no—
71#555—216no—
72#555—28no—
73#555—216no—
74#555—28no—
75#555—28no—
76#555—216no—
77#555C23116yes—
78#533—22no—
79#533—22no—
80#533—22no—
81#533C2314yes—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8D414yes—
87C2×C4⋊Q8D412yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2314yes—
92C2×#125—21no—
93C2×#124—216no—
94C2×#124—28no—
95C2×#124—28no—
96C2×#124—216no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×#121—28no—
100C2×#121—28no—
101C2×#121—24no—
102C2×#121—24no—
103C2×#121—24no—
104C2×#121—24no—
105C2×#121—28no—
106C2×#121—28no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4D412yes—
118C2×C4×D4D414yes—
119C2×C4×D4D414yes—
120C2×C4×D4D412yes—
121C2×C4×D4C2314yes—
122C2×C4×D4C2314yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×#117—28no—
126C2×#117—28no—
127C2×#117—24no—
128C2×#117—24no—
129C2×#117—24no—
130C2×#117—24no—
131C2×#117—28no—
132C2×#117—28no—
133C2×#117—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4D411yescharacteristic
137C22×C4⋊C4D411yescharacteristic
138C22×C22⋊C4—22no—
139C22×C22⋊C4—22no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—21no—
142C22×C42—22no—
143C22×C42C2311yescharacteristic
144#494—24no—
145#494C2318yescomplemented
146#487—24no—
147#487C2318yescomplemented
148#483—28no—
149#478—216no—
150#476—24no—
151#476—24no—
152#476C2318yescomplemented
153#476C2318yescomplemented
154#474—24no—
155#474C2318yescomplemented
156#462—216no—
157#461—24no—
158#461C2318yescomplemented
159#451—216no—
160#450—28no—
161C4×SD16—216no—
162#431—22no—
163#431—22no—
164#431C2314yes—
165#431C2314yes—
166#428—28no—
167C2×#107—22no—
168C2×#106—22no—
169C2×#106—22no—
170#423—22no—
171#423C2314yes—
172C2×C4⋊C8—21no—
173C2×C4⋊C8—21no—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2312yes—
176#418—28no—
177C2×#103—24no—
178C2×#103—22no—
179C2×#102—22no—
180C2×#102—22no—
181C2×#102—22no—
182C2×C22⋊C8—21no—
183C2×C22⋊C8—21no—
184C2×C4×C8—21no—
185C2×C4×C8C2312yes—
186#333C23116yescomplemented
187D4⋊C8C23116yescomplemented
188D4⋊C8C23116yescomplemented
189C25—21no—
190C25—21no—
191C22×Q8—24no—
192C22×Q8—22no—
193C22×Q8—21no—
194C22×Q8—21no—
195C22×Q8—84no—
196C22×Q8C2×D414yes—
197C22×Q8C2×D418yes—
198C22×D4—216no—
199C22×D4—28no—
200C22×D4—22no—
201C22×D4—21no—
202C22×D4—22no—
203C22×D4—21no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—44no—
207C22×D4—88no—
208C22×D4C2×D412yes—
209C22×D4C2×D412yes—
210C22×D4C2×D414yes—
211C22×D4C2×D414yes—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—48no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C2×Q16—216no—
230C2×Q16—88no—
231C2×SD16—28no—
232C2×SD16—48no—
233C2×M16—24no—
234C2×M16—44no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C4⋊Q8—24no—
239C4⋊Q8—24no—
240C4⋊Q8—24no—
241C4⋊Q8—24no—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8C2×D418yes—
252C4⋊Q8C2×D414yes—
253C4⋊Q8C2412yes—
254C4⋊Q8C2412yes—
255C4⋊Q8C2414yes—
256#125—24no—
257#125—42no—
258#124—264no—
259#124—232no—
260#124—232no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—42no—
264C22⋊Q8—42no—
265#121—232no—
266#121—232no—
267#121—216no—
268#121—216no—
269#121—216no—
270#121—216no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—42no—
274C4×Q8—42no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4C2×D418yes—
290C4×D4C2×D414yes—
291C4×D4C2×D414yes—
292C4×D4C2×D418yes—
293C4×D4C2414yes—
294C4×D4C2414yes—
295#117—232no—
296#117—232no—
297#117—216no—
298#117—216no—
299#117—216no—
300#117—216no—
301#117—24no—
302#117—42no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—41no—
320C2×C4⋊C4—41no—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D414yes—
325C2×C22⋊C4—216no—
326C2×C22⋊C4—216no—
327C2×C22⋊C4—28no—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—41no—
338C2×C42—216no—
339C2×C42—28no—
340C2×C42—41no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349#107—28no—
350#106—28no—
351#106—28no—
352C4⋊C8—24no—
353C4⋊C8—24no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8C2414yes—
357C4⋊C8C2414yes—
358#103—216no—
359#103—28no—
360#102—28no—
361#102—28no—
362#102—28no—
363C22⋊C8—24no—
364C22⋊C8—24no—
365C4×C8—24no—
366C4×C8—22no—
367C4×C8C2414yes—
368C24—24no—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—44no—
373C24—44no—
374C24—44no—
375C24—44no—
376C24—44no—
377C24—44no—
378C24#12011yescharacteristic
379C2×Q8—28no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—44no—
388C2×Q8—84no—
389C2×Q8—88no—
390C2×Q8—88no—
391C2×Q8C22×D418yes—
392C2×D4—232no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—48no—
404C2×D4—44no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—44no—
408C2×D4—832no—
409C2×D4—816no—
410C2×D4C22×D414yes—
411C2×D4C22×D414yes—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—48no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—48no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4#12012yes—
442C22×C4#12011yescharacteristic
443C22×C4#12011yescharacteristic
444C22×C4#12012yes—
445C22×C4#12012yes—
446C22×C4#12011yescharacteristic
447C22×C4#13514yes—
448C22×C4#13714yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455Q16—816no—
456SD16—416no—
457M16—48no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—44no—
462C2×C8—44no—
463C2×C8—44no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4C22×D414yes—
485C4⋊C4C22×D414yes—
486C22⋊C4—232no—
487C22⋊C4—232no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—44no—
495C42—232no—
496C42—44no—
497C42#12012yes—
498C42#12012yes—
499C42#12012yes—
500C42C22×D411yescharacteristic
501C42C22×D412yes—
502C42C22×D412yes—
503C42C22×D412yes—
504C42C22×D411yescharacteristic
505C42C22×D411yescharacteristic
506C42C2511yescharacteristic
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23—44no—
517C23—48no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—816no—
524C23—816no—
525C23#45214yes—
526C23C2×#12012yes—
527C23C2×#12011yescharacteristic
528Q8—48no—
529Q8—44no—
530Q8—44no—
531Q8—84no—
532Q8—88no—
533Q8—82no—
534Q8—82no—
535D4—24no—
536D4—24no—
537D4—24no—
538D4—24no—
539D4—48no—
540D4—48no—
541D4—44no—
542D4—44no—
543D4—832no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—416no—
555C2×C4—48no—
556C2×C4—416no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—416no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4#45218yes—
572C2×C4#45214yes—
573C2×C4C2×#12012yes—
574C2×C4C2×#12011yescharacteristic
575C2×C4C2×#12012yes—
576C2×C4C2×#12012yes—
577C2×C4C2×#12011yescharacteristic
578C2×C4C2×#12011yescharacteristic
579C2×C4C2×#13512yes—
580C2×C4C2×#13514yes—
581C2×C4C2×#13714yes—
582C2×C4C2×#13712yes—
583C2×C4#57514yes—
584C2×C4C23×D411yescharacteristic
585C2×C4C23×D411yescharacteristic
586C2×C4C23×D411yescharacteristic
587C8—44no—
588C8—44no—
589C8—44no—
590C22—28no—
591C22—22no—
592C22—22no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—816no—
600C22—816no—
601C22—88no—
602C22—88no—
603C22C2×#45214yes—
604C22C2×#45212yes—
605C22#300512yes—
606C22#301912yes—
607C22C22×#12011yescharacteristic
608C4—22no—
609C4—21no—
610C4—22no—
611C4—21no—
612C4—44no—
613C4—44no—
614C4—48no—
615C4—82no—
616C4—82no—
617C4C2×#45214yes—
618C4C2×#45212yes—
619C4#300512yes—
620C4C22×#13511yescharacteristic
621C4C22×#13711yescharacteristic
622C4C2×#57512yes—
623C2—24no—
624C2—84no—
625C2—84no—
626C2#1877314yesdirect product factor
627C2C22×#45211yescharacteristic
628C2C2×#300511yescharacteristic
629C2C2×#301911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC22×#555C2
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×#3209C2semidirect
3. C2×#3101C2semidirect
4. C2×#3077C2semidirect
5. #18773C2direct (N × H)
6. C2×#1646C2semidirect
7. C2×#1634C2semidirect
8. C2×#1548C2semidirect
9. C2×#1486C2semidirect
10. C2×#1478C2semidirect
29. #3209C22semidirect
30. C2×#494C22semidirect
31. C2×#487C22semidirect
33. #3101C22semidirect
36. #3077C22semidirect
38. C2×#476C22semidirect
39. C2×#476C22semidirect
40. C2×#474C22semidirect
42. C2×#461C22semidirect
51. #1646C22semidirect
52. #1634C22semidirect
53. #1548C22semidirect
54. C2×#333C22semidirect
55. #1486C22semidirect
56. #1478C22semidirect
57. C2×D4⋊C8C22semidirect
58. C2×D4⋊C8C22semidirect
145. #494C23semidirect
147. #487C23semidirect
152. #476C23semidirect
153. #476C23semidirect
155. #474C23semidirect
158. #461C23semidirect
186. #333C23semidirect
187. D4⋊C8C23semidirect
188. D4⋊C8C23semidirect
626. C2#18773direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5552
C2×#32092
C2×#31012
C2×#30772
#187732
C2×#16462
C2×#16342
C2×#15482
C2×#14862
C2×#14782