Finite Group Databasebeta

← C2×#18749 #6348662

Subgroups — C2×#18749 #6348662

Order 512.

3160 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#3209C211yesmaximal, characteristic, complemented
3C2×#3104C211yesmaximal, characteristic, complemented
4C2×#3077C211yesmaximal, characteristic
5#18749C2116yesmaximal, direct product factor
6C2×#1644C214yesmaximal, complemented
7C2×#1632C214yesmaximal, complemented
8C2×#1548C211yesmaximal, characteristic, complemented
9C2×#1487C211yesmaximal, characteristic, complemented
10C2×#1478C211yesmaximal, characteristic
11C2×#554—24no—
12C2×#554—22no—
13C2×#554—24no—
14C2×#554—22no—
15C2×#554—22no—
16C2×#554—24no—
17C2×#554C2214yes—
18C2×#554C2218yes—
19C2×#533C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125—21no—
22C22×C22⋊Q8—21no—
23C22×C22⋊Q8—22no—
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4—21no—
26C22×C4×D4C2211yescharacteristic
27C23×C4⋊C4—21no—
28#3209—24no—
29#3209C2218yescomplemented
30C2×#494C2212yescomplemented
31C2×#487C2212yescomplemented
32#3104—24no—
33#3104C2218yescomplemented
34C2×#484—21no—
35#3077—24no—
36#3077C2218yes—
37C2×#477C2212yescomplemented
38C2×#478—22no—
39C2×#476C2212yes—
40C2×#474C2212yes—
41C2×#462—22no—
42C2×#462C2212yescomplemented
43C2×#451—22no—
44C2×#451—21no—
45C2×C4×Q16—22no—
46C2×#431C2211yescharacteristic
47C2×#431C2211yescharacteristic
48C2×#428—21no—
49C2×#423C2211yescharacteristic
50C2×#418—21no—
51#1644C22132yescomplemented
52#1632C22132yescomplemented
53#1548C2218yescomplemented
54C2×#333C2214yescomplemented
55#1487C2218yescomplemented
56#1478C2218yes—
57C2×Q8⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yes—
59C23×Q8D411yescharacteristic
60C23×Q8D412yes—
61C23×D4D411yescharacteristic
62C24×C4—22no—
63C24×C4—21no—
64C22×Q16—22no—
65C22×Q16—24no—
66C22×M16—21no—
67#554—216no—
68#554—216no—
69#554—216no—
70#554—216no—
71#554—28no—
72#554—216no—
73#554—28no—
74#554—28no—
75#554—216no—
76#554C23116yes—
77#533—22no—
78#533—22no—
79#533—22no—
80#533C2314yes—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8D414yes—
86C2×C4⋊Q8D412yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2314yes—
91C2×#125—28no—
92C2×#125—24no—
93C2×#125—24no—
94C2×#125—28no—
95C2×#125—21no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—216no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—216no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8D414yes—
113C2×C4×Q8D412yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2314yes—
116C2×C4×D4—28no—
117C2×C4×D4—24no—
118C2×C4×D4—24no—
119C2×C4×D4—28no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4D412yes—
125C2×C4×D4D414yes—
126C2×C4×D4C2314yes—
127C2×C4×D4C2312yes—
128C2×#117—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—28no—
131C22×C4⋊C4—24no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—24no—
137C22×C4⋊C4—28no—
138C22×C4⋊C4D411yescharacteristic
139C22×C4⋊C4D411yescharacteristic
140C22×C22⋊C4—22no—
141C22×C22⋊C4—21no—
142C22×C42—21no—
143C22×C42C2311yescharacteristic
144#494—24no—
145#494C2318yescomplemented
146#487—24no—
147#487C2318yescomplemented
148#484—28no—
149#478—216no—
150#477—24no—
151#477C2318yescomplemented
152#476—24no—
153#476C2318yes—
154#474—24no—
155#474C2318yes—
156#462—216no—
157#462—24no—
158#462C2318yescomplemented
159#451—28no—
160#451—216no—
161C4×Q16—216no—
162#431—22no—
163#431—22no—
164#431C2314yes—
165#431C2314yes—
166#428—28no—
167C2×#107—22no—
168C2×#107—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—22no—
178C2×#103—24no—
179C2×#103—22no—
180C2×#102—22no—
181C2×#102—22no—
182C2×C22⋊C8—21no—
183C2×C22⋊C8—21no—
184C2×C4×C8—21no—
185C2×C4×C8C2312yes—
186#333C23116yescomplemented
187Q8⋊C8C23116yescomplemented
188D4⋊C8C23116yes—
189C25—21no—
190C22×Q8—22no—
191C22×Q8—21no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—21no—
195C22×Q8—21no—
196C22×Q8—44no—
197C22×Q8—84no—
198C22×Q8C2×D412yes—
199C22×Q8C2×D414yes—
200C22×Q8C2×D414yes—
201C22×Q8C2×D418yes—
202C22×D4—22no—
203C22×D4—21no—
204C22×D4—24no—
205C22×D4Q1618yescomplemented
206C22×D4C2×D412yes—
207C22×D4C2×D414yes—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—21no—
212C23×C4—216no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—48no—
219C23×C4—88no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C2×Q16—28no—
224C2×Q16—216no—
225C2×Q16—48no—
226C2×Q16—88no—
227C2×M16—24no—
228C2×M16—44no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8—24no—
236C4⋊Q8—22no—
237C4⋊Q8—24no—
238C4⋊Q8—22no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8C2×D418yes—
246C4⋊Q8C2×D414yes—
247C4⋊Q8C2412yes—
248C4⋊Q8C2412yes—
249C4⋊Q8C2414yes—
250#125—232no—
251#125—216no—
252#125—216no—
253#125—24no—
254#125—42no—
255C22⋊Q8—232no—
256C22⋊Q8—216no—
257C22⋊Q8—264no—
258C22⋊Q8—232no—
259C22⋊Q8—216no—
260C22⋊Q8—232no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—42no—
264C22⋊Q8—42no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—22no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—22no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—42no—
275C4×Q8—42no—
276C4×Q8C2×D418yes—
277C4×Q8C2×D414yes—
278C4×Q8C2414yes—
279C4×D4—232no—
280C4×D4—216no—
281C4×D4—216no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4C2×D414yes—
290C4×D4C2×D418yes—
291C4×D4C2414yes—
292#117—24no—
293#117—42no—
294C2×C4⋊C4—216no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—232no—
297C2×C4⋊C4—216no—
298C2×C4⋊C4—216no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—216no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—41no—
322C2×C4⋊C4—41no—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D414yes—
327C2×C22⋊C4—216no—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—41no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—41no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347#107—28no—
348#107—28no—
349#106—28no—
350C4⋊C8—24no—
351C4⋊C8—24no—
352C4⋊C8—22no—
353C4⋊C8—22no—
354C4⋊C8C2414yes—
355C4⋊C8C2414yes—
356#103—28no—
357#103—216no—
358#103—28no—
359#102—28no—
360#102—28no—
361C22⋊C8—24no—
362C22⋊C8—24no—
363C4×C8—24no—
364C4×C8—22no—
365C4×C8C2414yes—
366C24—24no—
367C24—22no—
368C24—28no—
369C24—44no—
370C24—44no—
371C24#12011yescharacteristic
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—28no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—48no—
381C2×Q8—44no—
382C2×Q8—48no—
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×D414yes—
392C2×Q8C22×D418yes—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—28no—
397C2×D4—216no—
398C2×D4—28no—
399C2×D4C2×Q1618yes—
400C2×D4C2×Q16116yescomplemented
401C2×D4C22×D414yes—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—232no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—42no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—832no—
427C22×C4—816no—
428C22×C4#12012yes—
429C22×C4#12011yescharacteristic
430C22×C4#12011yescharacteristic
431C22×C4#12012yes—
432C22×C4#12012yes—
433C22×C4#12011yescharacteristic
434C22×C4C2×Q1614yes—
435C22×C4#13714yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442Q16—416no—
443Q16—816no—
444M16—48no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—44no—
449C2×C8—44no—
450C2×C8—44no—
451C4⋊C4—232no—
452C4⋊C4—232no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4C22×D414yes—
477C4⋊C4C22×D414yes—
478C22⋊C4—232no—
479C22⋊C4—28no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—44no—
483C42—28no—
484C42—24no—
485C42—24no—
486C42—44no—
487C42#12012yes—
488C42#12012yes—
489C42#12012yes—
490C42C22×D412yes—
491C42C22×D411yescharacteristic
492C42C22×D412yes—
493C42C22×D411yescharacteristic
494C42C22×D412yes—
495C42C22×D411yescharacteristic
496C42C2511yescharacteristic
497C23—22no—
498C23—22no—
499C23—24no—
500C23—28no—
501C23—216no—
502C23—48no—
503C23—48no—
504C23—432no—
505C23—44no—
506C23—44no—
507C23#45114yes—
508C23C2×#12012yes—
509C23C2×#12011yescharacteristic
510Q8—24no—
511Q8—24no—
512Q8—48no—
513Q8—48no—
514Q8—44no—
515Q8—44no—
516Q8—44no—
517Q8—84no—
518Q8—88no—
519Q8—82no—
520Q8—82no—
521D4—28no—
522D4—216no—
523D4—24no—
524D4C22×Q1618yes—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—416no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—416no—
540C2×C4—48no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—832no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4#45118yes—
553C2×C4#45114yes—
554C2×C4C2×#12012yes—
555C2×C4C2×#12011yescharacteristic
556C2×C4C2×#12012yes—
557C2×C4C2×#12012yes—
558C2×C4C2×#12011yescharacteristic
559C2×C4C2×#12011yescharacteristic
560C2×C4C22×Q1612yes—
561C2×C4C22×Q1614yes—
562C2×C4C2×#13712yes—
563C2×C4C2×#13714yes—
564C2×C4#57514yes—
565C2×C4C23×D411yescharacteristic
566C2×C4C23×D411yescharacteristic
567C2×C4C23×D411yescharacteristic
568C8—44no—
569C8—44no—
570C8—44no—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—28no—
575C22—432no—
576C22—416no—
577C22—44no—
578C22—44no—
579C22C2×#45114yes—
580C22C2×#45112yes—
581C22#300512yes—
582C22#301912yes—
583C22C22×#12011yescharacteristic
584C4—22no—
585C4—21no—
586C4—22no—
587C4—21no—
588C4—44no—
589C4—44no—
590C4—44no—
591C4—48no—
592C4—82no—
593C4—82no—
594C4C2×#45114yes—
595C4C2×#45112yes—
596C4#300512yes—
597C4C23×Q1611yescharacteristic
598C4C22×#13711yescharacteristic
599C4C2×#57512yes—
600C2—24no—
601C2—48no—
602C2#1874914yesdirect product factor
603C2C22×#45111yescharacteristic
604C2C2×#300511yescharacteristic
605C2C2×#301911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC23×D4D4
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×#3104C2semidirect
5. #18749C2direct (N × H)
6. C2×#1644C2semidirect
7. C2×#1632C2semidirect
8. C2×#1548C2semidirect
9. C2×#1487C2semidirect
29. #3209C22semidirect
30. C2×#494C22semidirect
31. C2×#487C22semidirect
33. #3104C22semidirect
37. C2×#477C22semidirect
42. C2×#462C22semidirect
51. #1644C22semidirect
52. #1632C22semidirect
53. #1548C22semidirect
54. C2×#333C22semidirect
55. #1487C22semidirect
57. C2×Q8⋊C8C22semidirect
145. #494C23semidirect
147. #487C23semidirect
151. #477C23semidirect
158. #462C23semidirect
186. #333C23semidirect
187. Q8⋊C8C23semidirect
205. C22×D4Q16semidirect
400. C2×D4C2×Q16semidirect
602. C2#18749direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#32092
C2×#31042
C2×#30772
#187492
C2×#16442
C2×#16322
C2×#15482
C2×#14872
C2×#14782