Finite Group Databasebeta

← C2×#18753 #6348666

Subgroups — C2×#18753 #6348666

Order 512.

3160 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#3209C211yesmaximal, characteristic, complemented
3C2×#3101C211yesmaximal, characteristic
4C2×#3078C211yesmaximal, characteristic, complemented
5#18753C2116yesmaximal, direct product factor
6C2×#1643C214yesmaximal, complemented
7C2×#1632C214yesmaximal, complemented
8C2×#1548C211yesmaximal, characteristic, complemented
9C2×#1486C211yesmaximal, characteristic
10C2×#1479C211yesmaximal, characteristic, complemented
11C2×#554—24no—
12C2×#554—22no—
13C2×#554—22no—
14C2×#554—24no—
15C2×#554—24no—
16C2×#554—22no—
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#3101—24no—
33#3101C2218yes—
34C2×#483—21no—
35#3078—24no—
36#3078C2218yescomplemented
37C2×#479—22no—
38C2×#477C2212yescomplemented
39C2×#475C2212yescomplemented
40C2×#476C2212yes—
41C2×#462—22no—
42C2×#461C2212yes—
43C2×#451—22no—
44C2×#450—21no—
45C2×C4×SD16—22no—
46C2×#431C2211yescharacteristic
47C2×#431C2211yescharacteristic
48C2×#428—21no—
49C2×#423C2211yescharacteristic
50C2×#419—21no—
51#1643C22132yescomplemented
52#1632C22132yescomplemented
53#1548C2218yescomplemented
54C2×#333C2214yescomplemented
55#1486C2218yes—
56#1479C2218yescomplemented
57C2×Q8⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yes—
59C23×Q8D411yescharacteristic
60C23×Q8D412yes—
61C23×D4D411yescharacteristic
62C24×C4—22no—
63C24×C4—21no—
64C22×Q16—24no—
65C22×SD16—22no—
66C22×M16—21no—
67#554—216no—
68#554—216no—
69#554—216no—
70#554—216no—
71#554—28no—
72#554—28no—
73#554—216no—
74#554—216no—
75#554—28no—
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×Q8D412yes—
113C2×C4×Q8D414yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2314yes—
116C2×C4×D4—28no—
117C2×C4×D4—24no—
118C2×C4×D4—28no—
119C2×C4×D4—24no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—22no—
124C2×C4×D4D414yes—
125C2×C4×D4D412yes—
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#483—28no—
149#479—216no—
150#477—24no—
151#477C2318yescomplemented
152#476—24no—
153#476C2318yes—
154#475—24no—
155#475C2318yescomplemented
156#462—216no—
157#461—24no—
158#461C2318yes—
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#419—28no—
177C2×#103—22no—
178C2×#103—22no—
179C2×#103—24no—
180C2×#103—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—24no—
195C22×Q8—24no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—84no—
199C22×Q8C2×D412yes—
200C22×Q8C2×D414yes—
201C22×Q8C2×D414yes—
202C22×Q8C2×D418yes—
203C22×D4—28no—
204C22×D4—22no—
205C22×D4—21no—
206C22×D4C2×D414yes—
207C22×D4C2×D412yes—
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—44no—
219C23×C4—44no—
220C23×C4—88no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C2×Q16—216no—
225C2×Q16—88no—
226C2×SD16—28no—
227C2×SD16—48no—
228C2×M16—24no—
229C2×M16—44no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8—24no—
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8C2×D418yes—
247C4⋊Q8C2×D414yes—
248C4⋊Q8C2412yes—
249C4⋊Q8C2412yes—
250C4⋊Q8C2414yes—
251#125—232no—
252#125—216no—
253#125—216no—
254#125—24no—
255#125—42no—
256C22⋊Q8—232no—
257C22⋊Q8—216no—
258C22⋊Q8—264no—
259C22⋊Q8—232no—
260C22⋊Q8—216no—
261C22⋊Q8—232no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—42no—
265C22⋊Q8—42no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—22no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—42no—
276C4×Q8—42no—
277C4×Q8C2×D414yes—
278C4×Q8C2×D418yes—
279C4×Q8C2414yes—
280C4×D4—232no—
281C4×D4—216no—
282C4×D4—216no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4C2×D418yes—
291C4×D4C2×D414yes—
292C4×D4C2414yes—
293#117—24no—
294#117—42no—
295C2×C4⋊C4—216no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—232no—
298C2×C4⋊C4—216no—
299C2×C4⋊C4—216no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—216no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—41no—
323C2×C4⋊C4—41no—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D414yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D414yes—
328C2×C22⋊C4—216no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—41no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—41no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348#107—28no—
349#106—28no—
350#106—28no—
351C4⋊C8—24no—
352C4⋊C8—24no—
353C4⋊C8—22no—
354C4⋊C8—22no—
355C4⋊C8C2414yes—
356C4⋊C8C2414yes—
357#103—28no—
358#103—28no—
359#103—216no—
360#103—28no—
361#102—28no—
362C22⋊C8—24no—
363C22⋊C8—24no—
364C4×C8—24no—
365C4×C8—22no—
366C4×C8C2414yes—
367C24—28no—
368C24—24no—
369C24—22no—
370C24—44no—
371C24—44no—
372C24#12011yescharacteristic
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—28no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—28no—
379C2×Q8—24no—
380C2×Q8—28no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—48no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—84no—
391C2×Q8—88no—
392C2×Q8—88no—
393C2×Q8C22×D414yes—
394C2×Q8C22×D418yes—
395C2×D4—28no—
396C2×D4—216no—
397C2×D4—216no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4#13718yes—
402C2×D4C22×D414yes—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—232no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—416no—
426C22×C4—48no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—832no—
430C22×C4—816no—
431C22×C4#12012yes—
432C22×C4#12011yescharacteristic
433C22×C4#12011yescharacteristic
434C22×C4#12012yes—
435C22×C4#12012yes—
436C22×C4#12011yescharacteristic
437C22×C4#13514yes—
438C22×C4#13714yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445Q16—816no—
446SD16—416no—
447M16—48no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—44no—
452C2×C8—44no—
453C2×C8—44no—
454C4⋊C4—232no—
455C4⋊C4—232no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—28no—
459C4⋊C4—24no—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4C22×D414yes—
480C4⋊C4C22×D414yes—
481C22⋊C4—232no—
482C22⋊C4—28no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—44no—
486C42—28no—
487C42—24no—
488C42—24no—
489C42—44no—
490C42#12012yes—
491C42#12012yes—
492C42#12012yes—
493C42C22×D411yescharacteristic
494C42C22×D412yes—
495C42C22×D412yes—
496C42C22×D412yes—
497C42C22×D411yescharacteristic
498C42C22×D411yescharacteristic
499C42C2511yescharacteristic
500C23—22no—
501C23—22no—
502C23—216no—
503C23—24no—
504C23—28no—
505C23—48no—
506C23—432no—
507C23—48no—
508C23—44no—
509C23—44no—
510C23#45214yes—
511C23C2×#12012yes—
512C23C2×#12011yescharacteristic
513Q8—24no—
514Q8—24no—
515Q8—48no—
516Q8—44no—
517Q8—48no—
518Q8—44no—
519Q8—44no—
520Q8—84no—
521Q8—88no—
522Q8—82no—
523Q8—82no—
524D4—216no—
525D4—28no—
526D4—24no—
527D4C2×#13718yes—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—48no—
540C2×C4—416no—
541C2×C4—48no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—416no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—832no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#45218yes—
557C2×C4#45214yes—
558C2×C4C2×#12012yes—
559C2×C4C2×#12012yes—
560C2×C4C2×#12011yescharacteristic
561C2×C4C2×#12012yes—
562C2×C4C2×#12011yescharacteristic
563C2×C4C2×#12011yescharacteristic
564C2×C4C2×#13512yes—
565C2×C4C2×#13514yes—
566C2×C4C2×#13714yes—
567C2×C4C2×#13712yes—
568C2×C4C2×#13714yes—
569C2×C4C23×D411yescharacteristic
570C2×C4C23×D411yescharacteristic
571C2×C4C23×D411yescharacteristic
572C8—44no—
573C8—44no—
574C8—44no—
575C22—28no—
576C22—22no—
577C22—22no—
578C22—28no—
579C22—432no—
580C22—416no—
581C22—44no—
582C22—44no—
583C22C2×#45214yes—
584C22C2×#45212yes—
585C22#300412yes—
586C22#301912yes—
587C22C22×#12011yescharacteristic
588C4—22no—
589C4—21no—
590C4—22no—
591C4—21no—
592C4—44no—
593C4—44no—
594C4—44no—
595C4—48no—
596C4—82no—
597C4—82no—
598C4C2×#45214yes—
599C4C2×#45212yes—
600C4#300412yes—
601C4C22×#13511yescharacteristic
602C4C22×#13711yescharacteristic
603C4C22×#13712yes—
604C2—24no—
605C2—48no—
606C2#1875314yesdirect product factor
607C2C22×#45211yescharacteristic
608C2C2×#300411yescharacteristic
609C2C2×#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
4. C2×#3078C2semidirect
5. #18753C2direct (N × H)
6. C2×#1643C2semidirect
7. C2×#1632C2semidirect
8. C2×#1548C2semidirect
10. C2×#1479C2semidirect
29. #3209C22semidirect
30. C2×#494C22semidirect
31. C2×#487C22semidirect
36. #3078C22semidirect
38. C2×#477C22semidirect
39. C2×#475C22semidirect
51. #1643C22semidirect
52. #1632C22semidirect
53. #1548C22semidirect
54. C2×#333C22semidirect
56. #1479C22semidirect
57. C2×Q8⋊C8C22semidirect
145. #494C23semidirect
147. #487C23semidirect
151. #477C23semidirect
155. #475C23semidirect
186. #333C23semidirect
187. Q8⋊C8C23semidirect
606. C2#18753direct (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×#31012
C2×#30782
#187532
C2×#16432
C2×#16322
C2×#15482
C2×#14862
C2×#14792