Finite Group Databasebeta

← C2×#18760 #6348674

Subgroups — C2×#18760 #6348674

Order 512.

3046 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic, complemented
2C2×#3226C211yesmaximal, characteristic
3C2×#3103C211yesmaximal, characteristic
4C2×#3037C211yesmaximal, characteristic
5#18760C2116yesmaximal, direct product factor
6C2×#1648C214yesmaximal, complemented
7C2×#1626C214yesmaximal, complemented
8C2×#1546C211yesmaximal, characteristic, complemented
9C2×#1487C211yesmaximal, characteristic, complemented
10C2×#1479C211yesmaximal, characteristic, complemented
11C22×#137—21no—
12C2×#557—22no—
13C2×#557—24no—
14C2×#557—22no—
15C2×#557—24no—
16C2×#557—22no—
17C2×#557—24no—
18C2×#557C2214yes—
19C2×#557C2218yescomplemented
20C2×#532C2211yescharacteristic
21C22×C4⋊Q8—21no—
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125—21no—
24C22×#125—22no—
25C22×C4×Q8—21no—
26C22×C4×Q8—21no—
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29#3226—24no—
30#3226C2218yes—
31C2×#500C2212yes—
32C2×#486C2212yes—
33#3103—24no—
34#3103C2218yes—
35C2×#481—21no—
36C2×#480—22no—
37C2×#475C2212yes—
38C2×#474—22no—
39#3037—24no—
40#3037C2218yes—
41C2×#464—22no—
42C2×#462C2212yes—
43C2×#460C2212yes—
44C2×#460C2212yes—
45C2×#448—21no—
46C2×C4×SD16—22no—
47C2×#431C2211yescharacteristic
48C2×#431C2211yescharacteristic
49C2×#423C2211yescharacteristic
50C2×#417—21no—
51#1648C22132yescomplemented
52#1626C22132yescomplemented
53#1546C2218yescomplemented
54C2×#332C2214yes—
55#1487C2218yescomplemented
56#1479C2218yescomplemented
57C2×Q8⋊C8C2214yes—
58C2×Q8⋊C8C2214yes—
59C22×#41—21no—
60C23×Q8—21no—
61C23×Q8D411yescharacteristic
62C23×Q8D411yescharacteristic
63C2×#137—28no—
64C2×#137—48no—
65C22×Q16—22no—
66C22×SD16—22no—
67C22×SD16—22no—
68C22×M16—21no—
69#557—216no—
70#557—216no—
71#557—216no—
72#557—28no—
73#557—216no—
74#557—28no—
75#557—28no—
76#557—216no—
77#557—216no—
78#557C23116yes—
79#532—22no—
80#532—22no—
81#532—22no—
82#532C2314yes—
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8D414yes—
92C2×C4⋊Q8D412yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2314yes—
96C2×#127C2311yescharacteristic
97C2×#125—216no—
98C2×#125—28no—
99C2×#125—28no—
100C2×#125—24no—
101C2×#125—28no—
102C2×#125—24no—
103C2×#125—216no—
104C2×#125—28no—
105C2×#124—21no—
106C2×#121—21no—
107C2×#121—21no—
108C2×C4×Q8—28no—
109C2×C4×Q8—24no—
110C2×C4×Q8—28no—
111C2×C4×Q8—24no—
112C2×C4×Q8—24no—
113C2×C4×Q8—28no—
114C2×C4×Q8—24no—
115C2×C4×Q8—28no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8—22no—
120C2×C4×Q8—21no—
121C2×C4×Q8—22no—
122C2×C4×Q8—21no—
123C2×C4×Q8—22no—
124C2×C4×Q8D414yes—
125C2×C4×Q8D412yes—
126C2×C4×Q8D412yes—
127C2×C4×Q8D414yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2314yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2314yes—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4D412yes—
141C22×C4⋊C4D411yescharacteristic
142C22×C4⋊C4D411yescharacteristic
143C22×C42—22no—
144C22×C42—21no—
145C22×C42—21no—
146C22×C42C2311yescharacteristic
147#500—24no—
148#500C2318yes—
149#486—24no—
150#486C2318yes—
151#481—28no—
152#480—216no—
153#475—24no—
154#475C2318yes—
155#474—216no—
156#464—216no—
157#462—24no—
158#462C2318yes—
159#460—24no—
160#460—24no—
161#460C2318yes—
162#460C2318yes—
163#448—28no—
164C4×SD16—216no—
165#431—22no—
166#431—22no—
167#431C2314yes—
168#431C2314yes—
169C2×#107—24no—
170C2×#106—22no—
171#423—22no—
172#423C2314yes—
173C2×C4⋊C8—21no—
174C2×C4⋊C8—21no—
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2312yes—
177#417—28no—
178C2×#103—22no—
179C2×#103—22no—
180C2×#102—24no—
181C2×#102—22no—
182C2×#102—22no—
183C2×C22⋊C8—21no—
184C2×C22⋊C8—21no—
185C2×C4×C8—21no—
186C2×C4×C8C2312yes—
187#332C23116yes—
188Q8⋊C8C23116yes—
189Q8⋊C8C23116yes—
190C2×#41—24no—
191C2×#41—42no—
192C2×#41—44no—
193C2×#41—44no—
194C22×Q8—28no—
195C22×Q8—22no—
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—21no—
199C22×Q8—24no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—21no—
206C22×Q8—44no—
207C22×Q8—44no—
208C22×Q8C2×D414yes—
209C22×Q8C2×D412yes—
210C22×Q8C2×D414yes—
211C22×Q8C2×D412yes—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4—41no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—22no—
220C23×C4—21no—
221C23×C4—41no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225#137—432no—
226C2×Q16—28no—
227C2×Q16—48no—
228C2×SD16—28no—
229C2×SD16—28no—
230C2×SD16—48no—
231C2×SD16—48no—
232C2×M16—24no—
233C2×M16—44no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C4⋊Q8—216no—
238C4⋊Q8—232no—
239C4⋊Q8—216no—
240C4⋊Q8—24no—
241C4⋊Q8—24no—
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8C2×D418yes—
251C4⋊Q8C2×D414yes—
252C4⋊Q8C2412yes—
253C4⋊Q8C2414yes—
254#127—21no—
255#127—21no—
256#127—21no—
257#127C2412yes—
258#125—232no—
259#125—264no—
260#125—216no—
261#125—232no—
262#125—232no—
263#125—216no—
264#124—24no—
265#124—42no—
266#121—24no—
267#121—24no—
268#121—42no—
269#121—42no—
270C4×Q8—216no—
271C4×Q8—232no—
272C4×Q8—216no—
273C4×Q8—232no—
274C4×Q8—216no—
275C4×Q8—216no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—22no—
283C4×Q8—24no—
284C4×Q8—22no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8C2×D418yes—
291C4×Q8C2×D414yes—
292C4×Q8C2×D414yes—
293C4×Q8C2×D418yes—
294C4×Q8C2414yes—
295C4×Q8C2414yes—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—42no—
299C4×D4—42no—
300C2×C4⋊C4—216no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—216no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—216no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D414yes—
330C2×C4⋊C4C2×D414yes—
331C2×C4⋊C4C2×D418yes—
332C2×C22⋊C4—41no—
333C2×C22⋊C4—41no—
334C2×C42—216no—
335C2×C42—28no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352#107—216no—
353#106—28no—
354C4⋊C8—24no—
355C4⋊C8—24no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358C4⋊C8C2414yes—
359C4⋊C8C2414yes—
360#103—28no—
361#103—28no—
362#102—28no—
363#102—28no—
364#102—216no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367C4×C8—24no—
368C4×C8—22no—
369C4×C8C2414yes—
370C24—41no—
371C24#12011yescharacteristic
372#41—48no—
373#41—48no—
374#41—88no—
375C2×Q8—216no—
376C2×Q8—28no—
377C2×Q8—216no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—28no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—28no—
388C2×Q8—24no—
389C2×Q8—28no—
390C2×Q8—22no—
391C2×Q8—41no—
392C2×Q8—44no—
393C2×Q8—48no—
394C2×Q8—44no—
395C2×Q8—48no—
396C2×Q8—44no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—416no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8#13718yescomplemented
403C2×Q8C22×D414yes—
404C2×Q8C22×D414yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—84no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—41no—
435C22×C4—44no—
436C22×C4—84no—
437C22×C4#12012yes—
438C22×C4#12011yescharacteristic
439C22×C4#12011yescharacteristic
440C22×C4#12012yes—
441C22×C4#12012yes—
442C22×C4#12011yescharacteristic
443C22×C4#13514yes—
444C22×C4AGL(1,ℤ/8)14yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451Q16—416no—
452SD16—416no—
453SD16—416no—
454M16—48no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—44no—
459C2×C8—44no—
460C2×C8—44no—
461C4⋊C4—232no—
462C4⋊C4—232no—
463C4⋊C4—232no—
464C4⋊C4—28no—
465C4⋊C4—24no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—28no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4C22×D414yes—
484C4⋊C4C22×D414yes—
485C4⋊C4C22×D418yes—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C42—232no—
489C42—28no—
490C42—24no—
491C42—28no—
492C42—24no—
493C42—24no—
494C42—24no—
495C42#12012yes—
496C42#12012yes—
497C42#12012yes—
498C42C22×D412yes—
499C42C22×D411yescharacteristic
500C42C22×D412yes—
501C42C22×D412yes—
502C42C22×D411yescharacteristic
503C42C22×D411yescharacteristic
504C42C2511yescharacteristic
505C23—22no—
506C23—22no—
507C23—42no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23#44914yes—
512C23C2×#12012yes—
513C23C2×#12011yescharacteristic
514Q8—216no—
515Q8—28no—
516Q8—24no—
517Q8—24no—
518Q8—24no—
519Q8—48no—
520Q8—44no—
521Q8—416no—
522Q8—48no—
523Q8—44no—
524Q8—44no—
525Q8—42no—
526Q8—42no—
527Q8—82no—
528Q8C2×#13718yescomplemented
529D4—44no—
530D4—42no—
531D4—42no—
532D4—44no—
533D4—82no—
534D4—88no—
535D4—82no—
536D4—82no—
537C2×C4—216no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—416no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—82no—
561C2×C4—88no—
562C2×C4#44914yes—
563C2×C4#44918yes—
564C2×C4C2×#12012yes—
565C2×C4C2×#12012yes—
566C2×C4C2×#12011yescharacteristic
567C2×C4C2×#12012yes—
568C2×C4C2×#12011yescharacteristic
569C2×C4C2×#12011yescharacteristic
570C2×C4C2×#13512yes—
571C2×C4C2×#13514yes—
572C2×C4C2×AGL(1,ℤ/8)12yes—
573C2×C4C2×AGL(1,ℤ/8)14yes—
574C2×C4C2×#13714yes—
575C2×C4C23×D411yescharacteristic
576C2×C4C23×D411yescharacteristic
577C2×C4C23×D411yescharacteristic
578C8—44no—
579C8—44no—
580C8—44no—
581C22—28no—
582C22—22no—
583C22—22no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—164no—
588C22C2×#44914yes—
589C22C2×#44912yes—
590C22#300412yes—
591C22#302012yes—
592C22C22×#12011yescharacteristic
593C4—28no—
594C4—22no—
595C4—21no—
596C4—22no—
597C4—21no—
598C4—44no—
599C4—48no—
600C4—44no—
601C4—44no—
602C4—82no—
603C4C2×#44914yes—
604C4C2×#44912yes—
605C4#300412yes—
606C4C22×#13511yescharacteristic
607C4C22×AGL(1,ℤ/8)11yescharacteristic
608C4C22×#13712yes—
609C2—24no—
610C2—162no—
611C2#1876014yesdirect product factor
612C2C22×#44911yescharacteristic
613C2C2×#300411yescharacteristic
614C2C2×#302011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC2×#532C22
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×#557C2semidirect
5. #18760C2direct (N × H)
6. C2×#1648C2semidirect
7. C2×#1626C2semidirect
8. C2×#1546C2semidirect
9. C2×#1487C2semidirect
10. C2×#1479C2semidirect
19. C2×#557C22semidirect
51. #1648C22semidirect
52. #1626C22semidirect
53. #1546C22semidirect
55. #1487C22semidirect
56. #1479C22semidirect
402. C2×Q8#137semidirect
528. Q8C2×#137semidirect
611. C2#18760direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#32262
C2×#31032
C2×#30372
#187602
C2×#16482
C2×#16262
C2×#15462
C2×#14872
C2×#14792