Finite Group Databasebeta

← C2×#18653 #6348231

Subgroups — C2×#18653 #6348231

Order 512.

2986 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic
2C2×#3209C211yesmaximal, characteristic, complemented
3C22×#500C211yesmaximal, characteristic, complemented
4C2×#3079C211yesmaximal, characteristic
5#18653C2116yesmaximal, direct product factor
6C2×#1648C214yesmaximal, complemented
7C2×#1649C214yesmaximal, complemented
8C2×#1549C211yesmaximal, characteristic, complemented
9C2×#1493C211yesmaximal, characteristic, complemented
10C2×#1476C211yesmaximal, characteristic
11#3503—24no—
12#3503—24no—
13#3503—24no—
14#3503C2218yes—
15C2×Q82—21no—
16C2×#557C2212yes—
17C2×#554—21no—
18C2×#554—22no—
19C2×D4×Q8—22no—
20C2×D4×Q8C2212yes—
21C2×#544—22no—
22C2×#544—21no—
23C2×#533—21no—
24C2×#533C2211yescharacteristic
25C22×C4⋊Q8—21no—
26C22×C4⋊Q8C2211yescharacteristic
27C22×C4⋊Q8C2211yescharacteristic
28C2×#520—21no—
29C2×C4×#41C2211yescharacteristic
30#3209—24no—
31#3209C2218yescomplemented
32C2×#500—22no—
33C2×#500—24no—
34C2×#500C2214yes—
35C2×#500C2218yescomplemented
36C2×#494—21no—
37C2×#487—21no—
38#3079—24no—
39#3079C2218yes—
40C2×#477—21no—
41C2×#475C2212yes—
42C2×#476—21no—
43C2×#474C2212yes—
44C2×#462C2214yes—
45C2×#451—22no—
46C2×#431C2211yescharacteristic
47C2×#428—21no—
48C22×#107—22no—
49C2×#423C2211yescharacteristic
50C2×#417—21no—
51C22×C4×C8C2211yescharacteristic
52#1649C22132yescomplemented
53#1648C22132yescomplemented
54#1549C2218yescomplemented
55C2×#332C2214yes—
56#1493C2218yescomplemented
57#1476C2218yes—
58C2×Q8⋊C8C2212yes—
59C2×Q8⋊C8C2214yes—
60C2×D4⋊C8C2212yes—
61C22×#41D411yescharacteristic
62C23×Q8—22no—
63C23×Q8—21no—
64C23×Q8—21no—
65C22×Q16—24no—
66C22×M16—21no—
67C23×C8—21no—
68Q82—28no—
69Q82—28no—
70#557—24no—
71#557—24no—
72#557—24no—
73#557C2318yes—
74#554—28no—
75#554—216no—
76#554—28no—
77#554—216no—
78D4×Q8—216no—
79D4×Q8—24no—
80D4×Q8—216no—
81D4×Q8—24no—
82D4×Q8—24no—
83D4×Q8C2318yes—
84#544—216no—
85#544—28no—
86#544—216no—
87#544—28no—
88#533—28no—
89#533—28no—
90#533—22no—
91#533—22no—
92#533D414yes—
93#533C2314yes—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—22no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8D411yescharacteristic
112C2×C4⋊Q8D411yescharacteristic
113C2×C4⋊Q8D412yes—
114C2×C4⋊Q8D414yes—
115C2×C4⋊Q8C2312yes—
116C2×C4⋊Q8C2314yes—
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2314yes—
119C2×#125—22no—
120C2×#125—24no—
121C2×#125—22no—
122C2×#125—21no—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8—24no—
125C2×C22⋊Q8—24no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—22no—
128C2×C22⋊Q8—24no—
129C2×C22⋊Q8D412yes—
130#520—28no—
131#520—28no—
132C4×#41—22no—
133C4×#41—22no—
134C4×#41D414yes—
135C4×#41C2314yes—
136C2×C4×Q8—22no—
137C2×C4×Q8—22no—
138C2×C4×Q8—22no—
139C2×C4×Q8—21no—
140C2×C4×Q8D411yescharacteristic
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2312yes—
143C2×C4×D4—24no—
144C2×C4×D4—22no—
145C2×C4×D4—21no—
146C2×C4×D4—21no—
147C2×C4×D4D411yescharacteristic
148C2×C4×D4C2311yescharacteristic
149C2×#117—22no—
150C2×#117—21no—
151C2×#117—21no—
152C2×#117D411yescharacteristic
153C22×C4⋊C4—22no—
154C22×C4⋊C4—22no—
155C22×C4⋊C4—21no—
156C22×C4⋊C4—21no—
157C22×C4⋊C4—21no—
158C22×C4⋊C4D412yes—
159C22×C42—21no—
160C22×C42C2311yescharacteristic
161#500—28no—
162#500—216no—
163#500C23116yes—
164#494—28no—
165#487—28no—
166#477—28no—
167#476—28no—
168#475—24no—
169#475C2318yes—
170#474—24no—
171#474C2318yes—
172#462—28no—
173#462C23116yes—
174#451—216no—
175#431—22no—
176#431C2314yes—
177#428—28no—
178C2×#107—22no—
179C2×#107—28no—
180C2×#107—216no—
181C2×#106—22no—
182#423—22no—
183#423C2314yes—
184C2×C4⋊C8—21no—
185C2×C4⋊C8C2312yes—
186C2×C4⋊C8C2312yes—
187#417—28no—
188C2×#103—22no—
189C2×#103—24no—
190C2×#102—22no—
191C2×C22⋊C8—21no—
192C2×C4×C8—21no—
193C2×C4×C8—22no—
194C2×C4×C8—21no—
195C2×C4×C8C2312yes—
196C2×C4×C8C2314yes—
197#332C23116yes—
198Q8⋊C8C2318yes—
199Q8⋊C8C23116yes—
200D4⋊C8C2318yes—
201C2×#41—22no—
202C2×#41—24no—
203C2×#41Q1618yescomplemented
204C2×#41C2×D414yes—
205C22×Q8—24no—
206C22×Q8—28no—
207C22×Q8—28no—
208C22×Q8—21no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—22no—
212C22×Q8—24no—
213C22×Q8—22no—
214C22×Q8—24no—
215C22×Q8—42no—
216C22×Q8—41no—
217C22×Q8—84no—
218C22×Q8—84no—
219C22×Q8—88no—
220C22×Q8C2×D412yes—
221C22×Q8C2×D411yescharacteristic
222C22×D4—21no—
223C22×D4C2×D411yescharacteristic
224C23×C4—21no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—41no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C2×Q16—216no—
234C2×Q16—88no—
235C2×M16—24no—
236C2×M16—44no—
237C22×C8—28no—
238C22×C8—24no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C4⋊Q8—216no—
243C4⋊Q8—28no—
244C4⋊Q8—216no—
245C4⋊Q8—232no—
246C4⋊Q8—28no—
247C4⋊Q8—28no—
248C4⋊Q8—28no—
249C4⋊Q8—22no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—24no—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267C4⋊Q8C2×D414yes—
268C4⋊Q8C2×D418yes—
269C4⋊Q8C2×D414yes—
270C4⋊Q8C2×D414yes—
271C4⋊Q8C2414yes—
272C4⋊Q8C2414yes—
273#125—216no—
274#125—216no—
275#125—28no—
276#125—216no—
277#125—28no—
278#125—24no—
279#125—24no—
280C22⋊Q8—28no—
281C22⋊Q8—216no—
282C22⋊Q8—232no—
283C22⋊Q8—216no—
284C22⋊Q8—232no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—24no—
288C22⋊Q8—28no—
289C22⋊Q8—216no—
290C22⋊Q8C2×D418yes—
291C4×Q8—28no—
292C4×Q8—28no—
293C4×Q8—216no—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—24no—
297C4×Q8—22no—
298C4×Q8—24no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8C2×D414yes—
304C4×Q8C2×D412yes—
305C4×Q8C2×D414yes—
306C4×Q8C2412yes—
307C4×Q8C2414yes—
308C4×D4—232no—
309C4×D4—28no—
310C4×D4—28no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—22no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4C2×D414yes—
319C4×D4C2×D412yes—
320C4×D4C2412yes—
321#117—216no—
322#117—24no—
323#117—22no—
324#117—24no—
325#117—24no—
326#117—24no—
327#117C2×D414yes—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—216no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—216no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—41no—
354C2×C4⋊C4—41no—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D418yes—
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D412yes—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C42—24no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—21no—
370C2×C42—24no—
371C2×C42—24no—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D412yes—
375C2×C42C2×D412yes—
376C2×C42C2×D412yes—
377C2×C42C2×D411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2412yes—
380#107—232no—
381#107—28no—
382#106—28no—
383C4⋊C8—24no—
384C4⋊C8—22no—
385C4⋊C8—22no—
386C4⋊C8C2414yes—
387C4⋊C8C2414yes—
388#103—28no—
389#103—216no—
390#102—28no—
391C22⋊C8—24no—
392C4×C8—24no—
393C4×C8—22no—
394C4×C8—24no—
395C4×C8C2414yes—
396C24—21no—
397C24#12011yescharacteristic
398#41—28no—
399#41—216no—
400#41C2×Q16116yescomplemented
401C2×Q8—216no—
402C2×Q8—21no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—28no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—28no—
410C2×Q8—24no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—48no—
414C2×Q8—44no—
415C2×Q8—48no—
416C2×Q8—84no—
417C2×Q8—88no—
418C2×Q8—88no—
419C2×Q8—88no—
420C2×Q8—84no—
421C2×Q8—88no—
422C2×Q8—88no—
423C2×Q8—816no—
424C2×Q8—816no—
425C2×Q8C2×Q1614yes—
426C2×Q8C22×D414yes—
427C2×Q8C22×D412yes—
428C2×D4—24no—
429C2×D4—28no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4C2×Q1614yes—
433C2×D4C22×D412yes—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—216no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—84no—
454C22×C4—88no—
455C22×C4—84no—
456C22×C4#12012yes—
457C22×C4#12012yes—
458C22×C4#12011yescharacteristic
459C22×C4#12012yes—
460C22×C4#12011yescharacteristic
461C22×C4#12011yescharacteristic
462C22×C4C2×Q1614yes—
463C22×C4#13714yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470Q16—816no—
471M16—48no—
472C2×C8—28no—
473C2×C8—216no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—44no—
478C2×C8—44no—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—216no—
483C4⋊C4—24no—
484C4⋊C4—28no—
485C4⋊C4—24no—
486C4⋊C4—28no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4C22×D414yes—
507C4⋊C4C22×D412yes—
508C4⋊C4C22×D418yes—
509C4⋊C4C22×D412yes—
510C22⋊C4—216no—
511C22⋊C4—28no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C42—216no—
517C42—24no—
518C42—22no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—24no—
523C42#12012yes—
524C42#12012yes—
525C42#12012yes—
526C42C22×D412yes—
527C42C22×D412yes—
528C42C22×D411yescharacteristic
529C42C22×D411yescharacteristic
530C42C22×D411yescharacteristic
531C42C22×D412yes—
532C42C2511yescharacteristic
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—42no—
538C23—42no—
539C23#45114yes—
540C23C2×#12012yes—
541C23C2×#12011yescharacteristic
542Q8—24no—
543Q8—22no—
544Q8—28no—
545Q8—44no—
546Q8—48no—
547Q8—48no—
548Q8—48no—
549Q8—48no—
550Q8—88no—
551Q8—88no—
552Q8—816no—
553Q8—84no—
554Q8C22×Q1614yes—
555D4—216no—
556D4—24no—
557D4—22no—
558D4C22×Q1614yes—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—216no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—88no—
583C2×C4—816no—
584C2×C4—88no—
585C2×C4—84no—
586C2×C4#45114yes—
587C2×C4#45118yes—
588C2×C4#45614yes—
589C2×C4C2×#12011yescharacteristic
590C2×C4C2×#12012yes—
591C2×C4C2×#12011yescharacteristic
592C2×C4C2×#12012yes—
593C2×C4C2×#12011yescharacteristic
594C2×C4C2×#12012yes—
595C2×C4C22×Q1612yes—
596C2×C4C22×Q1614yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13714yes—
599C2×C4#57614yes—
600C2×C4C23×D411yescharacteristic
601C2×C4C23×D411yescharacteristic
602C2×C4C23×D411yescharacteristic
603C8—28no—
604C8—44no—
605C8—44no—
606C22—22no—
607C22—28no—
608C22—22no—
609C22—24no—
610C22—42no—
611C22—48no—
612C22—42no—
613C22C2×#45114yes—
614C22C2×#45112yes—
615C22#300812yes—
616C22C2×#45612yes—
617C22C22×#12011yescharacteristic
618C4—24no—
619C4—22no—
620C4—21no—
621C4—21no—
622C4—42no—
623C4—44no—
624C4—48no—
625C4—42no—
626C4—44no—
627C4—84no—
628C4C2×#45114yes—
629C4C2×#45112yes—
630C4#300812yes—
631C4C2×#45614yes—
632C4C23×Q1611yescharacteristic
633C4C22×#13711yescharacteristic
634C4C2×#57612yes—
635C2—24no—
636C2—44no—
637C2#1865314yesdirect product factor
638C2C22×#45111yescharacteristic
639C2C2×#300811yescharacteristic
640C2C22×#45611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC22×#41D4
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. C22×#500C2semidirect
5. #18653C2direct (N × H)
6. C2×#1648C2semidirect
7. C2×#1649C2semidirect
8. C2×#1549C2semidirect
9. C2×#1493C2semidirect
31. #3209C22semidirect
35. C2×#500C22semidirect
52. #1649C22semidirect
53. #1648C22semidirect
54. #1549C22semidirect
56. #1493C22semidirect
203. C2×#41Q16semidirect
400. #41C2×Q16semidirect
637. C2#18653direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#35032
C2×#32092
C22×#5002
C2×#30792
#186532
C2×#16482
C2×#16492
C2×#15492
C2×#14932
C2×#14762