Finite Group Databasebeta

← C2×#24621 #6453175

Subgroups — C2×#24621 #6453175

Order 512.

2012 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3095C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic
3C2×#2924C211yesmaximal, characteristic
4C22×#428C211yesmaximal, characteristic
5#24621C2116yesmaximal, direct product factor
6C2×#2090C212yesmaximal, complemented
7C2×#2064C212yesmaximal, complemented
8C2×#1951C212yesmaximal, complemented
9C2×#1930C212yesmaximal, complemented
10C2×#1901C211yesmaximal, characteristic
11C2×#1855C211yesmaximal, characteristic
12C2×#1821C211yesmaximal, characteristic
13C23×M16C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#3095—24no—
18#3095C2218yes—
19C2×#484—21no—
20C2×#485—21no—
21C2×#485—21no—
22C2×#483—21no—
23C2×#483—21no—
24C2×#483—22no—
25C2×#483—24no—
26C2×#483—21no—
27C2×#483C2218yes—
28#2924—24no—
29#2924C2218yes—
30C2×#433—22no—
31C2×#432—21no—
32C2×#431—21no—
33C22×#108C2212yes—
34C2×#428—21no—
35C2×#428—21no—
36C2×#428—24no—
37C2×#428C2218yes—
38C2×#428C2212yes—
39C2×#428C2212yes—
40C2×#428C2211yescharacteristic
41C2×#428C2211yescharacteristic
42C22×#107—21no—
43C22×#106—21no—
44C22×#106C2212yes—
45C2×#424C2212yes—
46C2×#419C2211yescharacteristic
47C2×#419C2211yescharacteristic
48C22×#103C2212yes—
49C22×#101C2212yes—
50C2×#407C2211yescharacteristic
51C2×#407C2211yescharacteristic
52C22×C22⋊C8C2211yescharacteristic
53C2×#405C2212yes—
54#2090C22116yescomplemented
55#2064C22116yescomplemented
56#1951C22116yescomplemented
57#1930C22116yescomplemented
58#1901C2218yes—
59#1855C2218yes—
60#1821C2218yes—
61C2×#343C2212yes—
62C2×#343C2212yes—
63C2×#343C2212yes—
64C2×#341C2212yes—
65C23×Q8C2311yescharacteristic
66C24×C4—21no—
67C24×C4C2311yescharacteristic
68C22×M16—21no—
69C22×M16—22no—
70C22×M16—28no—
71C22×M16C2312yes—
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2314yes—
75C22×M16C2312yes—
76C23×C8—21no—
77C23×C8C2312yes—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8C2314yes—
86C2×#117—21no—
87C2×#117—22no—
88C2×#117—21no—
89C2×#117—22no—
90C2×#117D411yescharacteristic
91C2×#117D411yescharacteristic
92C2×#117D414yes—
93C2×#117C2314yes—
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C22×C4⋊C4—21no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C42—21no—
113#485—28no—
114#485—28no—
115#484—28no—
116#483—28no—
117#483—28no—
118#483—28no—
119#483—216no—
120#433—216no—
121#432—28no—
122#431—28no—
123C2×#108—28no—
124C2×#108C2318yes—
125#428—22no—
126#428—22no—
127#428—24no—
128#428—24no—
129#428—28no—
130#428—28no—
131#428C2318yes—
132#428C2314yes—
133#428C2314yes—
134#428C2318yes—
135C2×#107—21no—
136C2×#107—21no—
137C2×#107—22no—
138C2×#107—22no—
139C2×#107—28no—
140C2×#106—22no—
141C2×#106—22no—
142C2×#106—24no—
143C2×#106—21no—
144C2×#106—21no—
145C2×#106—22no—
146C2×#106—22no—
147C2×#106—28no—
148C2×#106C2318yes—
149#424—24no—
150#424C2318yes—
151C2×C4⋊C8—22no—
152C2×C4⋊C8—22no—
153#419—22no—
154#419—22no—
155#419C2314yes—
156#419C2314yes—
157C2×#103—22no—
158C2×#103—22no—
159C2×#103—24no—
160C2×#103—21no—
161C2×#103—21no—
162C2×#103—21no—
163C2×#103—21no—
164C2×#103C2318yes—
165C2×#101—28no—
166C2×#101C2318yes—
167#407—22no—
168#407—22no—
169#407C2314yes—
170#407C2314yes—
171C2×C22⋊C8—21no—
172C2×C22⋊C8—21no—
173C2×C22⋊C8—21no—
174C2×C22⋊C8—21no—
175C2×C22⋊C8D411yescharacteristic
176C2×C22⋊C8D411yescharacteristic
177C2×C22⋊C8D412yes—
178C2×C22⋊C8D414yes—
179C2×C22⋊C8C2314yes—
180#405—24no—
181#405C2318yes—
182C2×#97—22no—
183C2×C4×C8—22no—
184#343C2318yes—
185#343C2318yes—
186#343C2318yes—
187#341C2318yes—
188C25#4111yescharacteristic
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—21no—
192C22×Q8—21no—
193C22×Q8—21no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8C2412yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—42no—
206C23×C4—41no—
207C23×C4—41no—
208C23×C4—84no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D412yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4#4111yescharacteristic
216C23×C4#4112yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C2×M16—216no—
220C2×M16—28no—
221C2×M16—24no—
222C2×M16—22no—
223C2×M16—24no—
224C2×M16—216no—
225C2×M16—22no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—24no—
229C2×M16—24no—
230C2×M16—24no—
231C2×M16—24no—
232C2×M16#4118yes—
233C2×M16C2412yes—
234C2×M16C2414yes—
235C2×M16C2412yes—
236C2×M16C2414yes—
237C22×C8—28no—
238C22×C8—24no—
239C22×C8—22no—
240C22×C8—24no—
241C22×C8—24no—
242C22×C8—22no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8C2×D412yes—
246C22×C8C2×D412yes—
247C22×C8C2×D414yes—
248C22×C8C2414yes—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—42no—
254C22⋊Q8—42no—
255C22⋊Q8—42no—
256C22⋊Q8—42no—
257#117—24no—
258#117—24no—
259#117—22no—
260#117—22no—
261#117—21no—
262#117—21no—
263#117—24no—
264#117—24no—
265#117—21no—
266#117—21no—
267#117C2×D412yes—
268#117C2×D412yes—
269#117C2×D414yes—
270#117C2×D414yes—
271#117C2412yes—
272#117C2412yes—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—41no—
299C2×C4⋊C4—41no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—41no—
302C2×C4⋊C4—41no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—41no—
320C2×C22⋊C4—41no—
321C2×C42—24no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—24no—
327#108—216no—
328#107—28no—
329#107—28no—
330#107—24no—
331#107—24no—
332#106—28no—
333#106—28no—
334#106—28no—
335#106—28no—
336#106—24no—
337#106—24no—
338C4⋊C8—28no—
339C4⋊C8—28no—
340#103—28no—
341#103—28no—
342#103—24no—
343#103—24no—
344#103—24no—
345#103—24no—
346#101—216no—
347C22⋊C8—24no—
348C22⋊C8—24no—
349C22⋊C8—24no—
350C22⋊C8—24no—
351C22⋊C8C2×D414yes—
352C22⋊C8C2×D414yes—
353C22⋊C8C2×D418yes—
354#97—28no—
355C4×C8—28no—
356C24—24no—
357C24—24no—
358C24—42no—
359C24#12111yescharacteristic
360C24#12112yes—
361C24#12111yescharacteristic
362C24C2×#4112yes—
363C24C2×#4111yescharacteristic
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—44no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8—44no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—44no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—82no—
410C22×C4—88no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—82no—
414C22×C4—84no—
415C22×C4—88no—
416C22×C4#12112yes—
417C22×C4#12112yes—
418C22×C4#12112yes—
419C22×C4#12112yes—
420C22×C4#12112yes—
421C22×C4#12111yescharacteristic
422C22×C4#12111yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4114yes—
434C22×C42−1+414yes—
435C22×C4C2511yescharacteristic
436M16—216no—
437M16—28no—
438M16—24no—
439M16—24no—
440M16—24no—
441M16—24no—
442M16—24no—
443M16—42no—
444M16—42no—
445M16—44no—
446M16C2×#4118yes—
447C2×C8—216no—
448C2×C8—28no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—28no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—44no—
461C2×C8C22×D414yes—
462C2×C8C22×D414yes—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4C22×D412yes—
482C4⋊C4C22×D412yes—
483C4⋊C4C22×D412yes—
484C4⋊C4C22×D412yes—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C42—24no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—42no—
511C23—42no—
512C23—416no—
513C23#47114yes—
514C23C2×#12112yes—
515C23C2×#12111yescharacteristic
516C23C2×#12112yes—
517C23C2×#12111yescharacteristic
518C23#55214yes—
519C23C22×#4111yescharacteristic
520Q8—42no—
521Q8—42no—
522Q8—44no—
523Q8—82no—
524Q8—82no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—84no—
553C2×C4—82no—
554C2×C4—84no—
555C2×C4—88no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—88no—
559C2×C4—84no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—84no—
563C2×C4C2×#12112yes—
564C2×C4C2×#12112yes—
565C2×C4C2×#12111yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4#52512yes—
568C2×C4#52512yes—
569C2×C4#53612yes—
570C2×C4#53612yes—
571C2×C4#54714yes—
572C2×C4#54714yes—
573C2×C4#55214yes—
574C2×C4C23×D411yescharacteristic
575C2×C4C23×D411yescharacteristic
576C2×C4C22×#4112yes—
577C2×C4C2×2−1+412yes—
578C8—28no—
579C8—24no—
580C8—24no—
581C8—42no—
582C8—42no—
583C8—44no—
584C22—28no—
585C22—28no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—22no—
590C22—21no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—48no—
596C22—48no—
597C22—44no—
598C22#241312yes—
599C22#241312yes—
600C22C2×#47114yes—
601C22C2×#47112yes—
602C22#306912yes—
603C22C22×#12111yescharacteristic
604C22C2×#55212yes—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—84no—
612C4—82no—
613C4—82no—
614C4#241312yes—
615C4#241312yes—
616C4C2×#52511yescharacteristic
617C4C2×#53611yescharacteristic
618C4C2×#54712yes—
619C2—22no—
620C2—22no—
621C2—24no—
622C2—42no—
623C2#2462114yesdirect product factor
624C2C2×#241311yescharacteristic
625C2C22×#47111yescharacteristic
626C2C2×#306911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #24621C2direct (N × H)
6. C2×#2090C2semidirect
7. C2×#2064C2semidirect
8. C2×#1951C2semidirect
9. C2×#1930C2semidirect
54. #2090C22semidirect
55. #2064C22semidirect
56. #1951C22semidirect
57. #1930C22semidirect
623. C2#24621direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#30952
C22×#4832
C2×#29242
C22×#4282
#246212
C2×#20902
C2×#20642
C2×#19512
C2×#19302
C2×#19012
C2×#18552
C2×#18212