Finite Group Databasebeta

← C2×#24514 #6451978

Subgroups — C2×#24514 #6451978

Order 512.

2752 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#451C211yesmaximal, characteristic
2C22×#433C211yesmaximal, characteristic
3C2×#2625C211yesmaximal, characteristic
4C2×#2310C211yesmaximal, characteristic
5#24514C2116yesmaximal, direct product factor
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2066C211yesmaximal, characteristic, complemented
8C2×#2056C211yesmaximal, characteristic, complemented
9C2×#2056C211yesmaximal, characteristic, complemented
10C2×#1971C211yesmaximal, characteristic, complemented
11C2×#1969C211yesmaximal, characteristic, complemented
12C2×#1935C211yesmaximal, characteristic, complemented
13C2×#1933C211yesmaximal, characteristic, complemented
14C2×#1873C211yesmaximal, characteristic
15C2×#1854C211yesmaximal, characteristic
16C2×#1787C211yesmaximal, characteristic
17C24×Q8C2211yescharacteristic
18C23×Q16C2211yescharacteristic
19C23×Q16C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C2×#451—28no—
24C2×#451C2218yes—
25C2×#433—28no—
26C2×#433C2218yes—
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40C22×C22⋊C8C2211yescharacteristic
41C22×#97C2211yescharacteristic
42C22×C4×C8C2211yescharacteristic
43#2625—24no—
44#2625C2218yes—
45C2×#400—21no—
46C2×#398—21no—
47C2×#397—21no—
48C2×#397—21no—
49#2310—24no—
50#2310C2218yes—
51C2×#391—21no—
52C2×#391—21no—
53C2×#391C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#391C2211yescharacteristic
56C2×#391C2211yescharacteristic
57C2×#391C2211yescharacteristic
58C2×#391C2211yescharacteristic
59C2×#389C2211yescharacteristic
60C2×#389C2211yescharacteristic
61C2×#381—21no—
62C2×#381—21no—
63C2×#380C2211yescharacteristic
64#2068C2218yescomplemented
65#2066C2218yescomplemented
66#2056C2218yescomplemented
67#2056C2218yescomplemented
68#1971C2218yescomplemented
69#1969C2218yescomplemented
70#1935C2218yescomplemented
71#1933C2218yescomplemented
72#1873C2218yes—
73#1854C2218yes—
74#1787C2218yes—
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—22no—
82C23×Q8—21no—
83C23×Q8—24no—
84C23×Q8—24no—
85C23×Q8—42no—
86C23×Q8—42no—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C23×Q8C2314yes—
91C24×C4—21no—
92C24×C4C2311yescharacteristic
93C22×Q16—24no—
94C22×Q16—24no—
95C22×Q16—24no—
96C22×Q16—24no—
97C22×Q16—42no—
98C22×Q16—42no—
99C22×Q16C2314yes—
100C22×Q16C2314yes—
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104C23×C8C2311yescharacteristic
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8C2314yes—
109C2×#117—24no—
110C2×#117—24no—
111C2×#117—22no—
112C2×#117—42no—
113C2×#117—42no—
114C2×#117D414yes—
115C2×#117C2314yes—
116C2×#117C2314yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4D411yescharacteristic
131C22×C22⋊C4D411yescharacteristic
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#451—232no—
136#433—232no—
137C2×#107—42no—
138C2×#107C2314yes—
139C2×#106—42no—
140C2×#106C2314yes—
141C2×C4⋊C8—24no—
142C2×C4⋊C8—24no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103—24no—
148C2×#103—24no—
149C2×#103—42no—
150C2×#103—42no—
151C2×#103—42no—
152C2×#103—42no—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×C22⋊C8—24no—
162C2×C22⋊C8D418yes—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165C2×#97—24no—
166C2×#97C2314yes—
167C2×C4×C8—24no—
168C2×C4×C8C2314yes—
169#400—28no—
170#398—28no—
171#397—28no—
172#397—28no—
173#391—28no—
174#391—28no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391—22no—
179#391—22no—
180#391—22no—
181#391C2314yes—
182#391C2314yes—
183#391C2314yes—
184#391C2314yes—
185#391C2314yes—
186#391C2314yes—
187#389—22no—
188#389—22no—
189#389C2314yes—
190#389C2314yes—
191#381—28no—
192#381—28no—
193#380—22no—
194#380C2314yes—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214C25C2×D411yescharacteristic
215C22×Q8—28no—
216C22×Q8—216no—
217C22×Q8—28no—
218C22×Q8—28no—
219C22×Q8—24no—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—28no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—28no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—416no—
244C22×Q8—416no—
245C22×Q8—416no—
246C22×Q8—416no—
247C22×Q8—44no—
248C22×Q8—48no—
249C22×Q8—44no—
250C22×Q8—48no—
251C22×Q8#4114yes—
252C22×Q8#4114yes—
253C22×Q8C2412yes—
254C22×Q8C2412yes—
255C22×Q8C2412yes—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—28no—
261C23×C4—24no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—44no—
271C23×C4—44no—
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D412yes—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D414yes—
279C23×C4#4114yes—
280C23×C4C2412yes—
281C23×C4C2411yescharacteristic
282C2×Q16—28no—
283C2×Q16—28no—
284C2×Q16—28no—
285C2×Q16—28no—
286C2×Q16—44no—
287C2×Q16—44no—
288C2×Q16—416no—
289C2×Q16—416no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8C2×D414yes—
301C22×C8C2×D414yes—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2412yes—
306C22⋊Q8—28no—
307C22⋊Q8—28no—
308C22⋊Q8—48no—
309#117—216no—
310#117—28no—
311#117—28no—
312#117—44no—
313#117—44no—
314#117—48no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—82no—
337C2×C4⋊C4—82no—
338C2×C4⋊C4C2×D414yes—
339C2×C4⋊C4C2×D414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—82no—
353C2×C22⋊C4C2×D414yes—
354C2×C22⋊C4C2×D414yes—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—24no—
360C2×C42—44no—
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2412yes—
364C2×C42C2412yes—
365#107—44no—
366#106—44no—
367C4⋊C8—28no—
368C4⋊C8—28no—
369#103—28no—
370#103—28no—
371#103—28no—
372#103—28no—
373#103—44no—
374#103—44no—
375#103—44no—
376#103—44no—
377C22⋊C8—28no—
378C22⋊C8C2×D4116yes—
379#97—28no—
380C4×C8—28no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24#12114yes—
403C24C22×D412yes—
404C24C22×D411yescharacteristic
405C2×Q8—216no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—28no—
411C2×Q8—28no—
412C2×Q8—28no—
413C2×Q8—28no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—416no—
429C2×Q8—416no—
430C2×Q8—416no—
431C2×Q8—416no—
432C2×Q8—432no—
433C2×Q8—48no—
434C2×Q8—464no—
435C2×Q8—48no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—44no—
451C2×Q8C2×#4114yes—
452C2×Q8C2×#4114yes—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—216no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—432no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4#12114yes—
514C22×C4#12118yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C22×D414yes—
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D412yes—
523C22×C4C2×#4114yes—
524C22×C4C2×#4112yes—
525C22×C4C2×#4112yes—
526C22×C4C2×#4112yes—
527C22×C4H(5,𝔽2)12yes—
528C22×C4H(5,𝔽2)12yes—
529C22×C4C2511yescharacteristic
530Q16—416no—
531Q16—416no—
532C2×C8—28no—
533C2×C8—28no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—44no—
545C2×C8—44no—
546C2×C8C22×D414yes—
547C2×C8C22×D414yes—
548C4⋊C4—28no—
549C4⋊C4—28no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—42no—
561C4⋊C4—48no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—84no—
567C4⋊C4—84no—
568C22⋊C4—28no—
569C22⋊C4—28no—
570C22⋊C4—48no—
571C22⋊C4—48no—
572C22⋊C4—84no—
573C42—24no—
574C42—24no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—48no—
580C23—28no—
581C23—22no—
582C23—22no—
583C23—28no—
584C23—28no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23C2×#12114yes—
594C23C2×#12112yes—
595C23#52512yes—
596C23#53512yes—
597C23#57712yes—
598C23#57812yes—
599C23#57812yes—
600C23#57812yes—
601C23C23×D411yescharacteristic
602Q8—24no—
603Q8—24no—
604Q8—24no—
605Q8—24no—
606Q8—432no—
607Q8—48no—
608Q8—48no—
609Q8—42no—
610Q8—42no—
611Q8—42no—
612Q8—42no—
613Q8—42no—
614Q8—42no—
615Q8—42no—
616Q8—42no—
617Q8—84no—
618C2×C4—28no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—216no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—432no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—48no—
644C2×C4—48no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4C2×#12114yes—
676C2×C4C2×#12112yes—
677C2×C4#52512yes—
678C2×C4#53512yes—
679C2×C4#54614yes—
680C2×C4#54614yes—
681C2×C4#55114yes—
682C2×C4#55114yes—
683C2×C4C23×D411yescharacteristic
684C2×C4C22×#4111yescharacteristic
685C2×C4C22×#4111yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C2×C4C2×H(5,𝔽2)11yescharacteristic
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C8—24no—
690C8—24no—
691C8—42no—
692C8—42no—
693C22—24no—
694C22—24no—
695C22—24no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—416no—
701C22#241114yes—
702C22#305914yes—
703C22#306314yes—
704C22#312114yes—
705C22#312714yes—
706C22#322314yes—
707C22#322514yes—
708C22C22×#12111yescharacteristic
709C22C2×#52511yescharacteristic
710C22C2×#53511yescharacteristic
711C22C2×#57711yescharacteristic
712C22C2×#57811yescharacteristic
713C22C2×#57811yescharacteristic
714C22C2×#57811yescharacteristic
715C4—24no—
716C4—42no—
717C4—42no—
718C4—42no—
719C4—42no—
720C4—44no—
721C4—44no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4#241114yes—
729C4C2×#54611yescharacteristic
730C4C2×#54611yescharacteristic
731C4C2×#55111yescharacteristic
732C4C2×#55111yescharacteristic
733C2—44no—
734C2#2451418yesdirect product factor
735C2C2×#241111yescharacteristic
736C2C2×#305911yescharacteristic
737C2C2×#306311yescharacteristic
738C2C2×#312111yescharacteristic
739C2C2×#312711yescharacteristic
740C2C2×#322311yescharacteristic
741C2C2×#322511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #24514C2direct (N × H)
6. C2×#2068C2semidirect
7. C2×#2066C2semidirect
8. C2×#2056C2semidirect
9. C2×#2056C2semidirect
10. C2×#1971C2semidirect
11. C2×#1969C2semidirect
12. C2×#1935C2semidirect
13. C2×#1933C2semidirect
64. #2068C22semidirect
65. #2066C22semidirect
66. #2056C22semidirect
67. #2056C22semidirect
68. #1971C22semidirect
69. #1969C22semidirect
70. #1935C22semidirect
71. #1933C22semidirect
734. C2#24514direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4512
C22×#4332
C2×#26252
C2×#23102
#245142
C2×#20682
C2×#20662
C2×#20562
C2×#19712
C2×#19692
C2×#19352
C2×#19332
C2×#18732
C2×#18542
C2×#17872