Finite Group Databasebeta

← C2×#14813 #607754

Subgroups — C2×#14813 #607754

Order 512.

5830 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#394C211yesmaximal, characteristic
2C22×#390C211yesmaximal, characteristic
3C22×#388C212yesmaximal, complemented
4C22×#387C211yesmaximal, characteristic, complemented
5C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
6#14813C2116yesmaximal, direct product factor
7C2×#1442C211yesmaximal, characteristic, complemented
8C2×#1441C211yesmaximal, characteristic, complemented
9C2×#1439C212yesmaximal, complemented
10C2×#1439C212yesmaximal, complemented
11C2×#1438C211yesmaximal, characteristic
12C2×#1437C212yesmaximal, complemented
13C24×D4C2211yescharacteristic
14C23×C4⋊C4C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2212yes—
23C23×C22⋊C4C2212yes—
24C23×C42C2212yes—
25C23×C42C2211yescharacteristic
26C2×#394—28no—
27C2×#394C2218yes—
28C2×#390—28no—
29C2×#390C2218yes—
30C2×#388—216no—
31C2×#388C22116yescomplemented
32C2×#387—28no—
33C2×#387C2218yescomplemented
34C2×C4×C4⋊C4—28no—
35C2×C4×C4⋊C4C2218yescomplemented
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2211yescharacteristic
46#1442C2218yescomplemented
47#1441C2218yescomplemented
48#1439C22116yescomplemented
49#1439C22116yescomplemented
50#1438C2218yes—
51#1437C22116yescomplemented
52C26C2312yes—
53C23×D4—28no—
54C23×D4—24no—
55C23×D4—24no—
56C23×D4—24no—
57C23×D4—44no—
58C23×D4C2314yes—
59C24×C4C2312yes—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2312yes—
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2312yes—
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C22×C4⋊C4—28no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—28no—
72C22×C4⋊C4—28no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4D418yescomplemented
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2318yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—28no—
87C22×C22⋊C4—28no—
88C22×C22⋊C4—28no—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4—44no—
100C22×C22⋊C4—44no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4C2318yes—
103C22×C22⋊C4C2318yes—
104C22×C22⋊C4C2318yes—
105C22×C22⋊C4C2318yes—
106C22×C22⋊C4C2318yes—
107C22×C22⋊C4C2318yes—
108C22×C42—24no—
109C22×C42—24no—
110C22×C42—28no—
111C22×C42—28no—
112C22×C42—44no—
113C22×C42D418yes—
114C22×C42C2318yes—
115C22×C42C2314yes—
116#394—232no—
117#390—232no—
118#388—264no—
119#387—232no—
120C4×C4⋊C4—232no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95—24no—
125C2×#95—24no—
126C2×#95—28no—
127C2×#95—28no—
128C2×#95—44no—
129C2×#95—44no—
130C2×#95D418yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2318yes—
137C2×#95C2318yes—
138C2×#95C2318yes—
139C2×#95C2318yes—
140C2×#95C2314yes—
141C25—24no—
142C25—24no—
143C25—24no—
144C25—24no—
145C25—24no—
146C25—24no—
147C25—24no—
148C25—24no—
149C25—24no—
150C25—24no—
151C25—24no—
152C25—24no—
153C25—24no—
154C25—24no—
155C25#4118yes—
156C25C2414yes—
157C25C2411yescharacteristic
158C22×D4—232no—
159C22×D4—216no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—48no—
164C22×D4—432no—
165C22×D4—432no—
166C22×D4—416no—
167C22×D4—416no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4C2×D418yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4#4118yes—
225C23×C4#4114yes—
226C23×C4#4114yes—
227C23×C4#4118yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C2×C4⋊C4—264no—
239C2×C4⋊C4—232no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—416no—
251C2×C4⋊C4—416no—
252C2×C4⋊C4C2×D4116yescomplemented
253C2×C22⋊C4—264no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—216no—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—216no—
258C2×C22⋊C4—216no—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—216no—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—216no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—432no—
273C2×C22⋊C4—432no—
274C2×C22⋊C4—432no—
275C2×C22⋊C4—432no—
276C2×C22⋊C4—432no—
277C2×C42—232no—
278C2×C42—28no—
279C2×C42—28no—
280C2×C42—216no—
281C2×C42—216no—
282C2×C42—48no—
283C2×C42—432no—
284C2×C42C2×D4116yes—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—216no—
291#95—216no—
292#95—48no—
293#95—48no—
294#95C2×D4116yes—
295C24—216no—
296C24—216no—
297C24—216no—
298C24—216no—
299C24—216no—
300C24—216no—
301C24—216no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—48no—
317C24—48no—
318C24—48no—
319C24—48no—
320C24—48no—
321C24—48no—
322C24—48no—
323C24—48no—
324C24—48no—
325C24—48no—
326C24—48no—
327C24—48no—
328C24—48no—
329C24—48no—
330C24—48no—
331C24—48no—
332C24—48no—
333C24—48no—
334C24—48no—
335C24—48no—
336C24—48no—
337C24—48no—
338C24—48no—
339C24—48no—
340C24—48no—
341C24—48no—
342C24—48no—
343C24—48no—
344C24C22×D412yes—
345C24C22×D412yes—
346C24C22×D412yes—
347C24C2×#4118yes—
348C24C2×#4112yes—
349C24C2×#4112yes—
350C24C2×#4114yes—
351C24C2×#4114yes—
352C24C2×#4114yes—
353C24H(5,𝔽2)14yes—
354C24H(5,𝔽2)12yes—
355C242−1+412yes—
356C24C2511yescharacteristic
357C2×D4—232no—
358C2×D4—432no—
359C2×D4—432no—
360C2×D4—4128no—
361C2×D4—416no—
362C2×D4—416no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—216no—
366C22×C4—28no—
367C22×C4—216no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—216no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4#121116yes—
469C22×C4#121116yes—
470C22×C4#124116yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D418yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C2×#4118yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4114yes—
482C22×C4C2×#4118yes—
483C4⋊C4—264no—
484C4⋊C4—232no—
485C4⋊C4—416no—
486C4⋊C4—416no—
487C22⋊C4—264no—
488C22⋊C4—432no—
489C22⋊C4—432no—
490C22⋊C4—432no—
491C22⋊C4—432no—
492C22⋊C4—432no—
493C42—232no—
494C42—432no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—432no—
503C23—432no—
504C23—432no—
505C23—44no—
506C23—44no—
507C23—432no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—432no—
535C23—432no—
536C23—432no—
537C23—816no—
538C23—816no—
539C23—816no—
540C23—816no—
541C23—816no—
542C23—816no—
543C23—816no—
544C23—816no—
545C23C2×#12114yes—
546C23C2×#12114yes—
547C23#52514yes—
548C23#52518yes—
549C23#52518yes—
550C23C2×#12414yes—
551C23#52918yes—
552C23#52918yes—
553C23#53214yes—
554C23#53514yes—
555C23#53714yes—
556C23#53818yes—
557C23#54218yes—
558C23D4214yes—
559C23#54618yes—
560C23#54714yes—
561C23#54814yes—
562C23#55018yes—
563C23#55118yes—
564C23#55218yes—
565C23#55318yes—
566C23#55314yes—
567C23#55918yes—
568C23C23×D411yescharacteristic
569C23C23×D411yescharacteristic
570C23C23×D411yescharacteristic
571C23C22×#4111yescharacteristic
572C23C22×#4111yescharacteristic
573C23C22×#4112yes—
574C23C22×#4112yes—
575C23C22×#4112yes—
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×2−1+411yescharacteristic
579D4—464no—
580C2×C4—232no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—232no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—232no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—416no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—432no—
626C2×C4—432no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4—816no—
647C2×C4C2×#12118yes—
648C2×C4C2×#12118yes—
649C2×C4C2×#12418yes—
650C22—48no—
651C22—48no—
652C22—48no—
653C22—48no—
654C22—48no—
655C22—48no—
656C22—48no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22—864no—
666C22#239218yes—
667C22#241118yes—
668C22#241618yes—
669C22#2456116yes—
670C22#2458116yes—
671C22#2479116yes—
672C22#254818yes—
673C22#256218yes—
674C22#2597116yes—
675C22#266218yes—
676C22#285018yes—
677C22C22×#12111yescharacteristic
678C22C22×#12111yescharacteristic
679C22C2×#52512yes—
680C22C2×#52511yescharacteristic
681C22C2×#52512yes—
682C22C22×#12411yescharacteristic
683C22C2×#52912yes—
684C22C2×#52912yes—
685C22C2×#53211yescharacteristic
686C22C2×#53511yescharacteristic
687C22C2×#53711yescharacteristic
688C22C2×#53812yes—
689C22C2×#54212yes—
690C22C2×D4211yescharacteristic
691C22C2×#54612yes—
692C22C2×#54711yescharacteristic
693C22C2×#54811yescharacteristic
694C22C2×#55012yes—
695C22C2×#55112yes—
696C22C2×#55212yes—
697C22C2×#55311yescharacteristic
698C22C2×#55312yes—
699C22C2×#55912yes—
700C4—28no—
701C4—28no—
702C4—28no—
703C4—48no—
704C4—48no—
705C4—44no—
706C4—44no—
707C4—82no—
708C4—84no—
709C4—82no—
710C2—88no—
711C2#14813116yesdirect product factor
712C2C2×#239211yescharacteristic
713C2C2×#266211yescharacteristic
714C2C2×#245612yes—
715C2C2×#241611yescharacteristic
716C2C2×#245812yes—
717C2C2×#247912yes—
718C2C2×#241111yescharacteristic
719C2C2×#259712yes—
720C2C2×#256211yescharacteristic
721C2C2×#254811yescharacteristic
722C2C2×#285011yescharacteristic
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#388C2semidirect
4. C22×#387C2semidirect
5. C22×C4×C4⋊C4C2semidirect
6. #14813C2direct (N × H)
7. C2×#1442C2semidirect
8. C2×#1441C2semidirect
9. C2×#1439C2semidirect
10. C2×#1439C2semidirect
12. C2×#1437C2semidirect
31. C2×#388C22semidirect
33. C2×#387C22semidirect
35. C2×C4×C4⋊C4C22semidirect
46. #1442C22semidirect
47. #1441C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
51. #1437C22semidirect
79. C22×C4⋊C4D4semidirect
252. C2×C4⋊C4C2×D4semidirect
711. C2#14813direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3942
C22×#3902
C22×#3882
C22×#3872
C22×C4×C4⋊C42
#148132
C2×#14422
C2×#14412
C2×#14392
C2×#14382
C2×#14372