Finite Group Databasebeta

← C2×#14812 #607753

Subgroups — C2×#14812 #607753

Order 512.

8198 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D4C211yesmaximal, characteristic
2C22×#396C212yesmaximal, complemented
3C22×#387C211yesmaximal, characteristic, complemented
4C22×#385C212yesmaximal, complemented
5#14812C2116yesmaximal, direct product factor
6C2×#1444C211yesmaximal, characteristic, complemented
7C2×#1442C211yesmaximal, characteristic, complemented
8C2×#1439C212yesmaximal, complemented
9C2×#1438C211yesmaximal, characteristic
10C2×#1438C211yesmaximal, characteristic
11C2×C4⋊#95C211yesmaximal, characteristic, complemented
12C2×C22⋊#95C212yesmaximal, complemented
13C24×D4C2211yescharacteristic
14C25×C4C2212yes—
15C22×C4×D4—28no—
16C22×C4×D4—216no—
17C22×C4×D4C2218yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C22⋊C4C2212yes—
24C23×C22⋊C4C2212yes—
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2212yes—
27C23×C22⋊C4C2212yes—
28C23×C42C2211yescharacteristic
29C2×#396—216no—
30C2×#396C22116yescomplemented
31C2×#387—28no—
32C2×#387C2218yescomplemented
33C2×#385—216no—
34C2×#385C22116yescomplemented
35C22×#95C2211yescharacteristic
36C22×#95C2211yescharacteristic
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2212yes—
44C22×#95C2212yes—
45#1444C2218yescomplemented
46#1442C2218yescomplemented
47#1439C22116yescomplemented
48#1438C2218yes—
49#1438C2218yes—
50C4⋊#95C2218yescomplemented
51C22⋊#95C22116yescomplemented
52C26C2312yes—
53C23×D4—28no—
54C23×D4—28no—
55C23×D4—24no—
56C23×D4—24no—
57C23×D4Q818yes—
58C23×D4C2314yes—
59C24×C4—28no—
60C24×C4—28no—
61C24×C4—28no—
62C24×C4—28no—
63C24×C4—28no—
64C24×C4—44no—
65C24×C4—44no—
66C24×C4C2311yescharacteristic
67C24×C4C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2318yes—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C2×C4×D4—2128no—
78C2×C4×D4—264no—
79C2×C4×D4—232no—
80C22×C4⋊C4—28no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—28no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4D418yescomplemented
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2318yes—
96C22×C4⋊C4C2318yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4D4116yes—
113C22×C22⋊C4C2318yes—
114C22×C22⋊C4C2318yes—
115C22×C22⋊C4C2318yes—
116C22×C22⋊C4C2318yes—
117C22×C22⋊C4C2318yes—
118C22×C42—28no—
119C22×C42D418yes—
120C22×C42C2314yes—
121#396—264no—
122#387—232no—
123#385—264no—
124C2×#95—28no—
125C2×#95—28no—
126C2×#95—28no—
127C2×#95—24no—
128C2×#95—24no—
129C2×#95—24no—
130C2×#95—24no—
131C2×#95—24no—
132C2×#95—24no—
133C2×#95—44no—
134C2×#95C2318yes—
135C2×#95C2314yes—
136C2×#95C2318yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C25—24no—
145C25—24no—
146C25—24no—
147C25—24no—
148C25—24no—
149C25—24no—
150C25—24no—
151C25—24no—
152C25—24no—
153C25—24no—
154C25—24no—
155C25—24no—
156C25C2×D418yes—
157C25C2×Q818yes—
158C25#4118yes—
159C25C2414yes—
160C25C2411yescharacteristic
161C22×D4—232no—
162C22×D4—232no—
163C22×D4—264no—
164C22×D4—264no—
165C22×D4—216no—
166C22×D4—216no—
167C22×D4—232no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4C2×Q8116yes—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—264no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—216no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—216no—
201C23×C4—216no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—216no—
220C23×C4—216no—
221C23×C4—48no—
222C23×C4—48no—
223C23×C4—432no—
224C23×C4—432no—
225C23×C4—432no—
226C23×C4—432no—
227C23×C4—432no—
228C23×C4—432no—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D418yes—
232C23×C4C2×D414yes—
233C23×C4C2×D418yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×Q814yes—
238C23×C4#4118yes—
239C23×C4#4114yes—
240C23×C4#4114yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2414yes—
247C23×C4C2414yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C4×D4—2256no—
252C2×C4⋊C4—232no—
253C2×C4⋊C4—232no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—216no—
261C2×C4⋊C4—216no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—216no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—432no—
267C2×C4⋊C4—432no—
268C2×C4⋊C4—416no—
269C2×C4⋊C4C2×D4116yescomplemented
270C2×C22⋊C4—264no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—264no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—432no—
287C2×C22⋊C4—432no—
288C2×C22⋊C4—432no—
289C2×C22⋊C4C2×D4132yes—
290C2×C42—232no—
291C2×C42—216no—
292C2×C42C2×D4116yes—
293#95—216no—
294#95—216no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—216no—
300#95—28no—
301#95—28no—
302#95—48no—
303C24—216no—
304C24—216no—
305C24—216no—
306C24—216no—
307C24—216no—
308C24—216no—
309C24—216no—
310C24—216no—
311C24—216no—
312C24—216no—
313C24—216no—
314C24—216no—
315C24—216no—
316C24—216no—
317C24—216no—
318C24—216no—
319C24—216no—
320C24—216no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—48no—
334C24—48no—
335C24—48no—
336C24—48no—
337C24—48no—
338C24—48no—
339C24—48no—
340C24—48no—
341C24—48no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—48no—
346C24—48no—
347C24—48no—
348C24—48no—
349C24C22⋊Q8132yes—
350C24C22×D418yes—
351C24C22×D412yes—
352C24C22×D414yes—
353C24C22×D412yes—
354C24C22×Q818yes—
355C24C22×Q812yes—
356C24C2×#4118yes—
357C24C2×#4112yes—
358C24C2×#4114yes—
359C24C2×#4112yes—
360C24C2×#4114yes—
361C24H(5,𝔽2)14yes—
362C24H(5,𝔽2)12yes—
363C242−1+412yes—
364C24C2511yescharacteristic
365C2×D4—232no—
366C2×D4—2256no—
367C2×D4—232no—
368C2×D4—264no—
369C2×D4—264no—
370C2×D4—232no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—216no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—264no—
391C22×C4—216no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
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—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—432no—
439C22×C4—432no—
440C22×C4—432no—
441C22×C4—432no—
442C22×C4—4256no—
443C22×C4—432no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—432no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4#120116yes—
477C22×C4#121116yes—
478C22×C4C22⋊Q8116yes—
479C22×C4C22×D414yes—
480C22×C4C22×D418yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D418yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×Q814yes—
488C22×C4C2×#4118yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C4⋊C4—232no—
492C4⋊C4—232no—
493C4⋊C4—432no—
494C4⋊C4—432no—
495C4⋊C4—416no—
496C22⋊C4—264no—
497C22⋊C4—264no—
498C22⋊C4—432no—
499C22⋊C4—432no—
500C22⋊C4—432no—
501C42—232no—
502C23—264no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—264no—
522C23—264no—
523C23—432no—
524C23—432no—
525C23—432no—
526C23—432no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—432no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—432no—
544C23—432no—
545C23—44no—
546C23—432no—
547C23—432no—
548C23—432no—
549C23—432no—
550C23—432no—
551C23C2×#12014yes—
552C23C2×#12114yes—
553C23C2×C22⋊Q8116yes—
554C23C2×C22⋊Q814yes—
555C23C2×C22⋊Q818yes—
556C23#52514yes—
557C23#52518yes—
558C23#52518yes—
559C23#52918yes—
560C23#53214yes—
561C23#53514yes—
562C23#53714yes—
563C23#53818yes—
564C23#53918yes—
565C23D4218yes—
566C23#54618yes—
567C23#54618yes—
568C23#54718yes—
569C23D4×Q814yes—
570C23#55014yes—
571C23#55318yes—
572C23#55418yes—
573C23#55414yes—
574C23#55614yes—
575C23C23×D411yescharacteristic
576C23C23×D412yes—
577C23C23×D411yescharacteristic
578C23C23×Q811yescharacteristic
579C23C22×#4111yescharacteristic
580C23C22×#4112yes—
581C23C22×#4111yescharacteristic
582C23C22×#4112yes—
583C23C2×H(5,𝔽2)12yes—
584C23C2×H(5,𝔽2)11yescharacteristic
585C23C2×2−1+411yescharacteristic
586D4—2128no—
587C2×C4—232no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—232no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—232no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—4128no—
613C2×C4—416no—
614C2×C4—42no—
615C2×C4—416no—
616C2×C4—432no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—432no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—88no—
649C2×C4—88no—
650C2×C4—816no—
651C2×C4C2×#12018yes—
652C2×C4C2×#12118yes—
653C2×C4C2×C22⋊Q818yes—
654C22—216no—
655C22—216no—
656C22—216no—
657C22—48no—
658C22—48no—
659C22—48no—
660C22—48no—
661C22—48no—
662C22—48no—
663C22—48no—
664C22—48no—
665C22—48no—
666C22—48no—
667C22—48no—
668C22—4128no—
669C22—48no—
670C22#239218yes—
671C22#240818yes—
672C22#243018yes—
673C22#2449116yes—
674C22#2450116yes—
675C22#2456116yes—
676C22#2466116yes—
677C22#253818yes—
678C22#256618yes—
679C22#263618yes—
680C22#284818yes—
681C22C22×#12011yescharacteristic
682C22C22×#12111yescharacteristic
683C22C22×C22⋊Q811yescharacteristic
684C22C22×C22⋊Q812yes—
685C22C2×#52511yescharacteristic
686C22C2×#52512yes—
687C22C2×#52512yes—
688C22C2×#52912yes—
689C22C2×#53211yescharacteristic
690C22C2×#53511yescharacteristic
691C22C2×#53711yescharacteristic
692C22C2×#53812yes—
693C22C2×#53912yes—
694C22C2×D4212yes—
695C22C2×#54612yes—
696C22C2×#54612yes—
697C22C2×#54712yes—
698C22C2×D4×Q811yescharacteristic
699C22C2×#55011yescharacteristic
700C22C2×#55312yes—
701C22C2×#55412yes—
702C22C2×#55411yescharacteristic
703C22C2×#55611yescharacteristic
704C4—28no—
705C4—28no—
706C4—28no—
707C4—48no—
708C4—44no—
709C4—48no—
710C4—44no—
711C4—82no—
712C4—84no—
713C4—82no—
714C2—416no—
715C2#14812116yesdirect product factor
716C2C2×#239211yescharacteristic
717C2C2×#263611yescharacteristic
718C2C2×#244912yes—
719C2C2×#245612yes—
720C2C2×#246612yes—
721C2C2×#245012yes—
722C2C2×#240811yescharacteristic
723C2C2×#243011yescharacteristic
724C2C2×#256611yescharacteristic
725C2C2×#253811yescharacteristic
726C2C2×#284811yescharacteristic
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
2. C22×#396C2semidirect
3. C22×#387C2semidirect
4. C22×#385C2semidirect
5. #14812C2direct (N × H)
6. C2×#1444C2semidirect
7. C2×#1442C2semidirect
8. C2×#1439C2semidirect
11. C2×C4⋊#95C2semidirect
12. C2×C22⋊#95C2semidirect
30. C2×#396C22semidirect
32. C2×#387C22semidirect
34. C2×#385C22semidirect
45. #1444C22semidirect
46. #1442C22semidirect
47. #1439C22semidirect
50. C4⋊#95C22semidirect
51. C22⋊#95C22semidirect
92. C22×C4⋊C4D4semidirect
269. C2×C4⋊C4C2×D4semidirect
715. C2#14812direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×D42
C22×#3962
C22×#3872
C22×#3852
#148122
C2×#14442
C2×#14422
C2×#14392
C2×#14382
C2×C4⋊#952
C2×C22⋊#952