Finite Group Databasebeta

← C2×#14877 #607856

Subgroups — C2×#14877 #607856

Order 512.

6362 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×Q8C211yesmaximal, characteristic, complemented
2C22×#396C212yesmaximal
3C22×#395C211yesmaximal, characteristic, complemented
4C22×#393C211yesmaximal, characteristic
5C22×#388C211yesmaximal, characteristic
6#14877C2116yesmaximal, direct product factor
7C2×#1441C211yesmaximal, characteristic, complemented
8C2×#1441C212yesmaximal, complemented
9C2×#1439C211yesmaximal, characteristic
10C2×#1437C212yesmaximal
11C2×#1436C212yesmaximal, complemented
12C2×C4⋊#95C211yesmaximal, characteristic, complemented
13C24×Q8C2211yescharacteristic
14C22×C4×Q8—28no—
15C22×C4×Q8—216no—
16C22×C4×Q8C2218yescomplemented
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2212yes—
21C23×C4⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2212yes—
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2211yescharacteristic
27C23×C42C2212yes—
28C23×C42C2211yescharacteristic
29C2×#396—216no—
30C2×#396C22116yes—
31C2×#395—28no—
32C2×#395C2218yescomplemented
33C2×#393—28no—
34C2×#393C2218yes—
35C2×#388—28no—
36C2×#388C2218yes—
37C22×#95C2212yes—
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2211yescharacteristic
46C22×#95C2212yes—
47#1441C2218yescomplemented
48#1441C22116yescomplemented
49#1439C2218yes—
50#1437C22116yes—
51#1436C22116yescomplemented
52C4⋊#95C2218yescomplemented
53C26C2311yescharacteristic
54C23×Q8—24no—
55C23×Q8—28no—
56C23×Q8—28no—
57C23×Q8—24no—
58C23×Q8D418yes—
59C23×Q8C2314yes—
60C24×C4C2311yescharacteristic
61C24×C4C2312yes—
62C24×C4C2311yescharacteristic
63C24×C4C2312yes—
64C24×C4C2311yescharacteristic
65C24×C4C2312yes—
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C2×C4×Q8—2128no—
71C2×C4×Q8—264no—
72C2×C4×Q8—232no—
73C22×C4⋊C4—28no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—28no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—28no—
79C22×C4⋊C4—28no—
80C22×C4⋊C4—28no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—28no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4D4116yes—
87C22×C4⋊C4C2318yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2318yes—
91C22×C4⋊C4C2318yes—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4Q818yescomplemented
106C22×C22⋊C4C2318yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2314yes—
111C22×C42—28no—
112C22×C42—28no—
113C22×C42—28no—
114C22×C42—28no—
115C22×C42D418yes—
116C22×C42C2318yes—
117C22×C42C2314yes—
118#396—264no—
119#395—232no—
120#393—232no—
121#388—232no—
122C2×#95—24no—
123C2×#95—28no—
124C2×#95—24no—
125C2×#95—28no—
126C2×#95—28no—
127C2×#95—24no—
128C2×#95—24no—
129C2×#95—28no—
130C2×#95—28no—
131C2×#95—44no—
132C2×#95C2314yes—
133C2×#95C2318yes—
134C2×#95C2314yes—
135C2×#95C2318yes—
136C2×#95C2318yes—
137C2×#95C2318yes—
138C2×#95C2314yes—
139C2×#95C2318yes—
140C2×#95C2314yes—
141C2×#95C2318yes—
142C25—22no—
143C25—24no—
144C25—22no—
145C25—22no—
146C25—24no—
147C25—24no—
148C25—22no—
149C25—22no—
150C25—24no—
151C25—22no—
152C25C2×Q814yes—
153C25C2412yes—
154C25C2411yescharacteristic
155C22×Q8—232no—
156C22×Q8—232no—
157C22×Q8—264no—
158C22×Q8—264no—
159C22×Q8—232no—
160C22×Q8—216no—
161C22×Q8—28no—
162C22×Q8—216no—
163C22×Q8—28no—
164C22×Q8C2×D4116yes—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4C2×D418yes—
212C23×C4C2×D418yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D418yes—
216C23×C4C2×D414yes—
217C23×C4C2×D418yes—
218C23×C4C2×D414yes—
219C23×C4C2×Q814yes—
220C23×C4C2×Q814yes—
221C23×C4#4118yes—
222C23×C4#4118yes—
223C23×C4#4114yes—
224C23×C4#4114yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C4×Q8—2256no—
236C2×C4⋊C4—232no—
237C2×C4⋊C4—264no—
238C2×C4⋊C4—216no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—216no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—432no—
252C2×C4⋊C4—432no—
253C2×C4⋊C4—416no—
254C2×C4⋊C4C2×D4132yes—
255C2×C22⋊C4—232no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—216no—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—432no—
270C2×C22⋊C4—416no—
271C2×C22⋊C4—432no—
272C2×C22⋊C4—416no—
273C2×C22⋊C4C2×Q8116yescomplemented
274C2×C42—264no—
275C2×C42—232no—
276C2×C42—216no—
277C2×C42—216no—
278C2×C42—216no—
279C2×C42—216no—
280C2×C42C2×D4116yes—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—216no—
285#95—28no—
286#95—216no—
287#95—216no—
288#95—216no—
289#95—216no—
290#95—48no—
291C24—28no—
292C24—216no—
293C24—216no—
294C24—28no—
295C24—28no—
296C24—22no—
297C24—24no—
298C24—22no—
299C24—22no—
300C24—24no—
301C24—24no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—22no—
306C24—44no—
307C24—48no—
308C24—44no—
309C24—44no—
310C24—48no—
311C24—48no—
312C24—44no—
313C24—44no—
314C24—44no—
315C24—48no—
316C24—48no—
317C24—48no—
318C24—48no—
319C24—48no—
320C24—48no—
321C24—44no—
322C24—48no—
323C24—44no—
324C24C22×D412yes—
325C24C22×D414yes—
326C24C22×D412yes—
327C24C22×Q814yes—
328C24C22×Q812yes—
329C24C2×#4114yes—
330C24C2×#4112yes—
331C24C2×#4112yes—
332C24C2×#4114yes—
333C24H(5,𝔽2)12yes—
334C242−1+414yes—
335C242−1+412yes—
336C24C2511yescharacteristic
337C2×Q8—232no—
338C2×Q8—2256no—
339C2×Q8—232no—
340C2×Q8—264no—
341C2×Q8—232no—
342C2×Q8—264no—
343C22×C4—28no—
344C22×C4—216no—
345C22×C4—28no—
346C22×C4—216no—
347C22×C4—216no—
348C22×C4—216no—
349C22×C4—28no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—216no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—216no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—216no—
373C22×C4—28no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—48no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4#120116yes—
446C22×C4#121132yes—
447C22×C4#121116yes—
448C22×C4C22⋊Q8116yes—
449C22×C4C22×D418yes—
450C22×C4C22×D414yes—
451C22×C4C22×D418yes—
452C22×C4C22×D418yes—
453C22×C4C22×D414yes—
454C22×C4C22×D418yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×Q814yes—
458C22×C4C22×Q814yes—
459C22×C4C2×#4118yes—
460C22×C4C2×#4118yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4114yes—
463C4⋊C4—232no—
464C4⋊C4—264no—
465C4⋊C4—432no—
466C4⋊C4—416no—
467C4⋊C4—432no—
468C22⋊C4—232no—
469C22⋊C4—432no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—432no—
473C42—264no—
474C42—232no—
475C23—24no—
476C23—28no—
477C23—24no—
478C23—28no—
479C23—24no—
480C23—416no—
481C23—42no—
482C23—42no—
483C23—44no—
484C23—416no—
485C23—44no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—432no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—432no—
502C23—416no—
503C23—88no—
504C23—88no—
505C23—816no—
506C23—88no—
507C23—816no—
508C23—88no—
509C23C2×#12014yes—
510C23C2×#12118yes—
511C23C2×#12114yes—
512C23C2×C22⋊Q814yes—
513C23#52514yes—
514C23#52518yes—
515C23#52918yes—
516C23#53214yes—
517C23#53318yes—
518C23#53614yes—
519C23#53714yes—
520C23#54118yes—
521C23#54218yes—
522C23#54618yes—
523C23#54718yes—
524C23#54818yes—
525C23#54818yes—
526C23D4×Q818yes—
527C23D4×Q814yes—
528C23#55014yes—
529C23#55414yes—
530C23#55618yes—
531C23#55614yes—
532C23C23×D411yescharacteristic
533C23C23×D412yes—
534C23C23×D411yescharacteristic
535C23C23×Q811yescharacteristic
536C23C22×#4112yes—
537C23C22×#4111yescharacteristic
538C23C22×#4111yescharacteristic
539C23C22×#4112yes—
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×2−1+412yes—
542C23C2×2−1+411yescharacteristic
543Q8—2128no—
544C2×C4—264no—
545C2×C4—232no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—232no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—232no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—432no—
581C2×C4—432no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—416no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—88no—
608C2×C4—816no—
609C2×C4—88no—
610C2×C4C2×#12018yes—
611C2×C4C2×#12118yes—
612C2×C4C2×#121116yes—
613C2×C4C2×C22⋊Q818yes—
614C22—44no—
615C22—44no—
616C22—48no—
617C22—48no—
618C22—44no—
619C22—82no—
620C22—832no—
621C22—82no—
622C22—82no—
623C22—84no—
624C22—84no—
625C22—82no—
626C22#239618yes—
627C22#240918yes—
628C22#2426116yes—
629C22#2429116yes—
630C22#243018yes—
631C22#2465116yes—
632C22#2469116yes—
633C22#253818yes—
634C22#254518yes—
635C22#263718yes—
636C22#284718yes—
637C22C22×#12011yescharacteristic
638C22C22×#12111yescharacteristic
639C22C22×#12112yes—
640C22C22×C22⋊Q811yescharacteristic
641C22C2×#52511yescharacteristic
642C22C2×#52512yes—
643C22C2×#52912yes—
644C22C2×#53211yescharacteristic
645C22C2×#53312yes—
646C22C2×#53611yescharacteristic
647C22C2×#53711yescharacteristic
648C22C2×#54112yes—
649C22C2×#54212yes—
650C22C2×#54612yes—
651C22C2×#54712yes—
652C22C2×#54812yes—
653C22C2×#54812yes—
654C22C2×D4×Q811yescharacteristic
655C22C2×D4×Q812yes—
656C22C2×#55011yescharacteristic
657C22C2×#55411yescharacteristic
658C22C2×#55612yes—
659C22C2×#55611yescharacteristic
660C4—216no—
661C4—28no—
662C4—28no—
663C4—28no—
664C4—44no—
665C4—48no—
666C4—48no—
667C4—82no—
668C4—82no—
669C4—84no—
670C2—84no—
671C2#14877116yesdirect product factor
672C2C2×#239611yescharacteristic
673C2C2×#263711yescharacteristic
674C2C2×#240911yescharacteristic
675C2C2×#246512yes—
676C2C2×#242912yes—
677C2C2×#246912yes—
678C2C2×#242612yes—
679C2C2×#243011yescharacteristic
680C2C2×#254511yescharacteristic
681C2C2×#253811yescharacteristic
682C2C2×#284711yescharacteristic
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C4×Q8C2semidirect
3. C22×#395C2semidirect
6. #14877C2direct (N × H)
7. C2×#1441C2semidirect
8. C2×#1441C2semidirect
11. C2×#1436C2semidirect
12. C2×C4⋊#95C2semidirect
16. C22×C4×Q8C22semidirect
32. C2×#395C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
51. #1436C22semidirect
52. C4⋊#95C22semidirect
105. C22×C22⋊C4Q8semidirect
273. C2×C22⋊C4C2×Q8semidirect
671. C2#14877direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×Q82
C22×#3962
C22×#3952
C22×#3932
C22×#3882
#148772
C2×#14412
C2×#14392
C2×#14372
C2×#14362
C2×C4⋊#952