Finite Group Databasebeta

← C2×#14893 #607889

Subgroups — C2×#14893 #607889

Order 512.

6262 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#117C211yesmaximal, characteristic
2C22×#400C211yesmaximal, characteristic, complemented
3C22×#399C211yesmaximal, characteristic
4C22×#389C211yesmaximal, characteristic, complemented
5#14893C2116yesmaximal, direct product factor
6C2×#1444C212yesmaximal, complemented
7C2×#1441C211yesmaximal, characteristic, complemented
8C2×#1439C212yesmaximal
9C2×#1439C211yesmaximal, characteristic
10C2×#1436C212yesmaximal, complemented
11C2×#1434C211yesmaximal, characteristic, complemented
12C2×C22⋊#95C212yesmaximal
13C25×C4C2211yescharacteristic
14C22×#117—28no—
15C22×#117—28no—
16C22×#117—28no—
17C22×#117C2218yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2212yes—
27C23×C22⋊C4C2211yescharacteristic
28C23×C42C2211yescharacteristic
29C23×C42C2211yescharacteristic
30C2×#400—28no—
31C2×#400C2218yescomplemented
32C2×#399—28no—
33C2×#399C2218yes—
34C2×#389—28no—
35C2×#389C2218yescomplemented
36C22×#95C2212yes—
37C22×#95C2212yes—
38C22×#95C2211yescharacteristic
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2211yescharacteristic
42C22×#95C2212yes—
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46#1444C22116yescomplemented
47#1441C2218yescomplemented
48#1439C22116yes—
49#1439C2218yes—
50#1436C22116yescomplemented
51#1434C2218yescomplemented
52C22⋊#95C22116yes—
53C26C2311yescharacteristic
54C24×C4—24no—
55C24×C4—24no—
56C24×C4—24no—
57C24×C4—24no—
58C24×C4—24no—
59C24×C4—24no—
60C24×C4Q818yes—
61C24×C4C2312yes—
62C24×C4C2312yes—
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2314yes—
67C24×C4C2312yes—
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C2×#117—2128no—
73C2×#117—232no—
74C2×#117—232no—
75C2×#117—232no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—28no—
83C22×C4⋊C4—28no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4D418yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2318yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2318yes—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
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⋊C4D418yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2318yes—
114C22×C22⋊C4C2314yes—
115C22×C42—24no—
116C22×C42—24no—
117C22×C42—24no—
118C22×C42—24no—
119C22×C42—24no—
120C22×C42D418yes—
121C22×C42C2314yes—
122C22×C42C2314yes—
123#400—232no—
124#399—232no—
125#389—232no—
126C2×#95—24no—
127C2×#95—28no—
128C2×#95—28no—
129C2×#95—24no—
130C2×#95—28no—
131C2×#95—28no—
132C2×#95—28no—
133C2×#95—28no—
134C2×#95—28no—
135C2×#95—44no—
136C2×#95C2314yes—
137C2×#95C2318yes—
138C2×#95C2314yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C25—22no—
147C25—22no—
148C25—24no—
149C25—24no—
150C25—22no—
151C25—22no—
152C25—24no—
153C25—24no—
154C25C2×D414yes—
155C25C2×Q814yes—
156C25#4114yes—
157C25C2412yes—
158C25C2411yescharacteristic
159C23×C4—232no—
160C23×C4—232no—
161C23×C4—232no—
162C23×C4—232no—
163C23×C4—232no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—28no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—28no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—232no—
177C23×C4—232no—
178C23×C4—28no—
179C23×C4—28no—
180C23×C4—28no—
181C23×C4—28no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×Q8116yes—
229C23×C4C2×Q814yes—
230C23×C4C2×Q814yes—
231C23×C4#4118yes—
232C23×C4#4118yes—
233C23×C4#4114yes—
234C23×C4#4114yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245#117—2256no—
246C2×C4⋊C4—232no—
247C2×C4⋊C4—232no—
248C2×C4⋊C4—232no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—216no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—216no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—432no—
266C2×C4⋊C4—432no—
267C2×C4⋊C4—416no—
268C2×C4⋊C4C2×D4116yes—
269C2×C22⋊C4—232no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—232no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—216no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—432no—
283C2×C22⋊C4—432no—
284C2×C22⋊C4C2×D4116yes—
285C2×C42—232no—
286C2×C42—232no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—28no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42C2×D4116yes—
293#95—216no—
294#95—28no—
295#95—28no—
296#95—216no—
297#95—216no—
298#95—216no—
299#95—216no—
300#95—216no—
301#95—216no—
302#95—48no—
303C24—28no—
304C24—28no—
305C24—28no—
306C24—28no—
307C24—28no—
308C24—28no—
309C24—216no—
310C24—216no—
311C24—216no—
312C24—216no—
313C24—216no—
314C24—216no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—24no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—44no—
324C24—48no—
325C24—44no—
326C24—44no—
327C24—44no—
328C24—48no—
329C24—48no—
330C24—48no—
331C24—48no—
332C24—48no—
333C24C22⋊Q8116yes—
334C24C22×D414yes—
335C24C22×D412yes—
336C24C22×D412yes—
337C24C22×D412yes—
338C24C22×Q814yes—
339C24C22×Q812yes—
340C24C2×#4114yes—
341C24C2×#4112yes—
342C24C2×#4112yes—
343C24C2×#4112yes—
344C24C2×#4114yes—
345C24C2×#4114yes—
346C24H(5,𝔽2)14yes—
347C242−1+414yes—
348C24C2511yescharacteristic
349C22×C4—232no—
350C22×C4—2256no—
351C22×C4—232no—
352C22×C4—232no—
353C22×C4—232no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—232no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—232no—
382C22×C4—232no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
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—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
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—24no—
433C22×C4—24no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4#120116yes—
465C22×C4C22⋊Q8116yes—
466C22×C4C22⋊Q8116yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×Q814yes—
476C22×C4C22×Q814yes—
477C22×C4C2×#4118yes—
478C22×C4C2×#4118yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C4⋊C4—232no—
482C4⋊C4—232no—
483C4⋊C4—232no—
484C4⋊C4—432no—
485C4⋊C4—432no—
486C4⋊C4—416no—
487C22⋊C4—232no—
488C22⋊C4—232no—
489C22⋊C4—432no—
490C22⋊C4—432no—
491C42—232no—
492C42—232no—
493C23—232no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—264no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—416no—
508C23—416no—
509C23—416no—
510C23—416no—
511C23—416no—
512C23—42no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—416no—
519C23—44no—
520C23—432no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—432no—
525C23—432no—
526C23C2×#12014yes—
527C23C2×C22⋊Q818yes—
528C23C2×C22⋊Q814yes—
529C23C2×C22⋊Q814yes—
530C23C2×C22⋊Q814yes—
531C23#52514yes—
532C23#52514yes—
533C23#52514yes—
534C23#52514yes—
535C23#52918yes—
536C23#53214yes—
537C23#53318yes—
538C23#53514yes—
539C23#53614yes—
540C23#54218yes—
541C23#54418yes—
542C23#54618yes—
543C23#54718yes—
544C23#54718yes—
545C23#55018yes—
546C23#55118yes—
547C23#55218yes—
548C23#55218yes—
549C23#55318yes—
550C23C23×D411yescharacteristic
551C23C23×D411yescharacteristic
552C23C23×D411yescharacteristic
553C23C23×Q811yescharacteristic
554C23C22×#4111yescharacteristic
555C23C22×#4111yescharacteristic
556C23C22×#4111yescharacteristic
557C23C22×#4112yes—
558C23C22×#4112yes—
559C23C2×H(5,𝔽2)12yes—
560C23C2×2−1+412yes—
561C2×C4—2128no—
562C2×C4—232no—
563C2×C4—232no—
564C2×C4—232no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—432no—
595C2×C4—416no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—432no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—816no—
630C2×C4—816no—
631C2×C4C2×#12018yes—
632C2×C4C2×C22⋊Q818yes—
633C2×C4C2×C22⋊Q818yes—
634C22—28no—
635C22—216no—
636C22—44no—
637C22—464no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—48no—
642C22—48no—
643C22—48no—
644C22—44no—
645C22—44no—
646C22#239618yes—
647C22#240618yes—
648C22#243118yes—
649C22#2438116yes—
650C22#2441116yes—
651C22#2470116yes—
652C22#2479116yes—
653C22#259818yes—
654C22#261118yes—
655C22#262818yes—
656C22#284818yes—
657C22C22×#12011yescharacteristic
658C22C22×C22⋊Q811yescharacteristic
659C22C22×C22⋊Q811yescharacteristic
660C22C22×C22⋊Q811yescharacteristic
661C22C2×#52511yescharacteristic
662C22C2×#52511yescharacteristic
663C22C2×#52511yescharacteristic
664C22C2×#52511yescharacteristic
665C22C2×#52912yes—
666C22C2×#53211yescharacteristic
667C22C2×#53312yes—
668C22C2×#53511yescharacteristic
669C22C2×#53611yescharacteristic
670C22C2×#54212yes—
671C22C2×#54412yes—
672C22C2×#54612yes—
673C22C2×#54712yes—
674C22C2×#54712yes—
675C22C2×#55012yes—
676C22C2×#55112yes—
677C22C2×#55212yes—
678C22C2×#55212yes—
679C22C2×#55312yes—
680C4—28no—
681C4—28no—
682C4—28no—
683C4—48no—
684C4—44no—
685C4—44no—
686C4—48no—
687C4—44no—
688C4—84no—
689C4—84no—
690C2—48no—
691C2#14893116yesdirect product factor
692C2C2×#239611yescharacteristic
693C2C2×#262811yescharacteristic
694C2C2×#243111yescharacteristic
695C2C2×#244112yes—
696C2C2×#247012yes—
697C2C2×#247912yes—
698C2C2×#243812yes—
699C2C2×#240611yescharacteristic
700C2C2×#259811yescharacteristic
701C2C2×#261111yescharacteristic
702C2C2×#284811yescharacteristic
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
2. C22×#400C2semidirect
4. C22×#389C2semidirect
5. #14893C2direct (N × H)
6. C2×#1444C2semidirect
7. C2×#1441C2semidirect
10. C2×#1436C2semidirect
11. C2×#1434C2semidirect
31. C2×#400C22semidirect
35. C2×#389C22semidirect
46. #1444C22semidirect
47. #1441C22semidirect
50. #1436C22semidirect
51. #1434C22semidirect
691. C2#14893direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#1172
C22×#4002
C22×#3992
C22×#3892
#148932
C2×#14442
C2×#14412
C2×#14392
C2×#14362
C2×#14342
C2×C22⋊#952