Finite Group Databasebeta

← C2×#14758 #607692

Subgroups — C2×#14758 #607692

Order 512.

8014 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C22×#395C211yesmaximal, characteristic, complemented
3C22×#385C211yesmaximal, characteristic
4C22×#380C212yesmaximal
5C23×#95C211yesmaximal, characteristic
6#14758C2116yesmaximal, direct product factor
7C2×#1446C211yesmaximal, characteristic, complemented
8C2×#1444C211yesmaximal, characteristic, complemented
9C2×#1444C212yesmaximal, complemented
10C2×#1441C211yesmaximal, characteristic, complemented
11C2×#1441C212yesmaximal, complemented
12C2×C22⋊#95C212yesmaximal
13C24×Q8C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C25×C4C2212yes—
16C22×C22⋊Q8—28no—
17C22×C22⋊Q8—216no—
18C22×C22⋊Q8C2218yes—
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23C23×C4⋊C4C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2212yes—
27C23×C22⋊C4C2211yescharacteristic
28C2×#395—28no—
29C2×#395C2218yescomplemented
30C2×#385—28no—
31C2×#385C2218yes—
32C2×#380—216no—
33C2×#380C22116yes—
34C22×#95—28no—
35C22×#95C2211yescharacteristic
36C22×#95C2212yes—
37C22×#95C2212yes—
38C22×#95C2211yescharacteristic
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2218yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46C22×#95C2211yescharacteristic
47#1446C2218yescomplemented
48#1444C22116yescomplemented
49#1444C2218yescomplemented
50#1441C2218yescomplemented
51#1441C22116yescomplemented
52C22⋊#95C22116yes—
53C26C2311yescharacteristic
54C23×Q8—28no—
55C23×Q8—24no—
56C23×Q8—24no—
57C23×Q8—24no—
58C23×Q8—44no—
59C23×Q8C2314yes—
60C24×C4—28no—
61C24×C4—28no—
62C24×C4—28no—
63C24×C4—24no—
64C24×C4—28no—
65C24×C4—24no—
66C24×C4—28no—
67C24×C4—44no—
68C24×C4—44no—
69C24×C4—44no—
70C24×C4D418yes—
71C24×C4Q8116yescomplemented
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2318yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C2×C22⋊Q8—2128no—
85C2×C22⋊Q8—232no—
86C2×C22⋊Q8—264no—
87C22×C4⋊C4—28no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2318yes—
106C22×C4⋊C4C2318yes—
107C22×C4⋊C4C2318yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—28no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4D4116yes—
116C22×C22⋊C4Q818yescomplemented
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2318yes—
119C22×C22⋊C4C2314yes—
120#395—232no—
121#385—232no—
122#380—264no—
123C2×#95—28no—
124C2×#95—24no—
125C2×#95—28no—
126C2×#95—28no—
127C2×#95—28no—
128C2×#95—24no—
129C2×#95—24no—
130C2×#95—28no—
131C2×#95—232no—
132C2×#95—44no—
133C2×#95—44no—
134C2×#95D418yes—
135C2×#95C2318yes—
136C2×#95C2318yes—
137C2×#95C2314yes—
138C2×#95C2318yes—
139C2×#95C2318yes—
140C2×#95C2314yes—
141C2×#95C2318yes—
142C2×#95C2314yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2314yes—
146C25—22no—
147C25—22no—
148C25—24no—
149C25—24no—
150C25—22no—
151C25—22no—
152C25C2×D414yes—
153C25C2×D418yes—
154C25C2×Q814yes—
155C25C2×Q818yes—
156C25#4114yes—
157C25C2412yes—
158C25C2411yescharacteristic
159C22×Q8—232no—
160C22×Q8—216no—
161C22×Q8—28no—
162C22×Q8—28no—
163C22×Q8—28no—
164C22×Q8—48no—
165C22×Q8—432no—
166C22×Q8—432no—
167C22×Q8—416no—
168C22×Q8—416no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—232no—
176C23×C4—264no—
177C23×C4—24no—
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—216no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—216no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—216no—
196C23×C4—22no—
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—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—232no—
213C23×C4—264no—
214C23×C4—28no—
215C23×C4—216no—
216C23×C4—28no—
217C23×C4—216no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—416no—
229C23×C4—432no—
230C23×C4—48no—
231C23×C4—48no—
232C23×C4—48no—
233C23×C4—432no—
234C23×C4—432no—
235C23×C4—432no—
236C23×C4—432no—
237C23×C4—432no—
238C23×C4—432no—
239C23×C4C2×D418yes—
240C23×C4C2×D418yes—
241C23×C4C2×D414yes—
242C23×C4C2×D4116yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2×D418yes—
246C23×C4C2×Q814yes—
247C23×C4C2×Q814yes—
248C23×C4C2×Q818yes—
249C23×C4C2×Q818yes—
250C23×C4C2×Q8132yescomplemented
251C23×C4#4114yes—
252C23×C4#4118yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2414yes—
256C23×C4C2414yes—
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2412yes—
260C23×C4C2414yes—
261C23×C4C2412yes—
262C23×C4C2414yes—
263C22⋊Q8—2256no—
264C2×C4⋊C4—232no—
265C2×C4⋊C4—264no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—216no—
269C2×C4⋊C4—216no—
270C2×C4⋊C4—216no—
271C2×C4⋊C4—216no—
272C2×C4⋊C4—216no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—216no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4—432no—
283C2×C4⋊C4—432no—
284C2×C4⋊C4—416no—
285C2×C4⋊C4—416no—
286C2×C22⋊C4—264no—
287C2×C22⋊C4—232no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—216no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—216no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—416no—
295C2×C22⋊C4C2×D4132yes—
296C2×C22⋊C4C2×Q8116yescomplemented
297#95—28no—
298#95—216no—
299#95—216no—
300#95—216no—
301#95—28no—
302#95—28no—
303#95—216no—
304#95—216no—
305#95—48no—
306#95—48no—
307#95C2×D4116yes—
308C24—28no—
309C24—216no—
310C24—28no—
311C24—216no—
312C24—28no—
313C24—28no—
314C24—216no—
315C24—216no—
316C24—216no—
317C24—28no—
318C24—216no—
319C24—216no—
320C24—216no—
321C24—28no—
322C24—216no—
323C24—28no—
324C24—216no—
325C24—28no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—24no—
330C24—22no—
331C24—22no—
332C24#120116yes—
333C24C22⋊Q8116yes—
334C24C22⋊Q8132yes—
335C24C4⋊Q8116yes—
336C24C4⋊Q8132yes—
337C24C22×D414yes—
338C24C22×D418yes—
339C24C22×D412yes—
340C24C22×D414yes—
341C24C22×D412yes—
342C24C22×Q814yes—
343C24C22×Q818yes—
344C24C22×Q814yes—
345C24C22×Q812yes—
346C24C2×#4114yes—
347C24C2×#4112yes—
348C24C2×#4112yes—
349C24C2×#4114yes—
350C24H(5,𝔽2)14yes—
351C24H(5,𝔽2)12yes—
352C242−1+412yes—
353C24C2511yescharacteristic
354C2×Q8—232no—
355C2×Q8—432no—
356C2×Q8—432no—
357C2×Q8—4128no—
358C2×Q8—416no—
359C2×Q8—416no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—28no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—28no—
379C22×C4—232no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—264no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—28no—
393C22×C4—216no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
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—22no—
427C22×C4—232no—
428C22×C4—264no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—416no—
440C22×C4—432no—
441C22×C4—4128no—
442C22×C4—4256no—
443C22×C4—432no—
444C22×C4—432no—
445C22×C4—432no—
446C22×C4—432no—
447C22×C4—432no—
448C22×C4—432no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4#121116yes—
481C22×C4C22⋊Q8132yescomplemented
482C22×C4C22×D418yes—
483C22×C4C22×D418yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D418yes—
488C22×C4C22×Q814yes—
489C22×C4C22×Q814yes—
490C22×C4C22×Q818yes—
491C22×C4C22×Q818yes—
492C22×C4C2×#4114yes—
493C22×C4C2×#4118yes—
494C4⋊C4—232no—
495C4⋊C4—264no—
496C4⋊C4—432no—
497C4⋊C4—432no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C22⋊C4—264no—
501C22⋊C4—232no—
502C22⋊C4—416no—
503C23—232no—
504C23—232no—
505C23—232no—
506C23—24no—
507C23—28no—
508C23—232no—
509C23—232no—
510C23—24no—
511C23—28no—
512C23—24no—
513C23—24no—
514C23—232no—
515C23—28no—
516C23—28no—
517C23—264no—
518C23—264no—
519C23—28no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—264no—
524C23—28no—
525C23—24no—
526C23—28no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23—264no—
531C23#395164yescomplemented
532C23C2×#12018yes—
533C23C2×#12014yes—
534C23C2×#12114yes—
535C23C2×C22⋊Q818yes—
536C23C2×C22⋊Q8116yes—
537C23C2×C22⋊Q814yes—
538C23C2×C22⋊Q818yes—
539C23C2×C22⋊Q818yes—
540C23C2×C4⋊Q8116yes—
541C23C2×C4⋊Q818yes—
542C23C2×C4⋊Q814yes—
543C23C2×C4⋊Q818yes—
544C23#53214yes—
545C23#53318yes—
546C23#53414yes—
547C23#53714yes—
548C23#54318yes—
549C23#54418yes—
550C23D4218yes—
551C23#54618yes—
552C23#54614yes—
553C23#54618yes—
554C23#54718yes—
555C23D4×Q814yes—
556C23#55118yes—
557C23#55214yes—
558C23#55418yes—
559C23#55414yes—
560C23C23×D411yescharacteristic
561C23C23×D412yes—
562C23C23×D411yescharacteristic
563C23C23×Q812yes—
564C23C23×Q811yescharacteristic
565C23C22×#4111yescharacteristic
566C23C22×#4111yescharacteristic
567C23C22×#4112yes—
568C23C2×H(5,𝔽2)12yes—
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×2−1+411yescharacteristic
571Q8—464no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—232no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—264no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—464no—
602C2×C4—4128no—
603C2×C4—416no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—416no—
607C2×C4—432no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—432no—
616C2×C4—432no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
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—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—88no—
641C2×C4—88no—
642C2×C4C2×#12118yes—
643C2×C4C2×C22⋊Q8116yescomplemented
644C22—216no—
645C22—28no—
646C22—28no—
647C22—28no—
648C22—28no—
649C22—216no—
650C22—216no—
651C22—28no—
652C22—2128no—
653C22—28no—
654C22—216no—
655C22C2×#39518yes—
656C22C2×#395116yescomplemented
657C22#241018yes—
658C22#241718yes—
659C22#243018yes—
660C22#2437116yes—
661C22#2438116yes—
662C22#2440116yes—
663C22#2549116yes—
664C22#256418yes—
665C22#265918yes—
666C22#285718yes—
667C22C22×#12011yescharacteristic
668C22C22×#12111yescharacteristic
669C22C22×C22⋊Q811yescharacteristic
670C22C22×C22⋊Q812yes—
671C22C22×C22⋊Q812yes—
672C22C22×C4⋊Q812yes—
673C22C22×C4⋊Q811yescharacteristic
674C22C2×#53211yescharacteristic
675C22C2×#53312yes—
676C22C2×#53411yescharacteristic
677C22C2×#53711yescharacteristic
678C22C2×#54312yes—
679C22C2×#54412yes—
680C22C2×D4212yes—
681C22C2×#54612yes—
682C22C2×#54612yes—
683C22C2×#54611yescharacteristic
684C22C2×#54712yes—
685C22C2×D4×Q811yescharacteristic
686C22C2×#55112yes—
687C22C2×#55211yescharacteristic
688C22C2×#55411yescharacteristic
689C22C2×#55412yes—
690C4—28no—
691C4—216no—
692C4—44no—
693C4—44no—
694C4—48no—
695C4—48no—
696C4—44no—
697C4—48no—
698C4—82no—
699C4—82no—
700C2—216no—
701C2#14758116yesdirect product factor
702C2C22×#39511yescharacteristic
703C2C2×#265911yescharacteristic
704C2C2×#241011yescharacteristic
705C2C2×#243712yes—
706C2C2×#244012yes—
707C2C2×#241711yescharacteristic
708C2C2×#243011yescharacteristic
709C2C2×#243812yes—
710C2C2×#254912yes—
711C2C2×#256411yescharacteristic
712C2C2×#285711yescharacteristic
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×#395C2semidirect
6. #14758C2direct (N × H)
7. C2×#1446C2semidirect
8. C2×#1444C2semidirect
9. C2×#1444C2semidirect
10. C2×#1441C2semidirect
11. C2×#1441C2semidirect
29. C2×#395C22semidirect
47. #1446C22semidirect
48. #1444C22semidirect
49. #1444C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
71. C24×C4Q8semidirect
116. C22×C22⋊C4Q8semidirect
250. C23×C4C2×Q8semidirect
296. C2×C22⋊C4C2×Q8semidirect
481. C22×C4C22⋊Q8semidirect
531. C23#395semidirect
643. C2×C4C2×C22⋊Q8semidirect
656. C22C2×#395semidirect
701. C2#14758direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C22×#3952
C22×#3852
C22×#3802
C23×#952
#147582
C2×#14462
C2×#14442
C2×#14412
C2×C22⋊#952