Finite Group Databasebeta

← C2×#14044 #514357

Subgroups — C2×#14044 #514357

Order 512.

6890 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2C22×#385C211yesmaximal, characteristic
3C22×#385C211yesmaximal, characteristic
4C22×C4×C22⋊C4C211yesmaximal, characteristic
5C23×#95C211yesmaximal, characteristic
6#14044C2116yesmaximal, direct product factor
7C2×#1442C212yesmaximal, complemented
8C2×#1441C212yesmaximal, complemented
9C2×#1436C212yesmaximal, complemented
10C2×C4⋊#95C212yesmaximal, complemented
11C2×C22⋊#95C211yesmaximal, characteristic
12C2×C22⋊#95C211yesmaximal, characteristic
13C25×C4C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C25×C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C42C2212yes—
25C2×#385—28no—
26C2×#385—28no—
27C2×#385C4116yescomplemented
28C2×#385C2218yes—
29C2×#385C2218yes—
30C2×#385C2218yes—
31C2×C4×C22⋊C4—28no—
32C2×C4×C22⋊C4C2218yes—
33C22×#95—28no—
34C22×#95C2212yes—
35C22×#95C2212yes—
36C22×#95C2212yes—
37C22×#95C2211yescharacteristic
38C22×#95C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2212yes—
41C22×#95C2218yes—
42C22×#95C2212yes—
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46#1442C22116yescomplemented
47#1441C22116yescomplemented
48#1436C22116yescomplemented
49C4⋊#95C22116yescomplemented
50C22⋊#95C2218yes—
51C22⋊#95C2218yes—
52C26C2311yescharacteristic
53C24×C4—24no—
54C24×C4—24no—
55C24×C4—24no—
56C24×C4—24no—
57C24×C4—24no—
58C24×C4—28no—
59C24×C4—28no—
60C24×C4—24no—
61C24×C4—24no—
62C24×C4—44no—
63C24×C4—44no—
64C24×C4—44no—
65C24×C4—44no—
66C24×C4C2×C418yes—
67C24×C4D418yes—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2312yes—
76C24×C4C2314yes—
77C24×C4C2314yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2314yes—
81C22×C4⋊C4—28no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4C2×C4116yescomplemented
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2318yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2318yes—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4C2×C418yes—
103C22×C22⋊C4C2×C418yes—
104C22×C22⋊C4D418yes—
105C22×C22⋊C4Q818yes—
106C22×C22⋊C4Q818yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42—28no—
112C22×C42—28no—
113C22×C42—44no—
114C22×C42C2318yes—
115#385—232no—
116#385—232no—
117#385C2×C4164yescomplemented
118C4×C22⋊C4—232no—
119C2×#95—28no—
120C2×#95—232no—
121C2×#95—28no—
122C2×#95—28no—
123C2×#95—24no—
124C2×#95—24no—
125C2×#95—28no—
126C2×#95—28no—
127C2×#95—44no—
128C2×#95—44no—
129C2×#95—44no—
130C2×#95C2×C4116yescomplemented
131C2×#95C2318yes—
132C2×#95C2314yes—
133C2×#95C2318yes—
134C2×#95C2318yes—
135C2×#95C2318yes—
136C2×#95C2318yes—
137C2×#95C2314yes—
138C2×#95C2318yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C25—24no—
143C25—24no—
144C25—24no—
145C25—24no—
146C25C22×C414yes—
147C25C2×D414yes—
148C25C2×D414yes—
149C25C2×Q814yes—
150C25C2×Q814yes—
151C25#4114yes—
152C25#4114yes—
153C25C2412yes—
154C25C2411yescharacteristic
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—232no—
163C23×C4—232no—
164C23×C4—232no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—28no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—28no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
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—232no—
201C23×C4—216no—
202C23×C4—216no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—28no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—432no—
216C23×C4—416no—
217C23×C4—48no—
218C23×C4—48no—
219C23×C4—48no—
220C23×C4—48no—
221C23×C4—432no—
222C23×C4—432no—
223C23×C4—432no—
224C23×C4—432no—
225C23×C4—416no—
226C23×C4—416no—
227C23×C4—416no—
228C23×C4—432no—
229C23×C4—416no—
230C23×C4C22×C414yes—
231C23×C4C22×C414yes—
232C23×C4C22×C418yes—
233C23×C4C22×C418yes—
234C23×C4C22×C4116yes—
235C23×C4C2×D414yes—
236C23×C4C2×D4116yes—
237C23×C4C2×D418yes—
238C23×C4C2×D414yes—
239C23×C4C2×Q818yes—
240C23×C4C2×Q818yes—
241C23×C4#4118yes—
242C23×C4#4118yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2412yes—
249C23×C4C2414yes—
250C23×C4C2414yes—
251C23×C4C2414yes—
252C23×C4C2412yes—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—216no—
257C2×C4⋊C4—216no—
258C2×C4⋊C4—432no—
259C2×C4⋊C4—48no—
260C2×C4⋊C4—48no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—432no—
264C2×C4⋊C4—416no—
265C2×C4⋊C4—416no—
266C2×C4⋊C4C22×C4132yescomplemented
267C2×C22⋊C4—232no—
268C2×C22⋊C4—232no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—416no—
277C2×C22⋊C4C4⋊C4164yescomplemented
278C2×C22⋊C4C22×C4116yes—
279C2×C22⋊C4C22×C4116yes—
280C2×C22⋊C4C2×D4116yes—
281C2×C22⋊C4C2×Q8116yes—
282C2×C22⋊C4C2×Q8116yes—
283C2×C42—216no—
284C2×C42—216no—
285C2×C42—48no—
286C2×C42—432no—
287#95—216no—
288#95—216no—
289#95—216no—
290#95—28no—
291#95—28no—
292#95—216no—
293#95—216no—
294#95—48no—
295#95—48no—
296#95—48no—
297#95C22×C4132yescomplemented
298C24—216no—
299C24—216no—
300C24—28no—
301C24—28no—
302C24—216no—
303C24—28no—
304C24—28no—
305C24—216no—
306C24—216no—
307C24—216no—
308C24—216no—
309C24—216no—
310C24—216no—
311C24—216no—
312C24—216no—
313C24—216no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24C2×C4⋊C4116yes—
319C24C4×D4116yes—
320C24C4×Q8116yes—
321C24#121116yes—
322C24C22⋊Q8116yes—
323C24#125116yes—
324C24C4⋊Q8116yes—
325C24C23×C414yes—
326C24C23×C412yes—
327C24C22×D414yes—
328C24C22×D414yes—
329C24C22×D412yes—
330C24C22×D412yes—
331C24C22×Q814yes—
332C24C22×Q814yes—
333C24C22×Q812yes—
334C24C22×Q812yes—
335C24C2×#4114yes—
336C24C2×#4114yes—
337C24C2×#4112yes—
338C24C2×#4112yes—
339C24C2×#4114yes—
340C24C2×#4114yes—
341C24H(5,𝔽2)14yes—
342C24H(5,𝔽2)14yes—
343C24C2511yescharacteristic
344C22×C4—216no—
345C22×C4—216no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—216no—
349C22×C4—216no—
350C22×C4—216no—
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—216no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—232no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—232no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—232no—
371C22×C4—28no—
372C22×C4—216no—
373C22×C4—28no—
374C22×C4—216no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
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—24no—
410C22×C4—232no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—432no—
422C22×C4—416no—
423C22×C4—4128no—
424C22×C4—432no—
425C22×C4—432no—
426C22×C4—432no—
427C22×C4—432no—
428C22×C4—4128no—
429C22×C4—4128no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—432no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—44no—
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—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4C2×C4⋊C4132yes—
466C22×C4#123116yes—
467C22×C4C23×C414yes—
468C22×C4C23×C418yes—
469C22×C4C23×C414yes—
470C22×C4C23×C418yes—
471C22×C4C22×D414yes—
472C22×C4C22×D418yes—
473C22×C4C22×D414yes—
474C22×C4C22×Q818yes—
475C22×C4C22×Q818yes—
476C22×C4C2×#4118yes—
477C22×C4C2×#4118yes—
478C4⋊C4—432no—
479C4⋊C4—432no—
480C4⋊C4—416no—
481C4⋊C4—416no—
482C22⋊C4—232no—
483C22⋊C4—232no—
484C22⋊C4—416no—
485C22⋊C4C2×C4⋊C4164yescomplemented
486C42—432no—
487C23—232no—
488C23—232no—
489C23—264no—
490C23—264no—
491C23—28no—
492C23—28no—
493C23—24no—
494C23—232no—
495C23—24no—
496C23—232no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—264no—
510C23—232no—
511C23—264no—
512C23—232no—
513C23#389164yes—
514C23C22×C4⋊C418yes—
515C23C22×C4⋊C414yes—
516C23C2×C4×D418yes—
517C23C2×C4×D414yes—
518C23C2×C4×Q818yes—
519C23C2×C4×Q814yes—
520C23C4×#4118yes—
521C23#51818yes—
522C23C2×#12118yes—
523C23C2×#12114yes—
524C23C2×C22⋊Q818yes—
525C23C2×C22⋊Q814yes—
526C23C2×#12314yes—
527C23#52514yes—
528C23C2×#12518yes—
529C23C2×#12514yes—
530C23#52918yes—
531C23C2×C4⋊Q818yes—
532C23C2×C4⋊Q814yes—
533C23#53414yes—
534C23#53514yes—
535C23#53818yes—
536C23#54618yes—
537C23#54618yes—
538C23#55118yes—
539C23#55118yes—
540C23#55418yes—
541C23#55418yes—
542C23#55418yes—
543C23#55418yes—
544C23C24×C411yescharacteristic
545C23C23×D411yescharacteristic
546C23C23×D411yescharacteristic
547C23C23×Q811yescharacteristic
548C23C23×Q811yescharacteristic
549C23C22×#4111yescharacteristic
550C23C22×#4111yescharacteristic
551C23C22×#4112yes—
552C23C22×#4112yes—
553C23C2×H(5,𝔽2)12yes—
554C23C2×H(5,𝔽2)12yes—
555C2×C4—264no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—232no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—464no—
584C2×C4—464no—
585C2×C4—464no—
586C2×C4—416no—
587C2×C4—432no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—432no—
591C2×C4—42no—
592C2×C4—416no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—432no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—88no—
623C2×C4—88no—
624C2×C4C22×C4⋊C4116yes—
625C2×C4C2×#12318yes—
626C22—28no—
627C22—28no—
628C22—216no—
629C22—216no—
630C22—28no—
631C22—216no—
632C22—28no—
633C22—216no—
634C22—28no—
635C22—2128no—
636C22—28no—
637C22C2×#389116yes—
638C22C2×#38918yes—
639C22C4×#12318yes—
640C22#232118yes—
641C22#2349116yes—
642C22#2368116yes—
643C22#241118yes—
644C22#241218yes—
645C22#2450116yes—
646C22#250218yes—
647C22#2505116yes—
648C22#258318yes—
649C22C23×C4⋊C411yescharacteristic
650C22C22×C4×D411yescharacteristic
651C22C22×C4×Q811yescharacteristic
652C22C2×C4×#4112yes—
653C22C2×#51812yes—
654C22C22×#12111yescharacteristic
655C22C22×C22⋊Q811yescharacteristic
656C22C22×#12311yescharacteristic
657C22C2×#52511yescharacteristic
658C22C22×#12511yescharacteristic
659C22C2×#52912yes—
660C22C22×C4⋊Q811yescharacteristic
661C22C2×#53411yescharacteristic
662C22C2×#53511yescharacteristic
663C22C2×#53812yes—
664C22C2×#54612yes—
665C22C2×#54612yes—
666C22C2×#55112yes—
667C22C2×#55112yes—
668C22C2×#55412yes—
669C22C2×#55412yes—
670C22C2×#55412yes—
671C22C2×#55412yes—
672C4—216no—
673C4—28no—
674C4—44no—
675C4—48no—
676C4—44no—
677C4—48no—
678C4—48no—
679C4—44no—
680C4—82no—
681C4—82no—
682C2—216no—
683C2#14044116yesdirect product factor
684C2C22×#38911yescharacteristic
685C2C2×C4×#12311yescharacteristic
686C2C2×#232111yescharacteristic
687C2C2×#234912yes—
688C2C2×#236812yes—
689C2C2×#241211yescharacteristic
690C2C2×#245012yes—
691C2C2×#241111yescharacteristic
692C2C2×#250512yes—
693C2C2×#250211yescharacteristic
694C2C2×#258311yescharacteristic
subgroupG/H
center, socleC25C24
commutatorC23C24×C4
Frattini, cosocleC24C25
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #14044C2direct (N × H)
7. C2×#1442C2semidirect
8. C2×#1441C2semidirect
9. C2×#1436C2semidirect
10. C2×C4⋊#95C2semidirect
27. C2×#385C4semidirect
46. #1442C22semidirect
47. #1441C22semidirect
48. #1436C22semidirect
49. C4⋊#95C22semidirect
90. C22×C4⋊C4C2×C4semidirect
117. #385C2×C4semidirect
130. C2×#95C2×C4semidirect
266. C2×C4⋊C4C22×C4semidirect
277. C2×C22⋊C4C4⋊C4semidirect
297. #95C22×C4semidirect
485. C22⋊C4C2×C4⋊C4semidirect
683. C2#14044direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3852
C22×C4×C22⋊C42
C23×#952
#140442
C2×#14422
C2×#14412
C2×#14362
C2×C4⋊#952
C2×C22⋊#952