Finite Group Databasebeta

← C2×#14043 #514356

Subgroups — C2×#14043 #514356

Order 512.

8458 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
1C24×C4⋊C4C211yesmaximal, characteristic
2C22×#389C212yesmaximal, complemented
3C22×#385C211yesmaximal, characteristic
4C22×#385C211yesmaximal, characteristic
5C22×#385C211yesmaximal, characteristic
6C22×C4×C22⋊C4C211yesmaximal, characteristic
7#14043C2116yesmaximal, direct product factor
8C2×#1444C212yesmaximal, complemented
9C2×#1441C212yesmaximal, complemented
10C2×#1436C212yesmaximal, complemented
11C2×C22⋊#95C211yesmaximal, characteristic
12C2×C22⋊#95C211yesmaximal, characteristic
13C25×C4C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C25×C4C2211yescharacteristic
16C23×C4⋊C4—28no—
17C23×C4⋊C4—216no—
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2218yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2212yes—
23C23×C4⋊C4C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29C23×C42C2212yes—
30C2×#389—216no—
31C2×#389C22116yescomplemented
32C2×#385—28no—
33C2×#385—28no—
34C2×#385C4116yescomplemented
35C2×#385C2218yes—
36C2×#385C2218yes—
37C2×#385C2218yes—
38C2×C4×C22⋊C4—28no—
39C2×C4×C22⋊C4C2218yes—
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42C22×#95C2212yes—
43C22×#95C2212yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2212yes—
48C22×#95C2212yes—
49#1444C22116yescomplemented
50#1441C22116yescomplemented
51#1436C22116yescomplemented
52C22⋊#95C2218yes—
53C22⋊#95C2218yes—
54C26C2311yescharacteristic
55C24×C4—24no—
56C24×C4—24no—
57C24×C4—28no—
58C24×C4—28no—
59C24×C4—24no—
60C24×C4—24no—
61C24×C4—24no—
62C24×C4—28no—
63C24×C4—28no—
64C24×C4—24no—
65C24×C4—24no—
66C24×C4—44no—
67C24×C4—44no—
68C24×C4C2×C418yes—
69C24×C4D418yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2314yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2314yes—
79C24×C4C2314yes—
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C22×C4⋊C4—2128no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—232no—
87C22×C4⋊C4—28no—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—264no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2×C4116yescomplemented
101C22×C4⋊C4D4116yes—
102C22×C4⋊C4C2318yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2318yes—
105C22×C4⋊C4C2318yes—
106C22×C4⋊C4C2318yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4C2×C418yes—
116C22×C22⋊C4C2×C418yes—
117C22×C22⋊C4D418yes—
118C22×C22⋊C4Q818yes—
119C22×C22⋊C4Q818yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C42—28no—
125C22×C42—28no—
126C22×C42—44no—
127C22×C42C2318yes—
128#389—264no—
129#385—232no—
130#385—232no—
131#385C2×C4164yescomplemented
132C4×C22⋊C4—232no—
133C2×#95—28no—
134C2×#95—24no—
135C2×#95—24no—
136C2×#95—28no—
137C2×#95—28no—
138C2×#95—28no—
139C2×#95—28no—
140C2×#95—44no—
141C2×#95C2×C4116yescomplemented
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2314yes—
146C2×#95C2318yes—
147C2×#95C2314yes—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C25—24no—
152C25—24no—
153C25—24no—
154C25—24no—
155C25C22×C414yes—
156C25C2×D414yes—
157C25C2×D414yes—
158C25C2×Q814yes—
159C25C2×Q814yes—
160C25#4114yes—
161C25#4114yes—
162C25C2412yes—
163C25C2411yescharacteristic
164C23×C4—232no—
165C23×C4—264no—
166C23×C4—232no—
167C23×C4—232no—
168C23×C4—28no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—28no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—264no—
182C23×C4—216no—
183C23×C4—216no—
184C23×C4—232no—
185C23×C4—232no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—22no—
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—232no—
214C23×C4—216no—
215C23×C4—216no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—28no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—28no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—416no—
229C23×C4—48no—
230C23×C4—48no—
231C23×C4—432no—
232C23×C4—432no—
233C23×C4—432no—
234C23×C4—432no—
235C23×C4—416no—
236C23×C4—416no—
237C23×C4C22×C414yes—
238C23×C4C22×C414yes—
239C23×C4C22×C4116yes—
240C23×C4C22×C418yes—
241C23×C4C22×C418yes—
242C23×C4C2×D414yes—
243C23×C4C2×D418yes—
244C23×C4C2×D4116yes—
245C23×C4C2×D418yes—
246C23×C4C2×D418yes—
247C23×C4C2×D414yes—
248C23×C4C2×D418yes—
249C23×C4C2×Q818yes—
250C23×C4C2×Q818yes—
251C23×C4#4118yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2414yes—
258C23×C4C2414yes—
259C23×C4C2414yes—
260C23×C4C2414yes—
261C23×C4C2412yes—
262C2×C4⋊C4—264no—
263C2×C4⋊C4—2256no—
264C2×C4⋊C4—264no—
265C2×C4⋊C4—264no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—216no—
268C2×C4⋊C4—216no—
269C2×C4⋊C4—216no—
270C2×C4⋊C4—216no—
271C2×C4⋊C4—216no—
272C2×C4⋊C4—216no—
273C2×C4⋊C4—216no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—216no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—432no—
281C2×C4⋊C4—416no—
282C2×C4⋊C4—416no—
283C2×C4⋊C4C22×C4132yescomplemented
284C2×C4⋊C4C2×D4132yes—
285C2×C22⋊C4—232no—
286C2×C22⋊C4—232no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—416no—
295C2×C22⋊C4C4⋊C4164yescomplemented
296C2×C22⋊C4C22×C4116yes—
297C2×C22⋊C4C22×C4116yes—
298C2×C22⋊C4C2×D4116yes—
299C2×C22⋊C4C2×Q8116yes—
300C2×C22⋊C4C2×Q8116yes—
301C2×C42—216no—
302C2×C42—216no—
303C2×C42—48no—
304C2×C42—432no—
305#95—216no—
306#95—28no—
307#95—28no—
308#95—216no—
309#95—216no—
310#95—216no—
311#95—216no—
312#95—48no—
313#95C22×C4132yescomplemented
314C24—216no—
315C24—216no—
316C24—28no—
317C24—28no—
318C24—216no—
319C24—28no—
320C24—28no—
321C24—216no—
322C24—216no—
323C24—216no—
324C24—216no—
325C24—216no—
326C24—216no—
327C24—216no—
328C24—216no—
329C24—216no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24C2×C4⋊C4116yes—
335C24C4×D4116yes—
336C24C4×Q8116yes—
337C24#121116yes—
338C24C22⋊Q8116yes—
339C24#125116yes—
340C24C4⋊Q8116yes—
341C24C23×C414yes—
342C24C23×C412yes—
343C24C22×D414yes—
344C24C22×D414yes—
345C24C22×D412yes—
346C24C22×D412yes—
347C24C22×D414yes—
348C24C22×Q814yes—
349C24C22×Q814yes—
350C24C22×Q812yes—
351C24C22×Q812yes—
352C24C2×#4114yes—
353C24C2×#4114yes—
354C24C2×#4112yes—
355C24C2×#4114yes—
356C24C2×#4112yes—
357C24H(5,𝔽2)14yes—
358C242−1+414yes—
359C24C2511yescharacteristic
360C22×C4—232no—
361C22×C4—264no—
362C22×C4—232no—
363C22×C4—2256no—
364C22×C4—216no—
365C22×C4—232no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—28no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—264no—
393C22×C4—232no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—232no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—28no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
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—232no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—416no—
444C22×C4—4128no—
445C22×C4—432no—
446C22×C4—432no—
447C22×C4—432no—
448C22×C4—432no—
449C22×C4—4128no—
450C22×C4—416no—
451C22×C4—44no—
452C22×C4—416no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4C2×C4⋊C4132yescomplemented
479C22×C4C4×D4132yes—
480C22×C4#120116yes—
481C22×C4C23×C414yes—
482C22×C4C23×C414yes—
483C22×C4C23×C418yes—
484C22×C4C23×C418yes—
485C22×C4C22×D414yes—
486C22×C4C22×D418yes—
487C22×C4C22×D418yes—
488C22×C4C22×D418yes—
489C22×C4C22×D414yes—
490C22×C4C22×D418yes—
491C22×C4C22×Q818yes—
492C22×C4C22×Q818yes—
493C22×C4C2×#4118yes—
494C4⋊C4—264no—
495C4⋊C4—264no—
496C4⋊C4—264no—
497C4⋊C4—432no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C22⋊C4—232no—
501C22⋊C4—232no—
502C22⋊C4—416no—
503C22⋊C4C2×C4⋊C4164yescomplemented
504C42—432no—
505C23—232no—
506C23—232no—
507C23—264no—
508C23—264no—
509C23—28no—
510C23—28no—
511C23—24no—
512C23—232no—
513C23—24no—
514C23—28no—
515C23—232no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—232no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—264no—
529C23—264no—
530C23—232no—
531C23#389164yescomplemented
532C23C22×C4⋊C418yes—
533C23C22×C4⋊C414yes—
534C23C2×C4×D418yes—
535C23C2×C4×D414yes—
536C23C2×C4×D418yes—
537C23C2×C4×Q818yes—
538C23C2×C4×Q814yes—
539C23#52018yes—
540C23C2×#12014yes—
541C23C2×#12118yes—
542C23C2×#12114yes—
543C23C2×C22⋊Q818yes—
544C23C2×C22⋊Q814yes—
545C23#52514yes—
546C23#52518yes—
547C23C2×#12518yes—
548C23C2×#12514yes—
549C23C2×C4⋊Q818yes—
550C23C2×C4⋊Q814yes—
551C23#53414yes—
552C23#53614yes—
553C23#53918yes—
554C23D4218yes—
555C23#54618yes—
556C23#54718yes—
557C23D4×Q818yes—
558C23D4×Q818yes—
559C23#55218yes—
560C23#55418yes—
561C23#55418yes—
562C23C24×C411yescharacteristic
563C23C23×D411yescharacteristic
564C23C23×D411yescharacteristic
565C23C23×D412yes—
566C23C23×Q811yescharacteristic
567C23C23×Q811yescharacteristic
568C23C22×#4111yescharacteristic
569C23C22×#4112yes—
570C23C22×#4111yescharacteristic
571C23C2×H(5,𝔽2)12yes—
572C23C2×2−1+412yes—
573C2×C4—2128no—
574C2×C4—264no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—232no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—264no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—464no—
607C2×C4—464no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—416no—
612C2×C4—432no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
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—432no—
628C2×C4—416no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—88no—
640C2×C4—88no—
641C2×C4C22×C4⋊C4116yescomplemented
642C2×C4C2×C4×D4116yes—
643C2×C4C2×#12018yes—
644C22—28no—
645C22—28no—
646C22—216no—
647C22—216no—
648C22—28no—
649C22—216no—
650C22—216no—
651C22—28no—
652C22—28no—
653C22—2128no—
654C22—28no—
655C22C2×#38918yes—
656C22C2×#389116yescomplemented
657C22C4×#12018yes—
658C22#232218yes—
659C22D4×C4⋊C4116yes—
660C22#2360116yes—
661C22#241018yes—
662C22#241318yes—
663C22#243418yes—
664C22#2449116yes—
665C22#2473116yes—
666C22#258318yes—
667C22C23×C4⋊C411yescharacteristic
668C22C22×C4×D411yescharacteristic
669C22C22×C4×D412yes—
670C22C22×C4×Q811yescharacteristic
671C22C2×#52012yes—
672C22C22×#12011yescharacteristic
673C22C22×#12111yescharacteristic
674C22C22×C22⋊Q811yescharacteristic
675C22C2×#52511yescharacteristic
676C22C2×#52512yes—
677C22C22×#12511yescharacteristic
678C22C22×C4⋊Q811yescharacteristic
679C22C2×#53411yescharacteristic
680C22C2×#53611yescharacteristic
681C22C2×#53912yes—
682C22C2×D4212yes—
683C22C2×#54612yes—
684C22C2×#54712yes—
685C22C2×D4×Q812yes—
686C22C2×D4×Q812yes—
687C22C2×#55212yes—
688C22C2×#55412yes—
689C22C2×#55412yes—
690C4—216no—
691C4—28no—
692C4—216no—
693C4—44no—
694C4—44no—
695C4—48no—
696C4—48no—
697C4—44no—
698C4—82no—
699C4—82no—
700C2—216no—
701C2#14043116yesdirect product factor
702C2C22×#38911yescharacteristic
703C2C2×C4×#12011yescharacteristic
704C2C2×#232211yescharacteristic
705C2C2×D4×C4⋊C412yes—
706C2C2×#236012yes—
707C2C2×#241011yescharacteristic
708C2C2×#244912yes—
709C2C2×#247312yes—
710C2C2×#243411yescharacteristic
711C2C2×#241311yescharacteristic
712C2C2×#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
2. C22×#389C2semidirect
7. #14043C2direct (N × H)
8. C2×#1444C2semidirect
9. C2×#1441C2semidirect
10. C2×#1436C2semidirect
31. C2×#389C22semidirect
34. C2×#385C4semidirect
49. #1444C22semidirect
50. #1441C22semidirect
51. #1436C22semidirect
100. C22×C4⋊C4C2×C4semidirect
131. #385C2×C4semidirect
141. C2×#95C2×C4semidirect
283. C2×C4⋊C4C22×C4semidirect
295. C2×C22⋊C4C4⋊C4semidirect
313. #95C22×C4semidirect
478. C22×C4C2×C4⋊C4semidirect
503. C22⋊C4C2×C4⋊C4semidirect
531. C23#389semidirect
641. C2×C4C22×C4⋊C4semidirect
656. C22C2×#389semidirect
701. C2#14043direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C4⋊C42
C22×#3892
C22×#3852
C22×C4×C22⋊C42
#140432
C2×#14442
C2×#14412
C2×#14362
C2×C22⋊#952