Finite Group Databasebeta

← C4×#2613 #515186

Subgroups — C4×#2613 #515186

Order 512.

7684 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D4C211yesmaximal, characteristic, complemented
2C2×#2613C211yesmaximal, characteristic
3C2×C4×#123C211yesmaximal, characteristic, complemented
4C2×C4×#120C212yesmaximal, complemented
5#14335C211yesmaximal, characteristic, complemented
6#14300C214yesmaximal, complemented
7#14299C214yesmaximal, complemented
8#14263C211yesmaximal, characteristic, complemented
9#14262C212yesmaximal, complemented
10#14256C211yesmaximal, characteristic, complemented
11#14255C212yesmaximal, complemented
12C4×#394C214yesmaximal, complemented
13C4×#392C214yesmaximal, complemented
14C4×#381C211yesmaximal, characteristic, complemented
15C4×#379C212yesmaximal, complemented
16C24×D4C2211yescharacteristic
17C25×C4C2212yes—
18C22×#123C2211yescharacteristic
19C22×#120C2212yes—
20C22×C4×D4—28no—
21C22×C4×D4—216no—
22C22×C4×D4C2214yes—
23C22×C4×D4C2214yes—
24C22×C4×D4C2211yescharacteristic
25C22×C4×D4C2214yescomplemented
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2212yes—
28C23×C42C2211yescharacteristic
29#2613C4116yesdirect product factor
30C2×#394C2214yes—
31C2×#392C2214yes—
32C4×#123—28no—
33C4×#120—216no—
34C2×#388C2212yescomplemented
35C2×#387C2212yescomplemented
36C2×#386C2211yescharacteristic
37C2×#386C2212yes—
38C2×#386C2214yescomplemented
39C2×#385C2211yescharacteristic
40C2×#385C2212yes—
41C2×#385C2214yescomplemented
42C2×#381C2211yescharacteristic
43C2×#381C2211yescharacteristic
44C2×#381C2211yescharacteristic
45C2×#379C2212yes—
46C2×#379C2212yes—
47C2×#379C2212yes—
48C2×C4×C4⋊C4C2212yescomplemented
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yescomplemented
52C2×C4×C22⋊C4C2214yescomplemented
53C2×C4×C22⋊C4C2218yescomplemented
54#1444C2214yescomplemented
55#1442C2214yescomplemented
56#1441C2214yescomplemented
57#1440C2218yescomplemented
58#1439C2214yescomplemented
59#1439C2218yescomplemented
60#1439C2218yescomplemented
61#1438C2214yes—
62#1438C2214yes—
63#1438C2218yescomplemented
64#1434C2214yescomplemented
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2218yescomplemented
69C4×#95C2214yescomplemented
70C26C2312yes—
71C23×D4—28no—
72C23×D4—28no—
73C23×D4—24no—
74C23×D4—24no—
75C23×D4C2×C418yescomplemented
76C23×D4C2314yes—
77C23×D4C2311yescharacteristic
78C23×D4C2314yes—
79C24×C4—28no—
80C24×C4—28no—
81C24×C4—416no—
82C24×C4D4116yescomplemented
83C24×C4C2311yescharacteristic
84C24×C4C2314yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2314yes—
89C24×C4C2311yescharacteristic
90C2×#123—24no—
91C2×#123—24no—
92C2×#123C2×C418yescomplemented
93C2×#120—28no—
94C2×#120—28no—
95C2×#120C2×C4116yescomplemented
96C2×C4×D4—2128no—
97C2×C4×D4—232no—
98C2×C4×D4—232no—
99C2×C4×D4—28no—
100C2×C4×D4—24no—
101C2×C4×D4—216no—
102C2×C4×D4—216no—
103C2×C4×D4—216no—
104C2×C4×D4—416no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—28no—
113C22×C22⋊C4—28no—
114C22×C22⋊C4—28no—
115C22×C22⋊C4C2318yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2318yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2318yes—
130C22×C42—28no—
131C22×C42D418yescomplemented
132C22×C42C2311yescharacteristic
133C22×C42C2314yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137#394C2×C4132yescomplemented
138#392C2×C4132yescomplemented
139#388—28no—
140#387—28no—
141#386—28no—
142#386—24no—
143#386—216no—
144#385—28no—
145#385—24no—
146#385—216no—
147#381—24no—
148#381—24no—
149#381C2×C418yescomplemented
150#379—28no—
151#379—28no—
152#379C2×C4116yescomplemented
153C4×C4⋊C4—28no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—216no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—416no—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2318yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2314yes—
172C25—24no—
173C25—24no—
174C25—24no—
175C25—24no—
176C25—48no—
177C25—48no—
178C25C22×C418yes—
179C25C2×D418yes—
180C25#4118yes—
181C25C2411yescharacteristic
182C25C2414yes—
183C25C2411yescharacteristic
184C22×D4—232no—
185C22×D4—232no—
186C22×D4—264no—
187C22×D4—264no—
188C22×D4—216no—
189C22×D4—216no—
190C22×D4—24no—
191C22×D4—28no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—22no—
196C22×D4—232no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—48no—
200C22×D4—48no—
201C22×D4C22×C414yes—
202C22×D4C22×C4116yes—
203C22×D4C22×C4116yescomplemented
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—264no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—216no—
216C23×C4—216no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—28no—
220C23×C4—464no—
221C23×C4—48no—
222C23×C4—48no—
223C23×C4—48no—
224C23×C4—44no—
225C23×C4—464no—
226C23×C4—464no—
227C23×C4C22×C414yes—
228C23×C4C2×D4116yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D418yes—
232C23×C4C2×D414yes—
233C23×C4C2×D418yes—
234C23×C4C2×D4116yes—
235C23×C4#4114yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2414yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248#123—232no—
249#120—264no—
250C4×D4—2128no—
251C4×D4—2128no—
252C4×D4—464no—
253C4×D4—416no—
254C2×C4⋊C4—232no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4C22×C418yescomplemented
266C2×C22⋊C4—264no—
267C2×C22⋊C4—216no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4C22×C418yes—
301C2×C22⋊C4C22×C414yes—
302C2×C22⋊C4C22×C4116yescomplemented
303C2×C22⋊C4C22×C418yescomplemented
304C2×C22⋊C4C22×C4132yescomplemented
305C2×C42—232no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—216no—
313C2×C42—28no—
314C2×C42C2×D418yes—
315C2×C42C2×D418yes—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—48no—
326#95—48no—
327#95—48no—
328#95C22×C4116yescomplemented
329C24—24no—
330C24—24no—
331C24—216no—
332C24—216no—
333C24—216no—
334C24—216no—
335C24—216no—
336C24—216no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—28no—
342C24—28no—
343C24—28no—
344C24—28no—
345C24—21no—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—416no—
352C24—432no—
353C24—416no—
354C24—432no—
355C24—416no—
356C24—416no—
357C24—416no—
358C24—416no—
359C24—432no—
360C24—432no—
361C24C4×D4132yescomplemented
362C24C23×C412yes—
363C24C23×C418yes—
364C24C23×C412yes—
365C24C22×D418yes—
366C24C22×D414yes—
367C24C22×D412yes—
368C24C2×#4118yes—
369C24C2×#4114yes—
370C24C2×#4112yes—
371C24C2511yescharacteristic
372C2×D4—232no—
373C2×D4—232no—
374C2×D4—264no—
375C2×D4—2256no—
376C2×D4—264no—
377C2×D4—216no—
378C2×D4—232no—
379C2×D4—28no—
380C2×D4—232no—
381C2×D4—432no—
382C2×D4—416no—
383C2×D4—416no—
384C2×D4—432no—
385C2×D4—44no—
386C2×D4—48no—
387C2×D4—44no—
388C22×C4—216no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—216no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—232no—
396C22×C4—216no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—232no—
419C22×C4—232no—
420C22×C4—232no—
421C22×C4—232no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—48no—
441C22×C4—432no—
442C22×C4—432no—
443C22×C4—4256no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—432no—
448C22×C4—432no—
449C22×C4—44no—
450C22×C4—416no—
451C22×C4—48no—
452C22×C4C4×D4116yescomplemented
453C22×C4#120116yes—
454C22×C4C23×C418yes—
455C22×C4C23×C414yes—
456C22×C4C23×C412yes—
457C22×C4C23×C414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D418yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4114yes—
466C22×C4H(5,𝔽2)18yes—
467C4⋊C4—232no—
468C4⋊C4—216no—
469C4⋊C4—48no—
470C4⋊C4—416no—
471C4⋊C4—44no—
472C22⋊C4—264no—
473C22⋊C4—216no—
474C22⋊C4—28no—
475C22⋊C4—232no—
476C22⋊C4—216no—
477C22⋊C4—432no—
478C22⋊C4—48no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—432no—
482C22⋊C4—416no—
483C22⋊C4—44no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C42—216no—
488C42—216no—
489C42—44no—
490C42—416no—
491C42—48no—
492C23—28no—
493C23—28no—
494C23—24no—
495C23—28no—
496C23—24no—
497C23—216no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—216no—
502C23—216no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—216no—
512C23—216no—
513C23—264no—
514C23—22no—
515C23—24no—
516C23—28no—
517C23—464no—
518C23—432no—
519C23—464no—
520C23—464no—
521C23—464no—
522C23—464no—
523C23—464no—
524C23—432no—
525C23—48no—
526C23—48no—
527C23—48no—
528C23—48no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—48no—
541C23—44no—
542C23C2×C4×D4116yes—
543C23C2×C4×D418yes—
544C23C2×C4×D414yes—
545C23C2×#12014yes—
546C23#52518yes—
547C23#52514yes—
548C23C24×C411yescharacteristic
549C23C23×D412yes—
550C23C23×D411yescharacteristic
551C23C22×#4112yes—
552C23C22×#4111yescharacteristic
553C23C2×H(5,𝔽2)14yes—
554C23#58514yes—
555D4—2128no—
556D4—464no—
557D4—416no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—232no—
565C2×C4—232no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—216no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—464no—
579C2×C4—416no—
580C2×C4—432no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—48no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—464no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—88no—
610C2×C4C2×C4×D418yes—
611C2×C4C2×#12014yes—
612C2×C4C2×#12014yes—
613C2×C4#534116yes—
614C2×C4#53518yes—
615C22—216no—
616C22—28no—
617C22—216no—
618C22—28no—
619C22—24no—
620C22—24no—
621C22—216no—
622C22—216no—
623C22—216no—
624C22—4128no—
625C22—48no—
626C22—48no—
627C22—416no—
628C22—416no—
629C22—416no—
630C22—416no—
631C22—48no—
632C22—48no—
633C22—48no—
634C22—44no—
635C22—42no—
636C22—84no—
637C22—82no—
638C22—81no—
639C22—81no—
640C22C4×#12018yes—
641C22C22×C4×D412yes—
642C22C22×C4×D411yescharacteristic
643C22C4×H(5,𝔽2)14yes—
644C22C22×#12011yescharacteristic
645C22C2×#52512yes—
646C22C2×#52511yescharacteristic
647C22C2×#53414yes—
648C22C2×#53512yes—
649C22#344914yes—
650C22#344912yes—
651C22#345918yes—
652C22#346014yes—
653C4—28no—
654C4—24no—
655C4—48no—
656C4—44no—
657C4—44no—
658C4—42no—
659C4—48no—
660C4—84no—
661C4—82no—
662C4—81no—
663C4—81no—
664C4#261318yesdirect product factor
665C2—28no—
666C2—416no—
667C2—82no—
668C2C2×C4×#12011yescharacteristic
669C2C4×#53414yes—
670C2C4×#53512yes—
671C2C2×#261311yescharacteristic
672C2#3456214yes—
673C2#3459312yes—
674C2#3463211yescharacteristic
subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC2×#2613C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C4×D4C2semidirect
3. C2×C4×#123C2semidirect
4. C2×C4×#120C2semidirect
5. #14335C2semidirect
6. #14300C2semidirect
7. #14299C2semidirect
8. #14263C2semidirect
9. #14262C2semidirect
10. #14256C2semidirect
11. #14255C2semidirect
12. C4×#394C2semidirect
13. C4×#392C2semidirect
14. C4×#381C2semidirect
15. C4×#379C2semidirect
25. C22×C4×D4C22semidirect
29. #2613C4direct (N × H)
34. C2×#388C22semidirect
35. C2×#387C22semidirect
38. C2×#386C22semidirect
41. C2×#385C22semidirect
48. C2×C4×C4⋊C4C22semidirect
51. C2×C4×C22⋊C4C22semidirect
52. C2×C4×C22⋊C4C22semidirect
53. C2×C4×C22⋊C4C22semidirect
54. #1444C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
57. #1440C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
63. #1438C22semidirect
64. #1434C22semidirect
68. C22⋊#95C22semidirect
69. C4×#95C22semidirect
75. C23×D4C2×C4semidirect
82. C24×C4D4semidirect
92. C2×#123C2×C4semidirect
95. C2×#120C2×C4semidirect
131. C22×C42D4semidirect
137. #394C2×C4semidirect
138. #392C2×C4semidirect
149. #381C2×C4semidirect
152. #379C2×C4semidirect
203. C22×D4C22×C4semidirect
265. C2×C4⋊C4C22×C4semidirect
302. C2×C22⋊C4C22×C4semidirect
303. C2×C22⋊C4C22×C4semidirect
304. C2×C22⋊C4C22×C4semidirect
328. #95C22×C4semidirect
361. C24C4×D4semidirect
452. C22×C4C4×D4semidirect
664. C4#2613direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×D42
C2×#26132
C2×C4×#1232
C2×C4×#1202
#143352
#143002
#142992
#142632
#142622
#142562
#142552
C4×#3942
C4×#3922
C4×#3812
C4×#3792