Finite Group Databasebeta

← C4×#1913 #354280

Subgroups — C4×#1913 #354280

Order 512.

2219 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#534C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic, complemented
3C2×#1913C212yesmaximal
4C2×C4×#99C211yesmaximal, characteristic, complemented
5C2×C4×#99C212yesmaximal, complemented
6#10938C212yesmaximal, complemented
7#10915C212yesmaximal, complemented
8#10754C212yesmaximal, complemented
9C4×#342C212yesmaximal, complemented
10C2×#534C2211yescharacteristic
11C2×#518C2212yes—
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#99C2212yes—
15C22×#99C2214yes—
16#2325—22no—
17#2322—21no—
18#2322C412yescomplemented
19#2321—21no—
20#2321C412yescomplemented
21#2315C2211yescharacteristic
22#2308C2211yescharacteristic
23#2308C2211yescharacteristic
24C4×#123—21no—
25C4×#123C412yescomplemented
26C4×#121—22no—
27C4×#120—21no—
28C4×#120C412yescomplemented
29#2275—24no—
30#2274—21no—
31#2273—21no—
32#2273C2212yes—
33C42×D4—24no—
34C4×#117—21no—
35C2×C4×C22⋊C4—21no—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39#1913C4116yesdirect product factor
40C4×#99—24no—
41C4×#99—24no—
42C4×#99—22no—
43C4×#99—22no—
44C2×#342C2214yes—
45C2×#342C2214yes—
46C2×#342C2212yes—
47C2×#342C2214yes—
48C23×D4C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4C2×C411yescharacteristic, complemented
51C24×C4D411yescharacteristic
52C24×C4C2311yescharacteristic
53#534—24no—
54#534C2×C418yes—
55C2×#123—21no—
56C2×#123C2×C412yes—
57C2×#121—22no—
58C2×#120—21no—
59C2×#120C2×C412yes—
60#518—24no—
61#518—28no—
62#518C2×C418yescomplemented
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—28no—
77C2×C4×D4C2×C412yes—
78C2×C4×D4C2×C412yes—
79C2×C4×D4D412yes—
80C2×C4×D4D412yes—
81C2×#117—22no—
82C22×C4⋊C4C2×C411yescharacteristic, complemented
83C22×C4⋊C4D411yescharacteristic
84C22×C22⋊C4—22no—
85C22×C22⋊C4C2×C411yescharacteristic, complemented
86C22×C22⋊C4D411yescharacteristic
87C22×C22⋊C4C2311yescharacteristic
88C22×C22⋊C4C2311yescharacteristic
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2311yescharacteristic
91C22×C22⋊C4C2311yescharacteristic
92C22×C22⋊C4C2314yes—
93C22×C42—22no—
94C22×C42—22no—
95C22×C42—21no—
96C22×C42—22no—
97C22×C42C2×C411yescharacteristic, complemented
98C22×C42D411yescharacteristic
99C2×#99—24no—
100C2×#99—24no—
101C2×#99—28no—
102C2×#99—28no—
103C2×#99C2×C418yescomplemented
104C2×#99C2×C4116yescomplemented
105#390—22no—
106#389—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#387—22no—
112#387—22no—
113#386—22no—
114#386—22no—
115#386—21no—
116#386—21no—
117#386—22no—
118#386—22no—
119#386C2×C412yes—
120#386C2×C412yes—
121#385—22no—
122#385—21no—
123#385—21no—
124#385—22no—
125#385C2×C412yescomplemented
126#385C2×C412yescomplemented
127#381—21no—
128#381C2×C412yescomplemented
129#380—22no—
130#379—21no—
131#379C2×C412yescomplemented
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—28no—
135C4×C4⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—28no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—28no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—22no—
152C4×C22⋊C4C2×C412yescomplemented
153C4×C22⋊C4C2×C412yescomplemented
154C4×C22⋊C4D414yescomplemented
155C4×C22⋊C4D414yescomplemented
156#376—22no—
157C2×#95—22no—
158C2×#95—21no—
159C2×#95—22no—
160C2×#95—22no—
161C43—22no—
162#342—24no—
163#342—28no—
164#342—28no—
165#342C2×C4116yescomplemented
166C25C22×C411yescharacteristic
167C25C2×D411yescharacteristic
168C25C2411yescharacteristic
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—24no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—24no—
178C22×D4—22no—
179C22×D4C22×C414yes—
180C22×D4C22×C412yes—
181C22×D4C22×C414yes—
182C22×D4C2×D412yes—
183C22×D4#4114yes—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—24no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—82no—
198C23×C4—82no—
199C23×C4—82no—
200C23×C4C22⋊C412yes—
201C23×C4C22⋊C412yes—
202C23×C4C22×C411yescharacteristic
203C23×C4C22×C411yescharacteristic
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D412yes—
216C23×C4#4112yes—
217C23×C4#4112yes—
218#123—24no—
219#123—28no—
220#123C4218yescomplemented
221#121—216no—
222#120—28no—
223#120—24no—
224#120C4218yescomplemented
225C4×D4—232no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—232no—
235C4×D4—41no—
236C4×D4—42no—
237C4×D4—41no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—42no—
241C4×D4—42no—
242C4×D4—41no—
243C4×D4—42no—
244C4×D4—41no—
245C4×D4—84no—
246#117—28no—
247#117—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4C22⋊C412yes—
259C2×C4⋊C4C22⋊C412yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4C22⋊C412yes—
294C2×C22⋊C4C22⋊C412yes—
295C2×C22⋊C4C22×C418yes—
296C2×C22⋊C4C22×C412yes—
297C2×C22⋊C4C22×C412yes—
298C2×C22⋊C4C22×C412yes—
299C2×C22⋊C4C22×C414yes—
300C2×C22⋊C4C22×C418yes—
301C2×C22⋊C4C2×D412yes—
302C2×C22⋊C4C2×D418yes—
303C2×C22⋊C4#4118yes—
304C2×C42—24no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—84no—
322C2×C42—84no—
323C2×C42—82no—
324C2×C42—84no—
325C2×C42C22⋊C412yes—
326C2×C42C22⋊C412yes—
327#99—216no—
328#99—232no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—42no—
342#95—42no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—21no—
347C24—21no—
348C24—22no—
349C24—22no—
350C24—81no—
351C24—81no—
352C24—81no—
353C24—81no—
354C24—81no—
355C24—81no—
356C24C2×C4212yescomplemented
357C24C2×C22⋊C412yes—
358C24C4×D414yes—
359C24C23×C411yescharacteristic
360C24C23×C412yes—
361C24C22×D411yescharacteristic
362C24C22×D411yescharacteristic
363C24C22×D412yes—
364C24C2×#4112yes—
365C24C2×#4112yes—
366C24C2×#4114yes—
367C2×D4—24no—
368C2×D4—28no—
369C2×D4—24no—
370C2×D4—28no—
371C2×D4—24no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—24no—
375C2×D4—28no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—41no—
381C2×D4—42no—
382C2×D4—41no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—42no—
387C2×D4—82no—
388C2×D4—82no—
389C2×D4C2×C4214yes—
390C2×D4#11714yes—
391C2×D4C4×D418yes—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—28no—
402C22×C4—21no—
403C22×C4—216no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—82no—
434C22×C4—84no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—82no—
438C22×C4—81no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—82no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4C2×C4212yescomplemented
460C22×C4C2×C22⋊C412yes—
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C2×C22⋊C412yes—
463C22×C4C2×C22⋊C412yes—
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4C2×C22⋊C411yescharacteristic
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C4×D414yes—
468C22×C4C4×D412yes—
469C22×C4C4×D412yes—
470C22×C4C4×D412yes—
471C22×C4C4×D412yes—
472C22×C4#12011yescharacteristic
473C22×C4#12011yescharacteristic
474C22×C4#12012yes—
475C22×C4#12112yes—
476C22×C4#12112yes—
477C22×C4#12112yes—
478C22×C4#12112yes—
479C22×C4#12311yescharacteristic
480C22×C4#12311yescharacteristic
481C22×C4#12312yes—
482C4⋊C4—216no—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C22⋊C4—216no—
492C22⋊C4—22no—
493C22⋊C4—28no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—24no—
506C22⋊C4—41no—
507C22⋊C4—42no—
508C22⋊C4—41no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—42no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4C2×C4214yescomplemented
520C22⋊C4C4×D4116yescomplemented
521C42—216no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—48no—
527C42—44no—
528C42—41no—
529C42—42no—
530C42—41no—
531C42—84no—
532C42—84no—
533C42—82no—
534C42—84no—
535C42—84no—
536C42—84no—
537C23—22no—
538C23—24no—
539C23—22no—
540C23—24no—
541C23—22no—
542C23—24no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—41no—
560C23—42no—
561C23—41no—
562C23—81no—
563C23—81no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23—84no—
568C23—82no—
569C23—84no—
570C23—82no—
571C23—82no—
572C23—84no—
573C23—84no—
574C23C4×C22⋊C414yes—
575C23C22×C4211yescharacteristic
576C23C22×C22⋊C411yescharacteristic
577C23C2×#11711yescharacteristic
578C23C2×C4×D412yes—
579C23C2×C4×D412yes—
580C23C2×C4×D412yes—
581C23C2×C4×D414yes—
582C23C4×#4112yes—
583C23C4×#4114yes—
584C23C2×#12011yescharacteristic
585C23C2×#12112yes—
586C23C2×#12311yescharacteristic
587C23#52514yes—
588C23#52512yes—
589C23#52914yes—
590C23#53212yes—
591D4—28no—
592D4—28no—
593D4—216no—
594D4—44no—
595D4—48no—
596D4—44no—
597D4—88no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—216no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—48no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—84no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—88no—
654C2×C4—88no—
655C2×C4—816no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—88no—
661C2×C4—82no—
662C2×C4—82no—
663C2×C4—82no—
664C2×C4—88no—
665C2×C4C4×C22⋊C414yes—
666C2×C4#38611yescharacteristic
667C2×C4#38611yescharacteristic
668C2×C4#38611yescharacteristic
669C2×C4#38612yes—
670C2×C4#38612yes—
671C2×C4#38611yescharacteristic
672C22—21no—
673C22—21no—
674C22—22no—
675C22—22no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—22no—
680C22—22no—
681C22—21no—
682C22—24no—
683C22—44no—
684C22—42no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—42no—
689C22—41no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—41no—
694C22—44no—
695C22—48no—
696C22—44no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—82no—
701C22—84no—
702C22—84no—
703C22—82no—
704C22—84no—
705C22—81no—
706C22—81no—
707C22C2×C4×C22⋊C411yescharacteristic
708C22C42×D412yes—
709C22C2×#38611yescharacteristic
710C22#229012yes—
711C22#229711yescharacteristic
712C22C4×#12012yes—
713C22C4×#12114yes—
714C22C4×#12312yes—
715C4—21no—
716C4—21no—
717C4—22no—
718C4—21no—
719C4—22no—
720C4—24no—
721C4—416no—
722C4—44no—
723C4—42no—
724C4—42no—
725C4—42no—
726C4—82no—
727C4—88no—
728C4—81no—
729C4—81no—
730C4#191312yesdirect product factor
731C2—22no—
732C2—21no—
733C2—21no—
734C2—24no—
735C2—22no—
736C2—22no—
737C2—22no—
738C2—44no—
739C2—82no—
740C2C4×#38611yescharacteristic
741C2C2×#191311yescharacteristic
742C2#2101411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC23C22×C42
Frattini, cosocleC25C24
socleC22C2×#386
prime radicalC2×#534C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#518C2semidirect
4. C2×C4×#99C2semidirect
5. C2×C4×#99C2semidirect
6. #10938C2semidirect
7. #10915C2semidirect
8. #10754C2semidirect
9. C4×#342C2semidirect
18. #2322C4semidirect
20. #2321C4semidirect
25. C4×#123C4semidirect
28. C4×#120C4semidirect
39. #1913C4direct (N × H)
50. C24×C4C2×C4semidirect
62. #518C2×C4semidirect
82. C22×C4⋊C4C2×C4semidirect
85. C22×C22⋊C4C2×C4semidirect
97. C22×C42C2×C4semidirect
103. C2×#99C2×C4semidirect
104. C2×#99C2×C4semidirect
125. #385C2×C4semidirect
126. #385C2×C4semidirect
128. #381C2×C4semidirect
131. #379C2×C4semidirect
152. C4×C22⋊C4C2×C4semidirect
153. C4×C22⋊C4C2×C4semidirect
154. C4×C22⋊C4D4semidirect
155. C4×C22⋊C4D4semidirect
165. #342C2×C4semidirect
220. #123C42semidirect
224. #120C42semidirect
356. C24C2×C42semidirect
459. C22×C4C2×C42semidirect
519. C22⋊C4C2×C42semidirect
520. C22⋊C4C4×D4semidirect
730. C4#1913direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#5342
C4×#5182
C2×#19132
C2×C4×#992
#109382
#109152
#107542
C4×#3422