Finite Group Databasebeta

← C4×#1507 #182613

Subgroups — C4×#1507 #182613

Order 512.

2052 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic, complemented
3C2×#1507C212yesmaximal
4C42×C22⋊C4C211yesmaximal, characteristic, complemented
5#8218C214yesmaximal, complemented
6C4×#327C214yesmaximal, complemented
7#8019C212yesmaximal, complemented
8C22×#121C2211yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C411yescharacteristic, complemented
11C23×C42C2211yescharacteristic
12C2×#407C2212yes—
13C2×C4×M16—21no—
14C4×#121—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121C412yescomplemented
19C4×#121C412yescomplemented
20C4×#121C2212yes—
21C4×#121C2212yes—
22C2×#390C411yescharacteristic, complemented
23C2×#389C411yescharacteristic, complemented
24C2×#388—21no—
25C2×#386—21no—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C411yescharacteristic, complemented
28C2×C4×C22⋊C4—21no—
29C2×C4×C22⋊C4—21no—
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4C2212yes—
32C2×C43—21no—
33#1749—21no—
34#1749C2212yescomplemented
35C4×C22⋊C8—21no—
36C4×C22⋊C8C2212yescomplemented
37#1728—21no—
38#1507C4116yesdirect product factor
39C2×#327C2218yes—
40C4×#95—21no—
41C4×#95C2212yescomplemented
42C4×#95C2212yescomplemented
43#1294C2218yescomplemented
44C23×D4—21no—
45C23×D4C2×C411yescharacteristic
46C24×C4—21no—
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—24no—
54C2×#121C2×C418yes—
55C2×#121C2×C412yes—
56C2×#121C2312yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—44no—
64C2×C4×D4—48no—
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C412yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—21no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—44no—
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#407—24no—
95#407C2×C418yescomplemented
96C2×C22⋊C8—22no—
97C2×C22⋊C8C2314yes—
98C4×M16—44no—
99C2×#97—22no—
100C2×C4×C8—22no—
101#390—21no—
102#390—21no—
103#390C2×C412yes—
104#390C2×C412yes—
105#389—21no—
106#389—21no—
107#389C2×C412yes—
108#389C2×C412yes—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#380—21no—
118#380—21no—
119#380C2×C412yes—
120#380C2312yes—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C412yes—
125C4×C22⋊C4—28no—
126C4×C22⋊C4—28no—
127C4×C22⋊C4—24no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4C2×C418yescomplemented
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—21no—
138C2×#95C2314yes—
139C43—44no—
140#336—22no—
141#336—22no—
142#327C2×C4132yescomplemented
143C25—21no—
144C25C2×D411yescharacteristic
145C22×D4—22no—
146C22×D4—28no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4C4214yescomplemented
155C22×D4C22×C412yes—
156C23×C4—21no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C414yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4#4114yes—
192C23×C4C2411yescharacteristic
193C2×M16—28no—
194C2×M16—44no—
195C2×M16—44no—
196C22×C8—24no—
197#121—24no—
198#121—28no—
199#121—28no—
200#121—24no—
201#121C4218yescomplemented
202#121C22×C418yes—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—48no—
207C4×D4—48no—
208C4×D4—48no—
209C4×D4—48no—
210C4×D4—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C4214yescomplemented
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—28no—
246C2×C42—28no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—28no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C22⋊C8—28no—
283C22⋊C8—24no—
284C22⋊C8C22×C418yescomplemented
285#97—44no—
286C4×C8—44no—
287#95—28no—
288#95—28no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95C22×C418yescomplemented
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—24no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24C2×C22⋊C411yescharacteristic
312C24C2×C22⋊C411yescharacteristic
313C24C4×D414yes—
314C24C22×D411yescharacteristic
315C2×D4—28no—
316C2×D4—28no—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—44no—
324C2×D4—432no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4C2×C4214yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—22no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—416no—
386C22×C4—416no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—816no—
406C22×C4#10414yes—
407C22×C4#10414yes—
408C22×C4C2×C4212yes—
409C22×C4C2×C4212yes—
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4#11712yes—
421C22×C4#11712yes—
422C22×C4C4×D414yes—
423C22×C4C4×D414yes—
424C22×C4C4×D414yes—
425C22×C4AGL(1,ℤ/8)14yes—
426C22×C4C23×C411yescharacteristic
427C22×C4C23×C412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4112yes—
430M16—416no—
431C2×C8—28no—
432C2×C8—44no—
433C2×C8—44no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4C2×C4214yes—
443C22⋊C4—232no—
444C22⋊C4—28no—
445C22⋊C4—28no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—416no—
470C42—416no—
471C42—416no—
472C42—44no—
473C42—44no—
474C42#9914yes—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—28no—
490C23—28no—
491C23—24no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—48no—
498C23—48no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—48no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23C4×C22⋊C412yes—
511C23C4×C22⋊C412yes—
512C23C2×#10412yes—
513C23C2×#10412yes—
514C23C22×C22⋊C411yescharacteristic
515C23C2×C4×D412yes—
516C23C2×AGL(1,ℤ/8)12yes—
517C23#57512yes—
518D4—44no—
519D4—44no—
520D4—416no—
521D4—416no—
522D4—48no—
523C2×C4—28no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—216no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—88no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—88no—
587C2×C4C4×C22⋊C412yes—
588C2×C4C4×C22⋊C412yes—
589C2×C4C2×#9912yes—
590C2×C4C2×#9912yes—
591C2×C4C2×#9912yes—
592C2×C4#41012yes—
593C2×C4#41812yes—
594C2×C4#41812yes—
595C2×C4C2×#10412yes—
596C2×C4C2×#10412yes—
597C2×C4C2×#10412yes—
598C2×C4C2×#10412yes—
599C2×C4C22×C4211yescharacteristic
600C2×C4C2×#11711yescharacteristic
601C2×C4C2×C4×D412yes—
602C8—48no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—416no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—816no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22C4×#10414yes—
632C22C2×C4×C22⋊C411yescharacteristic
633C22C22×#9911yescharacteristic
634C22C2×#41011yescharacteristic
635C22C2×#41811yescharacteristic
636C22#289611yescharacteristic
637C22C22×#10411yescharacteristic
638C22C22×#10411yescharacteristic
639C22C4×AGL(1,ℤ/8)14yes—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—21no—
647C4—22no—
648C4—21no—
649C4—416no—
650C4—41no—
651C4—41no—
652C4—44no—
653C4—82no—
654C4—82no—
655C4#150714yesdirect product factor
656C4C4×#9914yes—
657C2—24no—
658C2—24no—
659C2—42no—
660C2—84no—
661C2C2×#150711yescharacteristic
662C2#1753711yescharacteristic
663C2#1754211yescharacteristic
664C2#1754411yescharacteristic
665C2C2×C4×#9911yescharacteristic
666C2C2×C4×#10411yescharacteristic
667C2C4×#41811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#121C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#407C2semidirect
4. C42×C22⋊C4C2semidirect
5. #8218C2semidirect
6. C4×#327C2semidirect
7. #8019C2semidirect
10. C22×C4×D4C4semidirect
18. C4×#121C4semidirect
19. C4×#121C4semidirect
22. C2×#390C4semidirect
23. C2×#389C4semidirect
27. C2×C4×C4⋊C4C4semidirect
34. #1749C22semidirect
36. C4×C22⋊C8C22semidirect
38. #1507C4direct (N × H)
41. C4×#95C22semidirect
42. C4×#95C22semidirect
43. #1294C22semidirect
95. #407C2×C4semidirect
132. C4×C22⋊C4C2×C4semidirect
142. #327C2×C4semidirect
154. C22×D4C42semidirect
201. #121C42semidirect
220. C2×C4⋊C4C42semidirect
284. C22⋊C8C22×C4semidirect
296. #95C22×C4semidirect
655. C4#1507direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C4×#4072
C2×#15072
C42×C22⋊C42
#82182
C4×#3272
#80192