Finite Group Databasebeta

← C4×#1465 #182001

Subgroups — C4×#1465 #182001

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C42×D4C211yesmaximal, characteristic
2C4×#407C212yesmaximal, complemented
3C2×#1465C212yesmaximal
4#8153C212yesmaximal, complemented
5#8152C212yesmaximal, complemented
6C4×#324C212yesmaximal, complemented
7C4×#323C212yesmaximal, complemented
8#7911C212yesmaximal, complemented
9C22×C4×D4C411yescharacteristic
10C22×C4×D4C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C42C411yescharacteristic
13C23×C42C2211yescharacteristic
14C2×#407C2214yes—
15C2×C4×M16—22no—
16C42×D4—22no—
17C42×D4—22no—
18C42×D4—22no—
19C42×D4—22no—
20C2×C4×C4⋊C4C411yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C411yescharacteristic
24C2×C4×C22⋊C4C411yescharacteristic
25C2×C4×C22⋊C4C411yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C43C411yescharacteristic
30#1749—22no—
31#1749C2214yes—
32C4×C22⋊C8—22no—
33C4×C22⋊C8C2214yes—
34#1728—22no—
35#1465C4116yesdirect product factor
36C2×#324C2214yes—
37C2×#323C2214yes—
38#1280C2214yes—
39#1280C2214yes—
40C23×D4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—24no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—24no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4C2×C418yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42—44no—
74C22×C42—44no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42D412yes—
87C22×C42D412yes—
88C22×C42D412yes—
89C22×C42D412yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#407—28no—
93#407C2×C4116yescomplemented
94C2×C22⋊C8—24no—
95C2×C22⋊C8C2318yes—
96C4×M16—48no—
97C2×#97—24no—
98C2×C4×C8—24no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C2×#95C2×C411yescharacteristic
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C411yescharacteristic
120C2×#95C2×C411yescharacteristic
121C2×#95C2311yescharacteristic
122C2×#95C2311yescharacteristic
123C43—21no—
124C43—21no—
125C43—21no—
126C43—21no—
127#336—24no—
128#336—24no—
129#324C2×C4116yescomplemented
130#323C2×C4116yescomplemented
131C25C22×C411yescharacteristic
132C25C22×C411yescharacteristic
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4C4214yes—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4C4214yes—
167C23×C4C22⋊C412yes—
168C23×C4C22⋊C412yes—
169C23×C4C22⋊C412yes—
170C23×C4C22⋊C412yes—
171C23×C4M1614yescomplemented
172C23×C4C22×C411yescharacteristic
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×C4C22×C411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4#4114yes—
188C23×C4C2411yescharacteristic
189C2×M16—216no—
190C2×M16—48no—
191C2×M16—48no—
192C22×C8—28no—
193C4×D4—216no—
194C4×D4—216no—
195C4×D4—216no—
196C4×D4—216no—
197C4×D4—216no—
198C4×D4—216no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4M1614yescomplemented
207C2×C4⋊C4C22×C414yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4C4214yes—
224C2×C22⋊C4M1614yescomplemented
225C2×C22⋊C4C22×C414yes—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—28no—
237C2×C42—24no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—42no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42C4214yes—
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C412yes—
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C412yes—
272C2×C42C22⋊C411yescharacteristic
273C2×C42M1614yescomplemented
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C22⋊C8—216no—
279C22⋊C8—28no—
280C22⋊C8C22×C4116yes—
281#97—48no—
282C4×C8—48no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24C2×C4212yes—
306C24C2×C4212yes—
307C24C2×C22⋊C412yes—
308C24#11712yes—
309C24C2×M1612yes—
310C24C2×M1612yes—
311C24C23×C411yescharacteristic
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—28no—
316C2×D4—28no—
317C2×D4—28no—
318C2×D4—28no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4C2×C4212yes—
410C22×C4C2×C4212yes—
411C22×C4C2×C4212yes—
412C22×C4C2×C4212yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C412yes—
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4#11712yes—
422C22×C4C4×D418yes—
423C22×C4C4×D418yes—
424C22×C4C2×M1612yes—
425C22×C4C2×M1612yes—
426C22×C4C2×M1612yes—
427C22×C4C2×M1612yes—
428C22×C4C2×M1612yes—
429C22×C4C2×M1612yes—
430C22×C4C2×M1612yes—
431C22×C4C2×M1612yes—
432C22×C4C2×M1612yes—
433C22×C4C2×M1612yes—
434C22×C4C23×C412yes—
435C22×C4C22×D412yes—
436C22×C4C2×#4112yes—
437M16—432no—
438C2×C8—216no—
439C2×C8—48no—
440C2×C8—48no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—28no—
446C22⋊C4—28no—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C42—24no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—24no—
456C42—24no—
457C42—24no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—24no—
462C42—42no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42#9914yes—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C4×C22⋊C414yes—
527C23C4×M1614yescomplemented
528C23C22×C4211yescharacteristic
529C23C22×C22⋊C411yescharacteristic
530C23C2×#11711yescharacteristic
531C23C22×M1611yescharacteristic
532C23C22×M1611yescharacteristic
533C23C22×M1611yescharacteristic
534C23C22×M1611yescharacteristic
535D4—232no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—216no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4C4×C22⋊C412yes—
612C2×C4C4×C22⋊C412yes—
613C2×C4C4×M1614yescomplemented
614C2×C4#40712yes—
615C2×C4#40714yes—
616C2×C4#40712yes—
617C2×C4C2×#9912yes—
618C2×C4C2×#9912yes—
619C2×C4C2×#9912yes—
620C2×C4#41012yes—
621C2×C4#41312yes—
622C2×C4#41312yes—
623C2×C4C2×C4×D414yes—
624C8—416no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22C2×C4×C22⋊C411yescharacteristic
660C22C2×C4×M1611yescharacteristic
661C22C2×C4×M1611yescharacteristic
662C22C2×#40711yescharacteristic
663C22C2×#40711yescharacteristic
664C22C22×#9911yescharacteristic
665C22C2×#41011yescharacteristic
666C22C2×#41311yescharacteristic
667C22C2×#41311yescharacteristic
668C22#292011yescharacteristic
669C22#292011yescharacteristic
670C4—24no—
671C4—24no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—24no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4#146514yesdirect product factor
685C4C4×#9914yes—
686C2—21no—
687C2—21no—
688C2—21no—
689C2—21no—
690C2—44no—
691C2—44no—
692C2C2×#146511yescharacteristic
693C2#1736111yescharacteristic
694C2#1736211yescharacteristic
695C2#1736611yescharacteristic
696C2C4×#40711yescharacteristic
697C2C2×C4×#9911yescharacteristic
698C2C4×#41311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ 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. #8153C2semidirect
5. #8152C2semidirect
6. C4×#324C2semidirect
7. C4×#323C2semidirect
8. #7911C2semidirect
35. #1465C4direct (N × H)
93. #407C2×C4semidirect
129. #324C2×C4semidirect
130. #323C2×C4semidirect
171. C23×C4M16semidirect
206. C2×C4⋊C4M16semidirect
224. C2×C22⋊C4M16semidirect
273. C2×C42M16semidirect
527. C23C4×M16semidirect
613. C2×C4C4×M16semidirect
684. C4#1465direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C42×D42
C4×#4072
C2×#14652
#81532
#81522
C4×#3242
C4×#3232
#79112