Finite Group Databasebeta

← C4×#1510 #182624

Subgroups — C4×#1510 #182624

Order 512.

1868 conjugacy classes of subgroups.

The subgroup lattice has 661 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×#1510C212yesmaximal
4C4×#381C211yesmaximal, characteristic, complemented
5#8216C214yesmaximal, complemented
6C4×#327C214yesmaximal, complemented
7#8015C212yesmaximal, 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×#381C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4C411yescharacteristic, complemented
29C2×C4×C22⋊C4—21no—
30C2×C4×C22⋊C4—21no—
31#1749—21no—
32#1749C2212yescomplemented
33C4×C22⋊C8—21no—
34C4×C22⋊C8C2212yescomplemented
35#1728—21no—
36#1510C4116yesdirect product factor
37C2×#327C2218yes—
38#1434—21no—
39#1434C2212yescomplemented
40C22⋊#95—21no—
41C4×#95—21no—
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—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—21no—
78C22×C42—22no—
79C22×C42—44no—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#407—24no—
93#407C2×C418yescomplemented
94C2×C22⋊C8—22no—
95C2×C22⋊C8C2314yes—
96C4×M16—44no—
97C2×#97—22no—
98C2×C4×C8—22no—
99#390—21no—
100#390—21no—
101#390C2×C412yes—
102#390C2×C412yes—
103#389—21no—
104#389—21no—
105#389C2×C412yes—
106#389C2×C412yes—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#381—24no—
116#381C2×C418yescomplemented
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—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—44no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95C2314yes—
138#336—22no—
139#336—22no—
140#327C2×C4132yescomplemented
141C25—21no—
142C25C2×D411yescharacteristic
143C22×D4—22no—
144C22×D4—28no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—22no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4C4214yescomplemented
153C22×D4C22×C412yes—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4C22⋊C411yescharacteristic
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×C414yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4#4114yes—
190C23×C4C2411yescharacteristic
191C2×M16—28no—
192C2×M16—44no—
193C2×M16—44no—
194C22×C8—24no—
195#121—24no—
196#121—28no—
197#121—28no—
198#121—24no—
199#121C4218yescomplemented
200#121C22×C418yes—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—48no—
206C4×D4—48no—
207C4×D4—48no—
208C4×D4—48no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4C4214yescomplemented
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C22⋊C8—28no—
275C22⋊C8—24no—
276C22⋊C8C22×C418yescomplemented
277#97—44no—
278C4×C8—44no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—28no—
285#95—22no—
286#95—24no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95C22×C418yescomplemented
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—24no—
297C24—41no—
298C24—41no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24C2×C22⋊C411yescharacteristic
306C24C2×C22⋊C411yescharacteristic
307C24C4×D414yes—
308C24C22×D411yescharacteristic
309C2×D4—28no—
310C2×D4—28no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—42no—
314C2×D4—48no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—48no—
319C2×D4—44no—
320C2×D4—432no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4C2×C4214yes—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—28no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—48no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—48no—
377C22×C4—41no—
378C22×C4—416no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—416no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—816no—
401C22×C4#9914yes—
402C22×C4C2×C4212yes—
403C22×C4C2×C4212yes—
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4#11712yes—
415C22×C4#11712yes—
416C22×C4C4×D414yes—
417C22×C4C4×D414yes—
418C22×C4C4×D414yes—
419C22×C4AGL(1,ℤ/8)14yes—
420C22×C4C23×C411yescharacteristic
421C22×C4C23×C412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4112yes—
424M16—416no—
425C2×C8—28no—
426C2×C8—44no—
427C2×C8—44no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4C2×C4214yes—
437C22⋊C4—28no—
438C22⋊C4—28no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—416no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—44no—
465C42—44no—
466C42—88no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—48no—
490C23—48no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—48no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23—84no—
502C23C4×C22⋊C412yes—
503C23C4×C22⋊C412yes—
504C23C2×#9912yes—
505C23#41012yes—
506C23C22×C22⋊C411yescharacteristic
507C23C2×C4×D412yes—
508C23C2×AGL(1,ℤ/8)12yes—
509C23#57512yes—
510D4—44no—
511D4—44no—
512D4—48no—
513D4—416no—
514D4—416no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—48no—
575C2×C4—88no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—88no—
583C2×C4C4×C22⋊C412yes—
584C2×C4C4×C22⋊C412yes—
585C2×C4C2×#9912yes—
586C2×C4C2×#9912yes—
587C2×C4#41812yes—
588C2×C4#41812yes—
589C2×C4#42112yes—
590C2×C4#42112yes—
591C2×C4C22×C4211yescharacteristic
592C2×C4C2×#11711yescharacteristic
593C2×C4C2×C4×D412yes—
594C8—48no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—416no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—816no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22C4×#9914yes—
624C22C2×C4×C22⋊C411yescharacteristic
625C22C22×#9911yescharacteristic
626C22C2×#41011yescharacteristic
627C22C2×#41811yescharacteristic
628C22#289611yescharacteristic
629C22C2×#42111yescharacteristic
630C22C2×#42111yescharacteristic
631C22C4×AGL(1,ℤ/8)14yes—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—21no—
637C4—22no—
638C4—22no—
639C4—21no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—44no—
645C4—88no—
646C4—82no—
647C4—82no—
648C4#151014yesdirect product factor
649C2—24no—
650C2—24no—
651C2—42no—
652C2—84no—
653C2C2×#151011yescharacteristic
654C2#1755411yescharacteristic
655C2#1755911yescharacteristic
656C2#1756111yescharacteristic
657C2C2×C4×#9911yescharacteristic
658C2C4×#42111yescharacteristic
659C2C4×#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. C4×#381C2semidirect
5. #8216C2semidirect
6. C4×#327C2semidirect
7. #8015C2semidirect
10. C22×C4×D4C4semidirect
18. C4×#121C4semidirect
19. C4×#121C4semidirect
22. C2×#390C4semidirect
23. C2×#389C4semidirect
28. C2×C4×C4⋊C4C4semidirect
32. #1749C22semidirect
34. C4×C22⋊C8C22semidirect
36. #1510C4direct (N × H)
39. #1434C22semidirect
42. C4×#95C22semidirect
43. #1294C22semidirect
93. #407C2×C4semidirect
116. #381C2×C4semidirect
140. #327C2×C4semidirect
152. C22×D4C42semidirect
199. #121C42semidirect
218. C2×C4⋊C4C42semidirect
276. C22⋊C8C22×C4semidirect
290. #95C22×C4semidirect
648. C4#1510direct (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×#15102
C4×#3812
#82162
C4×#3272
#80152