Finite Group Databasebeta

← #360087

Subgroups — #360087

Order 512.

2000 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#423C211yesmaximal, characteristic, complemented
2C2×#2290C211yesmaximal, characteristic
3C2×#1870C212yesmaximal
4C2×#1798C212yesmaximal, complemented
5C2×#1797C211yesmaximal, characteristic, complemented
6#11202C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#10833C212yesmaximal, complemented
9#10715C212yesmaximal, complemented
10C23×M16C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C2×#423—24no—
15C2×#423—28no—
16C22×C4⋊C8C2212yes—
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2214yes—
21#2290—24no—
22#2290—24no—
23C2×#388C411yescharacteristic
24C2×#388C411yescharacteristic
25C2×#387C411yescharacteristic
26C2×#387C411yescharacteristic
27C2×#386C411yescharacteristic
28C2×#386C411yescharacteristic
29C2×#385C411yescharacteristic
30C2×#385C411yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×#376C2211yescharacteristic
35#1870—28no—
36#1798—28no—
37#1797—24no—
38C2×#336C2212yes—
39C2×#336C2214yes—
40C2×#336C2214yes—
41C2×#336C2214yes—
42C23×D4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—24no—
48C22×M16—24no—
49C22×M16—24no—
50C22×M16—28no—
51C22×M16—44no—
52C23×C8C2314yes—
53C23×C8C2314yes—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4D414yes—
59C2×C4×D4Q814yes—
60C2×#117—24no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—24no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C42—24no—
78C22×C42—24no—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D414yes—
82C22×C42D414yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#423—232no—
89C2×C4⋊C8—28no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8D418yescomplemented
97C2×C22⋊C8—24no—
98C2×C22⋊C8—28no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—28no—
102C2×C22⋊C8—44no—
103#388—24no—
104#388—24no—
105#387—24no—
106#387—24no—
107#386—24no—
108#386—24no—
109#385—24no—
110#385—24no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117#376—22no—
118#376—22no—
119C2×#95C2×C411yescharacteristic
120C2×#95C2×C411yescharacteristic
121C2×#95C2×C411yescharacteristic
122C2×#95C2×C411yescharacteristic
123C2×#95C2×C411yescharacteristic
124C2×#95C2×C411yescharacteristic
125C2×#95C2×C411yescharacteristic
126C2×#95C2×C411yescharacteristic
127C2×#95C2×C411yescharacteristic
128C2×#95C2×C411yescharacteristic
129#336—28no—
130#336—24no—
131#336—28no—
132#336—44no—
133C25C22×C411yescharacteristic
134C25C22×C411yescharacteristic
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4C4⋊C414yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C22⋊C414yes—
159C23×C4C22⋊C414yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×Q812yes—
177C23×C4C2×Q812yes—
178C23×C4#4112yes—
179C23×C4#4112yes—
180C23×C4C2411yescharacteristic
181C2×M16—216no—
182C2×M16—416no—
183C2×M16—416no—
184C2×M16—48no—
185C2×M16—48no—
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—24no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—24no—
195C22×C8—24no—
196C22×C8—24no—
197C22×C8—24no—
198C22×C8—24no—
199C22×C8C2×D418yes—
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—28no—
203C4×D4—28no—
204#117—44no—
205#117—48no—
206#117—44no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C4⋊C414yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4C4⋊C414yes—
235C2×C22⋊C4C4⋊C414yes—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—28no—
245C2×C42—216no—
246C2×C42—28no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42C22⋊C414yes—
256C2×C42C22⋊C414yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D414yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×Q812yes—
264C2×C42#4114yes—
265C4⋊C8—216no—
266C4⋊C8—216no—
267C4⋊C8—48no—
268C22⋊C8—416no—
269C22⋊C8—48no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24C2×C22⋊C412yes—
295C24C2×C4⋊C412yes—
296C24C2×C4⋊C412yes—
297C24#11712yes—
298C24C23×C411yescharacteristic
299C2×D4—28no—
300C2×D4—48no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—48no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
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—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
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—22no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—416no—
372C22×C4—416no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×C4⋊C412yes—
389C22×C4C2×C4⋊C412yes—
390C22×C4C2×C4⋊C412yes—
391C22×C4C2×C4⋊C412yes—
392C22×C4#11712yes—
393C22×C4C4×D414yes—
394C22×C4C4×D414yes—
395C22×C4#12114yes—
396C22×C4#12114yes—
397C22×C4C22⋊Q814yes—
398C22×C4C22⋊Q814yes—
399C22×C4#13114yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×Q811yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4112yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C42−1+412yes—
410M16—432no—
411C2×C8—28no—
412C2×C8—28no—
413C2×C8—28no—
414C2×C8—44no—
415C2×C8—44no—
416C2×C8—44no—
417C2×C8—44no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—44no—
422C2×C8—44no—
423C2×C8—44no—
424C2×C8—44no—
425C4⋊C4—28no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C22⋊C4—28no—
431C22⋊C4—28no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C42—22no—
439C42—24no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—28no—
446C42—28no—
447C42—41no—
448C42—42no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—42no—
453C42#12014yes—
454C42#12118yes—
455C42#12314yes—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—41no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—41no—
489C23#38014yes—
490C23C22×C22⋊C411yescharacteristic
491C23C22×C4⋊C411yescharacteristic
492C23C2×#11711yescharacteristic
493C23C2×C4×D412yes—
494C23#52012yes—
495C23C2×#13112yes—
496C23#56812yes—
497C23#56812yes—
498C23#56812yes—
499D4—416no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—416no—
529C2×C4—416no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4#38014yes—
563C2×C4#38614yes—
564C2×C4#38614yes—
565C2×C4#40814yes—
566C2×C4C2×#12011yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×#12112yes—
570C2×C4C2×C22⋊Q811yescharacteristic
571C2×C4C2×C22⋊Q811yescharacteristic
572C2×C4C2×#12311yescharacteristic
573C2×C4#53511yescharacteristic
574C2×C4#53611yescharacteristic
575C2×C4#53712yes—
576C2×C4D4212yes—
577C2×C4#54612yes—
578C2×C4#54712yes—
579C2×C4#54812yes—
580C2×C4D4×Q812yes—
581C2×C4#55212yes—
582C2×C4#55312yes—
583C2×C4#55412yes—
584C8—48no—
585C8—84no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—42no—
590C22—42no—
591C22—48no—
592C22—42no—
593C22—42no—
594C22—48no—
595C22—42no—
596C22—48no—
597C22—42no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—84no—
604C22C2×#38011yescharacteristic
605C22C2×#38611yescharacteristic
606C22#231111yescharacteristic
607C22D4×C4⋊C412yes—
608C22#235212yes—
609C22C2×#40811yescharacteristic
610C22#287911yescharacteristic
611C22#287911yescharacteristic
612C22#287911yescharacteristic
613C22#290512yes—
614C22#290612yes—
615C22#292112yes—
616C22#292512yes—
617C22#293014yes—
618C22#293314yes—
619C22#299314yes—
620C22#299514yes—
621C4—24no—
622C4—24no—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4#240811yescharacteristic
634C4#240911yescharacteristic
635C4#241612yes—
636C4#241712yes—
637C4#248611yescharacteristic
638C4#248711yescharacteristic
639C2—44no—
640C2—82no—
641C2#1387211yescharacteristic
642C2#2062711yescharacteristic
643C2#2062911yescharacteristic
644C2#2089611yescharacteristic
645C2#2089711yescharacteristic
646C2#2185011yescharacteristic
647C2#2185411yescharacteristic
648C2#2264914yes—
649C2#2275514yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#423C2semidirect
4. C2×#1798C2semidirect
5. C2×#1797C2semidirect
6. #11202C2semidirect
7. #11042C2semidirect
8. #10833C2semidirect
9. #10715C2semidirect
96. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4232
C2×#22902
C2×#18702
C2×#17982
C2×#17972
#112022
#110422
#108332
#107152