Finite Group Databasebeta

← #360103

Subgroups — #360103

Order 512.

2398 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C22×#384C211yesmaximal, characteristic
3C2×#1872C212yesmaximal
4C2×#1803C212yesmaximal, complemented
5C2×#1797C211yesmaximal, characteristic
6#11207C212yesmaximal, complemented
7#11050C212yesmaximal, complemented
8#10849C212yesmaximal, complemented
9#10730C212yesmaximal
10C22×C22⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C23×C4⋊C4—21no—
13C23×C4⋊C4—21no—
14C23×C4⋊C4C2212yes—
15C23×C42C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103C2214yes—
18C22×C22⋊C8C2212yes—
19C4×C22⋊Q8—24no—
20C4×C22⋊Q8C418yescomplemented
21C2×#391C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#387—21no—
24C2×#385—21no—
25C2×#384—21no—
26C2×#384—21no—
27C2×#384—28no—
28C2×#384C2212yes—
29C2×#384C2212yes—
30C2×#380C2211yescharacteristic
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34#1872—28no—
35#1803—28no—
36#1797—24no—
37C2×#340C2214yes—
38C2×#340C2214yes—
39C2×#340C2214yes—
40C2×#336C2212yes—
41C23×Q8C2311yescharacteristic
42C24×C4—22no—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8C2314yes—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8C2×C414yes—
51C2×C22⋊Q8D414yes—
52C2×C22⋊Q8Q814yes—
53C2×C4×Q8—22no—
54C2×C4×Q8—24no—
55C2×C4×Q8—24no—
56C2×C4×Q8C2×C414yescomplemented
57C22×C4⋊C4—21no—
58C22×C4⋊C4—28no—
59C22×C4⋊C4—28no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—28no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—44no—
71C22×C4⋊C4—44no—
72C22×C4⋊C4—44no—
73C22×C4⋊C4—44no—
74C22×C4⋊C4—44no—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C42—24no—
82C22×C42—24no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2×C414yes—
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D414yes—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×C4⋊C8—24no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—24no—
94C2×#103—28no—
95C2×#103—28no—
96C2×#103—44no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100#391—22no—
101#391C2×C414yescomplemented
102#389—22no—
103#389C2×C414yescomplemented
104#387—28no—
105#385—28no—
106#384—28no—
107#384—28no—
108#384—28no—
109#384—24no—
110#384D418yes—
111#380—22no—
112#380C2×C414yes—
113C4×C4⋊C4—28no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C414yescomplemented
116C4×C22⋊C4—28no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122#340—28no—
123#340—28no—
124#340—44no—
125#336—24no—
126C25C2×D411yescharacteristic
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8C22×C412yes—
135C22×Q8C2×D412yes—
136C22×Q8C2×Q812yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—28no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4C22⋊C414yes—
164C23×C4C22⋊C414yes—
165C23×C4SD1614yes—
166C23×C4Q1614yes—
167C23×C4C22×C412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×Q812yes—
176C23×C4#4112yes—
177C23×C4#4112yes—
178C23×C4C2411yescharacteristic
179C22×C8—24no—
180C22×C8—24no—
181C22×C8—24no—
182C22×C8—24no—
183C22×C8—24no—
184C22×C8—24no—
185C22×C8—24no—
186C22⋊Q8—28no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8C4⋊C4116yescomplemented
190C4×Q8—216no—
191C4×Q8—48no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—416no—
217C2×C4⋊C4—416no—
218C2×C4⋊C4—432no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—432no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C2×D418yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×Q812yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—216no—
253C2×C42—216no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42C22⋊C414yes—
261C2×C42C22⋊C414yes—
262C2×C42C22×C412yes—
263C2×C42C22×C412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D414yes—
266C2×C42C2×D412yes—
267C2×C42#4114yes—
268C4⋊C8—48no—
269#103—416no—
270C22⋊C8—48no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24C2×C22⋊C412yes—
285C24#12112yes—
286C24C22⋊Q812yes—
287C24C2×SD1612yes—
288C24C2×Q1612yes—
289C24AGL(1,ℤ/8)12yes—
290C24#13712yes—
291C24C22×D411yescharacteristic
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8C2×C4⋊C414yescomplemented
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—28no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—28no—
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—416no—
367C22×C4—416no—
368C22×C4—416no—
369C22×C4—41no—
370C22×C4—44no—
371C22×C4—41no—
372C22×C4—44no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—44no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4—88no—
390C22×C4—88no—
391C22×C4#103116yescomplemented
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C4⋊C414yes—
398C22×C4#11714yes—
399C22×C4#12012yes—
400C22×C4#12012yes—
401C22×C4#12114yes—
402C22×C4#12112yes—
403C22×C4C22⋊Q814yes—
404C22×C4C22⋊Q812yes—
405C22×C4#12312yes—
406C22×C4#12312yes—
407C22×C4C2×SD1612yes—
408C22×C4C2×SD1612yes—
409C22×C4C2×Q1612yes—
410C22×C4C2×Q1612yes—
411C22×C4AGL(1,ℤ/8)12yes—
412C22×C4AGL(1,ℤ/8)12yes—
413C22×C4#13712yes—
414C22×C4#13712yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×Q811yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4112yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C42−1+412yes—
425C2×C8—44no—
426C2×C8—44no—
427C2×C8—44no—
428C2×C8—44no—
429C2×C8—44no—
430C2×C8—44no—
431C2×C8—44no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—216no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—416no—
441C4⋊C4—416no—
442C4⋊C4—416no—
443C4⋊C4—416no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—416no—
451C4⋊C4C2×C4⋊C414yescomplemented
452C22⋊C4—216no—
453C22⋊C4—48no—
454C42—24no—
455C42—24no—
456C42—28no—
457C42—28no—
458C42—48no—
459C42—48no—
460C42C4×D418yes—
461C42#12118yes—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23#38014yes—
491C23C2×#10314yes—
492C23#41714yes—
493C23#46014yes—
494C23#46214yes—
495C23#47514yes—
496C23#47714yes—
497C23C22×C22⋊C411yescharacteristic
498C23C2×#12011yescharacteristic
499C23C2×#12111yescharacteristic
500C23C2×C22⋊Q811yescharacteristic
501C23C2×#12311yescharacteristic
502C23#53511yescharacteristic
503C23#53611yescharacteristic
504C23C22×SD1611yescharacteristic
505C23C22×Q1611yescharacteristic
506C23C2×AGL(1,ℤ/8)11yescharacteristic
507C23C2×AGL(1,ℤ/8)11yescharacteristic
508C23C2×AGL(1,ℤ/8)11yescharacteristic
509C23C2×#13711yescharacteristic
510C23C2×#13711yescharacteristic
511C23C2×#13711yescharacteristic
512Q8—28no—
513Q8—44no—
514Q8—44no—
515Q8—44no—
516C2×C4—232no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—232no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—832no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—84no—
573C2×C4—81no—
574C2×C4—84no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4#38014yes—
583C2×C4#38614yes—
584C2×C4#38614yes—
585C2×C4C2×#10314yes—
586C2×C4C2×#10314yes—
587C2×C4#41714yes—
588C2×C4#41714yes—
589C2×C4#45014yes—
590C2×C4#45114yes—
591C2×C4#48314yes—
592C2×C4#48414yes—
593C2×C4C22×C4⋊C411yescharacteristic
594C2×C4C2×#11711yescharacteristic
595C2×C4C2×C4×D412yes—
596C2×C4#52012yes—
597C2×C4C2×#12111yescharacteristic
598C2×C4C2×#12112yes—
599C2×C4C2×C22⋊Q811yescharacteristic
600C2×C4#53712yes—
601C2×C4D4212yes—
602C2×C4#54612yes—
603C2×C4#54712yes—
604C2×C4#54812yes—
605C2×C4D4×Q812yes—
606C2×C4#55212yes—
607C2×C4#55312yes—
608C2×C4#55412yes—
609C8—84no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22#180318yescomplemented
625C22C2×#38011yescharacteristic
626C22C2×#38611yescharacteristic
627C22#231111yescharacteristic
628C22#240811yescharacteristic
629C22#240911yescharacteristic
630C22#248611yescharacteristic
631C22#248711yescharacteristic
632C22C22×#10311yescharacteristic
633C22C2×#41711yescharacteristic
634C22C2×#41711yescharacteristic
635C22C2×#41711yescharacteristic
636C22C2×#45011yescharacteristic
637C22C2×#45111yescharacteristic
638C22#300311yescharacteristic
639C22#300411yescharacteristic
640C22C2×#46011yescharacteristic
641C22C2×#46211yescharacteristic
642C22#303811yescharacteristic
643C22#303911yescharacteristic
644C22C2×#47511yescharacteristic
645C22C2×#47711yescharacteristic
646C22#307911yescharacteristic
647C22#307911yescharacteristic
648C22C2×#48311yescharacteristic
649C22C2×#48411yescharacteristic
650C22#309211yescharacteristic
651C22#309211yescharacteristic
652C22#320811yescharacteristic
653C22#320911yescharacteristic
654C22#321711yescharacteristic
655C22#321811yescharacteristic
656C22#322911yescharacteristic
657C22#323011yescharacteristic
658C22#323811yescharacteristic
659C22#323911yescharacteristic
660C4—24no—
661C4—24no—
662C4—24no—
663C4—24no—
664C4—44no—
665C4—44no—
666C4—82no—
667C4—88no—
668C4—82no—
669C4#187218yescomplemented
670C4D4×C4⋊C412yes—
671C4#235212yes—
672C4#241612yes—
673C4#241712yes—
674C2—28no—
675C2#1387211yescharacteristic
676C2C2×#180311yescharacteristic
677C2#2064811yescharacteristic
678C2C2×#187211yescharacteristic
679C2#2090911yescharacteristic
680C2#2186011yescharacteristic
681C2#2186711yescharacteristic
682C2#2421111yescharacteristic
683C2#2421211yescharacteristic
684C2#2430111yescharacteristic
685C2#2430211yescharacteristic
686C2#2764711yescharacteristic
687C2#2764811yescharacteristic
688C2#2768011yescharacteristic
689C2#2768111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1803C2semidirect
6. #11207C2semidirect
7. #11050C2semidirect
8. #10849C2semidirect
20. C4×C22⋊Q8C4semidirect
56. C2×C4×Q8C2×C4semidirect
101. #391C2×C4semidirect
103. #389C2×C4semidirect
115. C4×C4⋊C4C2×C4semidirect
189. C22⋊Q8C4⋊C4semidirect
307. C2×Q8C2×C4⋊C4semidirect
391. C22×C4#103semidirect
451. C4⋊C4C2×C4⋊C4semidirect
624. C22#1803semidirect
669. C4#1872semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C22×#3842
C2×#18722
C2×#18032
C2×#17972
#112072
#110502
#108492
#107302