Finite Group Databasebeta

← #362761

Subgroups — #362761

Order 512.

3638 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic, complemented
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1984C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic
6C2×C4×#102C211yesmaximal, characteristic
7#13926C211yesmaximal, characteristic, complemented
8#11384C214yesmaximal, complemented
9#10788C214yesmaximal
10C24×Q8—21no—
11C2×D4×Q8—24no—
12C2×D4×Q8—24no—
13C2×D4×Q8—24no—
14C2×D4×Q8—24no—
15C2×D4×Q8—24no—
16C2×D4×Q8C418yescomplemented
17C22×C4⋊Q8C2212yes—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8—22no—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2212yes—
23C22×C4×D4C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2212yes—
27C22×C4×C8C2211yescharacteristic
28C2×#391—21no—
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1990—24no—
32#1988—24no—
33#1984—24no—
34#1802—24no—
35C4×#102—24no—
36C2×#340C2214yes—
37C2×#340C2214yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2214yes—
41#1441—22no—
42C4⋊#95—21no—
43C4⋊#95C2214yes—
44C23×Q8—28no—
45C23×Q8—28no—
46C23×Q8—28no—
47C23×Q8—22no—
48C23×Q8—21no—
49C23×Q8—44no—
50C23×Q8—44no—
51C23×Q8—44no—
52C23×Q8—44no—
53C23×Q8D411yescharacteristic
54C23×D4C2311yescharacteristic
55C24×C4—22no—
56C24×C4—21no—
57C23×C8C2312yes—
58C23×C8C2312yes—
59D4×Q8—232no—
60D4×Q8—232no—
61D4×Q8—232no—
62D4×Q8—232no—
63C2×C4⋊Q8—24no—
64C2×C4⋊Q8—24no—
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8—24no—
67C2×C4⋊Q8—24no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8C2×C418yes—
72C2×C4⋊Q8C2×C414yes—
73C2×C4⋊Q8D414yes—
74C2×C4⋊Q8D414yes—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—216no—
79C2×C22⋊Q8—28no—
80C2×C22⋊Q8—216no—
81C2×C4×Q8—22no—
82C2×C4×Q8—22no—
83C2×C4×Q8—22no—
84C2×C4×Q8—22no—
85C2×C4×Q8C2×C414yes—
86C2×C4×Q8D414yes—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—22no—
93C2×C4×D4—22no—
94C2×C4×D4—22no—
95C2×C4×D4C2×C418yes—
96C2×C4×D4C2×C414yes—
97C2×C4×D4D414yes—
98C2×C4×D4D414yes—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D412yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—22no—
110C22×C22⋊C4—21no—
111C22×C42—21no—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—22no—
117C2×#102—24no—
118C2×#102—24no—
119C2×#102—24no—
120C2×#102—28no—
121C2×#102Q818yescomplemented
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C2×C4×C8—22no—
125#391—44no—
126#384—22no—
127#384—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C2×#95—24no—
131C2×#95—21no—
132C2×#95—24no—
133#340—24no—
134#340—28no—
135#340—28no—
136#340—24no—
137#336—28no—
138C25—21no—
139C22×Q8—28no—
140C22×Q8—28no—
141C22×Q8—264no—
142C22×Q8—28no—
143C22×Q8—24no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—24no—
147C22×Q8—24no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—44no—
151C22×Q8—44no—
152C22×Q8—432no—
153C22×Q8—432no—
154C22×Q8—432no—
155C22×Q8—432no—
156C22×Q8—432no—
157C22×Q8—432no—
158C22×Q8—44no—
159C22×Q8—42no—
160C22×Q8—44no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8C22⋊C414yes—
165C22×Q8D814yes—
166C22×Q8SD1614yes—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4C22×C412yes—
173C22×D4C2×D412yes—
174C22×D4C2×D412yes—
175C22×D4C2×Q814yes—
176C22×D4#4114yes—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—22no—
180C23×C4—21no—
181C23×C4—28no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4C2×D412yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2411yescharacteristic
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—24no—
210C22×C8—24no—
211C22×C8C2×Q814yescomplemented
212C4⋊Q8—216no—
213C4⋊Q8—216no—
214C4⋊Q8—216no—
215C4⋊Q8—216no—
216C4⋊Q8—28no—
217C4⋊Q8—28no—
218C4⋊Q8—28no—
219C4⋊Q8C22⋊C4116yes—
220C22⋊Q8—264no—
221C22⋊Q8—2128no—
222C4×Q8—28no—
223C4×Q8—28no—
224C4×Q8—28no—
225C4×Q8—28no—
226C4×D4—216no—
227C4×D4—216no—
228C4×D4—216no—
229C4×D4—216no—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—28no—
233C4×D4C22⋊C4116yes—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4C22⋊C414yes—
265C2×C4⋊C4C22⋊C418yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×Q814yescomplemented
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42C22×C414yes—
287C2×C42C22×C412yes—
288C2×C42C2×D414yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C4⋊C8—44no—
295#102—216no—
296#102—48no—
297C4×C8—44no—
298#95—42no—
299#95—48no—
300#95—84no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—24no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—41no—
310C24—42no—
311C24—42no—
312C24C22×D411yescharacteristic
313C2×Q8—232no—
314C2×Q8—28no—
315C2×Q8—216no—
316C2×Q8—28no—
317C2×Q8—28no—
318C2×Q8—28no—
319C2×Q8—28no—
320C2×Q8—28no—
321C2×Q8—416no—
322C2×Q8—416no—
323C2×Q8—416no—
324C2×Q8—416no—
325C2×Q8—416no—
326C2×Q8—4256no—
327C2×Q8—416no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—48no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8#102116yescomplemented
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4C2×C22⋊C414yes—
349C2×D4C4×Q818yes—
350C2×D4C22⋊Q818yes—
351C2×D4#12414yes—
352C2×D4C4⋊Q814yes—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—232no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—216no—
371C22×C4—24no—
372C22×C4—416no—
373C22×C4—416no—
374C22×C4—416no—
375C22×C4—416no—
376C22×C4—416no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—416no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4C2×C22⋊C414yes—
404C22×C4C2×C22⋊C412yes—
405C22×C4C2×C22⋊C414yes—
406C22×C4C2×C22⋊C412yes—
407C22×C4#12012yes—
408C22×C4#12012yes—
409C22×C4#12012yes—
410C22×C4#12012yes—
411C22×C4C2×D814yes—
412C22×C4C2×D812yes—
413C22×C4C2×SD1614yes—
414C22×C4C2×SD1612yes—
415C22×C4AGL(1,ℤ/8)12yes—
416C22×C4AGL(1,ℤ/8)12yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×Q812yes—
424C22×C4C2×#4112yes—
425C22×C42−1+412yes—
426C22×C42−1+412yes—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—44no—
438C2×C8—44no—
439C4⋊C4—232no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—216no—
447C4⋊C4—216no—
448C4⋊C4—28no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4C2×C22⋊C418yes—
459C4⋊C4C2×C22⋊C414yes—
460C22⋊C4—232no—
461C22⋊C4—216no—
462C42—24no—
463C42—24no—
464C42—24no—
465C42—24no—
466C42—84no—
467C42C2×C22⋊C414yes—
468C42C2×C22⋊C414yes—
469C42#12014yes—
470C42#12014yes—
471C23—28no—
472C23—28no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23C22×C22⋊C411yescharacteristic
495C23C2×#12011yescharacteristic
496C23C2×#12011yescharacteristic
497C23#54812yes—
498C23D4×Q812yes—
499C23C22×D811yescharacteristic
500C23C22×SD1611yescharacteristic
501C23C2×AGL(1,ℤ/8)11yescharacteristic
502C23C2×AGL(1,ℤ/8)11yescharacteristic
503C23#57712yes—
504C23#57812yes—
505Q8—232no—
506Q8—464no—
507Q8—432no—
508D4—24no—
509D4—28no—
510D4—28no—
511D4—24no—
512D4—24no—
513D4#39118yescomplemented
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—464no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—432no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4#37914yes—
558C2×C4#37914yes—
559C2×C4C2×#10218yes—
560C2×C4C2×#10214yes—
561C2×C4#41814yes—
562C2×C4#44714yes—
563C2×C4#44714yes—
564C2×C4#45014yes—
565C2×C4#45014yes—
566C2×C4C22×C22⋊C411yescharacteristic
567C2×C4C22×C22⋊C411yescharacteristic
568C2×C4C2×C4×Q812yes—
569C2×C4#51912yes—
570C2×C4C2×#12011yescharacteristic
571C2×C4C2×#12011yescharacteristic
572C2×C4C2×C22⋊Q812yes—
573C2×C4C2×#12411yescharacteristic
574C2×C4C2×C4⋊Q811yescharacteristic
575C2×C4#53612yes—
576C2×C4#53611yescharacteristic
577C2×C4#53611yescharacteristic
578C2×C4#54812yes—
579C2×C4#54812yes—
580C2×C4#54812yes—
581C2×C4D4×Q812yes—
582C2×C4D4×Q812yes—
583C2×C4D4×Q812yes—
584C8—44no—
585C8—82no—
586C22—28no—
587C22—48no—
588C22—48no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—48no—
596C22—48no—
597C22C2×#37911yescharacteristic
598C22Q8×C22⋊C412yes—
599C22#240912yes—
600C22#243411yescharacteristic
601C22#243611yescharacteristic
602C22C22×#10211yescharacteristic
603C22C2×#41811yescharacteristic
604C22#289812yes—
605C22C2×#44711yescharacteristic
606C22C2×#44711yescharacteristic
607C22C2×#45011yescharacteristic
608C22C2×#45011yescharacteristic
609C22#301011yescharacteristic
610C22#301111yescharacteristic
611C22#301312yes—
612C22#335812yes—
613C22#335912yes—
614C22#336712yes—
615C22#337312yes—
616C22Q8×D812yes—
617C22Q8×SD1612yes—
618C22#338412yes—
619C22#338612yes—
620C4—28no—
621C4—24no—
622C4—48no—
623C4—44no—
624C4—44no—
625C4—84no—
626C4—162no—
627C4#178618yes—
628C4C2×#39111yescharacteristic
629C4#231011yescharacteristic
630C4Q8×C22⋊C412yes—
631C4#240912yes—
632C4#243411yescharacteristic
633C4#243611yescharacteristic
634C2—48no—
635C2#1392211yescharacteristic
636C2C2×#178611yescharacteristic
637C2#2060311yescharacteristic
638C2Q8×#10212yes—
639C2#2252312yes—
640C2#2432612yes—
641C2#2433012yes—
642C2#2583411yescharacteristic
643C2#2583811yescharacteristic
644C2#2592911yescharacteristic
645C2#2593311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1990C2semidirect
3. C2×#1988C2semidirect
4. C2×#1984C2semidirect
7. #13926C2semidirect
8. #11384C2semidirect
16. C2×D4×Q8C4semidirect
121. C2×#102Q8semidirect
211. C22×C8C2×Q8semidirect
272. C2×C4⋊C4C2×Q8semidirect
334. C2×Q8#102semidirect
513. D4#391semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Q82
C2×#19902
C2×#19882
C2×#19842
C2×#18022
C2×C4×#1022
#139262
#113842
#107882