Finite Group Databasebeta

← #466743

Subgroups — #466743

Order 512.

1622 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2056C211yesmaximal, characteristic, complemented
2C2×#2036C212yesmaximal, complemented
3C2×#2001C212yesmaximal, complemented
4C2×#1954C211yesmaximal, characteristic
5#15177C212yesmaximal
6#11232C211yesmaximal, characteristic, complemented
7#11200C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10926C212yesmaximal, complemented
10C23×Q16C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×C22⋊Q8C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#391C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25#2056—24no—
26#2036—28no—
27#2001—28no—
28#1954—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34#1444—22no—
35#1439—22no—
36#1439—22no—
37#1438—22no—
38C4⋊#95C2212yes—
39C23×Q8C2312yes—
40C23×D4D411yescharacteristic
41C24×C4C2312yes—
42C22×Q16—24no—
43C22×Q16—22no—
44C22×Q16—42no—
45C22×Q16—41no—
46C22×Q16—41no—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—42no—
55C2×C22⋊Q8—42no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8—42no—
59C2×C22⋊Q8—42no—
60C2×C22⋊Q8—42no—
61C2×C22⋊Q8—42no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—41no—
65C2×C4×D4—41no—
66C2×C4×D4—41no—
67C2×C4×D4—41no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D412yes—
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—41no—
82C2×C4⋊C8—41no—
83C2×C4⋊C8—41no—
84C2×C4⋊C8D414yes—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103D418yescomplemented
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100C2×C4×C8—22no—
101C2×C4×C8—41no—
102C2×C4×C8—41no—
103#391—24no—
104#391—24no—
105#380—24no—
106#380—24no—
107#380—42no—
108#380—42no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115#340—22no—
116#340—24no—
117#340—24no—
118#340—41no—
119#340—42no—
120C25C2×D412yes—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8C2×D414yes—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—41no—
138C22×D4—41no—
139C22×D4—41no—
140C22×D4—41no—
141C22×D4Q1614yescomplemented
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—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
164C2×Q16—48no—
165C2×Q16—44no—
166C2×Q16—44no—
167C2×Q16—48no—
168C2×Q16—44no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8C2×D412yes—
184C22×C8C2×D414yes—
185C22×C8C2×D412yes—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C22⋊Q8—48no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193C22⋊Q8—48no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—44no—
198C2×C4⋊C4—22no—
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⋊C4—22no—
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—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4#4114yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42C2×D412yes—
245C2×C42#4112yes—
246#107—44no—
247C4⋊C8—44no—
248#103—48no—
249#103—48no—
250#103—48no—
251C22⋊C8—48no—
252C22⋊C8—48no—
253C4×C8—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—82no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24C2×Q1614yes—
275C24C22×D411yescharacteristic
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×D4—28no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4#12112yes—
383C22×C4#12112yes—
384C22×C4C2×Q1612yes—
385C22×C4#13512yes—
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C4H(5,𝔽2)12yes—
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C4H(5,𝔽2)12yes—
396Q16—416no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—42no—
411C2×C8—41no—
412C2×C8—42no—
413C2×C8—41no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—44no—
419C4⋊C4—416no—
420C4⋊C4—416no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C22⋊C4—48no—
430C22⋊C4—416no—
431C22⋊C4—416no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23C2×#12111yescharacteristic
474C23#53411yescharacteristic
475C23#53411yescharacteristic
476C23D4212yes—
477C23#54612yes—
478C23C22×Q1611yescharacteristic
479C23C2×#13511yescharacteristic
480C23#57511yescharacteristic
481C23#57511yescharacteristic
482C23#57712yes—
483C23#57812yes—
484Q8—44no—
485Q8—44no—
486Q8—44no—
487Q8—44no—
488Q8—82no—
489Q8—82no—
490Q8—82no—
491Q8—82no—
492D4—416no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#46714yes—
548C2×C4#53212yes—
549C2×C4#53412yes—
550C2×C4#53411yescharacteristic
551C2×C4#53512yes—
552C2×C4#53511yescharacteristic
553C2×C4#53811yescharacteristic
554C2×C4#53811yescharacteristic
555C2×C4#53812yes—
556C2×C4#54012yes—
557C2×C4#54012yes—
558C2×C4D4212yes—
559C2×C4D4211yescharacteristic
560C2×C4#54612yes—
561C2×C4#55012yes—
562C2×C4#55511yescharacteristic
563C2×C4#56011yescharacteristic
564C2×C4#56012yes—
565C2×C4H(3,𝔽4)11yescharacteristic
566C8—42no—
567C8—42no—
568C8—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22#242812yes—
585C22#261711yescharacteristic
586C22#267812yes—
587C22#267911yescharacteristic
588C22#269911yescharacteristic
589C22C2×#46711yescharacteristic
590C22#305411yescharacteristic
591C22#306312yes—
592C22#318911yescharacteristic
593C22#318911yescharacteristic
594C22#319111yescharacteristic
595C22#319111yescharacteristic
596C22#320112yes—
597C22#320211yescharacteristic
598C22#320411yescharacteristic
599C22#327512yes—
600C22#327712yes—
601C22#328412yes—
602C22D4×Q1612yes—
603C22#329912yes—
604C22#330212yes—
605C22#332312yes—
606C22#332712yes—
607C4—24no—
608C4—42no—
609C4—84no—
610C4—84no—
611C4—82no—
612C4—84no—
613C4—162no—
614C4#255511yescharacteristic
615C4#266112yes—
616C4#266911yescharacteristic
617C4#267312yes—
618C4#273311yescharacteristic
619C4#282811yescharacteristic
620C2—84no—
621C2#1655111yescharacteristic
622C2#2538612yes—
623C2#2539112yes—
624C2#2973111yescharacteristic
625C2#2974611yescharacteristic
626C2#3101912yes—
627C2#3102912yes—
628C2#3105911yescharacteristic
629C2#3106211yescharacteristic
630C2#3172211yescharacteristic
631C2#3172311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2056C2semidirect
2. C2×#2036C2semidirect
3. C2×#2001C2semidirect
6. #11232C2semidirect
7. #11200C2semidirect
8. #10970C2semidirect
9. #10926C2semidirect
93. C2×#103D4semidirect
141. C22×D4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#20562
C2×#20362
C2×#20012
C2×#19542
#151772
#112322
#112002
#109702
#109262