Finite Group Databasebeta

← #441160

Subgroups — #441160

Order 512.

1486 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2429C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#1956C211yesmaximal, characteristic, complemented
4#15150C211yesmaximal, characteristic
5#11190C212yesmaximal, complemented
6#11010C211yesmaximal, characteristic, complemented
7#10970C212yesmaximal
8#10970C212yesmaximal
9#10921C211yesmaximal, characteristic
10#10880C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2429—24no—
23#2429—24no—
24C2×#398—21no—
25C2×#397—21no—
26C2×#397—21no—
27C2×#396—21no—
28C2×#391C2211yescharacteristic
29C2×#389—21no—
30C2×#389C2211yescharacteristic
31C2×#388—21no—
32C2×#384C2211yescharacteristic
33C2×#380C2212yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35#2099—24no—
36#1956—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×Q8C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8D414yes—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—42no—
63C2×C22⋊Q8—42no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—24no—
67C2×C4×Q8—42no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—42no—
103C2×C4×C8—22no—
104C2×C4×C8—42no—
105#398—28no—
106#397—44no—
107#397—44no—
108#396—28no—
109#391—22no—
110#391—22no—
111#389—28no—
112#389—22no—
113#389D414yes—
114#388—28no—
115#384—22no—
116#384—22no—
117#380—42no—
118#380—42no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131#340—24no—
132#340—24no—
133#340—24no—
134#340—42no—
135#336—22no—
136#336—42no—
137C25C2×D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8C2×D412yes—
152C22×Q8C2×D412yes—
153C22×Q8#4114yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8#4114yes—
188C4⋊Q8—28no—
189C4⋊Q8—44no—
190C4⋊Q8—48no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195C22⋊Q8—44no—
196C22⋊Q8—44no—
197C4×Q8—48no—
198C4×Q8—44no—
199C4×Q8—44no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4#4114yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4SD1614yes—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—44no—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262#103—48no—
263#103—48no—
264#103—48no—
265#103—48no—
266C22⋊C8—48no—
267C4×C8—44no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—22no—
272#95—22no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—44no—
277#95—42no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—22no—
287C24—22no—
288C24C2×SD1612yes—
289C24C22×D411yescharacteristic
290C2×Q8—22no—
291C2×Q8—24no—
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8#12314yes—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
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—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4#12012yes—
391C22×C4#12012yes—
392C22×C4C2×SD1612yes—
393C22×C4C2×SD1612yes—
394C22×C4#13512yes—
395C22×C4AGL(1,ℤ/8)12yes—
396C22×C4#13712yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)12yes—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—48no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4#12414yes—
441C22⋊C4—28no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—44no—
445C22⋊C4—88no—
446C42—22no—
447C42—22no—
448C42—24no—
449C42—48no—
450C42—41no—
451C42—41no—
452C42—42no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23C2×#12011yescharacteristic
477C23#53211yescharacteristic
478C23#53511yescharacteristic
479C23#54612yes—
480C23#54612yes—
481C23C22×SD1611yescharacteristic
482C23C2×#13511yescharacteristic
483C23C2×AGL(1,ℤ/8)11yescharacteristic
484C23C2×#13711yescharacteristic
485C23#57812yes—
486C23#57812yes—
487Q8—28no—
488Q8—44no—
489Q8—44no—
490Q8—44no—
491Q8—44no—
492Q8—44no—
493Q8—82no—
494Q8—82no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
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—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#44814yes—
557C2×C4#44914yes—
558C2×C4C2×#12311yescharacteristic
559C2×C4C2×#12411yescharacteristic
560C2×C4#52912yes—
561C2×C4#53411yescharacteristic
562C2×C4#53511yescharacteristic
563C2×C4#53812yes—
564C2×C4#53812yes—
565C2×C4#54012yes—
566C2×C4#54612yes—
567C2×C4#54612yes—
568C2×C4#54612yes—
569C2×C4#55012yes—
570C2×C4#55312yes—
571C2×C4#55312yes—
572C2×C4#55312yes—
573C2×C4#56012yes—
574C8—44no—
575C8—82no—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—82no—
585C22—82no—
586C22—81no—
587C22#240511yescharacteristic
588C22#243111yescharacteristic
589C22#243511yescharacteristic
590C22#255912yes—
591C22#270312yes—
592C22C2×#44811yescharacteristic
593C22C2×#44911yescharacteristic
594C22#301111yescharacteristic
595C22#301111yescharacteristic
596C22#316711yescharacteristic
597C22#317712yes—
598C22#318211yescharacteristic
599C22#320811yescharacteristic
600C22#321011yescharacteristic
601C22#322312yes—
602C22#329812yes—
603C22#329812yes—
604C22#330012yes—
605C22#330212yes—
606C22#330812yes—
607C22#331212yes—
608C22#332812yes—
609C22#333012yes—
610C4—24no—
611C4—24no—
612C4—44no—
613C4—42no—
614C4—44no—
615C4—81no—
616C4—81no—
617C4—82no—
618C4—82no—
619C4—162no—
620C4—162no—
621C4#244412yes—
622C4#249012yes—
623C4#268812yes—
624C4#271412yes—
625C2—82no—
626C2#1567411yescharacteristic
627C2#2403511yescharacteristic
628C2#2404111yescharacteristic
629C2#2559811yescharacteristic
630C2#2560911yescharacteristic
631C2#2588211yescharacteristic
632C2#2588411yescharacteristic
633C2#2904512yes—
634C2#2906612yes—
635C2#3183612yes—
636C2#3184112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2099C2semidirect
3. C2×#1956C2semidirect
5. #11190C2semidirect
6. #11010C2semidirect
10. #10880C2semidirect
11. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24292
C2×#20992
C2×#19562
#151502
#111902
#110102
#109702
#109212
#108802
#107892