Finite Group Databasebeta

← #404024

Subgroups — #404024

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2347C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic, complemented
3C2×#1981C211yesmaximal, characteristic, complemented
4C2×#1873C212yesmaximal
5#15178C211yesmaximal, characteristic
6#11227C211yesmaximal, characteristic, complemented
7#11067C212yesmaximal, complemented
8#11042C211yesmaximal, characteristic
9#10970C212yesmaximal
10#10932C212yesmaximal, complemented
11#10807C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#117C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2212yes—
21#2347—24no—
22#2347—24no—
23C2×#391C2212yes—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388—21no—
27C2×#388—21no—
28C2×#383—21no—
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#2043—24no—
35#1981—24no—
36#1873—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1444—21no—
44#1444—21no—
45#1439—21no—
46#1439—21no—
47C4⋊#95C2211yescharacteristic
48C23×Q8C2312yes—
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—42no—
61C2×C22⋊Q8—42no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117D414yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×C4⋊C8—22no—
89C2×C4⋊C8—42no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103—42no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—42no—
104C2×C22⋊C8—42no—
105#391—24no—
106#391—42no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389D414yes—
111#388—28no—
112#388—28no—
113#383—28no—
114#380—22no—
115#380—22no—
116C4×C4⋊C4—28no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4D414yes—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95D411yescharacteristic
133C2×#95D411yescharacteristic
134#340—24no—
135#340—24no—
136#340—24no—
137#340—42no—
138#336—22no—
139#336—42no—
140C25C2×D411yescharacteristic
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×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8#4114yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4#4112yes—
176C23×C4C2411yescharacteristic
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—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C4⋊Q8—28no—
192C4⋊Q8—48no—
193C4⋊Q8—44no—
194C22⋊Q8—216no—
195C22⋊Q8—48no—
196C22⋊Q8—48no—
197C22⋊Q8—48no—
198#117—28no—
199#117—28no—
200#117—28no—
201#117—28no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
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—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4#4114yes—
239C2×C4⋊C4#4112yes—
240C2×C4⋊C4#4112yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—82no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—82no—
267C2×C42—82no—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C4⋊C8—44no—
271#103—48no—
272#103—48no—
273#103—48no—
274#103—48no—
275C22⋊C8—48no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—22no—
282#95—22no—
283#95—44no—
284#95—44no—
285#95—42no—
286#95—42no—
287C24—21no—
288C24—21no—
289C24—22no—
290C24—22no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—22no—
296C24—22no—
297C24#12112yes—
298C24C22×D411yescharacteristic
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—42no—
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—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
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—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4#12012yes—
401C22×C4#12012yes—
402C22×C4#12112yes—
403C22×C4#13512yes—
404C22×C4#13712yes—
405C22×C4#13712yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4112yes—
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4112yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C4H(5,𝔽2)12yes—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—48no—
442C4⋊C4—416no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—48no—
451C4⋊C4#12114yes—
452C4⋊C4#12114yes—
453C22⋊C4—216no—
454C22⋊C4—44no—
455C22⋊C4—416no—
456C22⋊C4—44no—
457C22⋊C4—84no—
458C42—216no—
459C42—22no—
460C42—22no—
461C42—24no—
462C42—41no—
463C42—41no—
464C42—42no—
465C42—84no—
466C42—84no—
467C23—22no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23C2×#12011yescharacteristic
491C23C2×#12111yescharacteristic
492C23#52511yescharacteristic
493C23#54612yes—
494C23#54612yes—
495C23C2×#13511yescharacteristic
496C23C2×#13711yescharacteristic
497C23C2×#13711yescharacteristic
498C23#57511yescharacteristic
499C23#57712yes—
500C23#57812yes—
501Q8—44no—
502Q8—44no—
503Q8—44no—
504Q8—44no—
505Q8—82no—
506Q8—82no—
507Q8—82no—
508Q8—82no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4#45214yes—
576C2×C4C2×#12111yescharacteristic
577C2×C4C2×#12111yescharacteristic
578C2×C4#52511yescharacteristic
579C2×C4#52511yescharacteristic
580C2×C4#52512yes—
581C2×C4#52511yescharacteristic
582C2×C4#52912yes—
583C2×C4#53211yescharacteristic
584C2×C4#53511yescharacteristic
585C2×C4#53511yescharacteristic
586C2×C4#53512yes—
587C2×C4#54012yes—
588C2×C4#54612yes—
589C2×C4#55012yes—
590C2×C4#55112yes—
591C2×C4#55112yes—
592C2×C4#55312yes—
593C2×C4#55512yes—
594C8—82no—
595C8—82no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—44no—
600C22—44no—
601C22—82no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22#240411yescharacteristic
608C22#241112yes—
609C22#243111yescharacteristic
610C22#247612yes—
611C22#284711yescharacteristic
612C22C2×#45211yescharacteristic
613C22#300711yescharacteristic
614C22#300911yescharacteristic
615C22#301111yescharacteristic
616C22#303511yescharacteristic
617C22#303911yescharacteristic
618C22#304612yes—
619C22#310711yescharacteristic
620C22#311611yescharacteristic
621C22#312012yes—
622C22#330112yes—
623C22#330212yes—
624C22#330712yes—
625C22#331312yes—
626C22#332012yes—
627C22#332012yes—
628C22#332812yes—
629C22#332812yes—
630C4—24no—
631C4—42no—
632C4—44no—
633C4—44no—
634C4—42no—
635C4—42no—
636C4—84no—
637C4—82no—
638C4—84no—
639C4—81no—
640C4—82no—
641C4#239211yescharacteristic
642C4#241412yes—
643C4#247112yes—
644C4#259811yescharacteristic
645C4#261011yescharacteristic
646C4#261911yescharacteristic
647C2—82no—
648C2#1488611yescharacteristic
649C2#2396411yescharacteristic
650C2#2398011yescharacteristic
651C2#2448512yes—
652C2#2449612yes—
653C2#2557011yescharacteristic
654C2#2558211yescharacteristic
655C2#2740112yes—
656C2#2742012yes—
657C2#3290711yescharacteristic
658C2#3291611yescharacteristic
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×#2043C2semidirect
3. C2×#1981C2semidirect
6. #11227C2semidirect
7. #11067C2semidirect
10. #10932C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23472
C2×#20432
C2×#19812
C2×#18732
#151782
#112272
#110672
#110422
#109702
#109322
#108072