Finite Group Databasebeta

← #470230

Subgroups — #470230

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic, complemented
3C2×#1928C211yesmaximal, characteristic, complemented
4#15177C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11200C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10970C212yesmaximal, complemented
10#10964C211yesmaximal, characteristic
11#10807C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C4×D4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21#2563—24no—
22#2563—24no—
23C2×#393—22no—
24C2×#391C2211yescharacteristic
25C2×#390—21no—
26C2×#382—21no—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2024—24no—
32#1928—24no—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1444—21no—
42#1439—22no—
43#1438—21no—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×D4D411yescharacteristic
48C24×C4C2312yes—
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8D414yes—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
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—42no—
65C2×C4×D4—42no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D412yes—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C2×#107—22no—
82C2×#107—42no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—42no—
93C2×C22⋊C8—42no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97#393—216no—
98#391—22no—
99#391—22no—
100#390—28no—
101#382—28no—
102#380—24no—
103#380—24no—
104#380—24no—
105#380—24no—
106#380—24no—
107#380—42no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4D414yes—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—21no—
118#340—22no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#340—42no—
124#336—22no—
125#336—22no—
126C25C2×D412yes—
127C22×Q8—24no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—44no—
138C22×Q8—81no—
139C22×Q8—82no—
140C22×Q8—81no—
141C22×Q8#4112yes—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4Q1614yes—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
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×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8#4112yes—
188C4⋊Q8—48no—
189C4⋊Q8—44no—
190C4⋊Q8—44no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193C22⋊Q8—48no—
194C22⋊Q8—48no—
195C4×D4—48no—
196C4×D4—44no—
197C4×D4—44no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
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—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4#4114yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4—82no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—44no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252#107—44no—
253#103—28no—
254#103—48no—
255#103—44no—
256C22⋊C8—48no—
257C22⋊C8—48no—
258#95—28no—
259#95—28no—
260#95—24no—
261#95—44no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—81no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24C2×Q1614yes—
281C24C22×D411yescharacteristic
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—48no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—42no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—88no—
304C2×Q8—88no—
305C2×D4—28no—
306C2×D4—48no—
307C2×D4—44no—
308C2×D4—48no—
309C2×D4—44no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
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—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4#12012yes—
386C22×C4#12012yes—
387C22×C4C2×Q1612yes—
388C22×C4#13712yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4112yes—
394C22×C4H(5,𝔽2)12yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C4H(5,𝔽2)12yes—
397C22×C4H(5,𝔽2)11yescharacteristic
398C22×C42−1+411yescharacteristic
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—416no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C22⋊C4—48no—
438C22⋊C4—416no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C42—24no—
443C42—22no—
444C42—22no—
445C42—48no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
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—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23C2×#12011yescharacteristic
482C23#53412yes—
483C23#54612yes—
484C23#54611yescharacteristic
485C23#54711yescharacteristic
486C23C22×Q1611yescharacteristic
487C23C2×#13711yescharacteristic
488C23#57512yes—
489C23#57611yescharacteristic
490C23#57712yes—
491C23#57711yescharacteristic
492Q8—42no—
493Q8—42no—
494Q8—44no—
495Q8—816no—
496Q8—81no—
497Q8—81no—
498Q8—82no—
499D4—416no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—81no—
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—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4#45114yes—
559C2×C4#52511yescharacteristic
560C2×C4#53211yescharacteristic
561C2×C4#53412yes—
562C2×C4#53511yescharacteristic
563C2×C4#53511yescharacteristic
564C2×C4#53511yescharacteristic
565C2×C4#53711yescharacteristic
566C2×C4#53812yes—
567C2×C4#53812yes—
568C2×C4#53812yes—
569C2×C4#54212yes—
570C2×C4#54811yescharacteristic
571C2×C4#55012yes—
572C2×C4#55011yescharacteristic
573C2×C4#55112yes—
574C2×C4#55512yes—
575C2×C4H(3,𝔽4)12yes—
576C2×C4#56212yes—
577C8—42no—
578C8—81no—
579C8—82no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22#240711yescharacteristic
596C22#243111yescharacteristic
597C22#261311yescharacteristic
598C22#268412yes—
599C22#268612yes—
600C22C2×#45111yescharacteristic
601C22#300511yescharacteristic
602C22#300911yescharacteristic
603C22#301211yescharacteristic
604C22#318912yes—
605C22#319612yes—
606C22#319812yes—
607C22#320012yes—
608C22#329911yescharacteristic
609C22#329912yes—
610C22#331912yes—
611C22#332112yes—
612C22#332112yes—
613C22#333112yes—
614C22#333311yescharacteristic
615C22#333811yescharacteristic
616C22#334812yes—
617C22#335611yescharacteristic
618C4—24no—
619C4—42no—
620C4—82no—
621C4—82no—
622C4—84no—
623C4—82no—
624C4—84no—
625C4—82no—
626C4—161no—
627C4#261011yescharacteristic
628C4#263111yescharacteristic
629C4#266211yescharacteristic
630C4#273012yes—
631C4#273312yes—
632C4#281911yescharacteristic
633C2—84no—
634C2#1668011yescharacteristic
635C2#2412211yescharacteristic
636C2#2413311yescharacteristic
637C2#2557311yescharacteristic
638C2#2557811yescharacteristic
639C2#2959711yescharacteristic
640C2#2960411yescharacteristic
641C2#3126312yes—
642C2#3127912yes—
643C2#3132412yes—
644C2#3134712yes—
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
2. C2×#2024C2semidirect
3. C2×#1928C2semidirect
5. #11384C2semidirect
6. #11200C2semidirect
7. #11096C2semidirect
8. #10970C2semidirect
9. #10970C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#20242
C2×#19282
#151772
#113842
#112002
#110962
#109702
#109642
#108072