Finite Group Databasebeta

← #401670

Subgroups — #401670

Order 512.

1688 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×#2349C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#1864C212yesmaximal
4C2×#1802C211yesmaximal, characteristic
5#15151C211yesmaximal, characteristic, complemented
6#11228C211yesmaximal, characteristic, complemented
7#11220C212yesmaximal, complemented
8#11208C212yesmaximal, complemented
9#11202C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2212yes—
19C2×#398—21no—
20C2×#398—21no—
21#2349—24no—
22#2349—24no—
23C2×#389C2212yes—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#385—21no—
27C2×#385—21no—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C22×#95—21no—
33#2084—24no—
34#1864—28no—
35#1802—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1445—21no—
45#1445—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2211yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4—21no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—42no—
65C2×C4×D4—42no—
66C2×C4×D4D414yes—
67C2×C4×D4D414yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—24no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—42no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102Q818yes—
97#398—44no—
98#398—44no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—24no—
103#389—24no—
104#389D414yes—
105#385—28no—
106#385—28no—
107#384—22no—
108#384—22no—
109#382—22no—
110#382—22no—
111#380—28no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C2×#95—28no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95D411yescharacteristic
125C2×#95D411yescharacteristic
126#340—24no—
127#340—24no—
128#340—24no—
129#340—42no—
130#340—42no—
131#336—22no—
132#336—24no—
133#336—42no—
134C25—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4C2×D412yes—
141C22×D4C2×D412yes—
142C22×D4C2×Q814yes—
143C22×D4#4112yes—
144C22×D4#4114yes—
145C23×C4—21no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—82no—
162C23×C4—82no—
163C23×C4—82no—
164C23×C4—82no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
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×C8C2×Q814yes—
186C4×D4—28no—
187C4×D4—28no—
188C4×D4—28no—
189C4×D4—28no—
190C4×D4—28no—
191C4×D4—44no—
192C4×D4—48no—
193C4×D4—48no—
194C4×D4—44no—
195C4×D4—44no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
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—21no—
213C2×C4⋊C4—21no—
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—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×Q814yes—
240C2×C4⋊C4#4112yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42#4112yes—
262#106—48no—
263C4⋊C8—44no—
264#102—216no—
265#102—48no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—22no—
270#95—24no—
271#95—22no—
272#95—42no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—44no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—24no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—42no—
287C24—42no—
288C24C22×D411yescharacteristic
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4#12114yes—
303C2×D4C22⋊Q818yes—
304C2×D4#12314yes—
305C2×D4#12518yes—
306C2×D4C4⋊Q814yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—216no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—48no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—48no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—84no—
367C22×C4—88no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—88no—
376C22×C4#12012yes—
377C22×C4#12012yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4#13512yes—
381C22×C4AGL(1,ℤ/8)12yes—
382C22×C4AGL(1,ℤ/8)12yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×Q812yes—
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4C2×#4111yescharacteristic
391C22×C4H(5,𝔽2)12yes—
392C22×C42−1+412yes—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—44no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4—88no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4#12114yes—
432C22⋊C4—216no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—24no—
442C42—41no—
443C42—41no—
444C42—42no—
445C23—24no—
446C23—24no—
447C23—28no—
448C23—24no—
449C23—28no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23C2×#12011yescharacteristic
469C23C2×#12111yescharacteristic
470C23#52511yescharacteristic
471C23#54612yes—
472C23D4×Q812yes—
473C23C2×#13511yescharacteristic
474C23C2×AGL(1,ℤ/8)11yescharacteristic
475C23C2×AGL(1,ℤ/8)11yescharacteristic
476C23#57511yescharacteristic
477C23#57712yes—
478C23#57712yes—
479D4—24no—
480D4—24no—
481D4—28no—
482D4—28no—
483D4—24no—
484D4#39818yescomplemented
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—816no—
531C2×C4—816no—
532C2×C4—816no—
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—82no—
548C2×C4#44914yes—
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×#12111yescharacteristic
551C2×C4C2×C22⋊Q812yes—
552C2×C4C2×#12311yescharacteristic
553C2×C4#52511yescharacteristic
554C2×C4#52511yescharacteristic
555C2×C4C2×#12512yes—
556C2×C4C2×C4⋊Q811yescharacteristic
557C2×C4#53411yescharacteristic
558C2×C4#53611yescharacteristic
559C2×C4#53712yes—
560C2×C4#53912yes—
561C2×C4#54612yes—
562C2×C4D4×Q812yes—
563C2×C4#55212yes—
564C2×C4#55212yes—
565C2×C4#55412yes—
566C2×C4#55412yes—
567C8—44no—
568C8—82no—
569C22—28no—
570C22—44no—
571C22—44no—
572C22—48no—
573C22—48no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—48no—
578C22—44no—
579C22—48no—
580C22#240511yescharacteristic
581C22#241712yes—
582C22#243411yescharacteristic
583C22#247312yes—
584C22#284711yescharacteristic
585C22C2×#44911yescharacteristic
586C22#300611yescharacteristic
587C22#300911yescharacteristic
588C22#300911yescharacteristic
589C22#303511yescharacteristic
590C22#303811yescharacteristic
591C22#304212yes—
592C22#310611yescharacteristic
593C22#311511yescharacteristic
594C22#312312yes—
595C22#329612yes—
596C22#330612yes—
597C22#330612yes—
598C22#332312yes—
599C22#338112yes—
600C22#338412yes—
601C22#338412yes—
602C22#338812yes—
603C4—24no—
604C4—42no—
605C4—42no—
606C4—44no—
607C4—42no—
608C4—42no—
609C4—81no—
610C4—82no—
611C4—82no—
612C4—84no—
613C4—162no—
614C4C2×#39811yescharacteristic
615C4#241712yes—
616C4#247312yes—
617C4#248411yescharacteristic
618C4#258311yescharacteristic
619C4#261811yescharacteristic
620C2—48no—
621C2#1482111yescharacteristic
622C2#2403811yescharacteristic
623C2#2404811yescharacteristic
624C2#2482212yes—
625C2#2482912yes—
626C2#2583511yescharacteristic
627C2#2585111yescharacteristic
628C2#2730712yes—
629C2#2731812yes—
630C2#3290411yescharacteristic
631C2#3291411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
5. #15151C2semidirect
6. #11228C2semidirect
7. #11220C2semidirect
8. #11208C2semidirect
9. #11202C2semidirect
484. D4#398semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#20842
C2×#18642
C2×#18022
#151512
#112282
#112202
#112082
#112022
#107882