Finite Group Databasebeta

← #365759

Subgroups — #365759

Order 512.

1886 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1949C211yesmaximal, characteristic, complemented
4C2×#1836C211yesmaximal, characteristic, complemented
5#13905C211yesmaximal, characteristic
6#11096C212yesmaximal, complemented
7#11095C212yesmaximal, complemented
8#10849C211yesmaximal, characteristic, complemented
9#10730C212yesmaximal, complemented
10#10730C212yesmaximal, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×C4×D4—21no—
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2212yes—
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×#97C2211yescharacteristic
21D4×C4⋊C4—24no—
22D4×C4⋊C4—24no—
23C2×#389—21no—
24C2×#385—22no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1955C418yescomplemented
33#1949—24no—
34#1836—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1445—21no—
44#1438—21no—
45#1437—22no—
46C23×D4D411yescharacteristic
47C24×C4—22no—
48C24×C4C2312yes—
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×C4×D4—28no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—48no—
58C2×C4×D4C2×C414yes—
59C2×C4×D4Q814yes—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4—42no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D412yes—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C2×#107—22no—
84C2×#107—42no—
85C2×#106—22no—
86C2×#106—42no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8C2×C414yescomplemented
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8—42no—
93C2×C22⋊C8—42no—
94C2×C22⋊C8C2×C418yescomplemented
95C2×C22⋊C8C2×C418yescomplemented
96C2×#97—22no—
97C2×#97—22no—
98C2×#97C2×C414yescomplemented
99#389—28no—
100#385—216no—
101#384—22no—
102#384D414yes—
103#382—22no—
104#382—22no—
105#380—24no—
106#380—24no—
107#380—24no—
108#380—24no—
109#380—42no—
110#380—42no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4D414yes—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122#340—24no—
123#340—24no—
124#340—22no—
125#340—22no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340—42no—
130C25C2×D412yes—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C4⋊C414yes—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—28no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4C22×C414yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×Q814yes—
166C23×C4C2411yescharacteristic
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8C22×C414yes—
182C22×C8C22×C412yes—
183C22×C8C22×C412yes—
184C4×D4—28no—
185C4×D4—28no—
186C4×D4—28no—
187C4×D4—28no—
188C4×D4—432no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4—81no—
226C2×C4⋊C4C4⋊C414yes—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4#4114yes—
231C2×C4⋊C4#4114yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—44no—
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⋊C4C4⋊C418yes—
244C2×C42—24no—
245C2×C42—44no—
246C2×C42—42no—
247C2×C42—81no—
248C2×C42—81no—
249C2×C42C22×C412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×Q812yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255#107—44no—
256#106—44no—
257C4⋊C8—28no—
258C22⋊C8—48no—
259C22⋊C8—48no—
260#97—28no—
261#95—28no—
262#95—24no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—42no—
267#95—42no—
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×C4⋊C414yes—
281C24AGL(1,ℤ/8)14yes—
282C24#13714yes—
283C24C22×D411yescharacteristic
284C2×D4—28no—
285C2×D4—216no—
286C2×D4—28no—
287C2×D4—216no—
288C2×D4—28no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—416no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
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—416no—
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—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4—88no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4C2×C4⋊C414yes—
371C22×C4C2×C4⋊C412yes—
372C22×C4C2×C4⋊C412yes—
373C22×C4#12012yes—
374C22×C4#12012yes—
375C22×C4#12312yes—
376C22×C4#12312yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×Q811yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4112yes—
384C22×C4C2×#4112yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C4H(5,𝔽2)12yes—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—416no—
414C4⋊C4—48no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—84no—
427C4⋊C4#12414yes—
428C22⋊C4—216no—
429C22⋊C4—416no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C42—48no—
433C42—84no—
434C42C4×D414yes—
435C42C4×D414yes—
436C42C22⋊Q814yes—
437C42C22⋊Q814yes—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
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#42818yes—
473C23C22×C4⋊C411yescharacteristic
474C23C2×#12011yescharacteristic
475C23C2×#12311yescharacteristic
476C23#54612yes—
477C23#54612yes—
478C23C2×AGL(1,ℤ/8)12yes—
479C23C2×#13712yes—
480C23#57611yescharacteristic
481C23#57711yescharacteristic
482C23#57711yescharacteristic
483C23#57811yescharacteristic
484D4—232no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—832no—
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—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4#38514yes—
546C2×C4#38514yes—
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C4×#4112yes—
550C2×C4#51812yes—
551C2×C4C2×C22⋊Q811yescharacteristic
552C2×C4C2×C22⋊Q811yescharacteristic
553C2×C4C2×#12411yescharacteristic
554C2×C4#52912yes—
555C2×C4#53211yescharacteristic
556C2×C4#53511yescharacteristic
557C2×C4#53511yescharacteristic
558C2×C4#54012yes—
559C2×C4#54612yes—
560C2×C4#55012yes—
561C2×C4#55312yes—
562C2×C4#55412yes—
563C2×C4#55412yes—
564C2×C4#55512yes—
565C8—42no—
566C8—42no—
567C8—44no—
568C22—28no—
569C22—28no—
570C22—28no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22C2×#38511yescharacteristic
584C22#234912yes—
585C22#242012yes—
586C22#243111yescharacteristic
587C22#243511yescharacteristic
588C22C2×#42812yes—
589C22#291211yescharacteristic
590C22#291311yescharacteristic
591C22#300311yescharacteristic
592C22#300411yescharacteristic
593C22#301211yescharacteristic
594C22#301311yescharacteristic
595C22#309212yes—
596C22#309611yescharacteristic
597C22#309711yescharacteristic
598C22#330612yes—
599C22#330812yes—
600C22#331012yes—
601C22#331212yes—
602C22#331412yes—
603C22#331612yes—
604C22#331812yes—
605C22#332012yes—
606C4—24no—
607C4—24no—
608C4—44no—
609C4—44no—
610C4—84no—
611C4—84no—
612C4—82no—
613C4—161no—
614C4—161no—
615C4C4×#12411yescharacteristic
616C4#232811yescharacteristic
617C4#236312yes—
618C4#245412yes—
619C4#256311yescharacteristic
620C4#256611yescharacteristic
621C2—48no—
622C2#1397911yescharacteristic
623C2#2084411yescharacteristic
624C2#2084911yescharacteristic
625C2#2270912yes—
626C2#2271112yes—
627C2#2497712yes—
628C2#2497912yes—
629C2#2557711yescharacteristic
630C2#2557911yescharacteristic
631C2#2589111yescharacteristic
632C2#2589411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×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
3. C2×#1949C2semidirect
4. C2×#1836C2semidirect
6. #11096C2semidirect
7. #11095C2semidirect
8. #10849C2semidirect
9. #10730C2semidirect
10. #10730C2semidirect
11. #10703C2semidirect
32. #1955C4semidirect
89. C2×C4⋊C8C2×C4semidirect
94. C2×C22⋊C8C2×C4semidirect
95. C2×C22⋊C8C2×C4semidirect
98. C2×#97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#19552
C2×#19492
C2×#18362
#139052
#110962
#110952
#108492
#107302
#107032