Finite Group Databasebeta

← #403711

Subgroups — #403711

Order 512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic
4#14714C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11325C212yesmaximal
7#11207C211yesmaximal, characteristic
8#11207C211yesmaximal, characteristic
9#10908C212yesmaximal
10#10907C212yesmaximal
11#10795C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C4×#125—24no—
17C4×#125—24no—
18C2×#391—21no—
19C2×#391—21no—
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#387—21no—
23C2×#387—21no—
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#2058—24no—
32#2057—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C4⋊#95—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125Q814yescomplemented
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106—42no—
88#391—44no—
89#391—44no—
90#389—22no—
91#389—22no—
92#389D414yes—
93#389D414yes—
94#387—28no—
95#387—28no—
96#384—42no—
97#382—22no—
98#382—22no—
99C4×C4⋊C4—28no—
100C4×C4⋊C4—28no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C43—22no—
116C43—22no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#336—22no—
124#336—22no—
125#336—24no—
126#336Q818yescomplemented
127C22×Q8—24no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—42no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8C2×Q814yes—
160C22×C8C2×Q814yes—
161C4⋊Q8—48no—
162C4⋊Q8—48no—
163#125—28no—
164#125—28no—
165#125—28no—
166#125—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
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—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×Q812yes—
216C2×C4⋊C4C2×Q812yes—
217C2×C4⋊C4#4114yes—
218C2×C4⋊C4#4112yes—
219C2×C4⋊C4#4112yes—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×Q812yes—
250C2×C42C2×Q814yes—
251C2×C42#4112yes—
252#107—48no—
253#106—48no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264C24C22×D411yescharacteristic
265C2×Q8—48no—
266C2×Q8—48no—
267C2×Q8—48no—
268C2×Q8—84no—
269C2×Q8—84no—
270C2×Q8—84no—
271C2×Q8—84no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4#12012yes—
345C22×C4#12012yes—
346C22×C4C22⋊Q812yes—
347C22×C4C22⋊Q812yes—
348C22×C4C4⋊Q812yes—
349C22×C4C4⋊Q812yes—
350C22×C4#13512yes—
351C22×C4#13512yes—
352C22×C4AGL(1,ℤ/8)12yes—
353C22×C4AGL(1,ℤ/8)12yes—
354C22×C4#13712yes—
355C22×C4#13712yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×Q811yescharacteristic
359C22×C4C22×Q812yes—
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4112yes—
364C22×C4H(5,𝔽2)12yes—
365C22×C42−1+412yes—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C4⋊C4—216no—
381C4⋊C4—216no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—48no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C4⋊C4C22⋊Q814yes—
406C4⋊C4C22⋊Q814yes—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—24no—
412C42—24no—
413C42—24no—
414C42—24no—
415C42—44no—
416C42—44no—
417C42—48no—
418C42—48no—
419C42—44no—
420C42—44no—
421C23C2×#12011yescharacteristic
422C23C2×C22⋊Q811yescharacteristic
423C23#52511yescharacteristic
424C23#52511yescharacteristic
425C23C2×C4⋊Q811yescharacteristic
426C23#53511yescharacteristic
427C23#53611yescharacteristic
428C23C2×#13511yescharacteristic
429C23C2×#13511yescharacteristic
430C23C2×AGL(1,ℤ/8)11yescharacteristic
431C23C2×AGL(1,ℤ/8)11yescharacteristic
432C23C2×#13711yescharacteristic
433C23C2×#13711yescharacteristic
434C23#57511yescharacteristic
435C23#57511yescharacteristic
436Q8—816no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4#44914yes—
519C2×C4#45214yes—
520C2×C4#50114yes—
521C2×C4C2×C22⋊Q811yescharacteristic
522C2×C4C2×C22⋊Q811yescharacteristic
523C2×C4#52511yescharacteristic
524C2×C4#52511yescharacteristic
525C2×C4#53312yes—
526C2×C4#53312yes—
527C2×C4#54212yes—
528C2×C4#54212yes—
529C2×C4#54612yes—
530C2×C4#54612yes—
531C2×C4D4×Q812yes—
532C2×C4D4×Q812yes—
533C2×C4#55412yes—
534C2×C4#55612yes—
535C2×C4#55712yes—
536C2×C4Q8212yes—
537C8—44no—
538C8—44no—
539C22#239911yescharacteristic
540C22#240411yescharacteristic
541C22#243411yescharacteristic
542C22#256711yescharacteristic
543C22#258111yescharacteristic
544C22#263211yescharacteristic
545C22#284811yescharacteristic
546C22C2×#44911yescharacteristic
547C22C2×#45211yescharacteristic
548C22#300611yescharacteristic
549C22#300711yescharacteristic
550C22#307611yescharacteristic
551C22#307611yescharacteristic
552C22#307911yescharacteristic
553C22#307911yescharacteristic
554C22#310511yescharacteristic
555C22#310611yescharacteristic
556C22#310711yescharacteristic
557C22#310811yescharacteristic
558C22#311411yescharacteristic
559C22#311511yescharacteristic
560C22#311611yescharacteristic
561C22#311711yescharacteristic
562C22#316011yescharacteristic
563C22C2×#50111yescharacteristic
564C22#316511yescharacteristic
565C22#316611yescharacteristic
566C22#321111yescharacteristic
567C22#321111yescharacteristic
568C22#321711yescharacteristic
569C22#321811yescharacteristic
570C22#322811yescharacteristic
571C22#323111yescharacteristic
572C22#323411yescharacteristic
573C22#323611yescharacteristic
574C4—24no—
575C4—24no—
576C4—24no—
577C4—42no—
578C4—44no—
579C4—42no—
580C4—44no—
581C4—42no—
582C4—82no—
583C4—84no—
584C4—82no—
585C4—162no—
586C4#245112yes—
587C4#245112yes—
588C4#246512yes—
589C4#246512yes—
590C2#1488111yescharacteristic
591C2#2383811yescharacteristic
592C2#2384111yescharacteristic
593C2#2393611yescharacteristic
594C2#2393711yescharacteristic
595C2#2581211yescharacteristic
596C2#2581911yescharacteristic
597C2#2920911yescharacteristic
598C2#2921311yescharacteristic
599C2#2927811yescharacteristic
600C2#2928011yescharacteristic
601C2#3006611yescharacteristic
602C2#3006711yescharacteristic
603C2#3295511yescharacteristic
604C2#3295811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
59. C2×#125Q8semidirect
126. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1252
C2×#20582
C2×#20572
#147142
#113852
#113252
#112072
#109082
#109072
#107952