Finite Group Databasebeta

← #400751

Subgroups — #400751

Order 512.

1708 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic, complemented
3C2×#2093C211yesmaximal, characteristic
4C2×#1863C212yesmaximal, complemented
5C2×#1864C212yesmaximal
6#14712C211yesmaximal, characteristic, complemented
7#11325C212yesmaximal, complemented
8#11050C211yesmaximal, characteristic, complemented
9#11049C211yesmaximal, characteristic
10#11009C212yesmaximal
11#10795C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C4×#124—24no—
22C4×#124—24no—
23C2×#391C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389—22no—
26C2×#389C2212yes—
27C2×#388—22no—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×C4×C22⋊C4—22no—
32C2×C43C2211yescharacteristic
33#2096—24no—
34#2093—24no—
35#1864—28no—
36#1863—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C4⋊#95—22no—
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#124—22no—
51C2×#124—22no—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124Q814yes—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—24no—
59C2×C4×Q8—42no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—24no—
63C2×C4×D4—42no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C42—21no—
76C22×C42D412yes—
77C22×C42D412yes—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—42no—
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—42no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—42no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—42no—
99#391—22no—
100#391D414yes—
101#390—22no—
102#390D414yes—
103#389—24no—
104#389—24no—
105#389—48no—
106#388—216no—
107#384—24no—
108#384—22no—
109#384—22no—
110#382—22no—
111#382—22no—
112C4×C22⋊C4—216no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—24no—
117C43—22no—
118C43—22no—
119#340—24no—
120#340—24no—
121#336—22no—
122#336—22no—
123#336—24no—
124#336D418yes—
125C25—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C2×D412yes—
134C22×Q8C2×Q812yes—
135C22×Q8#4112yes—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C2×D412yes—
144C22×D4C2×Q812yes—
145C22×D4#4112yes—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—42no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—44no—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8C2×D414yes—
174C22×C8C2×D414yes—
175#124—28no—
176#124—28no—
177#124—28no—
178#124—28no—
179C4×Q8—48no—
180C4×Q8—48no—
181C4×D4—48no—
182C4×D4—48no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
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—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4#4114yes—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D414yes—
244C2×C42C2×Q812yes—
245C2×C42#4112yes—
246#107—48no—
247#106—48no—
248#103—216no—
249#103—48no—
250#102—216no—
251#102—48no—
252#95—28no—
253#95—28no—
254#95—48no—
255#95—44no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24C22×D411yescharacteristic
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8C22⋊Q814yes—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4C22⋊Q814yes—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—48no—
324C22×C4—416no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—84no—
347C22×C4—88no—
348C22×C4—84no—
349C22×C4#12014yes—
350C22×C4C22⋊Q814yes—
351C22×C4#12712yes—
352C22×C4#12712yes—
353C22×C4#13514yes—
354C22×C4AGL(1,ℤ/8)14yes—
355C22×C4#13714yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C22×Q811yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4112yes—
363C22×C4C2×#4111yescharacteristic
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—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—416no—
385C4⋊C4—48no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C22⋊C4—232no—
397C22⋊C4—48no—
398C22⋊C4—416no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—24no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—48no—
408C42—48no—
409C42—48no—
410C23—24no—
411C23—44no—
412C23—44no—
413C23—42no—
414C23—44no—
415C23—44no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—42no—
423C23—82no—
424C23—82no—
425C23—81no—
426C23—81no—
427C23—81no—
428C23—81no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23C2×#12011yescharacteristic
434C23C2×C22⋊Q811yescharacteristic
435C23#52511yescharacteristic
436C23#52511yescharacteristic
437C23C2×#12711yescharacteristic
438C23#53511yescharacteristic
439C23#53611yescharacteristic
440C23C2×#13512yes—
441C23C2×AGL(1,ℤ/8)12yes—
442C23C2×#13712yes—
443C23#57512yes—
444Q8—28no—
445Q8—44no—
446Q8—44no—
447Q8—44no—
448D4—28no—
449D4—44no—
450D4—44no—
451D4—44no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—48no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—816no—
488C2×C4—84no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—84no—
501C2×C4—84no—
502C2×C4#44918yes—
503C2×C4#49714yes—
504C2×C4C2×C22⋊Q811yescharacteristic
505C2×C4C2×C22⋊Q811yescharacteristic
506C2×C4#52512yes—
507C2×C4#52512yes—
508C2×C4#52511yescharacteristic
509C2×C4#52511yescharacteristic
510C2×C4#53912yes—
511C2×C4#53912yes—
512C2×C4D4212yes—
513C2×C4D4212yes—
514C2×C4#54712yes—
515C2×C4#54712yes—
516C2×C4#54712yes—
517C2×C4D4×Q812yes—
518C2×C4#55412yes—
519C2×C4#55912yes—
520C8—44no—
521C8—44no—
522C22—48no—
523C22—44no—
524C22—84no—
525C22—84no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—84no—
531C22—82no—
532C22—82no—
533C22#239711yescharacteristic
534C22#240611yescharacteristic
535C22#243311yescharacteristic
536C22#254211yescharacteristic
537C22#258011yescharacteristic
538C22#263311yescharacteristic
539C22#284811yescharacteristic
540C22C2×#44912yes—
541C22#300712yes—
542C22#307612yes—
543C22#307912yes—
544C22#310712yes—
545C22#310812yes—
546C22#311412yes—
547C22#311512yes—
548C22#314711yescharacteristic
549C22C2×#49711yescharacteristic
550C22#315111yescharacteristic
551C22#315311yescharacteristic
552C22#320712yes—
553C22#321512yes—
554C22#323212yes—
555C22#323812yes—
556C4—28no—
557C4—24no—
558C4—42no—
559C4—44no—
560C4—44no—
561C4—82no—
562C4—82no—
563C4—84no—
564C4—162no—
565C4#244912yes—
566C4#244912yes—
567C4#245512yes—
568C4#245512yes—
569C2—84no—
570C2#1480711yescharacteristic
571C2#2379711yescharacteristic
572C2#2380611yescharacteristic
573C2#2406612yes—
574C2#2572011yescharacteristic
575C2#2573011yescharacteristic
576C2#2851311yescharacteristic
577C2#2851911yescharacteristic
578C2#2926212yes—
579C2#3008312yes—
580C2#3295512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
2. C2×#2096C2semidirect
4. C2×#1863C2semidirect
6. #14712C2semidirect
7. #11325C2semidirect
8. #11050C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
C2×#20962
C2×#20932
C2×#18632
C2×#18642
#147122
#113252
#110502
#110492
#110092
#107952