Finite Group Databasebeta

← #399604

Subgroups — #399604

Order 512.

1784 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#1864C211yesmaximal, characteristic
4C2×#1862C212yesmaximal
5#15194C211yesmaximal, characteristic, complemented
6#11371C211yesmaximal, characteristic, complemented
7#11346C211yesmaximal, characteristic, complemented
8#11328C212yesmaximal, complemented
9#11220C212yesmaximal, complemented
10#10788C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2212yes—
14C23×C4⋊C4—21no—
15C22×#107C2212yes—
16C22×#106C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20D4×C4⋊C4—24no—
21D4×C4⋊C4—24no—
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#385—22no—
26C2×#384C2211yescharacteristic
27C2×#380—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2083—24no—
30#1864—24no—
31#1862—28no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—21no—
42#1444—21no—
43#1441—22no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—22no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4D414yes—
62C22×C4⋊C4—28no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—48no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—42no—
83C2×#106—22no—
84C2×#106—24no—
85C2×#102—24no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—24no—
89C2×#102—22no—
90C2×#102—24no—
91C2×#102D414yes—
92C2×#102Q818yes—
93#389—24no—
94#389—24no—
95#389—24no—
96#389—22no—
97#389—22no—
98#389—42no—
99#385—216no—
100#384—22no—
101#384D414yes—
102#380—28no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D412yes—
113#340—24no—
114#340—24no—
115#340—22no—
116#340—24no—
117#340—42no—
118#336—24no—
119#336—22no—
120#336—22no—
121#336—42no—
122C25—21no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4C2×D412yes—
129C22×D4C2×D412yes—
130C22×D4C2×Q814yes—
131C22×D4#4112yes—
132C22×D4#4114yes—
133C23×C4—28no—
134C23×C4—21no—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—28no—
140C23×C4—21no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—44no—
148C23×C4—82no—
149C23×C4—82no—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8C2×D412yes—
171C22×C8C2×Q814yes—
172C4×D4—216no—
173C4×D4—28no—
174C4×D4—28no—
175C4×D4—48no—
176C4×D4—48no—
177C4×D4—48no—
178C4×D4—48no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—432no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×Q814yes—
222C2×C4⋊C4#4114yes—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238#107—48no—
239#106—44no—
240#102—28no—
241#102—216no—
242#102—44no—
243#95—28no—
244#95—24no—
245#95—44no—
246#95—42no—
247#95—42no—
248#95—82no—
249#95—81no—
250#95—82no—
251C24—22no—
252C24—24no—
253C24—22no—
254C24—22no—
255C24—24no—
256C24—41no—
257C24—42no—
258C24—41no—
259C24—42no—
260C24—41no—
261C24—41no—
262C24C22×D411yescharacteristic
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4#12114yes—
277C2×D4C22⋊Q818yes—
278C2×D4#12314yes—
279C2×D4#12518yes—
280C2×D4C4⋊Q814yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—216no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—48no—
307C22×C4—416no—
308C22×C4—416no—
309C22×C4—416no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—48no—
314C22×C4—416no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
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—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4—88no—
339C22×C4—88no—
340C22×C4—88no—
341C22×C4#12012yes—
342C22×C4#12012yes—
343C22×C4#13512yes—
344C22×C4AGL(1,ℤ/8)14yes—
345C22×C4#13712yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×Q812yes—
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C42−1+412yes—
356C2×C8—22no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—42no—
372C2×C8—42no—
373C4⋊C4—28no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—48no—
383C4⋊C4—416no—
384C4⋊C4—42no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—84no—
395C4⋊C4—88no—
396C4⋊C4—84no—
397C4⋊C4#12314yes—
398C22⋊C4—216no—
399C22⋊C4—416no—
400C42—22no—
401C42—22no—
402C42—24no—
403C42—42no—
404C42—42no—
405C42—42no—
406C42—42no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—28no—
411C23—28no—
412C23—42no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—44no—
430C23C2×#12011yescharacteristic
431C23#52512yes—
432C23D4211yescharacteristic
433C23#54611yescharacteristic
434C23D4×Q812yes—
435C23C2×#13511yescharacteristic
436C23C2×AGL(1,ℤ/8)12yes—
437C23C2×#13711yescharacteristic
438C23#57612yes—
439C23#57711yescharacteristic
440C23#57811yescharacteristic
441D4—24no—
442D4—28no—
443D4—28no—
444D4—24no—
445D4—24no—
446D4#39818yes—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—432no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—832no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4#45214yes—
502C2×C4C2×#12111yescharacteristic
503C2×C4C2×C22⋊Q812yes—
504C2×C4C2×#12311yescharacteristic
505C2×C4C2×#12311yescharacteristic
506C2×C4#52512yes—
507C2×C4C2×#12512yes—
508C2×C4C2×C4⋊Q811yescharacteristic
509C2×C4#53511yescharacteristic
510C2×C4#53611yescharacteristic
511C2×C4#53912yes—
512C2×C4#53912yes—
513C2×C4D4211yescharacteristic
514C2×C4#54611yescharacteristic
515C2×C4#54712yes—
516C2×C4D4×Q812yes—
517C2×C4#55212yes—
518C2×C4#55412yes—
519C2×C4#55412yes—
520C8—44no—
521C8—42no—
522C8—81no—
523C22—28no—
524C22—48no—
525C22—48no—
526C22—48no—
527C22—44no—
528C22—48no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22#240811yescharacteristic
535C22#243411yescharacteristic
536C22#244912yes—
537C22#247312yes—
538C22#284911yescharacteristic
539C22C2×#45211yescharacteristic
540C22#300311yescharacteristic
541C22#300811yescharacteristic
542C22#301311yescharacteristic
543C22#310512yes—
544C22#311012yes—
545C22#311912yes—
546C22#312412yes—
547C22#326412yes—
548C22#327611yescharacteristic
549C22#328411yescharacteristic
550C22#330211yescharacteristic
551C22#331812yes—
552C22#333111yescharacteristic
553C22#338012yes—
554C22#338412yes—
555C22#338512yes—
556C22#338612yes—
557C4—24no—
558C4—44no—
559C4—44no—
560C4—42no—
561C4—84no—
562C4—82no—
563C4—82no—
564C4—84no—
565C4—161no—
566C4C2×#39811yescharacteristic
567C4#244912yes—
568C4#247312yes—
569C4#248611yescharacteristic
570C4#250211yescharacteristic
571C4#262311yescharacteristic
572C2—48no—
573C2#1478611yescharacteristic
574C2#2424111yescharacteristic
575C2#2424811yescharacteristic
576C2#2583911yescharacteristic
577C2#2584411yescharacteristic
578C2#2638212yes—
579C2#2638412yes—
580C2#2730112yes—
581C2#2731712yes—
582C2#3300911yescharacteristic
583C2#3301211yescharacteristic
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×#2083C2semidirect
5. #15194C2semidirect
6. #11371C2semidirect
7. #11346C2semidirect
8. #11328C2semidirect
9. #11220C2semidirect

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×#20832
C2×#18642
C2×#18622
#151942
#113712
#113462
#113282
#112202
#107882