Finite Group Databasebeta

← #464876

Subgroups — #464876

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2094C211yesmaximal, characteristic
3C2×#2059C212yesmaximal, complemented
4C2×#2025C212yesmaximal
5C2×#1925C211yesmaximal, characteristic
6#15148C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic, complemented
8#11346C211yesmaximal, characteristic, complemented
9#11327C212yesmaximal, complemented
10#11066C212yesmaximal
11#10806C211yesmaximal, characteristic, complemented
12C22×#125C2212yes—
13C22×#107C2211yescharacteristic
14C22×#106C2212yes—
15C22×#102C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18#2552—24no—
19#2552—24no—
20C2×#399—21no—
21C2×#398—22no—
22C2×#390C2211yescharacteristic
23C2×#390C2212yes—
24C2×#389—21no—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#384—21no—
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2094—24no—
32#2059—28no—
33#2025—28no—
34#1925—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1439—22no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C23×D4C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C2×#125—24no—
52C2×#125—24no—
53C2×#125—24no—
54C2×#125—24no—
55C2×#125—42no—
56C2×#125—42no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C42—21no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—22no—
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—42no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—22no—
87C2×#102—24no—
88C2×#102—42no—
89C2×#102Q814yes—
90#399—44no—
91#398—216no—
92#390—24no—
93#390—22no—
94#390—22no—
95#390—42no—
96#389—28no—
97#389—24no—
98#389—24no—
99#389—24no—
100#389—42no—
101#384—28no—
102#384—22no—
103#384D414yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4D414yes—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95D412yes—
114#340—24no—
115#340—22no—
116#340—24no—
117#340—24no—
118#340D418yes—
119#336—24no—
120#336—24no—
121#336—22no—
122#336—22no—
123C25—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4C2×Q812yes—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8C2×D414yes—
160C22×C8C2×Q812yes—
161#125—48no—
162#125—48no—
163#125—48no—
164#125—48no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
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—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—81no—
202C2×C4⋊C4—81no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×Q812yes—
212C2×C4⋊C4#4112yes—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237#107—44no—
238#106—48no—
239#102—28no—
240#102—48no—
241#102—44no—
242#95—28no—
243#95—24no—
244#95—24no—
245#95—44no—
246#95—42no—
247#95—81no—
248C24—22no—
249C24—41no—
250C24—42no—
251C24—42no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—41no—
257C24—42no—
258C24—41no—
259C24C22×D411yescharacteristic
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—41no—
276C2×D4—42no—
277C2×D4—42no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4#12012yes—
330C22×C4#12012yes—
331C22×C4#13512yes—
332C22×C4AGL(1,ℤ/8)12yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D412yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×Q811yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4H(5,𝔽2)12yes—
339C22×C42−1+412yes—
340C22×C42−1+411yescharacteristic
341C22×C42−1+411yescharacteristic
342C22×C42−1+412yes—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—416no—
372C4⋊C4—416no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—84no—
382C4⋊C4—88no—
383C4⋊C4—88no—
384C4⋊C4C22⋊Q814yes—
385C4⋊C4#12714yes—
386C22⋊C4—84no—
387C22⋊C4—88no—
388C22⋊C4—88no—
389C22⋊C4—84no—
390C42—22no—
391C42—22no—
392C42—24no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—48no—
398C23—42no—
399C23—42no—
400C23—44no—
401C23—42no—
402C23—44no—
403C23—81no—
404C23—81no—
405C23—82no—
406C23—81no—
407C23—82no—
408C23—81no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23C2×#12011yescharacteristic
422C23#53612yes—
423C23D4212yes—
424C23#54811yescharacteristic
425C23D4×Q811yescharacteristic
426C23C2×#13511yescharacteristic
427C23C2×AGL(1,ℤ/8)11yescharacteristic
428C23#57512yes—
429C23#57611yescharacteristic
430C23#57711yescharacteristic
431C23#57712yes—
432D4—42no—
433D4—44no—
434D4—42no—
435D4—81no—
436D4—81no—
437D4—82no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4#44914yes—
493C2×C4C2×C22⋊Q811yescharacteristic
494C2×C4C2×#12711yescharacteristic
495C2×C4#53511yescharacteristic
496C2×C4#53611yescharacteristic
497C2×C4#53611yescharacteristic
498C2×C4#53611yescharacteristic
499C2×C4#53612yes—
500C2×C4#53612yes—
501C2×C4#53612yes—
502C2×C4#54212yes—
503C2×C4#54412yes—
504C2×C4D4212yes—
505C2×C4#54712yes—
506C2×C4#54811yescharacteristic
507C2×C4D4×Q811yescharacteristic
508C2×C4D4×Q812yes—
509C2×C4#56312yes—
510C2×C4#56312yes—
511C8—44no—
512C8—42no—
513C8—81no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—84no—
518C22—84no—
519C22—162no—
520C22—162no—
521C22—161no—
522C22—161no—
523C22—161no—
524C22—161no—
525C22#240911yescharacteristic
526C22#243311yescharacteristic
527C22#261411yescharacteristic
528C22#267412yes—
529C22#267712yes—
530C22C2×#44911yescharacteristic
531C22#300511yescharacteristic
532C22#301011yescharacteristic
533C22#301211yescharacteristic
534C22#323212yes—
535C22#323412yes—
536C22#324112yes—
537C22#324412yes—
538C22#326812yes—
539C22#327112yes—
540C22#327212yes—
541C22#328412yes—
542C22#336011yescharacteristic
543C22#336611yescharacteristic
544C22#337112yes—
545C22#338011yescharacteristic
546C22#338411yescharacteristic
547C22#338812yes—
548C4—24no—
549C4—44no—
550C4—44no—
551C4—42no—
552C4—84no—
553C4—82no—
554C4—82no—
555C4—84no—
556C4—161no—
557C4#254211yescharacteristic
558C4#262811yescharacteristic
559C4#265811yescharacteristic
560C4#267412yes—
561C4#267712yes—
562C4#282111yescharacteristic
563C2—162no—
564C2#1651111yescharacteristic
565C2#2432711yescharacteristic
566C2#2434011yescharacteristic
567C2#2575411yescharacteristic
568C2#2576511yescharacteristic
569C2#2964011yescharacteristic
570C2#2964711yescharacteristic
571C2#3081012yes—
572C2#3083512yes—
573C2#3095112yes—
574C2#3097512yes—
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
1. C2×#2552C2semidirect
3. C2×#2059C2semidirect
7. #11357C2semidirect
8. #11346C2semidirect
9. #11327C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#20942
C2×#20592
C2×#20252
C2×#19252
#151482
#113572
#113462
#113272
#110662
#108062