Finite Group Databasebeta

← #470375

Subgroups — #470375

Order 512.

1296 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2071C211yesmaximal, characteristic
3C2×#1926C211yesmaximal, characteristic
4#15148C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11327C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#11066C212yesmaximal
10#11049C212yesmaximal
11#10807C211yesmaximal, characteristic, complemented
12C22×#125—21no—
13C22×#106C2211yescharacteristic
14C22×#102C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17#2552—24no—
18#2552—24no—
19C2×#399—21no—
20C2×#398—22no—
21C2×#390C2211yescharacteristic
22C2×#390C2212yes—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#384C2212yes—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2071—24no—
31#1926—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1439—22no—
43#1439—21no—
44C4⋊#95C2211yescharacteristic
45C23×D4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#125—28no—
50C2×#125—48no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#106—22no—
72C2×#106—42no—
73C2×#102—22no—
74C2×#102—22no—
75C2×#102—22no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—22no—
79C2×#102—42no—
80C2×#102—42no—
81#399—44no—
82#398—216no—
83#390—24no—
84#390—22no—
85#390—22no—
86#390—42no—
87#389—28no—
88#389—24no—
89#389—24no—
90#389—24no—
91#389—42no—
92#384—24no—
93#384—24no—
94#384—22no—
95#384D414yes—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4D414yes—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95D412yes—
106#340—24no—
107#340—24no—
108#340—22no—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—22no—
114#336—42no—
115#336—42no—
116C25—21no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4#4112yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8#4112yes—
154#125—432no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4SD1614yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D414yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4#4114yes—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—44no—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225#106—44no—
226#102—28no—
227#102—48no—
228#102—44no—
229#95—28no—
230#95—24no—
231#95—24no—
232#95—44no—
233#95—42no—
234#95—81no—
235C24—22no—
236C24—41no—
237C24—42no—
238C24—42no—
239C24—41no—
240C24—41no—
241C24—42no—
242C24—41no—
243C24—42no—
244C24—41no—
245C24—41no—
246C24C22×D411yescharacteristic
247C2×D4—22no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—24no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4#12012yes—
313C22×C4#12012yes—
314C22×C4C2×SD1614yes—
315C22×C4C2×SD1612yes—
316C22×C4AGL(1,ℤ/8)14yes—
317C22×C4#13712yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4112yes—
323C22×C4H(5,𝔽2)12yes—
324C22×C4H(5,𝔽2)11yescharacteristic
325C22×C42−1+412yes—
326C22×C42−1+412yes—
327C22×C42−1+411yescharacteristic
328C2×C8—22no—
329C2×C8—24no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—28no—
346C4⋊C4—28no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—48no—
356C4⋊C4—416no—
357C4⋊C4—416no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—84no—
369C4⋊C4—88no—
370C4⋊C4—88no—
371C22⋊C4—88no—
372C22⋊C4—88no—
373C22⋊C4—84no—
374C22⋊C4—84no—
375C42—22no—
376C42—22no—
377C42—24no—
378C42—48no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—42no—
383C23—42no—
384C23—44no—
385C23—42no—
386C23—42no—
387C23—44no—
388C23—82no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—82no—
393C23—81no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23C2×#12011yescharacteristic
407C23#53612yes—
408C23#54612yes—
409C23#54811yescharacteristic
410C23#55011yescharacteristic
411C23C22×SD1611yescharacteristic
412C23C2×AGL(1,ℤ/8)12yes—
413C23C2×#13711yescharacteristic
414C23#57612yes—
415C23#57711yescharacteristic
416C23#57811yescharacteristic
417D4—42no—
418D4—42no—
419D4—44no—
420D4—81no—
421D4—81no—
422D4—82no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4#44814yes—
476C2×C4#52511yescharacteristic
477C2×C4#53211yescharacteristic
478C2×C4#53511yescharacteristic
479C2×C4#53611yescharacteristic
480C2×C4#53611yescharacteristic
481C2×C4#53612yes—
482C2×C4#53711yescharacteristic
483C2×C4#53912yes—
484C2×C4#53912yes—
485C2×C4#53912yes—
486C2×C4#54112yes—
487C2×C4#54611yescharacteristic
488C2×C4#54711yescharacteristic
489C2×C4#55012yes—
490C2×C4#55212yes—
491C2×C4#55212yes—
492C2×C4#56212yes—
493C2×C4#56312yes—
494C8—42no—
495C8—81no—
496C8—82no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—84no—
501C22—84no—
502C22—162no—
503C22—162no—
504C22—161no—
505C22—161no—
506C22—161no—
507C22—161no—
508C22#240711yescharacteristic
509C22#243111yescharacteristic
510C22#261411yescharacteristic
511C22#269012yes—
512C22#269412yes—
513C22C2×#44811yescharacteristic
514C22#300311yescharacteristic
515C22#300811yescharacteristic
516C22#301311yescharacteristic
517C22#322712yes—
518C22#323912yes—
519C22#324112yes—
520C22#324312yes—
521C22#329512yes—
522C22#330612yes—
523C22#330812yes—
524C22#333312yes—
525C22#335911yescharacteristic
526C22#336612yes—
527C22#337211yescharacteristic
528C22#339211yescharacteristic
529C22#339811yescharacteristic
530C22#339912yes—
531C4—24no—
532C4—42no—
533C4—44no—
534C4—44no—
535C4—84no—
536C4—82no—
537C4—84no—
538C4—82no—
539C4—161no—
540C4#261111yescharacteristic
541C4#263311yescharacteristic
542C4#266411yescharacteristic
543C4#269412yes—
544C4#272112yes—
545C4#282011yescharacteristic
546C2—162no—
547C2#1668211yescharacteristic
548C2#2418711yescharacteristic
549C2#2419611yescharacteristic
550C2#2556911yescharacteristic
551C2#2557711yescharacteristic
552C2#2963911yescharacteristic
553C2#2964611yescharacteristic
554C2#3145812yes—
555C2#3147412yes—
556C2#3157412yes—
557C2#3159212yes—
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
5. #11371C2semidirect
6. #11346C2semidirect
7. #11327C2semidirect
8. #11207C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#20712
C2×#19262
#151482
#113712
#113462
#113272
#112072
#110662
#110492
#108072