Finite Group Databasebeta

← #470334

Subgroups — #470334

Order 512.

1162 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2094C211yesmaximal, characteristic
3C2×#1925C211yesmaximal, characteristic
4#15147C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11325C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#11066C212yesmaximal
10#11051C212yesmaximal
11#10806C211yesmaximal, characteristic, complemented
12C22×#107C2211yescharacteristic
13C22×#106C2212yes—
14C22×#102C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17#2571—24no—
18#2571—24no—
19C2×#400—22no—
20C2×#397—21no—
21C2×#390C2211yescharacteristic
22C2×#390C2212yes—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#382—21no—
27C2×#382C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2094—24no—
31#1925—24no—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1440—22no—
42#1439—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×D4C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C42—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66C2×#107—22no—
67C2×#107—42no—
68C2×#106—24no—
69C2×#106—24no—
70C2×#106—42no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—22no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—22no—
77C2×#102—42no—
78C2×#102—42no—
79#400—216no—
80#397—44no—
81#390—24no—
82#390—22no—
83#390—22no—
84#390—42no—
85#389—28no—
86#389—24no—
87#389—24no—
88#384—22no—
89#384D414yes—
90#382—28no—
91#382—24no—
92#382—24no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4D414yes—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95D412yes—
106#340—24no—
107#340—22no—
108#340—42no—
109#340—42no—
110#340—42no—
111#336—24no—
112#336—22no—
113#336—22no—
114#336—42no—
115C25—21no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4#4112yes—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8#4112yes—
152C22×C8#4114yes—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—81no—
186C2×C4⋊C4—81no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D414yes—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4—82no—
202C2×C22⋊C4—82no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216#107—44no—
217#106—48no—
218#102—28no—
219#102—48no—
220#102—44no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—24no—
225#95—24no—
226#95—44no—
227#95—42no—
228#95—81no—
229#95—82no—
230C24—22no—
231C24—41no—
232C24—42no—
233C24—41no—
234C24—42no—
235C24—41no—
236C24—41no—
237C24—42no—
238C24—41no—
239C24—42no—
240C24—41no—
241C24C22×D411yescharacteristic
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—24no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
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—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4#12012yes—
312C22×C4#12012yes—
313C22×C4#13512yes—
314C22×C4AGL(1,ℤ/8)12yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4112yes—
320C22×C4H(5,𝔽2)12yes—
321C22×C4H(5,𝔽2)11yescharacteristic
322C22×C42−1+412yes—
323C22×C42−1+412yes—
324C22×C42−1+411yescharacteristic
325C2×C8—22no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—44no—
349C4⋊C4—48no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—44no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—84no—
364C4⋊C4—88no—
365C22⋊C4—84no—
366C22⋊C4—88no—
367C22⋊C4—88no—
368C22⋊C4—84no—
369C42—22no—
370C42—22no—
371C42—24no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—48no—
377C23—42no—
378C23—44no—
379C23—42no—
380C23—44no—
381C23—42no—
382C23—81no—
383C23—82no—
384C23—81no—
385C23—81no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—81no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23C2×#12011yescharacteristic
401C23#53612yes—
402C23#54611yescharacteristic
403C23#54711yescharacteristic
404C23#55012yes—
405C23C2×#13511yescharacteristic
406C23C2×AGL(1,ℤ/8)11yescharacteristic
407C23#57512yes—
408C23#57611yescharacteristic
409C23#57712yes—
410C23#57711yescharacteristic
411D4—42no—
412D4—42no—
413D4—44no—
414D4—81no—
415D4—81no—
416D4—82no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4#44914yes—
472C2×C4#52511yescharacteristic
473C2×C4#53211yescharacteristic
474C2×C4#53511yescharacteristic
475C2×C4#53611yescharacteristic
476C2×C4#53611yescharacteristic
477C2×C4#53612yes—
478C2×C4#53711yescharacteristic
479C2×C4#53912yes—
480C2×C4#53912yes—
481C2×C4#53912yes—
482C2×C4#54112yes—
483C2×C4#54612yes—
484C2×C4#54811yescharacteristic
485C2×C4#55011yescharacteristic
486C2×C4#55212yes—
487C2×C4#55212yes—
488C2×C4#56212yes—
489C2×C4#56312yes—
490C8—42no—
491C8—44no—
492C8—81no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—84no—
497C22—84no—
498C22—161no—
499C22—161no—
500C22—162no—
501C22—162no—
502C22—161no—
503C22—161no—
504C22#240711yescharacteristic
505C22#243111yescharacteristic
506C22#261411yescharacteristic
507C22#269412yes—
508C22#272112yes—
509C22C2×#44911yescharacteristic
510C22#300511yescharacteristic
511C22#301011yescharacteristic
512C22#301211yescharacteristic
513C22#323212yes—
514C22#323412yes—
515C22#324112yes—
516C22#324412yes—
517C22#330311yescharacteristic
518C22#331112yes—
519C22#332511yescharacteristic
520C22#334111yescharacteristic
521C22#335012yes—
522C22#335411yescharacteristic
523C22#339412yes—
524C22#339712yes—
525C22#340112yes—
526C22#340312yes—
527C4—24no—
528C4—42no—
529C4—82no—
530C4—84no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—161no—
535C4—162no—
536C4#261111yescharacteristic
537C4#263311yescharacteristic
538C4#266411yescharacteristic
539C4#269012yes—
540C4#269412yes—
541C4#282011yescharacteristic
542C2—162no—
543C2#1668211yescharacteristic
544C2#2411711yescharacteristic
545C2#2413111yescharacteristic
546C2#2562811yescharacteristic
547C2#2563911yescharacteristic
548C2#2964011yescharacteristic
549C2#2964711yescharacteristic
550C2#3158112yes—
551C2#3159712yes—
552C2#3238512yes—
553C2#3240112yes—
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×#2571C2semidirect
5. #11357C2semidirect
6. #11346C2semidirect
7. #11325C2semidirect
8. #11220C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#20942
C2×#19252
#151472
#113572
#113462
#113252
#112202
#110662
#110512
#108062