Finite Group Databasebeta

← #461433

Subgroups — #461433

Order 512.

1272 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1880C211yesmaximal, characteristic
2C2×#1803C211yesmaximal, characteristic, complemented
3#15155C211yesmaximal, characteristic
4#13931C211yesmaximal, characteristic, complemented
5#11357C211yesmaximal, characteristic, complemented
6#11267C212yesmaximal, complemented
7#11255C211yesmaximal, characteristic, complemented
8#11189C212yesmaximal
9#11076C211yesmaximal, characteristic
10#11050C212yesmaximal, complemented
11#11023C212yesmaximal
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C2×#399—21no—
20C2×#399—22no—
21C2×#399—21no—
22C2×#391C2212yes—
23C2×#390C2211yescharacteristic
24C2×#390C2212yes—
25C2×#384C2211yescharacteristic
26C2×#384C2212yes—
27#1880—24no—
28#1803—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1445—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C22×SD16—22no—
46C22×SD16—22no—
47C22×SD16—24no—
48C22×SD16—22no—
49C22×SD16—42no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4×Q8—22no—
54C2×C4×Q8—24no—
55C2×C4×Q8—42no—
56C2×C4×Q8D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C4⋊C8—42no—
75C2×#103—24no—
76C2×#103—22no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#102—24no—
81C2×#102—22no—
82C2×#102—42no—
83C2×#102—42no—
84C2×#102—42no—
85#399—48no—
86#399—44no—
87#399—44no—
88#391—24no—
89#391—42no—
90#390—22no—
91#390—22no—
92#390—42no—
93#390—42no—
94#384—22no—
95#384—22no—
96#384—42no—
97#384—42no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95D411yescharacteristic
105C2×#95D412yes—
106C2×#95D411yescharacteristic
107#340—24no—
108#340—22no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336—22no—
114#336—42no—
115C25—21no—
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8C2×D412yes—
125C22×Q8#4114yes—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4#4112yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C2×SD16—28no—
147C2×SD16—48no—
148C2×SD16—48no—
149C2×SD16—44no—
150C2×SD16—44no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8#4112yes—
168C4×Q8—28no—
169C4×Q8—44no—
170C4×Q8—48no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
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—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C2×D412yes—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—44no—
225C2×C42C2×D412yes—
226C2×C42#4112yes—
227C4⋊C8—44no—
228#103—48no—
229#103—44no—
230#102—44no—
231#102—48no—
232#95—24no—
233#95—22no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—42no—
238#95—82no—
239#95—81no—
240#95SD1614yescomplemented
241C24—22no—
242C24—41no—
243C24—42no—
244C24—42no—
245C24—41no—
246C24—41no—
247C24—42no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24—42no—
252C24C22×D411yescharacteristic
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8#12314yes—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4C2×SD1614yes—
339C22×C4C2×SD1612yes—
340C22×C4AGL(1,ℤ/8)14yes—
341C22×C4AGL(1,ℤ/8)12yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4112yes—
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4H(5,𝔽2)12yes—
348C22×C4H(5,𝔽2)12yes—
349C22×C4H(5,𝔽2)12yes—
350C22×C42−1+411yescharacteristic
351C22×C42−1+411yescharacteristic
352SD16—416no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—88no—
386C4⋊C4—88no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C22⋊C4—88no—
390C22⋊C4—88no—
391C22⋊C4—84no—
392C22⋊C4—84no—
393C42—22no—
394C42—22no—
395C42—24no—
396C42—48no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C23—42no—
402C23—44no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—82no—
407C23—81no—
408C23—82no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—81no—
418C23—81no—
419C23—82no—
420C23—81no—
421C23—81no—
422C23—82no—
423C23—82no—
424C23C2×#12111yescharacteristic
425C23#53512yes—
426C23#54612yes—
427C23#54711yescharacteristic
428C23#54811yescharacteristic
429C23C22×SD1611yescharacteristic
430C23C2×AGL(1,ℤ/8)12yes—
431C23C2×AGL(1,ℤ/8)11yescharacteristic
432C23#57711yescharacteristic
433C23#57812yes—
434C23#57811yescharacteristic
435Q8—24no—
436Q8—24no—
437Q8—42no—
438Q8—42no—
439Q8—44no—
440Q8—44no—
441D4—42no—
442D4—42no—
443D4—44no—
444D4—81no—
445D4—81no—
446D4—82no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4#46014yes—
500C2×C4C2×#12311yescharacteristic
501C2×C4#52511yescharacteristic
502C2×C4#53411yescharacteristic
503C2×C4#53511yescharacteristic
504C2×C4#53511yescharacteristic
505C2×C4#53611yescharacteristic
506C2×C4#53812yes—
507C2×C4#53912yes—
508C2×C4#53912yes—
509C2×C4#54012yes—
510C2×C4#54012yes—
511C2×C4#55112yes—
512C2×C4#55112yes—
513C2×C4#55112yes—
514C2×C4#55211yescharacteristic
515C2×C4#55611yescharacteristic
516C2×C4#56212yes—
517C2×C4#56212yes—
518C8—42no—
519C8—81no—
520C8—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—84no—
525C22—84no—
526C22—161no—
527C22—161no—
528C22—162no—
529C22—161no—
530C22—162no—
531C22—161no—
532C22#241211yescharacteristic
533C22#241511yescharacteristic
534C22#261911yescharacteristic
535C22#269612yes—
536C22#270612yes—
537C22C2×#46011yescharacteristic
538C22#303811yescharacteristic
539C22#304411yescharacteristic
540C22#304611yescharacteristic
541C22#320812yes—
542C22#321712yes—
543C22#322312yes—
544C22#322512yes—
545C22#329812yes—
546C22#330812yes—
547C22#330812yes—
548C22#333212yes—
549C22#333511yescharacteristic
550C22#334312yes—
551C22#335311yescharacteristic
552C22#335911yescharacteristic
553C22#336712yes—
554C22#337311yescharacteristic
555C4—24no—
556C4—44no—
557C4—44no—
558C4—44no—
559C4—42no—
560C4—82no—
561C4—162no—
562C4—161no—
563C4—162no—
564C4#248211yescharacteristic
565C4#262311yescharacteristic
566C4#263511yescharacteristic
567C4#274012yes—
568C4#274312yes—
569C4#281611yescharacteristic
570C2—162no—
571C2#1639211yescharacteristic
572C2#2452811yescharacteristic
573C2#2454211yescharacteristic
574C2#2467811yescharacteristic
575C2#2469411yescharacteristic
576C2#2979611yescharacteristic
577C2#2980511yescharacteristic
578C2#3163612yes—
579C2#3165612yes—
580C2#3193212yes—
581C2#3196012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1803C2semidirect
4. #13931C2semidirect
5. #11357C2semidirect
6. #11267C2semidirect
7. #11255C2semidirect
10. #11050C2semidirect
240. #95SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#18802
C2×#18032
#151552
#139312
#113572
#112672
#112552
#111892
#110762
#110502
#110232