Finite Group Databasebeta

← #440887

Subgroups — #440887

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2454C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic, complemented
3C2×#2071C211yesmaximal, characteristic
4#13911C211yesmaximal, characteristic, complemented
5#11385C212yesmaximal, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11293C212yesmaximal, complemented
8#11189C212yesmaximal
9#10931C211yesmaximal, characteristic
10#10885C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16#2454—24no—
17#2454—24no—
18C2×#399—21no—
19C2×#396—21no—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#385—21no—
25C2×#384C2211yescharacteristic
26C2×#382—21no—
27C2×#380—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2099—24no—
31#2071—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1442—21no—
43#1442—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—24no—
54C2×#125—42no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—24no—
58C2×C4×D4—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—42no—
87#399—28no—
88#396—28no—
89#390—22no—
90#390—22no—
91#390—22no—
92#390D414yes—
93#389—22no—
94#389—22no—
95#389—22no—
96#389D414yes—
97#385—28no—
98#384—22no—
99#384—22no—
100#382—44no—
101#380—28no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#336—22no—
121#336—22no—
122#336—42no—
123#336—42no—
124C25—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4C2×D412yes—
132C22×D4#4114yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—44no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—82no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164#125—48no—
165#125—44no—
166#125—44no—
167C4×D4—48no—
168C4×D4—44no—
169C4×D4—44no—
170C2×C4⋊C4—21no—
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—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
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—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4#4114yes—
212C2×C4⋊C4#4114yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—44no—
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—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239#102—48no—
240#102—48no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—22no—
245#95—22no—
246#95—44no—
247#95—44no—
248#95—82no—
249#95—82no—
250C24—22no—
251C24—24no—
252C24—41no—
253C24—41no—
254C24—42no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—42no—
260C24—42no—
261C24C22×D411yescharacteristic
262C2×D4—24no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4#12414yes—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
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—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—88no—
337C22×C4—84no—
338C22×C4—88no—
339C22×C4#12012yes—
340C22×C4#12012yes—
341C22×C4#13512yes—
342C22×C4AGL(1,ℤ/8)12yes—
343C22×C4#13712yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C2×#4112yes—
347C22×C4C2×#4112yes—
348C22×C4C2×#4112yes—
349C22×C4H(5,𝔽2)12yes—
350C22×C42−1+412yes—
351C22×C42−1+412yes—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—44no—
373C4⋊C4—48no—
374C4⋊C4—48no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C4⋊C4#12414yes—
390C22⋊C4—48no—
391C22⋊C4—48no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C42—24no—
396C42—24no—
397C42—48no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—42no—
403C42—42no—
404C23—24no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—42no—
410C23—44no—
411C23—42no—
412C23—44no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—82no—
418C23—81no—
419C23—81no—
420C23—82no—
421C23—81no—
422C23—81no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23C2×#12011yescharacteristic
428C23#52511yescharacteristic
429C23#53611yescharacteristic
430C23#54612yes—
431C23#54812yes—
432C23C2×#13511yescharacteristic
433C23C2×AGL(1,ℤ/8)11yescharacteristic
434C23C2×#13711yescharacteristic
435C23#57511yescharacteristic
436C23#57612yes—
437C23#57712yes—
438D4—28no—
439D4—44no—
440D4—44no—
441D4—44no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—816no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4#44914yes—
504C2×C4C2×#12411yescharacteristic
505C2×C4C2×#12411yescharacteristic
506C2×C4#52912yes—
507C2×C4#53511yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#53912yes—
510C2×C4#54112yes—
511C2×C4#54212yes—
512C2×C4#54612yes—
513C2×C4#54712yes—
514C2×C4#54712yes—
515C2×C4#54812yes—
516C2×C4#55112yes—
517C2×C4#55212yes—
518C2×C4#55212yes—
519C2×C4#56312yes—
520C8—82no—
521C8—82no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—84no—
529C22—82no—
530C22—84no—
531C22—84no—
532C22—84no—
533C22#240611yescharacteristic
534C22#243511yescharacteristic
535C22#243611yescharacteristic
536C22#247712yes—
537C22#269012yes—
538C22C2×#44911yescharacteristic
539C22#300711yescharacteristic
540C22#300811yescharacteristic
541C22#300911yescharacteristic
542C22#310811yescharacteristic
543C22#311411yescharacteristic
544C22#311912yes—
545C22#323211yescharacteristic
546C22#323811yescharacteristic
547C22#324112yes—
548C22#330412yes—
549C22#331412yes—
550C22#331912yes—
551C22#333212yes—
552C22#336112yes—
553C22#336612yes—
554C22#337212yes—
555C22#337612yes—
556C4—24no—
557C4—42no—
558C4—42no—
559C4—44no—
560C4—82no—
561C4—81no—
562C4—81no—
563C4—82no—
564C4—81no—
565C4—84no—
566C4—162no—
567C4#248812yes—
568C4#248912yes—
569C4#269812yes—
570C4#272312yes—
571C2—84no—
572C2#1567011yescharacteristic
573C2#2406611yescharacteristic
574C2#2406911yescharacteristic
575C2#2588711yescharacteristic
576C2#2589711yescharacteristic
577C2#2593011yescharacteristic
578C2#2594511yescharacteristic
579C2#2742312yes—
580C2#2743812yes—
581C2#3146712yes—
582C2#3147012yes—
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×#2099C2semidirect
4. #13911C2semidirect
5. #11385C2semidirect
6. #11346C2semidirect
7. #11293C2semidirect
10. #10885C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24542
C2×#20992
C2×#20712
#139112
#113852
#113462
#112932
#111892
#109312
#108852
#107882