Finite Group Databasebeta

← #766925

Subgroups — #766925

Order 512.

2034 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
1#16179C214yesmaximal, complemented
2#16035C214yesmaximal
3#15826C214yesmaximal, complemented
4#15597C214yesmaximal, complemented
5#15235C211yesmaximal, characteristic, complemented
6#15181C212yesmaximal
7#15151C211yesmaximal, characteristic, complemented
8#13974C212yesmaximal, complemented
9#13973C212yesmaximal, complemented
10#13876C212yesmaximal, complemented
11#13875C212yesmaximal, complemented
12#13847C211yesmaximal, characteristic
13#13783C212yesmaximal, complemented
14C22×C4×D4C2211yescharacteristic
15C2×#398C2214yes—
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#388C2214yes—
19C2×#388C2214yes—
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C2×#376C2211yescharacteristic
32#1445C2212yes—
33#1445C2214yes—
34#1443C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1440C2214yes—
42#1440C2214yes—
43#1439C2212yes—
44#1439C2212yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1438C2212yes—
50#1438C2214yes—
51#1437C2214yes—
52#1437C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57C4⋊#95C2212yes—
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2214yes—
64#1434C2214yes—
65#1434C2214yes—
66#1434C2214yes—
67C23×D4C2311yescharacteristic
68C24×C4C2312yes—
69C2×C4×D4—24no—
70C2×C4×D4—42no—
71C2×C4×D4—42no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2314yes—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2312yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2312yes—
94C22×C42C2314yes—
95C22×C42C2314yes—
96#398—48no—
97#389—28no—
98#389—28no—
99#388—48no—
100#388—48no—
101#384—42no—
102#382—42no—
103#380—28no—
104#380—28no—
105#380—44no—
106#380—44no—
107C4×C4⋊C4—28no—
108C4×C4⋊C4—42no—
109C4×C4⋊C4—44no—
110C4×C22⋊C4—28no—
111C4×C22⋊C4—28no—
112#376—42no—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2312yes—
116C2×#95C2312yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2312yes—
122C2×#95C2312yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2311yescharacteristic
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C25C2412yes—
141C22×D4—24no—
142C22×D4—22no—
143C22×D4—41no—
144C22×D4—41no—
145C22×D4—42no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4C2414yes—
154C23×C4C2414yes—
155C23×C4C2412yes—
156C23×C4C2414yes—
157C23×C4C2412yes—
158C23×C4C2411yescharacteristic
159C23×C4C2414yes—
160C23×C4C2412yes—
161C23×C4C2412yes—
162C23×C4C2411yescharacteristic
163C23×C4C2412yes—
164C23×C4C2411yescharacteristic
165C4×D4—416no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—42no—
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#4118yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—28no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42#4114yes—
235C2×C42#4118yes—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—28no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—41no—
253#95—42no—
254#95—41no—
255#95—44no—
256#95—42no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C242−1+414yes—
278C24C2511yescharacteristic
279C2×D4—48no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—48no—
283C2×D4—44no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4C2×#4114yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4118yes—
361C22×C4C2×#4114yes—
362C22×C4C2×#4112yes—
363C22×C4C2×#4112yes—
364C22×C4C2×#4114yes—
365C22×C4C2×#4114yes—
366C22×C4H(5,𝔽2)14yes—
367C22×C4H(5,𝔽2)12yes—
368C22×C4H(5,𝔽2)12yes—
369C4⋊C4—216no—
370C4⋊C4—48no—
371C4⋊C4—44no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C4⋊C4—48no—
375C4⋊C4—88no—
376C4⋊C4—88no—
377C4⋊C4—88no—
378C22⋊C4—216no—
379C22⋊C4—48no—
380C22⋊C4—48no—
381C22⋊C4—416no—
382C22⋊C4—48no—
383C22⋊C4—48no—
384C22⋊C4—88no—
385C22⋊C4—88no—
386C22⋊C4—88no—
387C42—28no—
388C42—216no—
389C42—44no—
390C42—44no—
391C42—88no—
392C42—88no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23C22×#4111yescharacteristic
429C23C22×#4112yes—
430C23C22×#4112yes—
431C23C2×H(5,𝔽2)11yescharacteristic
432C23C2×H(5,𝔽2)11yescharacteristic
433C23C2×H(5,𝔽2)12yes—
434C23C2×2−1+412yes—
435C23#58512yes—
436C23#58512yes—
437D4—416no—
438C2×C4—28no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—816no—
512C2×C4#52918yes—
513C2×C4#53814yes—
514C2×C4#55318yes—
515C2×C4#55514yes—
516C2×C4#55914yes—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22C2×#52912yes—
533C22C2×#53811yescharacteristic
534C22C2×#55312yes—
535C22C2×#55511yescharacteristic
536C22C2×#55911yescharacteristic
537C22#349212yes—
538C22#349312yes—
539C22#349412yes—
540C22#349512yes—
541C22#349512yes—
542C22#350811yescharacteristic
543C22#351211yescharacteristic
544C22#351412yes—
545C22#351612yes—
546C22#352012yes—
547C22#352412yes—
548C22#353811yescharacteristic
549C22#353911yescharacteristic
550C22#354112yes—
551C22#354212yes—
552C22#354612yes—
553C4—42no—
554C4—44no—
555C4—44no—
556C4—44no—
557C4—42no—
558C4—48no—
559C4—82no—
560C4—84no—
561C4—82no—
562C4—84no—
563C4—84no—
564C4#259318yes—
565C2—84no—
566C2C2×#259311yescharacteristic
567C2#3555312yes—
568C2#3565712yes—
569C2#3731711yescharacteristic
570C2#3738011yescharacteristic
571C2#4498112yes—
572C2#4499312yes—
573C2#4537212yes—
574C2#4539812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #16179C2semidirect
3. #15826C2semidirect
4. #15597C2semidirect
5. #15235C2semidirect
7. #15151C2semidirect
8. #13974C2semidirect
9. #13973C2semidirect
10. #13876C2semidirect
11. #13875C2semidirect
13. #13783C2semidirect

Maximal subgroups

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

subgroupindex
#161792
#160352
#158262
#155972
#152352
#151812
#151512
#139742
#139732
#138762
#138752
#138472
#137832