Finite Group Databasebeta

← #405863

Subgroups — #405863

Order 512.

1428 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1853C211yesmaximal, characteristic
2C22×#340C211yesmaximal, characteristic
3#13907C211yesmaximal, characteristic
4#13881C211yesmaximal, characteristic
5#11358C212yesmaximal, complemented
6#11257C212yesmaximal, complemented
7#11220C212yesmaximal, complemented
8#11208C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#382C2212yes—
20C2×#380—21no—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1853—24no—
25C2×#340—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1442—22no—
37#1441—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C22⋊#95—21no—
41C22⋊#95—21no—
42C22⋊#95—21no—
43C24×C4—21no—
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—42no—
49C23×C8—42no—
50C23×C8Q814yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—42no—
63C22×C4⋊C4—42no—
64C22×C4⋊C4D414yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4D411yescharacteristic
74C22×C22⋊C4D411yescharacteristic
75C22×C42C2312yes—
76C2×#106—22no—
77C2×#106—22no—
78C2×#106—24no—
79C2×#106—42no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8Q814yes—
83#389—24no—
84#389—42no—
85#382—24no—
86#382—42no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380—44no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D412yes—
104#340—22no—
105#340—22no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—42no—
113#336—42no—
114C25C2×D411yescharacteristic
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—44no—
127C23×C4—44no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—82no—
134C23×C4—82no—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×Q812yes—
140C23×C4C2×Q812yes—
141C23×C4#4112yes—
142C23×C4#4112yes—
143C23×C4#4112yes—
144C23×C4C2411yescharacteristic
145C22×C8—28no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—48no—
161C22×C8—44no—
162C22×C8—44no—
163C22×C8—48no—
164C22×C8C2×Q814yes—
165C22×C8C2×Q812yes—
166C22×C8C2×Q812yes—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4#4114yes—
207C2×C4⋊C4#4114yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226#106—44no—
227#106—44no—
228C22⋊C8—28no—
229#95—24no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—42no—
236#95—82no—
237#95—82no—
238#95—82no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24#12112yes—
244C24#12312yes—
245C24C4⋊Q812yes—
246C24#13512yes—
247C24#13512yes—
248C24AGL(1,ℤ/8)12yes—
249C24#13712yes—
250C24C22×D411yescharacteristic
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—48no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—48no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—48no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—84no—
319C22×C4—88no—
320C22×C4—88no—
321C22×C4—88no—
322C22×C4—88no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4#12112yes—
326C22×C4C22⋊Q814yes—
327C22×C4C22⋊Q814yes—
328C22×C4#12312yes—
329C22×C4#12514yes—
330C22×C4#12514yes—
331C22×C4C4⋊Q812yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×Q811yescharacteristic
334C22×C4C22×Q811yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4112yes—
338C22×C4C2×#4112yes—
339C22×C4C2×#4111yescharacteristic
340C22×C4H(5,𝔽2)12yes—
341C22×C4H(5,𝔽2)12yes—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—416no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—88no—
382C4⋊C4#12314yes—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—84no—
386C22⋊C4—84no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—42no—
391C23—24no—
392C23—24no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—22no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23#39814yes—
408C23#46314yes—
409C23#46414yes—
410C23#48214yes—
411C23#48514yes—
412C23#49914yes—
413C23#50114yes—
414C23C2×#12111yescharacteristic
415C23C2×#12311yescharacteristic
416C23C2×C4⋊Q811yescharacteristic
417C23#54612yes—
418C23#54612yes—
419C23C2×#13511yescharacteristic
420C23C2×#13511yescharacteristic
421C23C2×AGL(1,ℤ/8)11yescharacteristic
422C23C2×#13711yescharacteristic
423C23#57712yes—
424C23#57712yes—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—28no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—816no—
461C2×C4—816no—
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—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#39814yes—
486C2×C4C2×C22⋊Q811yescharacteristic
487C2×C4C2×C22⋊Q811yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4C2×#12511yescharacteristic
491C2×C4C2×#12511yescharacteristic
492C2×C4#52912yes—
493C2×C4#53312yes—
494C2×C4#53411yescharacteristic
495C2×C4#53511yescharacteristic
496C2×C4#53812yes—
497C2×C4#54312yes—
498C2×C4#55112yes—
499C2×C4#55112yes—
500C2×C4#55312yes—
501C2×C4#55312yes—
502C2×C4#55412yes—
503C2×C4#55412yes—
504C8—42no—
505C8—44no—
506C8—42no—
507C22—24no—
508C22—28no—
509C22—28no—
510C22—28no—
511C22—28no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22#208518yes—
518C22C2×#39811yescharacteristic
519C22#241111yescharacteristic
520C22#241211yescharacteristic
521C22#244412yes—
522C22#254912yes—
523C22C2×#46311yescharacteristic
524C22C2×#46411yescharacteristic
525C22#304211yescharacteristic
526C22#304311yescharacteristic
527C22C2×#48211yescharacteristic
528C22C2×#48511yescharacteristic
529C22#309912yes—
530C22C2×#49911yescharacteristic
531C22C2×#50111yescharacteristic
532C22#316312yes—
533C22#329612yes—
534C22#329612yes—
535C22#330112yes—
536C22#330112yes—
537C22#330612yes—
538C22#331012yes—
539C22#331612yes—
540C22#332012yes—
541C4—24no—
542C4—44no—
543C4—82no—
544C4—82no—
545C4—84no—
546C4—82no—
547C4—162no—
548C4—161no—
549C4—161no—
550C4#248711yescharacteristic
551C4#250512yes—
552C4#256912yes—
553C4#257911yescharacteristic
554C4#262611yescharacteristic
555C4#285511yescharacteristic
556C2—28no—
557C2#1492011yescharacteristic
558C2C2×#208511yescharacteristic
559C2#2366111yescharacteristic
560C2#2447711yescharacteristic
561C2#2448311yescharacteristic
562C2#2452711yescharacteristic
563C2#2453311yescharacteristic
564C2#2620612yes—
565C2#2621212yes—
566C2#2877012yes—
567C2#2877612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11358C2semidirect
6. #11257C2semidirect
7. #11220C2semidirect
8. #11208C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18532
C22×#3402
#139072
#138812
#113582
#112572
#112202
#112082
#107302
#107212
#107142