Finite Group Databasebeta

← #467942

Subgroups — #467942

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2041C212yesmaximal
2#15196C211yesmaximal, characteristic, complemented
3#14154C211yesmaximal, characteristic
4#11202C211yesmaximal, characteristic, complemented
5#11189C212yesmaximal, complemented
6#11189C212yesmaximal, complemented
7#11076C212yesmaximal, complemented
8#11049C212yesmaximal, complemented
9#10885C212yesmaximal, complemented
10C22×#127C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×#102C2212yes—
13C22×#102C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C2×#390—21no—
17C2×#390—21no—
18C2×#390C2212yes—
19C2×#390C2212yes—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#2041—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32#1444—21no—
33#1444—21no—
34#1441—21no—
35#1441—21no—
36#1438—21no—
37#1438—21no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C23×D4—21no—
41C23×D4—21no—
42C23×D4C2312yes—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#127—22no—
46C2×#127—41no—
47C2×#127—41no—
48C2×#127—41no—
49C2×#127—41no—
50C2×#127D414yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—41no—
68C2×C4⋊C8—42no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—42no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102—42no—
81#390—24no—
82#390—24no—
83#390—42no—
84#390—42no—
85#390—44no—
86#390—44no—
87#389—24no—
88#389—24no—
89#384—41no—
90#384—41no—
91C4×C4⋊C4—41no—
92C4×C4⋊C4—41no—
93C4×C4⋊C4—41no—
94C4×C4⋊C4—41no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95D411yescharacteristic
102C2×#95D411yescharacteristic
103#340—24no—
104#340—24no—
105#340—42no—
106#340—42no—
107#336—24no—
108#336—41no—
109#336—42no—
110C25—22no—
111C22×D4—24no—
112C22×D4—24no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—42no—
127C22×D4—42no—
128C22×D4—42no—
129C22×D4—42no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—82no—
135C22×D4—82no—
136C22×D4C2×D414yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
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—
161#127—44no—
162#127—44no—
163#127—44no—
164#127—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
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—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4SD1614yes—
194C2×C4⋊C4#4114yes—
195C2×C4⋊C4#4114yes—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42C2×D412yes—
224C2×C42#4112yes—
225C4⋊C8—44no—
226#102—48no—
227#102—48no—
228#102—48no—
229#102—48no—
230#95—41no—
231#95—41no—
232#95—42no—
233#95—42no—
234#95—44no—
235#95—42no—
236#95—82no—
237#95—82no—
238C24—24no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24C22×D411yescharacteristic
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—48no—
261C2×D4—48no—
262C2×D4—48no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—48no—
286C2×D4—88no—
287C2×D4—84no—
288C2×D4—84no—
289C2×D4—84no—
290C2×D4—84no—
291C2×D4—84no—
292C2×D4—84no—
293C2×D4—88no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—42no—
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—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4#12112yes—
341C22×C4#12112yes—
342C22×C4C2×SD1612yes—
343C22×C4C2×SD1612yes—
344C22×C4C2×SD1612yes—
345C22×C4AGL(1,ℤ/8)12yes—
346C22×C4#13712yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4112yes—
350C22×C4C2×#4112yes—
351C22×C4H(5,𝔽2)11yescharacteristic
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C4H(5,𝔽2)12yes—
356C22×C42−1+412yes—
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—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—42no—
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—84no—
387C4⋊C4—84no—
388C22⋊C4—88no—
389C22⋊C4—88no—
390C22⋊C4—88no—
391C22⋊C4—88no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—42no—
397C42—42no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—84no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23C2×#12111yescharacteristic
439C23#53411yescharacteristic
440C23#53511yescharacteristic
441C23#54612yes—
442C23#54812yes—
443C23C22×SD1611yescharacteristic
444C23C2×AGL(1,ℤ/8)11yescharacteristic
445C23C2×#13711yescharacteristic
446C23#57511yescharacteristic
447C23#57612yes—
448C23#57612yes—
449D4—44no—
450D4—44no—
451D4—44no—
452D4—44no—
453D4—816no—
454D4—816no—
455D4—82no—
456D4—82no—
457D4—82no—
458D4—82no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4#46114yes—
512C2×C4#52912yes—
513C2×C4#53411yescharacteristic
514C2×C4#53411yescharacteristic
515C2×C4#53811yescharacteristic
516C2×C4#53811yescharacteristic
517C2×C4#53912yes—
518C2×C4#54012yes—
519C2×C4#54012yes—
520C2×C4#54212yes—
521C2×C4#54212yes—
522C2×C4#54612yes—
523C2×C4#54712yes—
524C2×C4#55312yes—
525C2×C4#55511yescharacteristic
526C2×C4#55911yescharacteristic
527C2×C4#56011yescharacteristic
528C2×C4#56212yes—
529C2×C4#56311yescharacteristic
530C8—82no—
531C8—82no—
532C22—84no—
533C22—84no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—162no—
540C22—162no—
541C22—162no—
542C22—162no—
543C22—162no—
544C22—162no—
545C22—162no—
546C22—162no—
547C22#242712yes—
548C22#261711yescharacteristic
549C22#270911yescharacteristic
550C22#271112yes—
551C22#273111yescharacteristic
552C22C2×#46111yescharacteristic
553C22#303611yescharacteristic
554C22#304112yes—
555C22#318711yescharacteristic
556C22#319311yescharacteristic
557C22#319911yescharacteristic
558C22#319911yescharacteristic
559C22#320811yescharacteristic
560C22#321511yescharacteristic
561C22#322012yes—
562C22#329512yes—
563C22#330912yes—
564C22#331812yes—
565C22#333312yes—
566C22#336212yes—
567C22#336612yes—
568C22#337012yes—
569C22#337512yes—
570C4—24no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—81no—
576C4—84no—
577C4—84no—
578C4—162no—
579C4#259311yescharacteristic
580C4#265112yes—
581C4#269911yescharacteristic
582C4#270112yes—
583C4#272411yescharacteristic
584C4#283011yescharacteristic
585C2—164no—
586C2#1659611yescharacteristic
587C2#2532212yes—
588C2#2533012yes—
589C2#2971411yescharacteristic
590C2#2972111yescharacteristic
591C2#3202811yescharacteristic
592C2#3203611yescharacteristic
593C2#3208012yes—
594C2#3210912yes—
595C2#3262911yescharacteristic
596C2#3264711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
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. #15196C2semidirect
4. #11202C2semidirect
5. #11189C2semidirect
6. #11189C2semidirect
7. #11076C2semidirect
8. #11049C2semidirect
9. #10885C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20412
#151962
#141542
#112022
#111892
#110762
#110492
#108852