Finite Group Databasebeta

← #468110

Subgroups — #468110

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic, complemented
3C2×#1862C211yesmaximal, characteristic
4#15156C211yesmaximal, characteristic, complemented
5#11384C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11325C212yesmaximal, complemented
8#11268C212yesmaximal, complemented
9#11189C212yesmaximal
10#11121C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#106C2212yes—
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19#2563—24no—
20#2563—24no—
21C2×#400—21no—
22C2×#393—22no—
23C2×#390C2212yes—
24C2×#389C2211yescharacteristic
25C2×#382C2212yes—
26C2×#380—21no—
27C2×#380—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2058—24no—
30#1862—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40#1443—22no—
41#1441—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8D414yes—
55C2×C4×D4—22no—
56C2×C4×D4—42no—
57C2×C4×D4—42no—
58C2×C4×D4D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×#107—22no—
74C2×#107—42no—
75C2×#106—24no—
76C2×#106—42no—
77C2×#106—42no—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—22no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84C2×#102—42no—
85C2×#102Q814yescomplemented
86#400—44no—
87#393—216no—
88#390—24no—
89#390—24no—
90#389—22no—
91#389—22no—
92#382—24no—
93#382—24no—
94#380—28no—
95#380—28no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95D412yes—
108#340—24no—
109#340—22no—
110#340—42no—
111#340—42no—
112#340—42no—
113#336—24no—
114#336—24no—
115#336—22no—
116#336—22no—
117C25—21no—
118C22×Q8—24no—
119C22×Q8—44no—
120C22×Q8—81no—
121C22×Q8—82no—
122C22×Q8—81no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4C2×D412yes—
131C22×D4C2×Q812yes—
132C22×D4#4112yes—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×Q812yes—
164C22×C8#4114yes—
165C4⋊Q8—48no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4—48no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
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—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225#107—44no—
226#106—48no—
227#102—28no—
228#102—48no—
229#102—44no—
230#95—28no—
231#95—28no—
232#95—24no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—42no—
237#95—42no—
238#95—81no—
239#95—82no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—42no—
244C24—42no—
245C24—41no—
246C24—41no—
247C24—42no—
248C24—41no—
249C24—41no—
250C24—42no—
251C24C22×D411yescharacteristic
252C2×Q8—48no—
253C2×Q8—84no—
254C2×Q8—84no—
255C2×Q8—88no—
256C2×Q8—88no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4C22⋊Q814yes—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—48no—
299C22×C4—48no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—48no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—88no—
330C22×C4—88no—
331C22×C4#12012yes—
332C22×C4#12012yes—
333C22×C4#13512yes—
334C22×C4AGL(1,ℤ/8)12yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4H(5,𝔽2)12yes—
341C22×C4H(5,𝔽2)12yes—
342C22×C42−1+411yescharacteristic
343C22×C42−1+412yes—
344C22×C42−1+411yescharacteristic
345C2×C8—22no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—41no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—44no—
369C4⋊C4—48no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—44no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—84no—
384C22⋊C4—48no—
385C22⋊C4—88no—
386C22⋊C4—88no—
387C42—22no—
388C42—22no—
389C42—24no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C23—24no—
395C23—42no—
396C23—42no—
397C23—44no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—44no—
402C23—42no—
403C23—44no—
404C23—44no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23C2×#12011yescharacteristic
418C23#53512yes—
419C23#54712yes—
420C23#54811yescharacteristic
421C23D4×Q811yescharacteristic
422C23C2×#13511yescharacteristic
423C23C2×AGL(1,ℤ/8)11yescharacteristic
424C23#57512yes—
425C23#57611yescharacteristic
426C23#57712yes—
427C23#57711yescharacteristic
428Q8—816no—
429D4—24no—
430D4—24no—
431D4—42no—
432D4—42no—
433D4—44no—
434D4—44no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—816no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4#44914yes—
491C2×C4C2×C22⋊Q811yescharacteristic
492C2×C4#53211yescharacteristic
493C2×C4#53411yescharacteristic
494C2×C4#53411yescharacteristic
495C2×C4#53412yes—
496C2×C4#53611yescharacteristic
497C2×C4#53611yescharacteristic
498C2×C4#53912yes—
499C2×C4#53912yes—
500C2×C4#54212yes—
501C2×C4#54412yes—
502C2×C4#54811yescharacteristic
503C2×C4#54812yes—
504C2×C4D4×Q811yescharacteristic
505C2×C4#55112yes—
506C2×C4#55412yes—
507C2×C4#56312yes—
508C2×C4#56312yes—
509C8—42no—
510C8—44no—
511C8—81no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—84no—
516C22—82no—
517C22—82no—
518C22—84no—
519C22—84no—
520C22—82no—
521C22—82no—
522C22—84no—
523C22#240911yescharacteristic
524C22#243211yescharacteristic
525C22#261311yescharacteristic
526C22#272012yes—
527C22#272312yes—
528C22C2×#44911yescharacteristic
529C22#300511yescharacteristic
530C22#300911yescharacteristic
531C22#301211yescharacteristic
532C22#321112yes—
533C22#321312yes—
534C22#322212yes—
535C22#322412yes—
536C22#334012yes—
537C22#334712yes—
538C22#334812yes—
539C22#335412yes—
540C22#336011yescharacteristic
541C22#336611yescharacteristic
542C22#337612yes—
543C22#338011yescharacteristic
544C22#338411yescharacteristic
545C22#338812yes—
546C4—24no—
547C4—42no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—84no—
553C4—161no—
554C4—162no—
555C4#256411yescharacteristic
556C4#262711yescharacteristic
557C4#266311yescharacteristic
558C4#272812yes—
559C4#272912yes—
560C4#281711yescharacteristic
561C2—84no—
562C2#1660111yescharacteristic
563C2#2432711yescharacteristic
564C2#2434311yescharacteristic
565C2#2570011yescharacteristic
566C2#2571011yescharacteristic
567C2#2962311yescharacteristic
568C2#2963211yescharacteristic
569C2#3235112yes—
570C2#3237112yes—
571C2#3243512yes—
572C2#3245712yes—
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×#2058C2semidirect
4. #15156C2semidirect
5. #11384C2semidirect
6. #11346C2semidirect
7. #11325C2semidirect
8. #11268C2semidirect
85. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#20582
C2×#18622
#151562
#113842
#113462
#113252
#112682
#111892
#111212
#107882