Finite Group Databasebeta

← #464877

Subgroups — #464877

Order 512.

1542 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×#2551C211yesmaximal, characteristic, complemented
2C2×#2094C211yesmaximal, characteristic
3C2×#2058C212yesmaximal, complemented
4C2×#2023C212yesmaximal
5C2×#1926C211yesmaximal, characteristic
6#15148C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic, complemented
8#11346C211yesmaximal, characteristic, complemented
9#11327C212yesmaximal, complemented
10#11066C212yesmaximal
11#10807C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2212yes—
14C22×#106C2211yescharacteristic
15C22×#106C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19#2551—24no—
20#2551—24no—
21C2×#399—21no—
22C2×#395—22no—
23C2×#390C2211yescharacteristic
24C2×#390C2212yes—
25C2×#389—21no—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2094—24no—
32#2058—28no—
33#2023—28no—
34#1926—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44#1439—22no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C23×Q8—22no—
48C23×D4C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—28no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8—48no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×#106—24no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—22no—
83C2×#106—42no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—22no—
89C2×#102—24no—
90C2×#102—42no—
91C2×#102Q814yescomplemented
92#399—44no—
93#395—216no—
94#390—24no—
95#390—22no—
96#390—22no—
97#390—42no—
98#389—28no—
99#389—24no—
100#389—24no—
101#389—24no—
102#389—42no—
103#384—22no—
104#384D414yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4D414yes—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95D412yes—
115#340—24no—
116#340—22no—
117#340—24no—
118#340—24no—
119#340D418yes—
120#336—24no—
121#336—24no—
122#336—22no—
123#336—22no—
124C25—21no—
125C22×Q8—28no—
126C22×Q8—44no—
127C22×Q8—44no—
128C22×Q8—44no—
129C22×Q8—44no—
130C22×Q8—82no—
131C22×Q8—82no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4C2×Q812yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8C2×D414yes—
168C22×C8C2×Q812yes—
169C4⋊Q8—48no—
170C4⋊Q8—48no—
171C4⋊Q8—432no—
172C4⋊Q8—48no—
173C4⋊Q8—48no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—81no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4SD1614yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×Q812yes—
218C2×C4⋊C4#4112yes—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—44no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242#106—48no—
243#106—44no—
244#102—28no—
245#102—48no—
246#102—44no—
247#95—28no—
248#95—24no—
249#95—24no—
250#95—44no—
251#95—42no—
252#95—81no—
253C24—22no—
254C24—41no—
255C24—42no—
256C24—42no—
257C24—41no—
258C24—41no—
259C24—42no—
260C24—41no—
261C24—41no—
262C24—42no—
263C24—41no—
264C24C22×D411yescharacteristic
265C2×Q8—416no—
266C2×Q8—416no—
267C2×Q8—88no—
268C2×Q8—88no—
269C2×Q8—88no—
270C2×Q8—88no—
271C2×Q8—88no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
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—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
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—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4#12012yes—
338C22×C4#12012yes—
339C22×C4C2×SD1614yes—
340C22×C4C2×SD1612yes—
341C22×C4AGL(1,ℤ/8)14yes—
342C22×C4#13712yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D412yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×Q811yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4H(5,𝔽2)12yes—
349C22×C42−1+412yes—
350C22×C42−1+411yescharacteristic
351C22×C42−1+411yescharacteristic
352C22×C42−1+412yes—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—28no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—48no—
382C4⋊C4—416no—
383C4⋊C4—416no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—84no—
392C4⋊C4—88no—
393C4⋊C4C22⋊Q814yes—
394C4⋊C4#12714yes—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C22⋊C4—84no—
398C22⋊C4—84no—
399C42—22no—
400C42—22no—
401C42—24no—
402C42—48no—
403C42—42no—
404C42—42no—
405C42—42no—
406C42—42no—
407C23—42no—
408C23—44no—
409C23—42no—
410C23—42no—
411C23—44no—
412C23—81no—
413C23—81no—
414C23—82no—
415C23—81no—
416C23—82no—
417C23—81no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—81no—
423C23—81no—
424C23—82no—
425C23—82no—
426C23—81no—
427C23—81no—
428C23—82no—
429C23—82no—
430C23C2×#12011yescharacteristic
431C23#53612yes—
432C23D4212yes—
433C23#54811yescharacteristic
434C23D4×Q811yescharacteristic
435C23C22×SD1611yescharacteristic
436C23C2×AGL(1,ℤ/8)12yes—
437C23C2×#13711yescharacteristic
438C23#57611yescharacteristic
439C23#57711yescharacteristic
440C23#57712yes—
441Q8—832no—
442D4—42no—
443D4—44no—
444D4—42no—
445D4—81no—
446D4—81no—
447D4—82no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4#44814yes—
501C2×C4C2×C22⋊Q811yescharacteristic
502C2×C4C2×#12711yescharacteristic
503C2×C4#53511yescharacteristic
504C2×C4#53611yescharacteristic
505C2×C4#53611yescharacteristic
506C2×C4#53611yescharacteristic
507C2×C4#53612yes—
508C2×C4#53612yes—
509C2×C4#53612yes—
510C2×C4#54212yes—
511C2×C4#54412yes—
512C2×C4D4212yes—
513C2×C4#54712yes—
514C2×C4#54811yescharacteristic
515C2×C4D4×Q811yescharacteristic
516C2×C4D4×Q812yes—
517C2×C4#56312yes—
518C2×C4#56312yes—
519C8—44no—
520C8—42no—
521C8—81no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—84no—
526C22—84no—
527C22—162no—
528C22—162no—
529C22—161no—
530C22—161no—
531C22—161no—
532C22—161no—
533C22#240911yescharacteristic
534C22#243311yescharacteristic
535C22#261411yescharacteristic
536C22#267412yes—
537C22#267712yes—
538C22C2×#44811yescharacteristic
539C22#300311yescharacteristic
540C22#301011yescharacteristic
541C22#301211yescharacteristic
542C22#322712yes—
543C22#323912yes—
544C22#324112yes—
545C22#324412yes—
546C22#326412yes—
547C22#326512yes—
548C22#327812yes—
549C22D4×SD1612yes—
550C22#336211yescharacteristic
551C22#336911yescharacteristic
552C22#337312yes—
553C22Q8×SD1611yescharacteristic
554C22#338412yes—
555C22#338511yescharacteristic
556C4—24no—
557C4—44no—
558C4—44no—
559C4—44no—
560C4—44no—
561C4—42no—
562C4—84no—
563C4—84no—
564C4—161no—
565C4#254211yescharacteristic
566C4#262811yescharacteristic
567C4#265811yescharacteristic
568C4#267412yes—
569C4#267712yes—
570C4#282111yescharacteristic
571C2—162no—
572C2#1651111yescharacteristic
573C2#2432911yescharacteristic
574C2#2433811yescharacteristic
575C2#2575611yescharacteristic
576C2#2576311yescharacteristic
577C2#2963911yescharacteristic
578C2#2964711yescharacteristic
579C2#3081312yes—
580C2#3082412yes—
581C2#3095412yes—
582C2#3096212yes—
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
1. C2×#2551C2semidirect
3. C2×#2058C2semidirect
7. #11357C2semidirect
8. #11346C2semidirect
9. #11327C2semidirect
11. #10807C2semidirect
91. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#20942
C2×#20582
C2×#20232
C2×#19262
#151482
#113572
#113462
#113272
#110662
#108072