Finite Group Databasebeta

← #410013

Subgroups — #410013

Order 512.

2138 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2C2×#2855C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C2×#1823C211yesmaximal, characteristic
6C2×#1817C211yesmaximal, characteristic
7#11327C214yesmaximal, complemented
8#11067C214yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C23×C4⋊C4—21no—
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#483—24no—
16C2×#483—24no—
17C2×#483—24no—
18C22×#106C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2855—24no—
24#2855—24no—
25C2×#391C2212yes—
26C2×#389—22no—
27C2×#389—24no—
28C2×#389—24no—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#387—22no—
32C2×#385—21no—
33C2×#381—21no—
34C2×#380C2211yescharacteristic
35C22×#95—21no—
36#1872—24no—
37#1823—24no—
38#1817—24no—
39C2×#340C2214yes—
40C2×#340C2214yes—
41C2×#340C2212yes—
42C2×#336C2214yes—
43C23×Q8C2311yescharacteristic
44C24×C4—21no—
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—28no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—28no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—44no—
71C22×C4⋊C4—44no—
72C22×C4⋊C4D414yes—
73C22×C4⋊C4D414yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C42—22no—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—22no—
84C22×C42—22no—
85C22×C42—22no—
86C22×C42D412yes—
87C22×C42C2312yes—
88#483—232no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—28no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8D414yes—
103#391—24no—
104#391—24no—
105#389—216no—
106#389—24no—
107#389—24no—
108#389—28no—
109#389—28no—
110#387—216no—
111#385—28no—
112#381—28no—
113#380—22no—
114#380D414yes—
115C2×#95—28no—
116C2×#95—24no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121#340—28no—
122#340—24no—
123#340—28no—
124#336—28no—
125C25C2×D411yescharacteristic
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—24no—
132C22×Q8—24no—
133C22×Q8C2×D412yes—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—28no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—42no—
155C23×C4—44no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4SD1614yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—24no—
178C22×C8—24no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—24no—
183C22×C8—24no—
184C22×C8C2×D414yes—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—216no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—216no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—216no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—432no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4#4118yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42—44no—
249#106—216no—
250#103—216no—
251#103—48no—
252C22⋊C8—28no—
253C22⋊C8—44no—
254#95—216no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—24no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24#12012yes—
264C24#12012yes—
265C24#12112yes—
266C24C2×SD1612yes—
267C24#13512yes—
268C24AGL(1,ℤ/8)12yes—
269C24#13712yes—
270C24C22×D411yescharacteristic
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—28no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—44no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—44no—
282C2×Q8—44no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—28no—
292C22×C4—28no—
293C22×C4—216no—
294C22×C4—28no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—28no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—416no—
319C22×C4—416no—
320C22×C4—44no—
321C22×C4—48no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—48no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4#12014yes—
359C22×C4#12012yes—
360C22×C4#12012yes—
361C22×C4#12114yes—
362C22×C4#12114yes—
363C22×C4#12112yes—
364C22×C4#12714yes—
365C22×C4C2×SD1614yes—
366C22×C4#13714yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4114yes—
374C22×C42−1+414yes—
375C2×C8—24no—
376C2×C8—28no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—44no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—44no—
384C2×C8—44no—
385C4⋊C4—216no—
386C4⋊C4—24no—
387C4⋊C4—28no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—28no—
393C4⋊C4—416no—
394C4⋊C4—432no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—44no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4#12318yes—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—48no—
410C23—24no—
411C23—22no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—22no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23#39214yes—
427C23#44814yes—
428C23#44914yes—
429C23#45014yes—
430C23#45214yes—
431C23#46514yes—
432C23#46814yes—
433C23C2×#12011yescharacteristic
434C23C2×#12011yescharacteristic
435C23C2×#12111yescharacteristic
436C23#52512yes—
437C23#53612yes—
438C23C22×SD1611yescharacteristic
439C23C2×#13511yescharacteristic
440C23C2×AGL(1,ℤ/8)11yescharacteristic
441C23C2×#13712yes—
442C23C2×#13711yescharacteristic
443C23#57512yes—
444Q8—44no—
445Q8—44no—
446Q8—82no—
447Q8—82no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—416no—
464C2×C4—432no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—416no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—816no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4#39214yes—
504C2×C4C2×#12011yescharacteristic
505C2×C4C2×#12111yescharacteristic
506C2×C4C2×#12111yescharacteristic
507C2×C4C2×#12312yes—
508C2×C4#52512yes—
509C2×C4C2×#12711yescharacteristic
510C2×C4#53612yes—
511C2×C4#53612yes—
512C2×C4#54714yes—
513C2×C4#54714yes—
514C2×C4#54714yes—
515C2×C4#55214yes—
516C8—44no—
517C8—82no—
518C22—28no—
519C22—28no—
520C22—28no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—28no—
528C22#200018yes—
529C22C2×#39211yescharacteristic
530C22#240612yes—
531C22#241312yes—
532C22#261411yescharacteristic
533C22#284911yescharacteristic
534C22C2×#45011yescharacteristic
535C22C2×#44811yescharacteristic
536C22C2×#44911yescharacteristic
537C22C2×#45211yescharacteristic
538C22#300411yescharacteristic
539C22#300511yescharacteristic
540C22#300712yes—
541C22C2×#46511yescharacteristic
542C22C2×#46811yescharacteristic
543C22#305612yes—
544C22#310112yes—
545C22#310812yes—
546C22#311312yes—
547C22#311412yes—
548C22#322912yes—
549C22#323212yes—
550C22#323612yes—
551C22#323812yes—
552C4—24no—
553C4—44no—
554C4—44no—
555C4—48no—
556C4—84no—
557C4—84no—
558C4—82no—
559C4—82no—
560C4#240612yes—
561C4#241312yes—
562C4#248612yes—
563C4#250112yes—
564C2—28no—
565C2#1506911yescharacteristic
566C2C2×#200011yescharacteristic
567C2#2339211yescharacteristic
568C2#2406312yes—
569C2#2406612yes—
570C2#2460112yes—
571C2#2460912yes—
572C2#2964111yescharacteristic
573C2#2964311yescharacteristic
574C2#3300711yescharacteristic
575C2#3300811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11327C2semidirect
8. #11067C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C2×#28552
C22×#3892
C2×#18722
C2×#18232
C2×#18172
#113272
#110672
#107302