Finite Group Databasebeta

← #406166

Subgroups — #406166

Order 512.

1956 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C212yesmaximal
2C2×#2857C211yesmaximal, characteristic
3C2×#1873C212yesmaximal
4#13756C211yesmaximal, characteristic
5#11324C211yesmaximal, characteristic, complemented
6#11323C211yesmaximal, characteristic, complemented
7#11124C212yesmaximal, complemented
8#11122C212yesmaximal, complemented
9#11010C211yesmaximal, characteristic, complemented
10#11010C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#485—28no—
18C2×#485—28no—
19C2×#485—28no—
20C22×#107C2212yes—
21C22×#106C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×#103C2212yes—
26C22×C22⋊C8C2212yes—
27#2857—28no—
28C2×#391C2212yes—
29C2×#389—22no—
30C2×#384C2211yescharacteristic
31C2×#380—22no—
32#1873—28no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C22⋊#95—22no—
43C4×#95—22no—
44C23×Q8C2312yes—
45C24×C4D412yes—
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—42no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8D418yes—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—24no—
63C2×#117—24no—
64C2×#117—24no—
65C2×#117—42no—
66C2×#117D414yes—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C42—24no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—42no—
84C22×C42D412yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#485—264no—
88C2×#107—24no—
89C2×#107—24no—
90C2×#107—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—42no—
105C2×#103—42no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8—24no—
108C2×C22⋊C8D418yes—
109#391—24no—
110#391—24no—
111#389—216no—
112#384—24no—
113#380—216no—
114C2×#95—24no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119#340—24no—
120#340—24no—
121#336—22no—
122#336—22no—
123#336—42no—
124C25C2×D411yescharacteristic
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C2×D414yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D414yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D414yes—
163C23×C4C2×D412yes—
164C23×C4#4112yes—
165C23×C4C2411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8C2×D414yes—
179C22×C8C2×D414yes—
180C4⋊Q8—48no—
181C4⋊Q8—48no—
182C22⋊Q8—216no—
183C22⋊Q8—216no—
184C22⋊Q8—48no—
185C22⋊Q8—48no—
186#117—216no—
187#117—48no—
188#117—48no—
189#117—48no—
190C2×C4⋊C4—22no—
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—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4#4112yes—
220C2×C4⋊C4#4112yes—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—48no—
241C2×C42—48no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42#4112yes—
248C2×C42#4112yes—
249#107—216no—
250#106—216no—
251#103—216no—
252#103—216no—
253#103—48no—
254#103—48no—
255C22⋊C8—216no—
256#95—216no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24#12012yes—
270C24#12112yes—
271C24#12712yes—
272C24C22×D411yescharacteristic
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—216no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4#12018yes—
369C22×C4#12012yes—
370C22×C4#12118yes—
371C22×C4#12112yes—
372C22×C4#12314yes—
373C22×C4#12712yes—
374C22×C4#13514yes—
375C22×C4#13714yes—
376C22×C4#13714yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4H(5,𝔽2)12yes—
386C22×C42−1+412yes—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—416no—
408C4⋊C4—416no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—416no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—416no—
419C42—24no—
420C42—24no—
421C42—48no—
422C42—48no—
423C42—42no—
424C42—42no—
425C42—48no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—22no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23#39214yes—
449C23C2×#12011yescharacteristic
450C23C2×#12111yescharacteristic
451C23C2×#12311yescharacteristic
452C23#52511yescharacteristic
453C23C2×#12711yescharacteristic
454C23#53411yescharacteristic
455C23#53611yescharacteristic
456C23C2×#13512yes—
457C23C2×#13712yes—
458C23C2×#13712yes—
459C23#57512yes—
460Q8—44no—
461Q8—44no—
462Q8—44no—
463Q8—44no—
464Q8—82no—
465Q8—82no—
466Q8—82no—
467Q8—82no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—432no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4—82no—
524C2×C4#39214yes—
525C2×C4#48518yescomplemented
526C2×C4C2×#12012yes—
527C2×C4C2×#12112yes—
528C2×C4#52911yescharacteristic
529C2×C4#52911yescharacteristic
530C2×C4#53211yescharacteristic
531C2×C4#53211yescharacteristic
532C2×C4#53712yes—
533C2×C4#53912yes—
534C2×C4#54612yes—
535C2×C4#54612yes—
536C2×C4#54612yes—
537C2×C4#54612yes—
538C2×C4#54712yes—
539C2×C4#54712yes—
540C2×C4#54712yes—
541C2×C4#54712yes—
542C8—44no—
543C8—44no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—42no—
553C22—42no—
554C22—44no—
555C22C2×#39211yescharacteristic
556C22#240511yescharacteristic
557C22#240611yescharacteristic
558C22#250111yescharacteristic
559C22#253911yescharacteristic
560C22#261811yescharacteristic
561C22#285411yescharacteristic
562C22#300212yes—
563C22#300412yes—
564C22#303512yes—
565C22#303912yes—
566C22C2×#48512yes—
567C22#309512yes—
568C22#310712yes—
569C22#311612yes—
570C22#314711yescharacteristic
571C22#315011yescharacteristic
572C22#315211yescharacteristic
573C22#315211yescharacteristic
574C22#319012yes—
575C22#319612yes—
576C22#323112yes—
577C22#323712yes—
578C4—24no—
579C4—28no—
580C4—44no—
581C4—84no—
582C4—82no—
583C4—84no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4#242512yes—
588C4#242512yes—
589C4#243012yes—
590C4#243012yes—
591C2—44no—
592C2#1492611yescharacteristic
593C2#2342111yescharacteristic
594C2#2343211yescharacteristic
595C2#2401012yes—
596C2#2406512yes—
597C2#2790811yescharacteristic
598C2#2791611yescharacteristic
599C2#2830811yescharacteristic
600C2#2831711yescharacteristic
601C2#2975912yes—
602C2#3311412yes—
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
5. #11324C2semidirect
6. #11323C2semidirect
7. #11124C2semidirect
8. #11122C2semidirect
9. #11010C2semidirect
10. #11010C2semidirect
525. C2×C4#485semidirect

Maximal subgroups

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

subgroupindex
C22×#4852
C2×#28572
C2×#18732
#137562
#113242
#113232
#111242
#111222
#110102
#107122