Finite Group Databasebeta

← #162819

Subgroups — #162819

Order 512.

1322 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
1#36625C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic
3C2×#1728C211yesmaximal, characteristic
4#7711C214yesmaximal, complemented
5#7668C212yesmaximal, complemented
6#7665C212yesmaximal, complemented
7#7632C214yesmaximal, complemented
8C23×M16—21no—
9C2×#533C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#423—21no—
12C2×#407—21no—
13#2857C411yescharacteristic
14#2857C411yescharacteristic
15#2857C2211yescharacteristic
16#2857C2211yescharacteristic
17#2855—21no—
18#2855—21no—
19#2851C412yes—
20#2851C2212yes—
21#2848—21no—
22#2848—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27#2296C2212yes—
28#2285C411yescharacteristic
29#2280C412yes—
30#1797—21no—
31#1797—21no—
32#1797C2212yes—
33#1797C2212yes—
34#1728—21no—
35#1728—21no—
36#1728C2212yes—
37#1728C2212yes—
38#1728C2212yes—
39C2×#336—21no—
40C2×#336C2212yes—
41#1271C2218yescomplemented
42#1271C2218yescomplemented
43C24×C4C2×C411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C22×M16—28no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—44no—
51C22×M16—44no—
52C23×C8—22no—
53#533—44no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×#117C2×C412yes—
66C2×#117C2×C412yes—
67C22×C42—44no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#423—44no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83#407—44no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86#391—21no—
87#391—21no—
88#391—21no—
89#391—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2312yes—
97#389C2312yes—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#382C2×C412yes—
111#381—22no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380C2×C412yes—
119#380C2×C412yes—
120#380C2×C412yes—
121#380C2×C412yes—
122#379—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336C2314yes—
134#336C2314yes—
135C25C22⋊C411yescharacteristic
136C22×Q8—21no—
137C22×Q8—21no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C2×M16—28no—
168C2×M16—28no—
169C2×M16—432no—
170C2×M16—44no—
171C2×M16—44no—
172C2×M16—44no—
173C2×M16—44no—
174C2×M16—44no—
175C2×M16—44no—
176C2×M16—44no—
177C22×C8—28no—
178C22×C8—24no—
179C22×C8—24no—
180C22×C8—44no—
181C22×C8—44no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184#125—44no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193C4×Q8—44no—
194C4×Q8—44no—
195#117—44no—
196#117—44no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2411yescharacteristic
249C4⋊C8—44no—
250C4⋊C8—44no—
251C22⋊C8—44no—
252C22⋊C8—44no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—41no—
269C24—41no—
270C24C2×C22⋊C411yescharacteristic
271C24C2×C22⋊C411yescharacteristic
272C24C2×C22⋊C411yescharacteristic
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C22×C4—216no—
280C22×C4—216no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
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—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—48no—
339C22×C4—416no—
340C22×C4—48no—
341C22×C4#9914yes—
342C22×C4#10114yes—
343C22×C4#10414yes—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4AGL(1,ℤ/8)12yes—
354C22×C4AGL(1,ℤ/8)12yes—
355C22×C4#13712yes—
356C22×C4#13712yes—
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360M16—416no—
361M16—416no—
362M16—88no—
363C2×C8—28no—
364C2×C8—44no—
365C2×C8—416no—
366C2×C8—44no—
367C2×C8—44no—
368C2×C8—44no—
369C2×C8—44no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—44no—
411C42—44no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—28no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23C2×#9912yes—
436C23C2×#10112yes—
437C23#41712yes—
438C23#41712yes—
439C23C2×#10412yes—
440C23C22×C22⋊C411yescharacteristic
441Q8—84no—
442Q8—84no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4C2×#9912yes—
504C2×C4C2×#9912yes—
505C2×C4C2×#10112yes—
506C2×C4C2×#10112yes—
507C2×C4#41712yes—
508C2×C4#41712yes—
509C2×C4C2×#10412yes—
510C2×C4C2×#10412yes—
511C2×C4#42112yes—
512C2×C4#56812yes—
513C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
514C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
515C2×C4C2×#13711yescharacteristic
516C2×C4C2×#13711yescharacteristic
517C8—48no—
518C8—84no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—416no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22#156114yes—
541C22#287911yescharacteristic
542C22C22×#9911yescharacteristic
543C22C22×#10111yescharacteristic
544C22C2×#41711yescharacteristic
545C22C2×#41711yescharacteristic
546C22C22×#10411yescharacteristic
547C22C2×#42111yescharacteristic
548C4—21no—
549C4—21no—
550C4—22no—
551C4—22no—
552C4—21no—
553C4—21no—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—41no—
560C4—41no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4#150414yes—
566C2—24no—
567C2—42no—
568C2—42no—
569C2#1739711yescharacteristic
570C2#1749811yescharacteristic
571C2#1750411yescharacteristic
572C2C2×#150411yescharacteristic
573C2#1759211yescharacteristic
574C2#1764811yescharacteristic
575C2C2×#156111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7711C2semidirect
5. #7668C2semidirect
6. #7665C2semidirect
7. #7632C2semidirect
41. #1271C22semidirect
42. #1271C22semidirect

Maximal subgroups

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

subgroupindex
#366252
#206262
C2×#17282
#77112
#76682
#76652
#76322