Finite Group Databasebeta

← #162812

Subgroups — #162812

Order 512.

1408 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4#7710C214yesmaximal, complemented
5#7670C214yesmaximal, complemented
6#7629C212yesmaximal, complemented
7#7626C212yesmaximal, complemented
8C22×#124C411yescharacteristic
9C22×C22⋊Q8C411yescharacteristic
10C22×#121C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#432—21no—
13C2×#407—21no—
14C2×C4×M16—21no—
15#2850—21no—
16#2850—21no—
17#2850—21no—
18#2850—21no—
19#2850C412yes—
20#2850C412yes—
21#2850C2212yes—
22#2850C2212yes—
23C2×#391C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#388—21no—
26C2×#388—21no—
27C2×#386—21no—
28C2×#386—21no—
29C2×#382C411yescharacteristic
30#1817C2212yescomplemented
31#1797—21no—
32#1797C2212yescomplemented
33#1779—21no—
34#1749—21no—
35#1749—21no—
36#1749C2212yes—
37#1749C2212yes—
38C4×C22⋊C8—21no—
39C4×C22⋊C8C2212yescomplemented
40#1728C2212yescomplemented
41#1272C2218yescomplemented
42#1270C2218yescomplemented
43C23×Q8C2×C411yescharacteristic
44C23×D4—21no—
45C23×D4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C22×M16—22no—
50C2×#124—21no—
51C2×#124—21no—
52C2×#124—44no—
53C2×#124C2×C412yes—
54C2×#124C2×C412yes—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—44no—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2×C412yes—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—44no—
63C2×#121C2×C412yes—
64C2×#121C2×C412yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—44no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#432—44no—
88C2×C4⋊C8—22no—
89#407—28no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C4×M16—44no—
96C2×#97—22no—
97C2×#97—22no—
98C2×C4×C8—22no—
99#391—21no—
100#391—21no—
101#391C2×C412yes—
102#391C2312yes—
103#390—21no—
104#390—21no—
105#390C2×C412yes—
106#390C2312yes—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#382—21no—
124#382—21no—
125#382C2×C412yes—
126#382C2×C412yes—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131#336—22no—
132#336—22no—
133#336C2314yes—
134#336C2314yes—
135C25—21no—
136C25C22⋊C411yescharacteristic
137C22×Q8—21no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8C22×C412yes—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4C22×C412yes—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C2×M16—28no—
183C2×M16—44no—
184C2×M16—44no—
185C22×C8—24no—
186C22×C8—24no—
187#124—44no—
188#124—44no—
189#124—44no—
190#124—44no—
191C22⋊Q8—44no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195#121—44no—
196#121—44no—
197#121—44no—
198#121—44no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
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×C22⋊C4—22no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2411yescharacteristic
254C4⋊C8—44no—
255C22⋊C8—28no—
256C22⋊C8—28no—
257C22⋊C8—44no—
258C22⋊C8—44no—
259C22⋊C8—44no—
260#97—44no—
261#97—44no—
262C4×C8—44no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24#10414yes—
286C24C2×C22⋊C411yescharacteristic
287C24C2×C22⋊C411yescharacteristic
288C24C2×C22⋊C411yescharacteristic
289C2×Q8—24no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—88no—
296C2×D4—24no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—42no—
308C2×D4—88no—
309C2×D4—816no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
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—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—416no—
367C22×C4—416no—
368C22×C4—416no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4#10014yes—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C23×C411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391M16—416no—
392C2×C8—28no—
393C2×C8—44no—
394C2×C8—44no—
395C2×C8—44no—
396C2×C8—44no—
397C2×C8—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42#9914yes—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—28no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—84no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23—84no—
467C23—84no—
468C23—84no—
469C23C2×#10012yes—
470C23C2×#10412yes—
471C23C2×#10412yes—
472C23C2×#10412yes—
473C23#42112yes—
474C23C22×C22⋊C411yescharacteristic
475Q8—84no—
476Q8—84no—
477D4—84no—
478D4—84no—
479D4—88no—
480D4—88no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4C2×#9912yes—
540C2×C4C2×#9912yes—
541C2×C4C2×#9912yes—
542C2×C4C2×#10012yes—
543C2×C4C2×#10012yes—
544C2×C4#56812yes—
545C2×C4#57512yes—
546C2×C4#57512yes—
547C8—48no—
548C8—84no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—416no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—168no—
578C22#152714yes—
579C22#287911yescharacteristic
580C22C22×#9911yescharacteristic
581C22C22×#10011yescharacteristic
582C22#289611yescharacteristic
583C22#289611yescharacteristic
584C22C22×#10411yescharacteristic
585C22C2×#42111yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—82no—
599C4—82no—
600C4—82no—
601C2—24no—
602C2—42no—
603C2—42no—
604C2—162no—
605C2#1739611yescharacteristic
606C2#1749711yescharacteristic
607C2#1750211yescharacteristic
608C2#1754211yescharacteristic
609C2#1758211yescharacteristic
610C2C2×#152711yescharacteristic
611C2#1773811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
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
2. #20715C2semidirect
3. C4×#407C2semidirect
4. #7710C2semidirect
5. #7670C2semidirect
6. #7629C2semidirect
7. #7626C2semidirect
30. #1817C22semidirect
32. #1797C22semidirect
39. C4×C22⋊C8C22semidirect
40. #1728C22semidirect
41. #1272C22semidirect
42. #1270C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
#207152
C4×#4072
#77102
#76702
#76292
#76262