Finite Group Databasebeta

← #191007

Subgroups — #191007

Order 512.

1756 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3#13753C211yesmaximal, characteristic, complemented
4#8712C212yesmaximal, complemented
5#8575C214yesmaximal, complemented
6#8328C212yesmaximal
7#8218C214yesmaximal, complemented
8C22×#121C2211yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C411yescharacteristic, complemented
11C23×C42C2211yescharacteristic
12C2×#432—21no—
13C4×#121—21no—
14C4×#121—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121C412yescomplemented
18C4×#121C412yescomplemented
19C4×#121C2212yes—
20C4×#121C2212yes—
21C2×#390C411yescharacteristic, complemented
22C2×#389C411yescharacteristic, complemented
23C2×#388—21no—
24C2×#386—21no—
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4C411yescharacteristic, complemented
27C2×C4×C22⋊C4—21no—
28C2×C4×C22⋊C4—21no—
29C2×#376—21no—
30#1817—21no—
31#1797—21no—
32#1797C2212yescomplemented
33#1779C2212yes—
34#1749—21no—
35#1749C2212yescomplemented
36C2×#327C2218yes—
37#1434—21no—
38#1434C2212yescomplemented
39C22⋊#95—21no—
40C22⋊#95C2212yes—
41C4×#95C2212yescomplemented
42#1294C2218yescomplemented
43C23×D4—21no—
44C23×D4C2×C411yescharacteristic
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C2×#121—21no—
50C2×#121—21no—
51C2×#121—28no—
52C2×#121C2×C412yes—
53C2×#121C2312yes—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—44no—
61C2×C4×D4—48no—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2×C412yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—21no—
75C22×C42—22no—
76C22×C42—44no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#432—44no—
90C2×C4⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8C2314yes—
94C2×#97—22no—
95#390—21no—
96#390—21no—
97#390C2×C412yes—
98#390C2×C412yes—
99#389—21no—
100#389—21no—
101#389C2×C412yes—
102#389C2×C412yes—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#380—21no—
112#380—21no—
113#380C2×C412yes—
114#380C2312yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C22⋊C4—28no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124#376—44no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95C2314yes—
133#336—22no—
134#336—22no—
135#327—216no—
136C25—21no—
137C25C2×D411yescharacteristic
138C22×D4—28no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—24no—
142C22×D4—21no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4C22×C412yes—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
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
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C22×C8—24no—
185#121—28no—
186#121—28no—
187#121—24no—
188#121—24no—
189#121—24no—
190#121#4118yes—
191C4×D4—44no—
192C4×D4—44no—
193C4×D4—44no—
194C4×D4—48no—
195C4×D4—48no—
196C4×D4—48no—
197C4×D4—48no—
198C4×D4—44no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—28no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C4⋊C8—44no—
264C22⋊C8—28no—
265C22⋊C8—44no—
266C22⋊C8—44no—
267#97—44no—
268#95—28no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—28no—
276#95—44no—
277#95—44no—
278#95—44no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—24no—
284C24—22no—
285C24—41no—
286C24—41no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24C2×C22⋊C411yescharacteristic
294C24C2×C22⋊C411yescharacteristic
295C24AGL(1,ℤ/8)14yes—
296C24C22×D411yescharacteristic
297C2×D4—24no—
298C2×D4—28no—
299C2×D4—28no—
300C2×D4—22no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—416no—
304C2×D4—42no—
305C2×D4—432no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—42no—
311C2×D4#11714yes—
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—24no—
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—22no—
345C22×C4—22no—
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—22no—
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—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—416no—
370C22×C4—416no—
371C22×C4—416no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—816no—
393C22×C4#10414yes—
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C2×#4114yes—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—44no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4#11714yes—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—416no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—416no—
446C42—44no—
447C42—44no—
448C42#9914yes—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—28no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—28no—
464C23—21no—
465C23—24no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—48no—
472C23—48no—
473C23—48no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—84no—
481C23—84no—
482C23—84no—
483C23—84no—
484C23#41012yes—
485C23#41812yes—
486C23#41812yes—
487C23C2×#10412yes—
488C23C22×C22⋊C411yescharacteristic
489C23#52512yes—
490C23C2×AGL(1,ℤ/8)12yes—
491C23#57512yes—
492D4—44no—
493D4—44no—
494D4—48no—
495D4—416no—
496D4—416no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
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—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—88no—
557C2×C4—88no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4C2×#9912yes—
565C2×C4C2×#9912yes—
566C2×C4C2×#9912yes—
567C2×C4C2×#10412yes—
568C2×C4C2×#10412yes—
569C2×C4#42112yes—
570C2×C4C2×#11712yes—
571C2×C4#52512yes—
572C8—84no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—416no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—816no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22#150514yes—
602C22#228011yescharacteristic
603C22C22×#9911yescharacteristic
604C22C2×#41011yescharacteristic
605C22C2×#41811yescharacteristic
606C22#289611yescharacteristic
607C22C22×#10411yescharacteristic
608C22C2×#42111yescharacteristic
609C22#311414yes—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—44no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—88no—
624C4—82no—
625C4—82no—
626C2—24no—
627C2—24no—
628C2—42no—
629C2—84no—
630C2C2×#150511yescharacteristic
631C2#1754211yescharacteristic
632C2#1757711yescharacteristic
633C2#1758011yescharacteristic
634C2#2072211yescharacteristic
635C2#2073911yescharacteristic
636C2#2074711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. #13753C2semidirect
4. #8712C2semidirect
5. #8575C2semidirect
7. #8218C2semidirect
10. C22×C4×D4C4semidirect
17. C4×#121C4semidirect
18. C4×#121C4semidirect
21. C2×#390C4semidirect
22. C2×#389C4semidirect
26. C2×C4×C4⋊C4C4semidirect
32. #1797C22semidirect
35. #1749C22semidirect
38. #1434C22semidirect
41. C4×#95C22semidirect
42. #1294C22semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#207152
#137532
#87122
#85752
#83282
#82182