Finite Group Databasebeta

← #208416

Subgroups — #208416

Order 512.

1628 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3C2×#1434C211yesmaximal, characteristic, complemented
4#9911C212yesmaximal, complemented
5#9715C212yesmaximal, complemented
6#9147C214yesmaximal, complemented
7#8216C214yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C2×#432—21no—
12#2857C411yescharacteristic, complemented
13#2857C2211yescharacteristic
14#2856C411yescharacteristic, complemented
15#2856C2211yescharacteristic
16#2854—21no—
17#2854C412yescomplemented
18#2850—21no—
19#2850—21no—
20#2847—21no—
21#2847C2212yes—
22C4×#121—21no—
23C4×#121—21no—
24C4×#121C412yescomplemented
25C4×#121C2212yes—
26#2297—21no—
27#2285C411yescharacteristic, complemented
28C22×#95—21no—
29C22×#95—21no—
30C22×#95C2212yes—
31#1817—21no—
32#1797—21no—
33#1797C2212yes—
34#1779C2212yes—
35#1749—21no—
36#1749C2212yes—
37C2×#327C2218yes—
38#1434—21no—
39#1434—21no—
40#1434C2212yes—
41#1434C2212yes—
42#1294C2218yes—
43C22×#41—21no—
44C24×C4—22no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48#532—44no—
49C2×C4⋊Q8C2×C411yescharacteristic, complemented
50C2×#127C2×C411yescharacteristic, complemented
51C2×#124—21no—
52#525—48no—
53C2×#123—22no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121C2×C412yes—
62C2×#121C2×C412yes—
63C2×#121C2312yes—
64C2×#120—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yescomplemented
69C2×#117—21no—
70C22×C42—44no—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#432—44no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8C2314yes—
88C2×#97—22no—
89#391—21no—
90#391—21no—
91#390—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390C2×C412yescomplemented
96#390C2×C412yescomplemented
97#389—21no—
98#389—21no—
99#389C2×C412yescomplemented
100#389C2×C412yescomplemented
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#384C2×C411yescharacteristic, complemented
112#384C2×C411yescharacteristic, complemented
113#382—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380C2×C412yes—
118#380C2×C412yes—
119#380C2312yes—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C412yescomplemented
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C2×#95—22no—
125C2×#95—28no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—44no—
131C2×#95—44no—
132C2×#95C2314yes—
133#336—22no—
134#336—22no—
135#327—216no—
136C25C2×D411yescharacteristic
137C2×#41—44no—
138C2×#41—44no—
139C2×#41—44no—
140C22×Q8—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C22×C412yes—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—28no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
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—
185C4⋊Q8—42no—
186#127—42no—
187#124—44no—
188#123—48no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191#121—28no—
192#121—44no—
193#121—44no—
194#121—44no—
195#121—44no—
196#121—44no—
197#121—44no—
198#121—42no—
199#121—42no—
200#120—48no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—48no—
204C4×D4—48no—
205#117—44no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C412yes—
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—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
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×C42—42no—
239C2×C42—42no—
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C4⋊C8—44no—
257C22⋊C8—28no—
258C22⋊C8—44no—
259C22⋊C8—44no—
260#97—44no—
261#95—28no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—24no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95#4118yescomplemented
274C24—22no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24C2×C22⋊C411yescharacteristic
283C24C2×C22⋊C411yescharacteristic
284C24AGL(1,ℤ/8)14yes—
285C24C22×D411yescharacteristic
286#41—816no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×D4—24no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—44no—
298C2×D4—44no—
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—42no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
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—28no—
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—416no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—416no—
371C22×C4—44no—
372C22×C4—416no—
373C22×C4—416no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—416no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—88no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4#9914yes—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4AGL(1,ℤ/8)12yes—
397C22×C4#13712yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C2×#4114yes—
401C2×C8—44no—
402C2×C8—44no—
403C2×C8—44no—
404C4⋊C4—24no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—44no—
442C23—28no—
443C23—24no—
444C23—28no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—28no—
461C23—28no—
462C23—21no—
463C23—21no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23C2×#9912yes—
472C23#41012yes—
473C23#41812yes—
474C23#41812yes—
475C23#42112yes—
476C23#42112yes—
477C23C22×C22⋊C411yescharacteristic
478C23#52512yes—
479C23C2×AGL(1,ℤ/8)11yescharacteristic
480C23C2×AGL(1,ℤ/8)12yes—
481C23C2×#13711yescharacteristic
482Q8—84no—
483D4—84no—
484D4—84no—
485D4—84no—
486D4—84no—
487D4—84no—
488D4—88no—
489D4—88no—
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—28no—
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—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
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—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—88no—
553C2×C4—832no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4C2×#9912yes—
559C2×C4C2×#9912yes—
560C2×C4#41712yes—
561C2×C4#41712yes—
562C2×C4C4×#4112yes—
563C2×C4#53212yes—
564C8—84no—
565C22—216no—
566C22—216no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—21no—
580C22—21no—
581C22—22no—
582C22—22no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—22no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22#151014yes—
595C22#151414yes—
596C22#229711yescharacteristic
597C22C22×#9911yescharacteristic
598C22C2×#41011yescharacteristic
599C22C2×#41811yescharacteristic
600C22C2×#41711yescharacteristic
601C22C2×#42111yescharacteristic
602C22C2×#42111yescharacteristic
603C22#310912yes—
604C22#311212yes—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—88no—
618C4—82no—
619C4—82no—
620C4—82no—
621C2—22no—
622C2—22no—
623C2—24no—
624C2—24no—
625C2—162no—
626C2C2×#151011yescharacteristic
627C2C2×#151411yescharacteristic
628C2#1758811yescharacteristic
629C2#1759211yescharacteristic
630C2#2139511yescharacteristic
631C2#2141711yescharacteristic
632C2#2142011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36624C2
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. C2×#1434C2semidirect
4. #9911C2semidirect
5. #9715C2semidirect
6. #9147C2semidirect
7. #8216C2semidirect
8. C2×#532C4semidirect
12. #2857C4semidirect
14. #2856C4semidirect
17. #2854C4semidirect
24. C4×#121C4semidirect
27. #2285C4semidirect
49. C2×C4⋊Q8C2×C4semidirect
50. C2×#127C2×C4semidirect
68. C2×C4×D4C2×C4semidirect
95. #390C2×C4semidirect
96. #390C2×C4semidirect
99. #389C2×C4semidirect
100. #389C2×C4semidirect
111. #384C2×C4semidirect
112. #384C2×C4semidirect
121. C4×C4⋊C4C2×C4semidirect
273. #95#41semidirect

Maximal subgroups

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

subgroupindex
#366242
#207152
C2×#14342
#99112
#97152
#91472
#82162