Finite Group Databasebeta

← #216488

Subgroups — #216488

Order 512.

1612 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3#13756C211yesmaximal, characteristic, complemented
4#9908C212yesmaximal, complemented
5#9712C212yesmaximal, complemented
6#9117C214yesmaximal, complemented
7#8577C214yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525—21no—
10C22×#117—21no—
11C23×C42C2211yescharacteristic
12C2×#423—21no—
13C2×#407C2212yes—
14#2857C411yescharacteristic, complemented
15#2857C2211yescharacteristic
16#2856C411yescharacteristic, complemented
17#2856C2211yescharacteristic
18#2854—21no—
19#2854C412yescomplemented
20#2850—21no—
21#2850—21no—
22#2847—21no—
23#2847C2212yes—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121C412yescomplemented
27C4×#121C2212yes—
28#2297—21no—
29#2285C411yescharacteristic, complemented
30#1797—21no—
31#1797—21no—
32#1797C2212yes—
33#1797C2212yes—
34#1728—21no—
35C2×#327C2218yes—
36C4⋊#95—21no—
37C4⋊#95C2212yes—
38C22⋊#95—21no—
39C22⋊#95C2212yes—
40C4×#95—21no—
41C4×#95C2212yes—
42#1294C2218yes—
43C22×#41—21no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48#532—44no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#127C2×C411yescharacteristic
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×C412yes—
69C2×#117—28no—
70C2×#117—21no—
71C2×#117—44no—
72C2×#117—44no—
73C22×C4⋊C4—22no—
74C22×C22⋊C4—22no—
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#423—44no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92#407—44no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8C2314yes—
95#391—21no—
96#391—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390C2×C412yes—
102#390C2×C412yes—
103#389—21no—
104#389—21no—
105#389C2×C412yes—
106#389C2×C412yes—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#384C2×C411yescharacteristic
118#384C2×C411yescharacteristic
119#382—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380C2×C412yes—
124#380C2×C412yes—
125#380C2312yes—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2×C412yes—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95C2314yes—
134#336—22no—
135#336—22no—
136#327—48no—
137C25C2×D411yescharacteristic
138C2×#41—44no—
139C2×#41—44no—
140C2×#41—44no—
141C22×Q8—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C22×C412yes—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2411yescharacteristic
179C2×M16—44no—
180C2×M16—44no—
181C2×M16—44no—
182C22×C8—24no—
183C4⋊Q8—42no—
184#127—42no—
185#124—44no—
186#123—48no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189#121—24no—
190#121—24no—
191#121—44no—
192#121—44no—
193#121—44no—
194#121—44no—
195#121—44no—
196#121—42no—
197#121—42no—
198#121—44no—
199#120—48no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—48no—
203C4×D4—48no—
204#117—432no—
205#117—44no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C22⋊C4—28no—
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×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—28no—
236C2×C42—22no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—44no—
248C2×C42—44no—
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×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C4⋊C8—44no—
266C4⋊C8—44no—
267C22⋊C8—28no—
268C22⋊C8—44no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—24no—
276#95—24no—
277#95—44no—
278#95—44no—
279#95—44no—
280C24—21no—
281C24—21no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—41no—
288C24—41no—
289C24C2×C22⋊C411yescharacteristic
290C24C2×C22⋊C411yescharacteristic
291C24C22×D411yescharacteristic
292#41—816no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
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—42no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C22×C4—216no—
316C22×C4—216no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
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—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
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—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—416no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4#9914yes—
386C22×C4#10414yes—
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×C4C2×C22⋊C411yescharacteristic
397C22×C4AGL(1,ℤ/8)12yes—
398C22×C4AGL(1,ℤ/8)14yes—
399C22×C4#13712yes—
400C22×C4C23×C411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404M16—88no—
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—44no—
408C4⋊C4—24no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—416no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—416no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—416no—
449C42—44no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—24no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23C2×#9912yes—
484C23C2×#10412yes—
485C23C22×C22⋊C411yescharacteristic
486C23#53712yes—
487C23C2×AGL(1,ℤ/8)12yes—
488C23C2×AGL(1,ℤ/8)11yescharacteristic
489C23C2×#13711yescharacteristic
490Q8—84no—
491D4—84no—
492D4—84no—
493D4—88no—
494D4—88no—
495D4—84no—
496D4—84no—
497D4—84no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—28no—
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—24no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—88no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4C2×#9912yes—
563C2×C4C2×#9912yes—
564C2×C4#41012yes—
565C2×C4#41712yes—
566C2×C4#41712yes—
567C2×C4#41812yes—
568C2×C4#41812yes—
569C2×C4C2×#10412yes—
570C2×C4C2×#10412yes—
571C2×C4#42112yes—
572C2×C4#52012yes—
573C2×C4#53511yescharacteristic
574C2×C4#53611yescharacteristic
575C8—84no—
576C22—22no—
577C22—22no—
578C22—21no—
579C22—21no—
580C22—22no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—48no—
591C22—48no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22#231111yescharacteristic
608C22C22×#9911yescharacteristic
609C22C2×#41011yescharacteristic
610C22C2×#41811yescharacteristic
611C22C2×#41711yescharacteristic
612C22C22×#10411yescharacteristic
613C22C2×#42111yescharacteristic
614C22#325312yes—
615C22#325412yes—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—41no—
627C4—41no—
628C4—88no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4#150414yes—
633C2—22no—
634C2—22no—
635C2—42no—
636C2—42no—
637C2—162no—
638C2C2×#150411yescharacteristic
639C2#1754411yescharacteristic
640C2#1756311yescharacteristic
641C2#1757711yescharacteristic
642C2#2185611yescharacteristic
643C2#2187411yescharacteristic
644C2#2189311yescharacteristic
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. #20626C2semidirect
3. #13756C2semidirect
4. #9908C2semidirect
5. #9712C2semidirect
6. #9117C2semidirect
7. #8577C2semidirect
8. C2×#532C4semidirect
14. #2857C4semidirect
16. #2856C4semidirect
19. #2854C4semidirect
26. C4×#121C4semidirect
29. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#366242
#206262
#137562
#99082
#97122
#91172
#85772