Finite Group Databasebeta

← #216736

Subgroups — #216736

Order 512.

1504 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3#14686C211yesmaximal, characteristic, complemented
4#9884C212yesmaximal, complemented
5#9680C212yesmaximal, complemented
6#9155C214yesmaximal, complemented
7#8571C214yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C2×#431—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
28C2×#384—21no—
29C2×#380—21no—
30C2×#380—21no—
31C2×#380C2212yes—
32#1817—21no—
33#1797—21no—
34#1797C2212yes—
35#1779C2212yes—
36C4×C22⋊C8—21no—
37C4×C22⋊C8C2212yes—
38C2×#327C2218yes—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42#1294C2218yes—
43C22×#41—21no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47#532—44no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#127C2×C411yescharacteristic
50C2×#124—21no—
51#525—44no—
52#525—44no—
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—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C42—44no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#431—44no—
88C2×C4⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8C2314yes—
92C2×C4×C8—22no—
93#391—21no—
94#391—21no—
95#390—21no—
96#390—21no—
97#390—21no—
98#390—21no—
99#390C2×C412yes—
100#390C2×C412yes—
101#389—21no—
102#389—21no—
103#389C2×C412yes—
104#389C2×C412yes—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#384—44no—
116#384C2×C411yescharacteristic
117#384C2×C411yescharacteristic
118#382—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—44no—
123#380—44no—
124#380—44no—
125#380C2×C412yes—
126#380C2×C412yes—
127#380C2312yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×C412yes—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95C2314yes—
135#336—22no—
136#336—22no—
137#327—48no—
138C25C2×D411yescharacteristic
139C2×#41—44no—
140C2×#41—44no—
141C2×#41—44no—
142C22×Q8—21no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—42no—
156C23×C4—42no—
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×C4D814yes—
170C23×C4SD1614yes—
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2411yescharacteristic
181C22×C8—24no—
182C4⋊Q8—42no—
183#127—42no—
184#124—44no—
185#123—48no—
186C22⋊Q8—44no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189#121—24no—
190#121—24no—
191#121—44no—
192#121—44no—
193#121—44no—
194#121—42no—
195#121—42no—
196#121—42no—
197#121—42no—
198#121—44no—
199#121—44no—
200#120—44no—
201#120—44no—
202C4×D4—44no—
203C4×D4—48no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—44no—
207#117—44no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C4⋊C8—44no—
271C22⋊C8—24no—
272C22⋊C8—24no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275C4×C8—44no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—28no—
283#95—44no—
284#95—44no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—22no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24C2×C22⋊C411yescharacteristic
295C24C2×C22⋊C411yescharacteristic
296C24C2×D812yes—
297C24C2×SD1612yes—
298C24C22×D411yescharacteristic
299#41—816no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×D4—24no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C22×C4—21no—
325C22×C4—21no—
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—21no—
337C22×C4—21no—
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—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—48no—
358C22×C4—48no—
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—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4#10214yes—
392C22×C4#10214yes—
393C22×C4#10214yes—
394C22×C4#10214yes—
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×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×D812yes—
406C22×C4C2×D812yes—
407C22×C4C2×SD1612yes—
408C22×C4C2×SD1612yes—
409C22×C4AGL(1,ℤ/8)12yes—
410C22×C4#13712yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4H(5,𝔽2)12yes—
414C22×C42−1+412yes—
415C2×C8—44no—
416C2×C8—44no—
417C2×C8—44no—
418C4⋊C4—24no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—88no—
429C4⋊C4—88no—
430C4⋊C4—88no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—88no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—44no—
460C42#9914yes—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23C2×#10212yes—
501C23C2×#10212yes—
502C23C22×C22⋊C411yescharacteristic
503C23#53511yescharacteristic
504C23#53611yescharacteristic
505C23C22×D811yescharacteristic
506C23C22×SD1611yescharacteristic
507C23C2×AGL(1,ℤ/8)11yescharacteristic
508C23C2×#13711yescharacteristic
509Q8—84no—
510D4—84no—
511D4—84no—
512D4—88no—
513D4—88no—
514D4—82no—
515D4—82no—
516D4—82no—
517D4—82no—
518D4—84no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—88no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4C2×#9912yes—
583C2×C4C2×#9912yes—
584C2×C4C2×#9912yes—
585C2×C4#41012yes—
586C2×C4C2×#10212yes—
587C2×C4C2×#10212yes—
588C2×C4C2×#10212yes—
589C2×C4C2×#10212yes—
590C2×C4#41712yes—
591C2×C4#41712yes—
592C2×C4#42112yes—
593C2×C4#42112yes—
594C2×C4#52012yes—
595C2×C4#53712yes—
596C8—84no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22#231111yescharacteristic
635C22C22×#9911yescharacteristic
636C22C2×#41011yescharacteristic
637C22C22×#10211yescharacteristic
638C22C2×#41711yescharacteristic
639C22C2×#42111yescharacteristic
640C22C2×#42111yescharacteristic
641C22#320511yescharacteristic
642C22#320611yescharacteristic
643C22#322611yescharacteristic
644C22#322711yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—41no—
654C4—41no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—164no—
659C4#151314yes—
660C4#151914yes—
661C2—21no—
662C2—21no—
663C2—21no—
664C2—21no—
665C2—42no—
666C2—42no—
667C2—162no—
668C2C2×#151311yescharacteristic
669C2C2×#151911yescharacteristic
670C2#1757411yescharacteristic
671C2#1758511yescharacteristic
672C2#2185911yescharacteristic
673C2#2187411yescharacteristic
674C2#2190011yescharacteristic
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. #20714C2semidirect
3. #14686C2semidirect
4. #9884C2semidirect
5. #9680C2semidirect
6. #9155C2semidirect
7. #8571C2semidirect
8. C2×#532C4semidirect
12. #2857C4semidirect
14. #2856C4semidirect
17. #2854C4semidirect
24. C4×#121C4semidirect
27. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#366242
#207142
#146862
#98842
#96802
#91552
#85712