Finite Group Databasebeta

← #358124

Subgroups — #358124

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×C4×#101C212yesmaximal, complemented
4C4×#407C211yesmaximal, characteristic
5#11204C212yesmaximal, complemented
6#11198C212yesmaximal
7#11045C212yesmaximal, complemented
8#10968C212yesmaximal
9C4×#343C212yesmaximal, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C23×C42C2211yescharacteristic
13#2920—28no—
14C2×#432—21no—
15C2×#432—21no—
16C2×#431—21no—
17C2×#431—21no—
18C2×#423C2212yes—
19C22×#101C2214yes—
20C2×#407C2212yes—
21C2×#407C2212yes—
22C2×C4×M16C2212yes—
23C2×C4×M16C2212yes—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27C4×C22⋊Q8—21no—
28C4×C22⋊Q8—21no—
29C4×C22⋊Q8—21no—
30C4×C22⋊Q8—21no—
31C4×C22⋊Q8—21no—
32C2×#391C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#387C412yes—
35C2×#385C412yes—
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C412yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C412yes—
40C4×#101—28no—
41#1749—21no—
42#1749—21no—
43C4×C22⋊C8—21no—
44C4×C22⋊C8—21no—
45C2×#343C2214yes—
46C2×#343C2214yes—
47C2×#343C2214yes—
48#1728C2212yes—
49C23×Q8C2311yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2311yescharacteristic
53C22×M16C2314yes—
54C22×M16C2314yes—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—24no—
61C2×C4×Q8—24no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—24no—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C42—24no—
76C22×C42—24no—
77C22×C42C2×C412yes—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C412yes—
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#432—28no—
92#432—28no—
93#431—28no—
94#431—28no—
95#423—28no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×#101—28no—
99C2×#101—28no—
100C2×#101—28no—
101#407—28no—
102#407—28no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C4×M16—24no—
108C4×M16—24no—
109C4×M16—24no—
110C4×M16D418yes—
111C2×#97—22no—
112C2×#97—22no—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115#391—21no—
116#391—21no—
117#391—21no—
118#391—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#387—22no—
124#387—22no—
125#387—22no—
126#387—22no—
127#385—22no—
128#385—22no—
129#385—22no—
130#385—22no—
131#380—21no—
132#380—21no—
133#380—21no—
134#380—21no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C2×#95C2×C412yes—
148C2×#95C2×C412yes—
149C2×#95C2×C412yes—
150C2×#95C2×C412yes—
151C2×#95C2×C412yes—
152#343—28no—
153#343—28no—
154#343—28no—
155#336—22no—
156#336—22no—
157C25C22×C411yescharacteristic
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—24no—
164C22×Q8#4114yes—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C412yes—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C412yes—
190C23×C4C22×C412yes—
191C23×C4C22×C412yes—
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2411yescharacteristic
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—24no—
201C2×M16—24no—
202C2×M16—24no—
203C2×M16—24no—
204C2×M16—24no—
205C2×M16—24no—
206C2×M16—24no—
207C2×M16—24no—
208C2×M16C2×D418yes—
209C2×M16#4118yes—
210C22×C8—24no—
211C22×C8—24no—
212C22⋊Q8—28no—
213C22⋊Q8—28no—
214C22⋊Q8—28no—
215C22⋊Q8—28no—
216C4×Q8—42no—
217C4×Q8—42no—
218C4×Q8—44no—
219C4×Q8—44no—
220C4×Q8—42no—
221C4×Q8—42no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4#4114yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—44no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—28no—
271C2×C42—44no—
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D414yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42#4114yes—
290C4⋊C8—28no—
291C4⋊C8—28no—
292#101—232no—
293C22⋊C8—28no—
294C22⋊C8—28no—
295C22⋊C8—28no—
296C22⋊C8—28no—
297#97—28no—
298#97—44no—
299C4×C8—28no—
300C4×C8—44no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—41no—
318C24—41no—
319C24C2×C22⋊C411yescharacteristic
320C24C2×C22⋊C411yescharacteristic
321C24C23×C411yescharacteristic
322C2×Q8—28no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—28no—
326C2×Q8—41no—
327C2×Q8—42no—
328C2×Q8—41no—
329C2×Q8—42no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—42no—
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—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—416no—
395C22×C4—416no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4#10114yescomplemented
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4#11714yes—
418C22×C4#11714yes—
419C22×C4C4×D414yes—
420C22×C4C4×D414yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4114yes—
428M16—28no—
429M16—28no—
430M16—28no—
431M16—28no—
432M16—44no—
433M16—44no—
434M16—44no—
435M16—44no—
436C2×C8—28no—
437C2×C8—28no—
438C2×C8—44no—
439C2×C8—44no—
440C2×C8—44no—
441C2×C8—44no—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—44no—
447C4⋊C4—41no—
448C4⋊C4—42no—
449C4⋊C4—41no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C22⋊C4—28no—
456C22⋊C4—28no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C42—21no—
460C42—21no—
461C42—22no—
462C42—22no—
463C42—21no—
464C42—21no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—44no—
474C42—44no—
475C42—44no—
476C42—44no—
477C42#12012yes—
478C42#12012yes—
479C42#12114yes—
480C42#12114yes—
481C42#12312yes—
482C42#12312yes—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—24no—
488C23—24no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23C2×#10112yes—
510C23#41312yes—
511C23C22×C22⋊C411yescharacteristic
512C23C2×#11711yescharacteristic
513C23C2×#11711yescharacteristic
514C23C2×C4×D412yes—
515C23C4×#4112yes—
516Q8—44no—
517Q8—48no—
518Q8—44no—
519C2×C4—28no—
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—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4#38612yes—
580C2×C4#38612yes—
581C2×C4#38612yes—
582C2×C4#38612yes—
583C2×C4C2×#10112yes—
584C2×C4C2×#10112yes—
585C2×C4#41312yes—
586C2×C4#41312yes—
587C2×C4C2×#12011yescharacteristic
588C2×C4C2×#12112yes—
589C2×C4C2×#12311yescharacteristic
590C2×C4#52514yes—
591C2×C4#52512yes—
592C2×C4#52512yes—
593C2×C4#52512yes—
594C2×C4#52914yes—
595C2×C4#52914yes—
596C2×C4#53214yes—
597C2×C4#53212yes—
598C8—48no—
599C8—48no—
600C22—22no—
601C22—22no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—48no—
615C22—48no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22#228011yescharacteristic
625C22C2×#38611yescharacteristic
626C22#229012yes—
627C22#229312yes—
628C22#229711yescharacteristic
629C22C22×#10111yescharacteristic
630C22C2×#41311yescharacteristic
631C22C2×#41311yescharacteristic
632C22C2×#41311yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—44no—
641C4—44no—
642C4—41no—
643C4—41no—
644C4—82no—
645C4—82no—
646C4#190114yes—
647C4#238812yes—
648C4#239214yes—
649C4#239512yes—
650C2—22no—
651C2—22no—
652C2—42no—
653C2—42no—
654C2#1383911yescharacteristic
655C2#2072711yescharacteristic
656C2#2072811yescharacteristic
657C2C2×#190111yescharacteristic
658C2#2098911yescharacteristic
659C2#2142511yescharacteristic
660C2#2142611yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C4×#101C2semidirect
5. #11204C2semidirect
7. #11045C2semidirect
9. C4×#343C2semidirect
406. C22×C4#101semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×C4×C22⋊Q82
C2×C4×#1012
C4×#4072
#112042
#111982
#110452
#109682
C4×#3432