Finite Group Databasebeta

← #196360

Subgroups — #196360

Order 512.

1607 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#431C211yesmaximal, characteristic, complemented
2#36621C211yesmaximal, characteristic
3C2×#1500C212yesmaximal, complemented
4#13753C211yesmaximal, characteristic, complemented
5#9114C212yesmaximal, complemented
6#9113C212yesmaximal
7#9112C212yesmaximal, complemented
8#9111C212yesmaximal, complemented
9#8013C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#431—21no—
13C2×#431—21no—
14C2×#431—28no—
15C2×#431C2212yes—
16C2×#431C2212yes—
17C22×C4⋊C8—21no—
18C22×C22⋊C8C2212yes—
19C22×C4×C8—21no—
20#2855C2211yescharacteristic
21#2854C2211yescharacteristic
22#2853—21no—
23#2852—21no—
24#2852—21no—
25#2851C411yescharacteristic, complemented
26#2851C411yescharacteristic, complemented
27#2850C411yescharacteristic, complemented
28#2850C411yescharacteristic, complemented
29#2849—21no—
30#2848C2211yescharacteristic
31#2847C2211yescharacteristic
32C4×C22⋊Q8C411yescharacteristic, complemented
33C4×C22⋊Q8C411yescharacteristic, complemented
34C4×#123—21no—
35C4×#121C411yescharacteristic, complemented
36C4×#121C411yescharacteristic, complemented
37C4×#120—21no—
38#2290—21no—
39#2290—21no—
40#2283C2211yescharacteristic
41C2×C4×C22⋊C4—21no—
42C2×#376C2212yes—
43#1500—28no—
44C2×#328C2214yes—
45C2×#327C2214yes—
46#1434—21no—
47#1434C2212yes—
48C22⋊#95—21no—
49C22⋊#95C2212yes—
50C4×#95—21no—
51#1294C2214yes—
52#1294C2214yes—
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4C2311yescharacteristic
56C23×C8—22no—
57#529—24no—
58#529—24no—
59C2×#126—21no—
60C2×#125C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#123—21no—
63C2×#123—21no—
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8C2311yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×#121C2×C411yescharacteristic
69C2×#121C2311yescharacteristic
70C2×#120—21no—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—21no—
77C22×C22⋊C4—22no—
78C22×C42—24no—
79C22×C42—24no—
80C22×C42—22no—
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#431—28no—
94#431—28no—
95#431—24no—
96#431—28no—
97#431D418yescomplemented
98C2×C4⋊C8—28no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—28no—
103C2×C22⋊C8C2314yes—
104C2×C4×C8—28no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107#391C2×C411yescharacteristic
108#391C2×C411yescharacteristic
109#390C2×C411yescharacteristic
110#390C2×C411yescharacteristic
111#389C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#389C2×C411yescharacteristic
115#388—21no—
116#388—21no—
117#388—21no—
118#388—21no—
119#388—21no—
120#388—21no—
121#388—21no—
122#388—21no—
123#387—21no—
124#387—21no—
125#387—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#386—21no—
132#386—21no—
133#386—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#385—21no—
147#385—21no—
148#385—21no—
149#385—21no—
150#385—21no—
151#383—21no—
152#382C2×C411yescharacteristic
153#382C2×C411yescharacteristic
154#381—21no—
155#381—21no—
156#380C2×C411yescharacteristic
157#380C2×C411yescharacteristic
158#380C2×C411yescharacteristic
159#380C2×C411yescharacteristic
160#380C2311yescharacteristic
161#380C2311yescharacteristic
162#379—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4C2×C411yescharacteristic
166C4×C4⋊C4C2×C411yescharacteristic
167C4×C22⋊C4—28no—
168C4×C22⋊C4—21no—
169C4×C22⋊C4—21no—
170C4×C22⋊C4—21no—
171C4×C22⋊C4—21no—
172C4×C22⋊C4—21no—
173#376—21no—
174#376—24no—
175#376—24no—
176C2×#95—22no—
177C2×#95—22no—
178C2×#95—22no—
179C2×#95—22no—
180C2×#95C2314yes—
181#328—28no—
182#327—28no—
183C25C2×D411yescharacteristic
184C22×Q8C22×C411yescharacteristic
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C22×C411yescharacteristic
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—21no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C22⋊C411yescharacteristic
205C23×C4C22⋊C411yescharacteristic
206C23×C4C22⋊C411yescharacteristic
207C23×C4C22⋊C411yescharacteristic
208C23×C4C22⋊C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2411yescharacteristic
219C22×C8—28no—
220C22×C8—24no—
221C22×C8—44no—
222C22×C8—44no—
223#126—42no—
224#126—42no—
225#125—24no—
226#124—24no—
227#123—44no—
228#123—44no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—42no—
232C22⋊Q8—42no—
233#121—22no—
234#121—22no—
235#121—42no—
236#121—42no—
237#120—44no—
238C4×Q8—24no—
239C4×D4—24no—
240C4×D4—44no—
241C4×D4—44no—
242C4×D4—44no—
243#117—42no—
244#117—42no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C22⋊C4—28no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—28no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42C22⋊C411yescharacteristic
306C2×C42C22⋊C411yescharacteristic
307C2×C42C22⋊C411yescharacteristic
308C2×C42C22⋊C411yescharacteristic
309C2×C42C22⋊C411yescharacteristic
310C2×C42C22⋊C411yescharacteristic
311C2×C42C22⋊C411yescharacteristic
312C2×C42C22⋊C411yescharacteristic
313C2×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D414yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42#4114yes—
323C4⋊C8—28no—
324C4⋊C8—28no—
325C22⋊C8—28no—
326C22⋊C8—24no—
327C22⋊C8C2×D418yes—
328C4×C8—28no—
329C4×C8—28no—
330#95—28no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—24no—
352#95—24no—
353#95—44no—
354#95—44no—
355#95—44no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—41no—
362C24—41no—
363C24C2×C22⋊C411yescharacteristic
364C24C2×C22⋊C411yescharacteristic
365C24#13514yes—
366C24C22×D411yescharacteristic
367C2×Q8—22no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×D4—22no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C22×C4—232no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4#10414yes—
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4#12114yes—
466C22×C4#12114yes—
467C22×C4#13514yes—
468C22×C4C23×C411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C2×#4112yes—
472C2×C8—28no—
473C2×C8—416no—
474C2×C8—44no—
475C2×C8—44no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—416no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—416no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42C4×D414yes—
533C42C4×D414yes—
534C42#12012yes—
535C42#12012yes—
536C42#12312yes—
537C42#12312yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—28no—
552C23—28no—
553C23—21no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23#41012yes—
567C23#41612yes—
568C23#41612yes—
569C23C2×#10412yes—
570C23C22×C22⋊C411yescharacteristic
571C23C2×#12112yes—
572C23C2×#13512yes—
573C23C2×#13512yes—
574Q8—82no—
575D4—82no—
576D4—84no—
577D4—84no—
578D4—84no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4#38612yes—
654C2×C4#38612yes—
655C2×C4#38612yes—
656C2×C4#38612yes—
657C2×C4#41012yes—
658C2×C4#41612yes—
659C2×C4#41612yes—
660C2×C4C2×#10412yes—
661C2×C4C2×#10412yes—
662C2×C4#42112yes—
663C2×C4C2×C4×D412yes—
664C2×C4C2×#12011yescharacteristic
665C2×C4C2×#12311yescharacteristic
666C8—48no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—416no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22#150014yescomplemented
695C22C2×#38611yescharacteristic
696C22C2×#41011yescharacteristic
697C22C2×#41011yescharacteristic
698C22C2×#41611yescharacteristic
699C22C2×#41611yescharacteristic
700C22C22×#10411yescharacteristic
701C22C2×#42111yescharacteristic
702C22#305014yes—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—21no—
710C4—21no—
711C4—22no—
712C4—22no—
713C4—41no—
714C4—41no—
715C4—88no—
716C4—81no—
717C4—81no—
718C4—81no—
719C4—81no—
720C4—81no—
721C4—81no—
722C4—81no—
723C4#187114yes—
724C2—24no—
725C2—24no—
726C2—42no—
727C2—161no—
728C2C2×#150011yescharacteristic
729C2#1753311yescharacteristic
730C2#1756711yescharacteristic
731C2#1757111yescharacteristic
732C2C2×#187111yescharacteristic
733C2#2096711yescharacteristic
734C2#2098311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#431C2semidirect
3. C2×#1500C2semidirect
4. #13753C2semidirect
5. #9114C2semidirect
7. #9112C2semidirect
8. #9111C2semidirect
9. #8013C2semidirect
25. #2851C4semidirect
26. #2851C4semidirect
27. #2850C4semidirect
28. #2850C4semidirect
32. C4×C22⋊Q8C4semidirect
33. C4×C22⋊Q8C4semidirect
35. C4×#121C4semidirect
36. C4×#121C4semidirect
97. #431D4semidirect
694. C22#1500semidirect

Maximal subgroups

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

subgroupindex
C22×#4312
#366212
C2×#15002
#137532
#91142
#91132
#91122
#91112
#80132