Finite Group Databasebeta

← #196369

Subgroups — #196369

Order 512.

1537 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
1C2×#2920C211yesmaximal, characteristic, complemented
2#36621C211yesmaximal, characteristic
3C2×#1500C212yesmaximal, complemented
4#13753C211yesmaximal, characteristic, complemented
5#9118C212yesmaximal, complemented
6#9117C212yesmaximal
7#9116C212yesmaximal, complemented
8#9115C212yesmaximal, complemented
9#8013C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C23×C42C2211yescharacteristic
12#2920—28no—
13C2×#432—21no—
14C2×#432C2212yes—
15C2×#431—21no—
16C2×#431C2212yes—
17C2×#423—21no—
18C2×#407C2212yes—
19C2×C4×M16—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
56C22×M16—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#432—28no—
94#432—28no—
95#431—28no—
96#431—24no—
97#431D418yescomplemented
98#423—28no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101#407—28no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8C2314yes—
104C4×M16—28no—
105C2×#97—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—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—21no—
196C23×C4—42no—
197C23×C4—42no—
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
219C2×M16—28no—
220C2×M16—44no—
221C2×M16—44no—
222C22×C8—24no—
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—
328#97—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—22no—
361C24—21no—
362C24—41no—
363C24—41no—
364C24C2×C22⋊C411yescharacteristic
365C24C2×C22⋊C411yescharacteristic
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—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
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—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
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—416no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4#10414yes—
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×C4C2×C22⋊C411yescharacteristic
466C22×C4#12114yes—
467C22×C4#12114yes—
468C22×C4C23×C411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C2×#4112yes—
472M16—416no—
473C2×C8—28no—
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—42no—
485C4⋊C4—42no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—416no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—41no—
508C22⋊C4—41no—
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—28no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23#41012yes—
569C23C2×#10412yes—
570C23C22×C22⋊C411yescharacteristic
571C23C2×#12112yes—
572C23#57512yes—
573C23#57512yes—
574Q8—82no—
575D4—82no—
576D4—84no—
577D4—84no—
578D4—84no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
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—28no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4—48no—
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—41no—
649C2×C4—41no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4#38612yes—
656C2×C4#38612yes—
657C2×C4#38612yes—
658C2×C4#38612yes—
659C2×C4#41012yes—
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—416no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22C2×#38611yescharacteristic
696C22C2×#41011yescharacteristic
697C22C2×#41011yescharacteristic
698C22#289611yescharacteristic
699C22#289611yescharacteristic
700C22C22×#10411yescharacteristic
701C22C2×#42111yescharacteristic
702C22#305414yes—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—21no—
708C4—21no—
709C4—22no—
710C4—22no—
711C4—41no—
712C4—41no—
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—42no—
726C2—42no—
727C2—161no—
728C2#1753711yescharacteristic
729C2#1754111yescharacteristic
730C2#1755411yescharacteristic
731C2#1755811yescharacteristic
732C2C2×#187111yescharacteristic
733C2#2097011yescharacteristic
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. C2×#2920C2semidirect
3. C2×#1500C2semidirect
4. #13753C2semidirect
5. #9118C2semidirect
7. #9116C2semidirect
8. #9115C2semidirect
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

Maximal subgroups

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

subgroupindex
C2×#29202
#366212
C2×#15002
#137532
#91182
#91172
#91162
#91152
#80132