Finite Group Databasebeta

← #193476

Subgroups — #193476

Order 512.

1561 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#423C211yesmaximal, characteristic, complemented
2#36621C211yesmaximal, characteristic
3#20714C211yesmaximal, characteristic, complemented
4C2×#1495C212yesmaximal, complemented
5#8944C212yesmaximal, complemented
6#8942C212yesmaximal, complemented
7#8939C212yesmaximal, complemented
8#8937C212yesmaximal, complemented
9#7899C212yesmaximal, complemented
10C23×M16—21no—
11C2×#529C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#431C2212yes—
14C2×#423—21no—
15C2×#423—21no—
16C2×#423—28no—
17C2×#423C2212yes—
18C2×#423C2212yes—
19C22×C4⋊C8—21no—
20C22×C4⋊C8C2212yes—
21#2855C411yescharacteristic
22#2854C411yescharacteristic
23#2853—21no—
24#2852—21no—
25#2852—21no—
26#2851C411yescharacteristic
27#2851C411yescharacteristic
28#2850C411yescharacteristic
29#2850C411yescharacteristic
30#2849—21no—
31#2848C411yescharacteristic
32#2847C411yescharacteristic
33C4×C22⋊Q8C2211yescharacteristic
34C4×C22⋊Q8C2211yescharacteristic
35C4×#123—21no—
36C4×#121C2211yescharacteristic
37C4×#121C2211yescharacteristic
38C4×#120—21no—
39#2290—21no—
40#2290—21no—
41#2283C2211yescharacteristic
42#1817—21no—
43#1797—21no—
44#1797—21no—
45#1779—21no—
46C4×C22⋊C8C2212yes—
47C4×C22⋊C8C2212yes—
48#1495—28no—
49C2×Q8⋊C8C2214yescomplemented
50C2×D4⋊C8C2214yescomplemented
51#1279C2214yescomplemented
52#1279C2214yescomplemented
53C24×C4C2×C411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C22×M16—28no—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16—44no—
60C22×M16—44no—
61C23×C8—22no—
62#529—24no—
63#529—24no—
64C2×#126—21no—
65C2×#125C2×C411yescharacteristic
66C2×#124C2×C411yescharacteristic
67C2×#123—21no—
68C2×#123—21no—
69C2×C22⋊Q8C2×C411yescharacteristic
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×#121C2×C411yescharacteristic
75C2×#120—21no—
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4C2311yescharacteristic
81C2×#117—21no—
82C22×C42—24no—
83C22×C42—24no—
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#431—24no—
97#431—24no—
98#423—28no—
99#423—28no—
100#423—24no—
101#423—28no—
102#423D418yescomplemented
103C2×C4⋊C8—28no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—28no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8C2314yes—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C4×C8C2314yes—
114#391C2×C411yescharacteristic
115#391C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#389C2×C411yescharacteristic
119#389C2×C411yescharacteristic
120#389C2×C411yescharacteristic
121#389C2×C411yescharacteristic
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#387—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#386—21no—
143#386—21no—
144#386—21no—
145#386—21no—
146#386—21no—
147#386—21no—
148#385—21no—
149#385—21no—
150#385—21no—
151#385—21no—
152#385—21no—
153#385—21no—
154#385—21no—
155#385—21no—
156#385—21no—
157#385—21no—
158#383—21no—
159#382C2×C411yescharacteristic
160#382C2×C411yescharacteristic
161#381—21no—
162#381—21no—
163#380C2×C411yescharacteristic
164#380C2×C411yescharacteristic
165#380C2×C411yescharacteristic
166#380C2×C411yescharacteristic
167#380C2×C411yescharacteristic
168#380C2×C411yescharacteristic
169#379—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4C2311yescharacteristic
173C4×C4⋊C4C2311yescharacteristic
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—21no—
179#376—21no—
180#336—22no—
181#336—22no—
182Q8⋊C8—28no—
183D4⋊C8—28no—
184C25C22⋊C411yescharacteristic
185C22×Q8C22×C411yescharacteristic
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4C22×C411yescharacteristic
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—21no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
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×C4C22⋊C411yescharacteristic
212C23×C4C22×C411yescharacteristic
213C23×C4C22×C411yescharacteristic
214C23×C4C22×C411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C2×M16—28no—
222C2×M16—28no—
223C2×M16—432no—
224C2×M16—44no—
225C2×M16—44no—
226C2×M16—44no—
227C2×M16—44no—
228C22×C8—28no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8—44no—
232C22×C8—44no—
233#126—44no—
234#125—24no—
235#124—24no—
236#123—44no—
237#123—44no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—42no—
241#121—24no—
242#121—24no—
243#121—42no—
244#120—44no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—42no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—42no—
253C4×D4—42no—
254C4×D4—42no—
255#117—42no—
256#117—42no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
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—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42C22⋊C411yescharacteristic
312C2×C42C22⋊C411yescharacteristic
313C2×C42C22⋊C411yescharacteristic
314C2×C42C22⋊C411yescharacteristic
315C2×C42C22⋊C411yescharacteristic
316C2×C42C22⋊C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D414yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42#4114yes—
328C2×C42C2411yescharacteristic
329C4⋊C8—28no—
330C4⋊C8—28no—
331C4⋊C8—28no—
332C4⋊C8—24no—
333C4⋊C8—44no—
334C4⋊C8C2×D418yes—
335C22⋊C8—44no—
336C22⋊C8—44no—
337C22⋊C8—44no—
338C4×C8—24no—
339C4×C8—24no—
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—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—21no—
366C24—41no—
367C24—41no—
368C24C2×C22⋊C411yescharacteristic
369C24C2×C22⋊C411yescharacteristic
370C24C2×C22⋊C411yescharacteristic
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
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—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
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×C4C2×C22⋊C411yescharacteristic
467C22×C4C2×C22⋊C411yescharacteristic
468C22×C4C4×D414yes—
469C22×C4#12012yes—
470C22×C4#12114yes—
471C22×C4#12312yes—
472C22×C4#13114yes—
473C22×C4AGL(1,ℤ/8)12yes—
474C22×C4AGL(1,ℤ/8)12yes—
475C22×C4#13712yes—
476C22×C4#13712yes—
477C22×C4C23×C411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480M16—416no—
481M16—416no—
482C2×C8—28no—
483C2×C8—416no—
484C2×C8—44no—
485C2×C8—44no—
486C2×C8—44no—
487C2×C8—44no—
488C2×C8—44no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C42—22no—
525C42—22no—
526C42—21no—
527C42—21no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—21no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42C4×D414yes—
545C42#12012yes—
546C42#12114yes—
547C42#12312yes—
548C42C22×D412yes—
549C42C2×#4112yes—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—28no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23#38612yes—
581C23#40812yes—
582C23#41712yes—
583C23#41712yes—
584C23#42112yes—
585C23C22×C22⋊C411yescharacteristic
586Q8—44no—
587D4—44no—
588D4—84no—
589D4—84no—
590D4—84no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
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—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
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—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4—81no—
664C2×C4—81no—
665C2×C4#38612yes—
666C2×C4#38612yes—
667C2×C4#38612yes—
668C2×C4#40812yes—
669C2×C4#41712yes—
670C2×C4#41712yes—
671C2×C4#42112yes—
672C2×C4C2×C4×D412yes—
673C2×C4C2×#12011yescharacteristic
674C2×C4C2×#12112yes—
675C2×C4C2×#12311yescharacteristic
676C2×C4C2×#13112yes—
677C2×C4#56812yes—
678C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
679C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
680C2×C4C2×#13711yescharacteristic
681C2×C4C2×#13711yescharacteristic
682C8—48no—
683C8—84no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—416no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—81no—
706C22—81no—
707C22—81no—
708C22—81no—
709C22—81no—
710C22—81no—
711C22—81no—
712C22#149514yescomplemented
713C22C2×#38611yescharacteristic
714C22C2×#40811yescharacteristic
715C22#287911yescharacteristic
716C22C2×#41711yescharacteristic
717C22C2×#41711yescharacteristic
718C22C2×#42111yescharacteristic
719C22C2×#42111yescharacteristic
720C4—22no—
721C4—22no—
722C4—22no—
723C4—22no—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—22no—
728C4—41no—
729C4—41no—
730C4—81no—
731C4—81no—
732C4—81no—
733C4—81no—
734C4—81no—
735C4—81no—
736C4—81no—
737C4#293314yes—
738C4#300311yescharacteristic
739C4#300411yescharacteristic
740C4#309211yescharacteristic
741C4#309211yescharacteristic
742C2—24no—
743C2—42no—
744C2—42no—
745C2—161no—
746C2C2×#149511yescharacteristic
747C2#1748111yescharacteristic
748C2#1749211yescharacteristic
749C2#1749811yescharacteristic
750C2#2089711yescharacteristic
751C2#2090211yescharacteristic
752C2#2090911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2847C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#423C2semidirect
3. #20714C2semidirect
4. C2×#1495C2semidirect
5. #8944C2semidirect
6. #8942C2semidirect
7. #8939C2semidirect
8. #8937C2semidirect
9. #7899C2semidirect
49. C2×Q8⋊C8C22semidirect
50. C2×D4⋊C8C22semidirect
51. #1279C22semidirect
52. #1279C22semidirect
102. #423D4semidirect
712. C22#1495semidirect

Maximal subgroups

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

subgroupindex
C22×#4232
#366212
#207142
C2×#14952
#89442
#89422
#89392
#89372
#78992