Finite Group Databasebeta

← #195872

Subgroups — #195872

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2C4×#525C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic, complemented
4C2×#1488C212yesmaximal, complemented
5#9078C212yesmaximal, complemented
6#9077C212yesmaximal, complemented
7#9074C212yesmaximal, complemented
8#9073C212yesmaximal, complemented
9#7902C212yesmaximal, complemented
10C2×#525C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C23×C42C2211yescharacteristic
13#2920—28no—
14C2×#432—21no—
15C2×#432—21no—
16C2×#432C2212yes—
17C2×#431C2212yes—
18C2×#431C2212yes—
19C2×#423—21no—
20C2×#407—21no—
21C2×C4×M16C2212yes—
22C4×C22⋊Q8C411yescharacteristic
23C4×C22⋊Q8C411yescharacteristic
24C4×C22⋊Q8C2211yescharacteristic
25C4×C22⋊Q8C2211yescharacteristic
26C4×#123—21no—
27C4×#123—21no—
28C4×#121C411yescharacteristic
29C4×#121C411yescharacteristic
30C4×#121C2211yescharacteristic
31C4×#121C2211yescharacteristic
32C4×#120—21no—
33C4×#120—21no—
34#2297C411yescharacteristic
35#2296C411yescharacteristic
36#2290—21no—
37#2290—21no—
38#2280C411yescharacteristic
39C42×D4—21no—
40C42×D4—21no—
41C4×#117C2211yescharacteristic
42#1817—21no—
43#1797C2212yes—
44#1797C2212yes—
45#1779—21no—
46#1749—21no—
47#1749—21no—
48#1488—28no—
49C2×Q8⋊C8C2214yescomplemented
50C2×D4⋊C8C2214yescomplemented
51#1279C2214yescomplemented
52#1279C2214yescomplemented
53C22×#41C2×C411yescharacteristic
54C24×C4C2×C411yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4D411yescharacteristic
57C22×M16—22no—
58#525—24no—
59#525—24no—
60C2×#123—21no—
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×#121C2×C411yescharacteristic
64C2×#121C2×C411yescharacteristic
65C2×#120—21no—
66C4×#41—24no—
67C4×#41—24no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×#117—21no—
81C2×#117C2×C411yescharacteristic
82C2×#117C2×C411yescharacteristic
83C2×#117C2×C411yescharacteristic
84C22×C42—24no—
85C22×C42—24no—
86C22×C42—21no—
87C22×C42C2×C411yescharacteristic
88C22×C42C2×C411yescharacteristic
89C22×C42C2×C411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#432—28no—
100#432—28no—
101#432—24no—
102#432—24no—
103#431—24no—
104#431—28no—
105#431D418yescomplemented
106#423—28no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8C2314yes—
110#407—28no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C4×M16—28no—
117C2×#97—22no—
118C2×#97—22no—
119C2×C4×C8C2314yes—
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#390C2×C411yescharacteristic
123#390C2×C411yescharacteristic
124#389C2×C411yescharacteristic
125#389C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#388—21no—
129#388—21no—
130#388—21no—
131#388—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#385—21no—
147#385—21no—
148#381—21no—
149#380C2×C411yescharacteristic
150#380C2×C411yescharacteristic
151#380C2×C411yescharacteristic
152#380C2×C411yescharacteristic
153#379—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2×C411yescharacteristic
161C4×C4⋊C4C2×C411yescharacteristic
162C4×C4⋊C4C2311yescharacteristic
163C4×C4⋊C4C2311yescharacteristic
164C4×C22⋊C4—21no—
165C4×C22⋊C4—21no—
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168C4×C22⋊C4—21no—
169C4×C22⋊C4—21no—
170C4×C22⋊C4—21no—
171C4×C22⋊C4—21no—
172C4×C22⋊C4—21no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178#376—21no—
179C43—21no—
180#336—22no—
181#336—22no—
182Q8⋊C8—28no—
183D4⋊C8—28no—
184C25C22⋊C411yescharacteristic
185C2×#41—22no—
186C2×#41—22no—
187C2×#41—24no—
188C2×#41—42no—
189C22×Q8C22×C411yescharacteristic
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4C22×C411yescharacteristic
194C23×C4—21no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4C22⋊C411yescharacteristic
209C23×C4C22⋊C411yescharacteristic
210C23×C4C22⋊C411yescharacteristic
211C23×C4C22⋊C411yescharacteristic
212C23×C4C22⋊C411yescharacteristic
213C23×C4C22⋊C411yescharacteristic
214C23×C4C22⋊C411yescharacteristic
215C23×C4C22⋊C411yescharacteristic
216C23×C4C22⋊C411yescharacteristic
217C23×C4C22×C411yescharacteristic
218C23×C4C22×C411yescharacteristic
219C23×C4C22×C411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C2×M16—28no—
227C2×M16—44no—
228C2×M16—44no—
229C22×C8—24no—
230C22×C8—24no—
231#123—28no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8M1614yescomplemented
236#121—22no—
237#121—22no—
238#121—22no—
239#121M1614yescomplemented
240#120—28no—
241C4×Q8—24no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×D4—28no—
245C4×D4—28no—
246C4×D4—24no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—44no—
250C4×D4—44no—
251C4×D4—44no—
252C4×D4—44no—
253C4×D4—42no—
254C4×D4—44no—
255C4×D4—42no—
256#117—22no—
257#117—22no—
258#117—24no—
259#117—42no—
260#117—42no—
261#117—42no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
293C2×C42—21no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—44no—
319C2×C42C22⋊C411yescharacteristic
320C2×C42C22⋊C411yescharacteristic
321C2×C42C22⋊C411yescharacteristic
322C2×C42C22⋊C411yescharacteristic
323C2×C42C22⋊C411yescharacteristic
324C2×C42C22⋊C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D414yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42#4114yes—
336C2×C42C2411yescharacteristic
337C4⋊C8—28no—
338C4⋊C8—28no—
339C4⋊C8—24no—
340C4⋊C8—24no—
341C22⋊C8—28no—
342C22⋊C8—28no—
343C22⋊C8—44no—
344C22⋊C8—44no—
345C22⋊C8—44no—
346#97—28no—
347#97—44no—
348C4×C8—24no—
349C4×C8C2×D418yes—
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—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—41no—
371C24—41no—
372C24—41no—
373C24—41no—
374C24C2×C22⋊C411yescharacteristic
375C24C2×C22⋊C411yescharacteristic
376C24C2×C22⋊C411yescharacteristic
377#41—48no—
378#41—48no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8C2×M1612yes—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4C2×M1612yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
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—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
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—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4C2×C22⋊C411yescharacteristic
472C22×C4C2×C22⋊C411yescharacteristic
473C22×C4C2×C22⋊C411yescharacteristic
474C22×C4C2×C22⋊C411yescharacteristic
475C22×C4C2×C22⋊C411yescharacteristic
476C22×C4C2×C22⋊C411yescharacteristic
477C22×C4C2×C22⋊C411yescharacteristic
478C22×C4C2×C22⋊C411yescharacteristic
479C22×C4C2×C22⋊C411yescharacteristic
480C22×C4C4×D414yes—
481C22×C4#12012yes—
482C22×C4#12114yes—
483C22×C4#12312yes—
484C22×C4C2×M1612yes—
485C22×C4C2×M1612yes—
486C22×C4C23×C411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489M16—416no—
490C2×C8—28no—
491C2×C8—44no—
492C2×C8—44no—
493C2×C8—44no—
494C2×C8—44no—
495C2×C8—44no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4C2×M1612yes—
512C4⋊C4C2×M1612yes—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—21no—
535C42—21no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—42no—
553C42—42no—
554C42C4×D414yes—
555C42#12012yes—
556C42#12114yes—
557C42#12312yes—
558C42C22×D412yes—
559C42C2×#4112yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—22no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23#38612yes—
593C23#40712yes—
594C23#42112yes—
595C23C22×C22⋊C411yescharacteristic
596Q8—24no—
597Q8—42no—
598D4—24no—
599D4—48no—
600D4—48no—
601D4—48no—
602D4—42no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—28no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—48no—
650C2×C4—44no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4#38612yes—
676C2×C4#38612yes—
677C2×C4#38612yes—
678C2×C4#40712yes—
679C2×C4#42112yes—
680C2×C4C2×C4×D412yes—
681C2×C4C2×#12011yescharacteristic
682C2×C4C2×#12112yes—
683C2×C4C2×#12311yescharacteristic
684C2×C4C22×M1611yescharacteristic
685C2×C4C22×M1611yescharacteristic
686C2×C4#56812yes—
687C2×C4#57512yes—
688C2×C4#57512yes—
689C8—48no—
690C8—84no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—416no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—81no—
717C22—81no—
718C22—81no—
719C22—81no—
720C22C2×#38611yescharacteristic
721C22C2×#40711yescharacteristic
722C22#287911yescharacteristic
723C22#289611yescharacteristic
724C22#289611yescharacteristic
725C22C2×#42111yescharacteristic
726C22C2×#42111yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—21no—
731C4—21no—
732C4—22no—
733C4—22no—
734C4—42no—
735C4—42no—
736C4—41no—
737C4—42no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—81no—
742C4—81no—
743C4—81no—
744C4—81no—
745C4D4×M1612yes—
746C4D4×M1612yes—
747C4#305414yes—
748C2—42no—
749C2—42no—
750C2—42no—
751C2—82no—
752C2#1745011yescharacteristic
753C2#1745211yescharacteristic
754C2#1749311yescharacteristic
755C2#1749411yescharacteristic
756C2#2089811yescharacteristic
757C2#2097011yescharacteristic
758C2#2097911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#525C4
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. C2×#2920C2semidirect
3. #20715C2semidirect
4. C2×#1488C2semidirect
5. #9078C2semidirect
6. #9077C2semidirect
7. #9074C2semidirect
8. #9073C2semidirect
9. #7902C2semidirect
49. C2×Q8⋊C8C22semidirect
50. C2×D4⋊C8C22semidirect
51. #1279C22semidirect
52. #1279C22semidirect
105. #431D4semidirect
235. C22⋊Q8M16semidirect
239. #121M16semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C4×#5252
#207152
C2×#14882
#90782
#90772
#90742
#90732
#79022