Finite Group Databasebeta

← #181986

Subgroups — #181986

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 763 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic, complemented
3C2×#1749C211yesmaximal, characteristic, complemented
4C2×#1468C212yesmaximal
5#8155C212yesmaximal, complemented
6#8154C212yesmaximal, complemented
7#8153C212yesmaximal, complemented
8#8152C212yesmaximal, complemented
9#7913C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×C4×D4C411yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#407C2212yes—
14C22×C22⋊C8C2212yes—
15C2×C4×M16—21no—
16C22×#97—21no—
17C4×#123—21no—
18C4×#123—21no—
19C4×#123—21no—
20C4×#123—21no—
21C4×#123—21no—
22C4×#123—21no—
23C4×#123—21no—
24C4×#123—21no—
25C2×#388C411yescharacteristic
26C2×#388C2211yescharacteristic
27C2×#387C411yescharacteristic
28C2×#387C2211yescharacteristic
29C2×#386C411yescharacteristic
30C2×#385C411yescharacteristic
31C2×#381C2211yescharacteristic
32C2×C4×C4⋊C4C411yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C411yescharacteristic
35C2×C4×C22⋊C4C411yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37#1749—21no—
38#1749—21no—
39#1749—21no—
40#1749C2212yes—
41#1749C2212yes—
42#1749C2212yes—
43C4×C22⋊C8—21no—
44C4×C22⋊C8C2212yes—
45#1728—21no—
46C2×#336—21no—
47#1468C4116yescomplemented
48C2×#324C2214yes—
49C2×#323C2214yes—
50#1280C2214yes—
51#1280C2214yes—
52C23×D4C2×C411yescharacteristic
53C24×C4C2×C411yescharacteristic
54C24×C4C2×C411yescharacteristic
55C24×C4C2×C411yescharacteristic
56C24×C4C2311yescharacteristic
57C22×M16—22no—
58C23×C8—22no—
59C2×#123—21no—
60C2×#123—21no—
61C2×#123—21no—
62C2×#123—21no—
63C2×#123—24no—
64C2×#123C2×C418yes—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—44no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2311yescharacteristic
83C22×C42—44no—
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42C2×C411yescharacteristic
88C22×C42C2×C411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#407—24no—
100#407C2×C418yescomplemented
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8C2×C418yescomplemented
105C2×C22⋊C8C2314yes—
106C2×C22⋊C8C2314yes—
107C4×M16—44no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—44no—
112C2×C4×C8—22no—
113#388—21no—
114#388—21no—
115#388—21no—
116#388—21no—
117#388—21no—
118#388—21no—
119#388—21no—
120#388—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#387—21no—
125#387—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#386—21no—
130#386—21no—
131#386—21no—
132#386—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#381—21no—
138#381—21no—
139#381—21no—
140#381—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C2×#95C2×C411yescharacteristic
162C2×#95C2×C411yescharacteristic
163C2×#95C2×C411yescharacteristic
164C2×#95C2×C411yescharacteristic
165C2×#95C2×C411yescharacteristic
166C2×#95C2×C411yescharacteristic
167C2×#95C2×C411yescharacteristic
168C2×#95C2×C411yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171#336—22no—
172#336—22no—
173#336—22no—
174#336—22no—
175#324C2×C4116yescomplemented
176#323C2×C4116yescomplemented
177C25C22×C411yescharacteristic
178C25C22×C411yescharacteristic
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—22no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4C4214yes—
185C23×C4—24no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
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×C4C22×C411yescharacteristic
212C23×C4C22×C411yescharacteristic
213C23×C4C22×C411yescharacteristic
214C23×C4C22×C411yescharacteristic
215C23×C4C22×C411yescharacteristic
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C411yescharacteristic
218C23×C4C22×C414yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4#4114yes—
224C23×C4C2411yescharacteristic
225C2×M16—28no—
226C2×M16—44no—
227C2×M16—44no—
228C22×C8—28no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8—44no—
232C22×C8—44no—
233#123—28no—
234#123—28no—
235#123—28no—
236#123—28no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—44no—
240C4×D4—44no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
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—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C4214yes—
257C2×C4⋊C4C22×C414yes—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
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—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4C4214yes—
283C2×C22⋊C4C4214yes—
284C2×C22⋊C4C22×C414yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—42no—
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×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C22⋊C8—28no—
323C22⋊C8—28no—
324C22⋊C8—24no—
325C22⋊C8—24no—
326C22⋊C8C22×C418yes—
327C22⋊C8C22×C418yes—
328#97—44no—
329#97—44no—
330#97—44no—
331C4×C8—44no—
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—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—41no—
361C24—41no—
362C24—41no—
363C24—41no—
364C24—41no—
365C24—41no—
366C24C2×C4212yes—
367C24C2×C4212yes—
368C24C2×C22⋊C411yescharacteristic
369C24C2×C22⋊C411yescharacteristic
370C24#11712yes—
371C24C23×C411yescharacteristic
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—42no—
375C2×D4—48no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—48no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
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—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
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—48no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—48no—
444C22×C4—41no—
445C22×C4—416no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—416no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—88no—
466C22×C4#9914yes—
467C22×C4C2×C4212yes—
468C22×C4C2×C4212yes—
469C22×C4C2×C4212yes—
470C22×C4C2×C4212yes—
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×C4C2×C22⋊C411yescharacteristic
481C22×C4#11712yes—
482C22×C4C4×D414yes—
483C22×C4C4×D414yes—
484C22×C4C4×D414yes—
485C22×C4C4×D414yes—
486C22×C4C23×C412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4112yes—
490M16—416no—
491C2×C8—28no—
492C2×C8—28no—
493C2×C8—44no—
494C2×C8—44no—
495C2×C8—44no—
496C2×C8—416no—
497C2×C8—44no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—24no—
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—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—42no—
541C42—42no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—42no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—24no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—22no—
565C23—24no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—42no—
583C23—82no—
584C23—82no—
585C23—82no—
586C23—82no—
587C23C4×C22⋊C412yes—
588C23C4×C22⋊C412yes—
589C23C2×#9912yes—
590C23#41012yes—
591C23C22×C4211yescharacteristic
592C23C22×C22⋊C411yescharacteristic
593C23C2×#11711yescharacteristic
594C23#56812yes—
595C23#56812yes—
596D4—48no—
597D4—48no—
598C2×C4—28no—
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—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—24no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
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—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
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—82no—
676C2×C4—82no—
677C2×C4—82no—
678C2×C4—82no—
679C2×C4C4×C22⋊C412yes—
680C2×C4C4×C22⋊C412yes—
681C2×C4C2×#9912yes—
682C2×C4C2×#9912yes—
683C2×C4#41312yes—
684C2×C4#41312yes—
685C2×C4C2×C4×D412yes—
686C2×C4C2×C4×D412yes—
687C8—48no—
688C8—48no—
689C22—22no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—48no—
707C22—48no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—84no—
716C22—84no—
717C22—81no—
718C22—81no—
719C22—81no—
720C22—81no—
721C22C4×#9914yes—
722C22C2×C4×C22⋊C411yescharacteristic
723C22#287512yes—
724C22#287911yescharacteristic
725C22#287911yescharacteristic
726C22C22×#9911yescharacteristic
727C22C2×#41011yescharacteristic
728C22C2×#41311yescharacteristic
729C22C2×#41311yescharacteristic
730C22#292312yes—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—42no—
740C4—42no—
741C4—42no—
742C4—41no—
743C4—41no—
744C4—41no—
745C4—41no—
746C4—81no—
747C4—81no—
748C4—81no—
749C4—81no—
750C2—22no—
751C2—22no—
752C2—24no—
753C2—42no—
754C2—82no—
755C2#1739211yescharacteristic
756C2#1739511yescharacteristic
757C2#1740011yescharacteristic
758C2#1740211yescharacteristic
759C2#2037511yescharacteristic
760C2C2×C4×#9911yescharacteristic
761C2C4×#41311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#407C2semidirect
3. C2×#1749C2semidirect
5. #8155C2semidirect
6. #8154C2semidirect
7. #8153C2semidirect
8. #8152C2semidirect
9. #7913C2semidirect
47. #1468C4semidirect
100. #407C2×C4semidirect
104. C2×C22⋊C8C2×C4semidirect
175. #324C2×C4semidirect
176. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C4×#4072
C2×#17492
C2×#14682
#81552
#81542
#81532
#81522
#79132