Finite Group Databasebeta

← #360086

Subgroups — #360086

Order 512.

2406 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#423C211yesmaximal, characteristic, complemented
2C2×C42×D4C211yesmaximal, characteristic
3C2×#1870C212yesmaximal
4C2×#1797C211yesmaximal, characteristic, complemented
5C2×#1797C212yesmaximal, complemented
6#11202C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#10832C212yesmaximal, complemented
9#10715C212yesmaximal, complemented
10C23×M16C2212yes—
11C22×C4×D4C411yescharacteristic
12C22×C4×D4C411yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#423—24no—
17C2×#423—28no—
18C22×C4⋊C8C2212yes—
19C22×C4⋊C8C2212yes—
20C22×C4⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2214yes—
23C42×D4—24no—
24C42×D4—24no—
25C2×C4×C4⋊C4C411yescharacteristic
26C2×C4×C4⋊C4C411yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C411yescharacteristic
29C2×C4×C22⋊C4C411yescharacteristic
30C2×C4×C22⋊C4C411yescharacteristic
31C2×C4×C22⋊C4C411yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#1870—28no—
36#1797—24no—
37#1797—28no—
38C2×#336C2212yes—
39C2×#336C2214yes—
40C2×#336C2214yes—
41C2×#336C2214yes—
42C23×D4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—24no—
50C22×M16—24no—
51C22×M16—24no—
52C22×M16—28no—
53C22×M16—44no—
54C23×C8C2314yes—
55C23×C8C2314yes—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4D414yes—
67C2×C4×D4Q814yes—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C42—24no—
78C22×C42—24no—
79C22×C42—24no—
80C22×C42—22no—
81C22×C42—22no—
82C22×C42—24no—
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42C2×C411yescharacteristic
88C22×C42C2×C411yescharacteristic
89C22×C42C2×C411yescharacteristic
90C22×C42C2×C411yescharacteristic
91C22×C42D414yes—
92C22×C42D414yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#423—232no—
99C2×C4⋊C8—28no—
100C2×C4⋊C8—24no—
101C2×C4⋊C8—24no—
102C2×C4⋊C8—24no—
103C2×C4⋊C8—24no—
104C2×C4⋊C8—24no—
105C2×C4⋊C8—24no—
106C2×C4⋊C8D418yescomplemented
107C2×C22⋊C8—24no—
108C2×C22⋊C8—28no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—24no—
111C2×C22⋊C8—28no—
112C2×C22⋊C8—44no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C22⋊C4—24no—
118C4×C22⋊C4—24no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—24no—
122C4×C22⋊C4—24no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C2×#95C2×C411yescharacteristic
126C2×#95C2×C411yescharacteristic
127C2×#95C2×C411yescharacteristic
128C2×#95C2×C411yescharacteristic
129C2×#95C2×C411yescharacteristic
130C2×#95C2×C411yescharacteristic
131C43—22no—
132C43—22no—
133#336—28no—
134#336—24no—
135#336—28no—
136#336—44no—
137C25C22×C411yescharacteristic
138C25C22×C411yescharacteristic
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4C4⋊C414yes—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4C22⋊C414yes—
176C23×C4C22⋊C414yes—
177C23×C4M1614yescomplemented
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2×Q812yes—
195C23×C4C2×Q812yes—
196C23×C4#4112yes—
197C23×C4#4112yes—
198C23×C4C2411yescharacteristic
199C2×M16—216no—
200C2×M16—416no—
201C2×M16—416no—
202C2×M16—48no—
203C2×M16—48no—
204C22×C8—24no—
205C22×C8—24no—
206C22×C8—24no—
207C22×C8—24no—
208C22×C8—24no—
209C22×C8—24no—
210C22×C8—24no—
211C22×C8—24no—
212C22×C8—24no—
213C22×C8—24no—
214C22×C8—24no—
215C22×C8—24no—
216C22×C8—24no—
217C22×C8C2×D418yes—
218C4×D4—232no—
219C4×D4—232no—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—28no—
223C4×D4—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4C4⋊C414yes—
232C2×C4⋊C4M1614yescomplemented
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4C4⋊C414yes—
249C2×C22⋊C4C4⋊C414yes—
250C2×C22⋊C4M1614yescomplemented
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—28no—
265C2×C42—216no—
266C2×C42—28no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42—48no—
285C2×C42—44no—
286C2×C42C22⋊C414yes—
287C2×C42C22⋊C414yes—
288C2×C42M1614yescomplemented
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D414yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×Q812yes—
296C2×C42#4114yes—
297C4⋊C8—216no—
298C4⋊C8—216no—
299C4⋊C8—48no—
300C22⋊C8—416no—
301C22⋊C8—48no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24C2×C22⋊C412yes—
325C24C2×C4⋊C412yes—
326C24C2×C4⋊C412yes—
327C24#11712yes—
328C24C2×M1612yes—
329C24C2×M1612yes—
330C24C23×C411yescharacteristic
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—28no—
335C2×D4—28no—
336C2×D4—28no—
337C2×D4—28no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
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—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4C2×C22⋊C412yes—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C412yes—
432C22×C4C2×C22⋊C412yes—
433C22×C4C2×C22⋊C412yes—
434C22×C4C2×C4⋊C412yes—
435C22×C4C2×C4⋊C412yes—
436C22×C4C2×C4⋊C412yes—
437C22×C4C2×C4⋊C412yes—
438C22×C4#11712yes—
439C22×C4C4×D414yes—
440C22×C4C4×D414yes—
441C22×C4#12114yes—
442C22×C4#12114yes—
443C22×C4C22⋊Q814yes—
444C22×C4C22⋊Q814yes—
445C22×C4C2×M1612yes—
446C22×C4C2×M1612yes—
447C22×C4C2×M1612yes—
448C22×C4C2×M1612yes—
449C22×C4C2×M1612yes—
450C22×C4C2×M1612yes—
451C22×C4C2×M1612yes—
452C22×C4C2×M1612yes—
453C22×C4C2×M1612yes—
454C22×C4C2×M1612yes—
455C22×C4#13114yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×Q811yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4111yescharacteristic
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+412yes—
466M16—432no—
467C2×C8—28no—
468C2×C8—28no—
469C2×C8—28no—
470C2×C8—44no—
471C2×C8—44no—
472C2×C8—44no—
473C2×C8—44no—
474C2×C8—44no—
475C2×C8—44no—
476C2×C8—44no—
477C2×C8—44no—
478C2×C8—44no—
479C2×C8—44no—
480C2×C8—44no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C22⋊C4—28no—
485C22⋊C4—28no—
486C22⋊C4—28no—
487C22⋊C4—28no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C42—28no—
491C42—22no—
492C42—24no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—28no—
499C42—28no—
500C42—28no—
501C42—44no—
502C42—41no—
503C42—42no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—42no—
508C42—44no—
509C42—44no—
510C42—44no—
511C42#12014yes—
512C42#12118yes—
513C42#12314yes—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23#38014yes—
559C23#42314yescomplemented
560C23C22×C22⋊C411yescharacteristic
561C23C22×C4⋊C411yescharacteristic
562C23C2×#11711yescharacteristic
563C23C2×C4×D412yes—
564C23#52012yes—
565C23C22×M1611yescharacteristic
566C23C22×M1611yescharacteristic
567C23C22×M1611yescharacteristic
568C23C22×M1611yescharacteristic
569C23C2×#13112yes—
570C23#56812yes—
571D4—232no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—416no—
612C2×C4—416no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—416no—
634C2×C4—416no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4#38014yes—
642C2×C4#38614yes—
643C2×C4#38614yes—
644C2×C4#40714yes—
645C2×C4#40714yes—
646C2×C4#40814yes—
647C2×C4#42314yescomplemented
648C2×C4#43518yes—
649C2×C4#43618yes—
650C2×C4C2×#12011yescharacteristic
651C2×C4C2×#12111yescharacteristic
652C2×C4C2×#12111yescharacteristic
653C2×C4C2×#12112yes—
654C2×C4C2×C22⋊Q811yescharacteristic
655C2×C4C2×C22⋊Q811yescharacteristic
656C2×C4C2×#12311yescharacteristic
657C2×C4#53511yescharacteristic
658C2×C4#53611yescharacteristic
659C2×C4#53712yes—
660C2×C4D4212yes—
661C2×C4#54612yes—
662C2×C4#54712yes—
663C2×C4#54812yes—
664C2×C4D4×Q812yes—
665C2×C4#55212yes—
666C2×C4#55312yes—
667C2×C4#55412yes—
668C8—48no—
669C8—84no—
670C22—24no—
671C22—28no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22C2×#38011yescharacteristic
693C22C2×#38611yescharacteristic
694C22#231111yescharacteristic
695C22D4×C4⋊C412yes—
696C22#235212yes—
697C22C2×#40711yescharacteristic
698C22C2×#40711yescharacteristic
699C22C2×#40811yescharacteristic
700C22#287911yescharacteristic
701C22C2×#42311yescharacteristic
702C22C2×#42311yescharacteristic
703C22#290612yes—
704C22#292011yescharacteristic
705C22#292011yescharacteristic
706C22#292512yes—
707C22C2×#43512yes—
708C22C2×#43612yes—
709C22D4×M1612yes—
710C22D4×M1612yes—
711C22#298912yes—
712C22#298912yes—
713C22#299012yes—
714C22#299112yes—
715C4—24no—
716C4—24no—
717C4—24no—
718C4—24no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—42no—
725C4—42no—
726C4#187018yescomplemented
727C4#240811yescharacteristic
728C4#240911yescharacteristic
729C4#241612yes—
730C4#241712yes—
731C4#248611yescharacteristic
732C4#248711yescharacteristic
733C2—44no—
734C2—44no—
735C2#1387211yescharacteristic
736C2#2062611yescharacteristic
737C2#2062911yescharacteristic
738C2C2×#187011yescharacteristic
739C2#2089811yescharacteristic
740C2#2184911yescharacteristic
741C2#2185311yescharacteristic
742C2#2264812yes—
743C2#2265012yes—
744C2#2274912yes—
745C2#2275112yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#423C2semidirect
4. C2×#1797C2semidirect
5. C2×#1797C2semidirect
6. #11202C2semidirect
7. #11042C2semidirect
8. #10832C2semidirect
9. #10715C2semidirect
106. C2×C4⋊C8D4semidirect
177. C23×C4M16semidirect
232. C2×C4⋊C4M16semidirect
250. C2×C22⋊C4M16semidirect
288. C2×C42M16semidirect
559. C23#423semidirect
647. C2×C4#423semidirect
726. C4#1870semidirect

Maximal subgroups

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

subgroupindex
C22×#4232
C2×C42×D42
C2×#18702
C2×#17972
#112022
#110422
#108322
#107152