Finite Group Databasebeta

← #6496013

Subgroups — #6496013

Order 512.

3360 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic
2C2×#3230C212yesmaximal, complemented
3C2×#2857C211yesmaximal, characteristic
4#32957C214yesmaximal, complemented
5#29691C214yesmaximal, complemented
6#25856C212yesmaximal, complemented
7#25813C212yesmaximal, complemented
8#25674C214yesmaximal, complemented
9#24064C214yesmaximal, complemented
10#24006C214yesmaximal
11#21860C212yesmaximal, complemented
12#20714C211yesmaximal, characteristic
13C2×#533—21no—
14C2×#533—21no—
15C2×#533—21no—
16C2×#533—21no—
17C2×#533—216no—
18C2×#533C2212yes—
19C2×#533C2212yes—
20C2×#533C2212yes—
21C2×#533C2212yes—
22C22×C4⋊Q8C2211yescharacteristic
23C22×C4⋊Q8C2211yescharacteristic
24C22×C4⋊Q8C2211yescharacteristic
25C22×#125—21no—
26C22×C22⋊Q8—21no—
27C22×C22⋊Q8—21no—
28C22×C22⋊Q8C2212yes—
29C22×C4×Q8—21no—
30C22×C4×Q8C2212yes—
31C22×#117—21no—
32C23×C42C2211yescharacteristic
33#3230—216no—
34C2×#500C2214yescomplemented
35C2×#487C2214yescomplemented
36C2×#484C2218yes—
37C2×#477C2218yescomplemented
38C2×#431C2212yes—
39#2857—21no—
40#2857—21no—
41#2857—21no—
42#2857—21no—
43#2857C2212yes—
44#2857C2212yes—
45#2857C2212yes—
46#2857C2212yes—
47C2×#391—21no—
48C2×#389—21no—
49C2×#389C2212yes—
50C2×#384C2211yescharacteristic
51C2×#380—21no—
52C2×#380C2212yes—
53#2057C2218yescomplemented
54#2043C2218yescomplemented
55#2036C22116yes—
56#1990C2214yescomplemented
57#1956C2214yescomplemented
58#1872C2218yes—
59#1863C22116yescomplemented
60#1852C2218yes—
61#1817C2212yes—
62#1803C2218yescomplemented
63#1797C2212yes—
64#1797C2212yes—
65#1787C2218yes—
66#1779C2212yes—
67C4×C22⋊C8C2212yes—
68C4×C22⋊C8C2212yes—
69C23×Q8—21no—
70C23×Q8—21no—
71C23×Q8C2312yes—
72C24×C4D411yescharacteristic
73C24×C4D411yescharacteristic
74C24×C4C2311yescharacteristic
75#533—216no—
76#533—216no—
77#533—216no—
78#533—216no—
79#533—28no—
80#533—28no—
81#533—216no—
82#533—216no—
83#533—28no—
84#533D4116yes—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2312yes—
100C2×#125—216no—
101C2×#125—22no—
102C2×#125—22no—
103C2×#125—22no—
104C2×#125—22no—
105C2×C22⋊Q8—216no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—216no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—48no—
118C2×C22⋊Q8C2314yes—
119C2×C22⋊Q8C2314yes—
120C2×C4×Q8—216no—
121C2×C4×Q8—216no—
122C2×C4×Q8—22no—
123C2×C4×Q8—22no—
124C2×C4×Q8—22no—
125C2×C4×Q8—22no—
126C2×C4×Q8—22no—
127C2×C4×Q8—22no—
128C2×C4×Q8C2314yes—
129C2×C4×Q8C2314yes—
130C2×#117—216no—
131C2×#117—22no—
132C2×#117—22no—
133C2×#117—22no—
134C2×#117—22no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4—22no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—22no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4C2312yes—
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2312yes—
147C22×C22⋊C4—22no—
148C22×C22⋊C4—22no—
149C22×C42—22no—
150C22×C42—28no—
151C22×C42D411yescharacteristic
152C22×C42D411yescharacteristic
153C22×C42D411yescharacteristic
154C22×C42D411yescharacteristic
155C22×C42D411yescharacteristic
156C22×C42D411yescharacteristic
157C22×C42C2311yescharacteristic
158C22×C42C2311yescharacteristic
159C22×C42C2311yescharacteristic
160C22×C42C2311yescharacteristic
161C22×C42C2311yescharacteristic
162C22×C42C2311yescharacteristic
163#500—216no—
164#487—216no—
165#484—232no—
166#477—232no—
167#431—28no—
168C2×#107C23116yes—
169C2×C4⋊C8C2314yes—
170C2×#103C23116yes—
171C2×#103C23116yes—
172C2×C22⋊C8C2314yes—
173C2×C22⋊C8C2314yes—
174C2×C22⋊C8C2314yes—
175C2×C4×C8C2314yes—
176#391—22no—
177#391—22no—
178#391—22no—
179#391—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389C2314yes—
187#389C2314yes—
188#384—21no—
189#384—21no—
190#384C2312yes—
191#384C2312yes—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380C2314yes—
199#380C2314yes—
200C2×#95—22no—
201C2×#95—22no—
202#340C23116yes—
203#336C2314yes—
204#336C2314yes—
205C25C2×D411yescharacteristic
206C22×Q8—28no—
207C22×Q8—28no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—28no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8—44no—
219C22×Q8—44no—
220C22×Q8C2414yes—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4Q1618yes—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C22×C8C2418yes—
253C4⋊Q8—28no—
254C4⋊Q8—28no—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—28no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8C2×D418yes—
266C4⋊Q8C2×D418yes—
267#125—216no—
268#125—216no—
269#125—216no—
270#125—216no—
271C22⋊Q8—216no—
272C22⋊Q8—216no—
273C22⋊Q8—216no—
274C22⋊Q8—216no—
275C22⋊Q8—216no—
276C22⋊Q8—28no—
277C22⋊Q8—216no—
278C22⋊Q8—28no—
279C22⋊Q8—28no—
280C22⋊Q8—48no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283C22⋊Q8—48no—
284C22⋊Q8C2×D4116yes—
285C4×Q8—216no—
286C4×Q8—216no—
287C4×Q8—216no—
288C4×Q8—216no—
289C4×Q8—216no—
290C4×Q8—28no—
291C4×Q8—28no—
292C4×Q8—216no—
293C4×Q8—28no—
294C4×Q8C2×D4116yes—
295#117—216no—
296#117—216no—
297#117—216no—
298#117—216no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—216no—
302C2×C4⋊C4—216no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2414yes—
335C2×C22⋊C4—216no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—48no—
341C2×C42—216no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D418yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#107—232no—
373C4⋊C8—28no—
374#103—232no—
375#103—416no—
376C22⋊C8—28no—
377C22⋊C8—28no—
378C22⋊C8—44no—
379C4×C8—28no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—41no—
389C24—41no—
390C24C2×Q1614yes—
391C24C22×D411yescharacteristic
392C24C22×D411yescharacteristic
393C24C22×D411yescharacteristic
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—28no—
397C2×Q8—24no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—432no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—432no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—816no—
415C2×Q8C22×D418yes—
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—232no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4#12014yes—
478C22×C4#12014yes—
479C22×C4C2×Q1614yes—
480C22×C4C2×Q1614yes—
481C22×C4C2×Q1614yes—
482C22×C4C2×Q1614yes—
483C22×C4C2×Q1614yes—
484C22×C4C2×Q1614yes—
485C22×C4AGL(1,ℤ/8)14yes—
486C22×C4#13714yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D414yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C42−1+414yes—
500C22×C4C2511yescharacteristic
501C2×C8—28no—
502C2×C8—28no—
503C2×C8—28no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—216no—
507C4⋊C4—216no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—216no—
514C4⋊C4—216no—
515C4⋊C4—24no—
516C4⋊C4—28no—
517C4⋊C4—28no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4C22×D418yes—
529C22⋊C4—216no—
530C22⋊C4—216no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C42—216no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—216no—
545C42—22no—
546C42—22no—
547C42#12014yes—
548C42#12014yes—
549C42C22×D414yes—
550C42C22×D414yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—28no—
563C23—28no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23C2×#12012yes—
573C23#53612yes—
574C23C22×Q1612yes—
575C23C22×Q1612yes—
576C23C22×Q1612yes—
577C23C2×AGL(1,ℤ/8)12yes—
578C23C2×#13712yes—
579C23C23×D411yescharacteristic
580Q8—216no—
581Q8—416no—
582Q8—416no—
583Q8—48no—
584Q8—416no—
585Q8—88no—
586Q8—88no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—28no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—416no—
623C2×C4—416no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4#451116yes—
639C2×C4C2×#12012yes—
640C2×C4C2×#12012yes—
641C2×C4C2×#12012yes—
642C2×C4C2×#12012yes—
643C2×C4C2×#12012yes—
644C2×C4#53612yes—
645C2×C4C22×Q1612yes—
646C2×C4C22×Q1612yes—
647C2×C4C22×Q1612yes—
648C2×C4C22×Q1612yes—
649C2×C4C2×AGL(1,ℤ/8)12yes—
650C2×C4C2×AGL(1,ℤ/8)12yes—
651C2×C4C2×#13712yes—
652C2×C4C2×#13712yes—
653C2×C4#57512yes—
654C2×C4C23×D412yes—
655C2×C4C2×2−1+412yes—
656C2×C4#58512yes—
657C8—48no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—416no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22C2×#45114yes—
679C22#323014yescomplemented
680C22C22×#12011yescharacteristic
681C22C2×#53611yescharacteristic
682C22#344911yescharacteristic
683C22C23×Q1611yescharacteristic
684C22C22×AGL(1,ℤ/8)11yescharacteristic
685C22C22×#13711yescharacteristic
686C22C2×#57511yescharacteristic
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—41no—
694C4—48no—
695C4—41no—
696C4—48no—
697C4—84no—
698C4—84no—
699C4C2×#45114yes—
700C4C2×#45114yes—
701C4#300614yes—
702C4#348914yes—
703C2—24no—
704C2—24no—
705C2—42no—
706C2#3530511yescharacteristic
707C2C22×#45111yescharacteristic
708C2C2×#300611yescharacteristic
709C2C2×#323011yescharacteristic
710C2#6086511yescharacteristic
711C2#6094311yescharacteristic
712C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3230C2semidirect
4. #32957C2semidirect
5. #29691C2semidirect
6. #25856C2semidirect
7. #25813C2semidirect
8. #25674C2semidirect
9. #24064C2semidirect
11. #21860C2semidirect
34. C2×#500C22semidirect
35. C2×#487C22semidirect
37. C2×#477C22semidirect
53. #2057C22semidirect
54. #2043C22semidirect
56. #1990C22semidirect
57. #1956C22semidirect
59. #1863C22semidirect
62. #1803C22semidirect
679. C22#3230semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#32302
C2×#28572
#329572
#296912
#258562
#258132
#256742
#240642
#240062
#218602
#207142