Finite Group Databasebeta

← #6388743

Subgroups — #6388743

Order 512.

2148 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#34126C211yesmaximal, characteristic
4#23075C214yesmaximal, complemented
5#22975C214yesmaximal, complemented
6#22615C212yesmaximal, complemented
7#22609C212yesmaximal, complemented
8#22570C212yesmaximal
9#22562C212yesmaximal
10#22117C212yesmaximal, complemented
11#22111C212yesmaximal, complemented
12#21962C211yesmaximal, characteristic, complemented
13#21840C211yesmaximal, characteristic, complemented
14#21614C212yesmaximal, complemented
15C4×#476C212yesmaximal, complemented
16#20738C211yesmaximal, characteristic
17#20546C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#519—21no—
20C22×C4×D4C2211yescharacteristic
21C2×#476C4116yescomplemented
22C2×#476C2211yescharacteristic
23C2×#476C2211yescharacteristic
24C2×#476C2212yes—
25C2×#476C2212yes—
26C2×#476C2211yescharacteristic
27C2×#476C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C22×#106C2212yes—
30C22×C4⋊C8C2211yescharacteristic
31C22×#102C2212yes—
32#2365—22no—
33#2365C2214yescomplemented
34#2357C2212yes—
35#2353—22no—
36#2349C2212yes—
37#2345C2211yescharacteristic
38#2345C2211yescharacteristic
39#2344—21no—
40#2344—21no—
41#2344C2212yes—
42#2344C2212yes—
43#2342—21no—
44#2342—21no—
45D4×C22⋊C4C2211yescharacteristic
46D4×C22⋊C4C2211yescharacteristic
47#2327—22no—
48#2310—21no—
49C4×C4⋊Q8C2212yescomplemented
50C4×C4⋊Q8C2212yescomplemented
51#2293C2212yes—
52#2290—22no—
53#2285C2211yescharacteristic
54#2283C2211yescharacteristic
55#2280—21no—
56C42×D4C2212yes—
57C4×#117C2211yescharacteristic
58C2×C4×C22⋊C4—21no—
59#1993C2214yescomplemented
60#1987C2212yescomplemented
61#1987C2212yescomplemented
62#1949C2214yescomplemented
63#1948C2212yescomplemented
64#1947C2212yescomplemented
65#1944C2214yescomplemented
66#1941C2214yescomplemented
67#1936C2214yes—
68#1934C2214yes—
69#1926C2214yes—
70#1925C2214yes—
71#1920C2214yescomplemented
72#1920C2214yescomplemented
73#1919C2214yescomplemented
74#1919C2214yescomplemented
75#1864C2214yes—
76#1862C2214yes—
77#1824C2212yes—
78#1823C2212yes—
79#1823C2212yes—
80#1808C2212yes—
81#1807C2212yes—
82#1802C2212yes—
83#1802C2212yes—
84#1786C2212yes—
85#1776C2214yescomplemented
86C4×#106C2214yescomplemented
87#1771C2214yescomplemented
88#1767C2212yescomplemented
89C4×C4⋊C8C2212yescomplemented
90#1762C2212yes—
91#1762C2212yes—
92#1762C2214yes—
93C4×#102C2214yes—
94#1752C2212yes—
95C8×C22⋊C4C2212yes—
96C23×Q8—21no—
97C23×D4C2311yescharacteristic
98C24×C4—21no—
99C23×C8C2312yes—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8C2×C418yescomplemented
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107#519—48no—
108C2×C4×Q8—24no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—24no—
114C2×C4×D4C2×C418yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—21no—
122C2×#117—22no—
123C2×#117—22no—
124C2×#117—22no—
125C2×#117D411yescharacteristic
126C2×#117D411yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C42—22no—
134C22×C42C2311yescharacteristic
135#476C2×C418yes—
136#476C2×C418yes—
137#476C2×C4116yes—
138#476C2×C4116yescomplemented
139#476C2×C418yescomplemented
140#476C2×C418yescomplemented
141#432—24no—
142C2×#107C2314yes—
143C2×#106C2×C4116yescomplemented
144C2×#106C2314yes—
145C2×#106C2312yes—
146C2×#106C2312yes—
147C2×#106C2312yes—
148C2×#106C2312yes—
149C2×C4⋊C8C2×C418yescomplemented
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2311yescharacteristic
153C2×C4⋊C8C2312yes—
154C2×#102C2×C4116yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×C22⋊C8C2312yes—
162C2×#97C2312yes—
163C2×#97C2314yes—
164C2×C4×C8C2314yes—
165#391—24no—
166#391—22no—
167#391—22no—
168#390C2312yes—
169#390C2312yes—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—24no—
181#387—24no—
182#387—24no—
183#387—24no—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—22no—
188#385—22no—
189#385—22no—
190#384C2312yes—
191#383—21no—
192#382—21no—
193#382C2312yes—
194#381—21no—
195#381—22no—
196#381—22no—
197#381—22no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380—22no—
202#380C2311yescharacteristic
203#380C2312yes—
204#380C2311yescharacteristic
205#379—21no—
206#379—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D411yescharacteristic
227C4×C22⋊C4D411yescharacteristic
228C4×C22⋊C4C2311yescharacteristic
229C4×C22⋊C4C2311yescharacteristic
230#376—21no—
231#376D412yes—
232C2×#95—22no—
233C43C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#336C2314yes—
243#336C2314yes—
244C25—21no—
245C22×Q8—28no—
246C22×Q8—22no—
247C22×Q8—21no—
248C22×Q8—21no—
249C22×Q8—44no—
250C22×Q8—44no—
251C22×D4—24no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—42no—
255C22×D4C22×C414yes—
256C22×D4C2411yescharacteristic
257C22×D4C2411yescharacteristic
258C23×C4—22no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—28no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4—84no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2411yescharacteristic
271C22×C8—44no—
272C22×C8C22×C418yes—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C4⋊Q8—28no—
277C4⋊Q8—28no—
278C4⋊Q8—24no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8C22×C414yes—
282C4⋊Q8C22×C414yes—
283C4⋊Q8C22×C418yes—
284C4×Q8—48no—
285C4×Q8—48no—
286C4×Q8—416no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—42no—
296C4×D4—42no—
297C4×D4C22×C414yes—
298C4×D4C22×C414yes—
299C4×D4C22×C414yes—
300C4×D4C22×C414yes—
301#117—24no—
302#117—42no—
303#117—48no—
304#117—44no—
305#117—48no—
306#117—84no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4C22×C414yes—
322C2×C4⋊C4C22×C418yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4C2×D411yescharacteristic
349C2×C22⋊C4C2×D411yescharacteristic
350C2×C22⋊C4C2×D411yescharacteristic
351C2×C22⋊C4C2×D411yescharacteristic
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—84no—
359C2×C42C22×C414yes—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42#4114yes—
367C2×C42C2412yes—
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—44no—
372#106—44no—
373#106C22×C418yes—
374#106C22×C418yes—
375#106C22×C418yes—
376#106C22×C418yes—
377C4⋊C8—24no—
378C4⋊C8C22×C414yes—
379C4⋊C8C22×C414yes—
380C4⋊C8C22×C418yes—
381#102—28no—
382#102—44no—
383#102C22×C418yes—
384#102C22×C418yes—
385#102C22×C418yes—
386#102C22×C418yes—
387C22⋊C8—24no—
388#97—24no—
389#97—44no—
390C4×C8—44no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409#95C2×D412yes—
410C24—21no—
411C24—21no—
412C24—24no—
413C24—44no—
414C24—42no—
415C24—42no—
416C24—82no—
417C24C22×D411yescharacteristic
418C2×Q8—28no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—432no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—44no—
428C2×D4—28no—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—44no—
436C2×D4—48no—
437C2×D4—44no—
438C2×D4C23×C412yes—
439C2×D4C23×C412yes—
440C2×D42−1+414yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—816no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4C4×D414yes—
474C22×C4C4×D414yes—
475C22×C4C23×C412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4112yes—
486C22×C4C2511yescharacteristic
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—48no—
493C2×C8C23×C414yes—
494C2×C8C23×C414yes—
495C4⋊C4—28no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—28no—
500C4⋊C4—24no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4C23×C414yes—
519C4⋊C4C23×C414yes—
520C4⋊C4C23×C412yes—
521C4⋊C4C23×C412yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—44no—
534C22⋊C4—42no—
535C22⋊C4—44no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4AGL(1,ℤ/8)14yes—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—42no—
547C42—44no—
548C42—48no—
549C42—48no—
550C42—44no—
551C42—82no—
552C42—82no—
553C42C4×D414yes—
554C42C4×D414yes—
555C42C23×C412yes—
556C42C23×C412yes—
557C42C2×#4112yes—
558C42C2×#4112yes—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—48no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—48no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23—84no—
574C23—88no—
575C23—88no—
576C23C2×C4×D412yes—
577C23C2×AGL(1,ℤ/8)12yes—
578C23#57512yes—
579C23C23×D411yescharacteristic
580Q8—48no—
581Q8—48no—
582Q8—416no—
583D4—22no—
584D4—22no—
585D4—22no—
586D4—24no—
587D4—48no—
588D4—42no—
589D4#51914yescomplemented
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—416no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4—84no—
638C2×C4—88no—
639C2×C4C2×C4×D412yes—
640C2×C4C2×C4×D412yes—
641C2×C4C2×C4×D412yes—
642C2×C4C2×C4×D412yes—
643C2×C4C2×C4×D412yes—
644C2×C4C2×AGL(1,ℤ/8)14yes—
645C2×C4#57714yes—
646C2×C4C24×C411yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C2×2−1+412yes—
649C2×C4#58512yes—
650C8—24no—
651C8—24no—
652C8—44no—
653C22—22no—
654C22—22no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—42no—
659C22—44no—
660C22—44no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—816no—
666C22—82no—
667C22—82no—
668C22C4×AGL(1,ℤ/8)14yes—
669C22C22×C4×D411yescharacteristic
670C22#348912yes—
671C22C22×AGL(1,ℤ/8)11yescharacteristic
672C22C2×#57511yescharacteristic
673C22C2×#57712yes—
674C4—22no—
675C4—22no—
676C4—24no—
677C4—22no—
678C4—22no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—44no—
686C4—82no—
687C4—84no—
688C4—82no—
689C4#294914yes—
690C4C2×#51911yescharacteristic
691C4#342811yescharacteristic
692C4#349512yes—
693C2—24no—
694C2—44no—
695C2—82no—
696C2#3420311yescharacteristic
697C2C2×C4×AGL(1,ℤ/8)11yescharacteristic
698C2C2×#294911yescharacteristic
699C2#6212612yes—
700C2#6212812yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34209C2semidirect
4. #23075C2semidirect
5. #22975C2semidirect
6. #22615C2semidirect
7. #22609C2semidirect
10. #22117C2semidirect
11. #22111C2semidirect
12. #21962C2semidirect
13. #21840C2semidirect
14. #21614C2semidirect
15. C4×#476C2semidirect
17. #20546C2semidirect
21. C2×#476C4semidirect
33. #2365C22semidirect
49. C4×C4⋊Q8C22semidirect
50. C4×C4⋊Q8C22semidirect
59. #1993C22semidirect
60. #1987C22semidirect
61. #1987C22semidirect
62. #1949C22semidirect
63. #1948C22semidirect
64. #1947C22semidirect
65. #1944C22semidirect
66. #1941C22semidirect
71. #1920C22semidirect
72. #1920C22semidirect
73. #1919C22semidirect
74. #1919C22semidirect
85. #1776C22semidirect
86. C4×#106C22semidirect
87. #1771C22semidirect
88. #1767C22semidirect
89. C4×C4⋊C8C22semidirect
103. C2×C4⋊Q8C2×C4semidirect
138. #476C2×C4semidirect
139. #476C2×C4semidirect
140. #476C2×C4semidirect
143. C2×#106C2×C4semidirect
149. C2×C4⋊C8C2×C4semidirect
589. D4#519semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
#342092
#341262
#230752
#229752
#226152
#226092
#225702
#225622
#221172
#221112
#219622
#218402
#216142
C4×#4762
#207382
#205462