Finite Group Databasebeta

← #6802508

Subgroups — #6802508

Order 512.

1823 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36599C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic
3#32906C211yesmaximal, characteristic
4#32168C214yesmaximal, complemented
5#30744C214yesmaximal, complemented
6#29715C212yesmaximal
7#28462C212yesmaximal, complemented
8#27749C212yesmaximal, complemented
9#25722C212yesmaximal
10#25722C211yesmaximal, characteristic
11#25370C212yesmaximal, complemented
12#24632C212yesmaximal, complemented
13#24212C212yesmaximal
14#24212C211yesmaximal, characteristic
15#22625C211yesmaximal, characteristic
16#22044C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C2×#494C2214yes—
22C2×#494C2212yes—
23C2×#462C2214yes—
24C2×#462C2212yes—
25C2×#462C2212yes—
26C2×#451C2212yes—
27C2×#451C2211yescharacteristic
28C2×#451C2212yes—
29C2×#451C2212yes—
30C2×#451C2212yes—
31C2×C4×Q16C2212yes—
32#2857C2212yes—
33#2857C2211yescharacteristic
34#2585—22no—
35#2569—22no—
36#2552C2211yescharacteristic
37#2546—22no—
38#2546—21no—
39#2546—21no—
40#2546C2214yes—
41#2546C2212yes—
42#2546C2211yescharacteristic
43#2546C2212yes—
44#2511C2212yes—
45#2509—21no—
46#2446C2212yes—
47#2442—21no—
48C2×#398—22no—
49#2370—21no—
50#2365C2211yescharacteristic
51#2355C2211yescharacteristic
52#2345—21no—
53C4×C22⋊Q8C2212yes—
54C4×C22⋊Q8C2211yescharacteristic
55C2×#391C2212yes—
56C2×#389—21no—
57C2×#389—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59#2072C2214yes—
60#2056C2214yes—
61#2056C2214yes—
62#2043C2214yes—
63#2043C2212yes—
64#2016C2214yes—
65#2016C2212yes—
66#2016C2214yes—
67#2016C2214yes—
68#2016C2214yes—
69#2016C2214yes—
70#1981C2212yes—
71#1981C2214yes—
72#1969C2214yes—
73#1956C2212yes—
74#1956C2214yes—
75#1939C2212yes—
76#1933C2214yes—
77#1928C2214yes—
78#1916C2214yes—
79#1878C2212yes—
80#1878C2214yes—
81#1878C2214yes—
82#1872C2211yescharacteristic
83#1872C2212yes—
84#1863C2212yes—
85#1852C2211yescharacteristic
86#1803C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2212yes—
90#1787C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C23×Q8C2312yes—
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C22×Q16C2314yes—
100C22×Q16C2314yes—
101C22×Q16C2314yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—44no—
107C2×C4⋊Q8—48no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2314yes—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2312yes—
114C2×#125—21no—
115C2×C22⋊Q8C2311yescharacteristic
116C2×C22⋊Q8C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—44no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#494—44no—
136#494—48no—
137#462—48no—
138#462—44no—
139#462—44no—
140#451—24no—
141#451—28no—
142#451—44no—
143#451—44no—
144#451—44no—
145C4×Q16—44no—
146C2×#107C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×C22⋊C8C2311yescharacteristic
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2311yescharacteristic
163C2×C4×C8C2312yes—
164C2×C4×C8C2314yes—
165#400—22no—
166#400—22no—
167#400—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#398—24no—
172#398—24no—
173#398—24no—
174#398—24no—
175#391—22no—
176#391—22no—
177#391C2312yes—
178#391C2312yes—
179#391C2314yes—
180#391C2314yes—
181#391C2312yes—
182#391C2311yescharacteristic
183#391C2312yes—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—22no—
189#389—24no—
190#389—22no—
191#389—24no—
192#389—22no—
193#389—22no—
194#389—21no—
195#389—22no—
196#389—22no—
197#389—21no—
198#389C2311yescharacteristic
199#389C2312yes—
200#387—22no—
201#387—22no—
202#387—21no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#385—21no—
209#385—22no—
210#384C2311yescharacteristic
211#384C2312yes—
212#382—21no—
213#380—21no—
214#380—22no—
215#380—22no—
216#380—21no—
217#380C2311yescharacteristic
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4D412yes—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2311yescharacteristic
230C4×C22⋊C4—22no—
231C4×C22⋊C4—21no—
232C2×#95—22no—
233C2×#95—22no—
234C2×#95D412yes—
235#340C2312yes—
236#340C2314yes—
237#340C2312yes—
238#336C2314yes—
239#336C2312yes—
240#336C2312yes—
241#336C2314yes—
242C22×Q8—24no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—22no—
248C22×Q8—24no—
249C22×Q8—42no—
250C22×Q8—44no—
251C22×Q8—44no—
252C22×Q8C2411yescharacteristic
253C22×Q8C2412yes—
254C22×Q8C2414yes—
255C22×Q8C2412yes—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264C2×Q16—24no—
265C2×Q16—28no—
266C2×Q16—24no—
267C2×Q16—28no—
268C2×Q16—42no—
269C2×Q16—44no—
270C2×Q16—44no—
271C2×Q16—44no—
272C2×Q16—44no—
273C2×Q16—44no—
274C2×Q16—44no—
275C2×Q16—44no—
276C2×Q16—44no—
277C2×Q16—44no—
278C2×Q16—44no—
279C22×C8C2412yes—
280C22×C8C2414yes—
281C22×C8C2412yes—
282C4⋊Q8—28no—
283C4⋊Q8—28no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—24no—
287C4⋊Q8—44no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290C4⋊Q8—44no—
291C4⋊Q8—44no—
292C4⋊Q8—44no—
293C4⋊Q8—48no—
294C4⋊Q8—48no—
295#125—44no—
296C22⋊Q8—22no—
297C22⋊Q8—24no—
298C22⋊Q8—22no—
299C22⋊Q8—24no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—42no—
304C4×Q8—42no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8—44no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—48no—
337C2×C4⋊C4—48no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4—44no—
340C2×C4⋊C4Q1618yescomplemented
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—44no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372#107—42no—
373C4⋊C8—24no—
374C4⋊C8—44no—
375#103—24no—
376#103—24no—
377#103—28no—
378#103—42no—
379#103—42no—
380#103—44no—
381#103—44no—
382#103—44no—
383#103—44no—
384#103—44no—
385#103—44no—
386C22⋊C8—22no—
387C22⋊C8—24no—
388C22⋊C8#4114yes—
389C4×C8—24no—
390C4×C8—44no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—22no—
404#95—22no—
405#95—21no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95C2×D412yes—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D412yes—
414C24C22×D411yescharacteristic
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—48no—
436C2×Q8—48no—
437C2×Q8—42no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—88no—
443C2×Q8—88no—
444C2×Q8H(5,𝔽2)14yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4C2×Q1614yes—
476C22×C4C2×Q1618yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4C2511yescharacteristic
489Q16—48no—
490Q16—88no—
491Q16—88no—
492Q16—88no—
493Q16—88no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8C2×#4114yes—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—81no—
533C4⋊C4—82no—
534C4⋊C4—81no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4C2×Q1618yes—
539C4⋊C4C2×Q1614yes—
540C4⋊C4C2×Q1614yes—
541C4⋊C4#13714yes—
542C22⋊C4—44no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C42—22no—
546C42—22no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—42no—
551C42—42no—
552C42—44no—
553C42—44no—
554C42—42no—
555C42—42no—
556C42—82no—
557C42—82no—
558C42H(5,𝔽2)14yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23#55012yes—
564C23C22×Q1612yes—
565C23#57812yes—
566C23C23×D411yescharacteristic
567Q8—24no—
568Q8—24no—
569Q8—48no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—44no—
574Q8—44no—
575Q8—44no—
576Q8—44no—
577Q8—42no—
578Q8—44no—
579Q8—44no—
580Q8—88no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4#53512yes—
636C2×C4#53512yes—
637C2×C4#55012yes—
638C2×C4C22×Q1612yes—
639C2×C4C22×Q1614yes—
640C2×C4C22×Q1614yes—
641C2×C4C22×Q1612yes—
642C2×C4C2×#13714yes—
643C2×C4C2×#13712yes—
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)12yes—
646C2×C4C2×H(5,𝔽2)12yes—
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C8—24no—
649C8—42no—
650C8—44no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22#339514yes—
658C22C2×#53511yescharacteristic
659C22C2×#55011yescharacteristic
660C22#348411yescharacteristic
661C22C23×Q1611yescharacteristic
662C22C22×#13711yescharacteristic
663C22C2×#57811yescharacteristic
664C22#358611yescharacteristic
665C4—22no—
666C4—24no—
667C4—24no—
668C4—22no—
669C4—24no—
670C4—22no—
671C4—41no—
672C4—42no—
673C4—44no—
674C4—81no—
675C4—82no—
676C4—81no—
677C4—84no—
678C4—81no—
679C4—82no—
680C4—82no—
681C4—81no—
682C4#320914yes—
683C4#351912yes—
684C4#353712yes—
685C2—24no—
686C2#3913211yescharacteristic
687C2C2×#320911yescharacteristic
688C2#6081411yescharacteristic
689C2C2×#339511yescharacteristic
690C2#6140411yescharacteristic
691C2#6156311yescharacteristic
692C2#6160211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32168C2semidirect
5. #30744C2semidirect
7. #28462C2semidirect
8. #27749C2semidirect
11. #25370C2semidirect
12. #24632C2semidirect
340. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#365992
C2×#25462
#329062
#321682
#307442
#297152
#284622
#277492
#257222
#253702
#246322
#242122
#226252
#220442
C4×#4512