Finite Group Databasebeta

← #6591588

Subgroups — #6591588

Order 512.

2125 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3299C212yesmaximal
2#36180C211yesmaximal, characteristic
3C4×D4×Q8C211yesmaximal, characteristic
4#33198C211yesmaximal, characteristic, complemented
5#32906C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28255C211yesmaximal, characteristic, complemented
8#27648C212yesmaximal, complemented
9#25324C212yesmaximal, complemented
10#25206C214yesmaximal, complemented
11#24817C212yesmaximal, complemented
12#24752C212yesmaximal, complemented
13#24532C214yesmaximal, complemented
14D4×C4⋊C8C211yesmaximal, characteristic
15#22487C212yesmaximal
16C4×#462C211yesmaximal, characteristic, complemented
17C2×#1803C212yesmaximal, complemented
18C2×#554C2212yes—
19C2×D4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#3299—216no—
22C2×#487C2212yes—
23C2×#484C2214yes—
24C2×#477C2212yes—
25C2×#467C2214yes—
26C2×#462C2212yes—
27C2×#462C2211yescharacteristic
28C2×#462C2212yes—
29C2×#451C2214yes—
30C2×C4×Q16C2212yes—
31C2×C8×D4C2212yes—
32C22×C4⋊C8C2212yes—
33C22×#103C2214yes—
34#2868C2211yescharacteristic
35#2857C2212yes—
36#2567C2212yes—
37#2552—21no—
38#2552—21no—
39#2502C2214yes—
40#2440—22no—
41#2440—22no—
42#2365—21no—
43#2365C2211yescharacteristic
44#2360—22no—
45#2360C2212yes—
46Q8×C4⋊C4C2211yescharacteristic
47D4×C4⋊C4—21no—
48D4×C4⋊C4—21no—
49D4×C4⋊C4—21no—
50D4×C4⋊C4C2211yescharacteristic
51Q8×C22⋊C4C2212yes—
52C4×C4⋊Q8—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C22⋊Q8—22no—
56C4×C22⋊Q8C2212yes—
57C2×#384C2212yes—
58C42×Q8C2211yescharacteristic
59C42×D4—21no—
60C42×D4C2211yescharacteristic
61#2083C2214yes—
62#2072C2214yes—
63#2072C2214yes—
64#2068C2214yes—
65#2056C2214yes—
66#2056C2214yes—
67#1990C2212yes—
68#1987C2212yes—
69#1981C2211yescharacteristic
70#1947C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1939C2212yes—
74#1928C2212yes—
75#1892C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1878C2212yes—
79#1872C2214yes—
80#1850C2214yes—
81#1842C2211yescharacteristic
82#1824C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2214yes—
88#1803C2212yes—
89#1803C2211yescharacteristic
90#1803C2214yes—
91#1803C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C2×#340C2214yes—
96C23×Q8C2312yes—
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C23×C8C2314yes—
100#554—28no—
101#554—28no—
102D4×Q8—24no—
103D4×Q8D418yes—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×#125—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2312yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—28no—
118C2×C4×Q8—44no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4—21no—
127C2×C4×D4—22no—
128C2×C4×D4—21no—
129C2×C4×D4D411yescharacteristic
130C2×C4×D4D411yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C22×C4⋊C4—22no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4C2314yes—
136C22×C4⋊C4C2312yes—
137C22×C42—22no—
138C22×C42C2312yes—
139#487—28no—
140#484—216no—
141#477—28no—
142#467—216no—
143#462—28no—
144#462—24no—
145#462—44no—
146#451—216no—
147C4×Q16—28no—
148C8×D4—28no—
149C2×#107C2314yes—
150C2×#107C2312yes—
151C2×C4⋊C8—44no—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2314yes—
158C2×#103—216no—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×C22⋊C8C2314yes—
169C2×C4×C8C2312yes—
170#398—24no—
171#398—24no—
172#395—24no—
173#391—22no—
174#391—22no—
175#391C2312yes—
176#391C2312yes—
177#391C2314yes—
178#391C2312yes—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—22no—
191#389C2312yes—
192#387—22no—
193#387—22no—
194#387—22no—
195#385—22no—
196#385—22no—
197#385—24no—
198#385—24no—
199#385—22no—
200#384—21no—
201#384—21no—
202#384—21no—
203#384—22no—
204#384—22no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2314yes—
208#384C2312yes—
209#384C2312yes—
210#380—22no—
211#380—22no—
212#380—22no—
213#380C2312yes—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2311yescharacteristic
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4D412yes—
234C4×C22⋊C4D412yes—
235C4×C22⋊C4C2312yes—
236C43—21no—
237C43C2311yescharacteristic
238#340C2314yes—
239#340C2314yes—
240#340C2312yes—
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2312yes—
245#336C2314yes—
246#336C2312yes—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8—24no—
252C22×Q8—42no—
253C22×Q8—42no—
254C22×Q8C2×D418yes—
255C22×Q8C2412yes—
256C22×Q8C2412yes—
257C22×Q8C2411yescharacteristic
258C22×D4C2×D411yescharacteristic
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—24no—
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2412yes—
265C2×Q16—24no—
266C2×Q16—24no—
267C2×Q16—24no—
268C2×Q16—28no—
269C2×Q16—44no—
270C2×Q16—44no—
271C22×C8—28no—
272C22×C8—44no—
273C22×C8—44no—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C4⋊Q8—28no—
278C4⋊Q8—24no—
279C4⋊Q8—22no—
280C4⋊Q8—24no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—82no—
284C4⋊Q8—82no—
285C4⋊Q8C2×D414yes—
286#125—48no—
287C22⋊Q8—24no—
288C22⋊Q8—216no—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C22⋊Q8—416no—
292C22⋊Q8C2×D418yes—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—24no—
296C4×Q8—22no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—48no—
303C4×Q8—42no—
304C4×Q8—42no—
305C4×Q8—41no—
306C4×Q8—48no—
307C4×Q8C2×D414yes—
308C4×Q8#4114yes—
309C4×D4—28no—
310C4×D4—24no—
311C4×D4—48no—
312C4×D4—44no—
313C4×D4Q1618yescomplemented
314C4×D4C2×D414yes—
315C4×D4#4114yes—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—42no—
347C2×C4⋊C4—48no—
348C2×C4⋊C4—84no—
349C2×C4⋊C4—84no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2414yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—48no—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D412yes—
380C2×C42C2×D412yes—
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2412yes—
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388#107—24no—
389#107—28no—
390C4⋊C8—24no—
391C4⋊C8—22no—
392C4⋊C8—22no—
393C4⋊C8—24no—
394C4⋊C8—44no—
395C4⋊C8—42no—
396#103—24no—
397#103—28no—
398#103—28no—
399#103—28no—
400#103—28no—
401#103—24no—
402#103—24no—
403#103—42no—
404#103—44no—
405C22⋊C8—28no—
406C4×C8—24no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—24no—
419#95—24no—
420#95C2×D412yes—
421#95C2×D412yes—
422#95C2×D412yes—
423C24C22×D412yes—
424C2×Q8—28no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—216no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—48no—
443C2×Q8—48no—
444C2×Q8—48no—
445C2×Q8—44no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8C22×D414yes—
449C2×Q8C22×D412yes—
450C2×Q8C2×#4112yes—
451C2×Q82−1+412yes—
452C2×D4#12114yes—
453C2×D4C2×Q1614yes—
454C2×D4#13714yes—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—28no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—48no—
479C22×C4—82no—
480C22×C4—84no—
481C22×C4—84no—
482C22×C4C2×Q1618yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4114yes—
494C22×C4C2511yescharacteristic
495Q16—48no—
496Q16—88no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—88no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—28no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—48no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—48no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—81no—
540C4⋊C4—82no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4#12114yes—
546C4⋊C4C2×Q1614yes—
547C4⋊C4C22×D412yes—
548C4⋊C4C2×#4112yes—
549C22⋊C4—28no—
550C22⋊C4—24no—
551C22⋊C4—48no—
552C22⋊C4—44no—
553C22⋊C4#12118yes—
554C22⋊C4C2×Q1618yes—
555C42—24no—
556C42—24no—
557C42—24no—
558C42—24no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—21no—
564C42—21no—
565C42—41no—
566C42—44no—
567C42—44no—
568C42—42no—
569C42—44no—
570C42—44no—
571C42—42no—
572C42—42no—
573C42—81no—
574C42C2×Q1614yes—
575C42C22×D412yes—
576C42C2×#4112yes—
577C42C2×#4112yes—
578C42H(5,𝔽2)12yes—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23C2×#12114yes—
584C23C22×Q1614yes—
585C23C2×#13714yes—
586C23C23×D411yescharacteristic
587Q8—22no—
588Q8—22no—
589Q8—22no—
590Q8—24no—
591Q8—24no—
592Q8—416no—
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—42no—
597Q8—42no—
598Q8—42no—
599Q8—164no—
600Q8#54714yes—
601D4#462116yescomplemented
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—216no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—48no—
630C2×C4—416no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—88no—
653C2×C4—168no—
654C2×C4C2×#12112yes—
655C2×C4C2×#12114yes—
656C2×C4C2×#12112yes—
657C2×C4#54612yes—
658C2×C4#54612yes—
659C2×C4C22×Q1612yes—
660C2×C4C22×Q1614yes—
661C2×C4C22×Q1612yes—
662C2×C4C22×Q1612yes—
663C2×C4C2×#13712yes—
664C2×C4#57712yes—
665C2×C4C23×D411yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×2−1+411yescharacteristic
670C2×C4#58511yescharacteristic
671C8—42no—
672C8—42no—
673C8—44no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22C2×#46218yes—
681C22C22×#12111yescharacteristic
682C22C2×#54611yescharacteristic
683C22#348911yescharacteristic
684C22C23×Q1611yescharacteristic
685C22C22×#13711yescharacteristic
686C22C2×#57711yescharacteristic
687C22#358511yescharacteristic
688C4—22no—
689C4—24no—
690C4—24no—
691C4—22no—
692C4—24no—
693C4—22no—
694C4—42no—
695C4—44no—
696C4—41no—
697C4—42no—
698C4—42no—
699C4—82no—
700C4—84no—
701C4—82no—
702C4—81no—
703C4—82no—
704C4C2×#46214yes—
705C4#329914yescomplemented
706C4C2×#54711yescharacteristic
707C4#347511yescharacteristic
708C4#349611yescharacteristic
709C4#352011yescharacteristic
710C2—28no—
711C2#3615311yescharacteristic
712C2C22×#46211yescharacteristic
713C2#6043111yescharacteristic
714C2C2×#329911yescharacteristic
715C2#6116211yescharacteristic
716C2#6210011yescharacteristic
717C2#6213111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33198C2semidirect
5. #32906C2semidirect
6. #28717C2semidirect
7. #28255C2semidirect
8. #27648C2semidirect
9. #25324C2semidirect
10. #25206C2semidirect
11. #24817C2semidirect
12. #24752C2semidirect
13. #24532C2semidirect
16. C4×#462C2semidirect
17. C2×#1803C2semidirect
313. C4×D4Q16semidirect
601. D4#462semidirect
705. C4#3299semidirect

Maximal subgroups

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

subgroupindex
C2×#32992
#361802
C4×D4×Q82
#331982
#329062
#287172
#282552
#276482
#253242
#252062
#248172
#247522
#245322
D4×C4⋊C82
#224872
C4×#4622
C2×#18032