Finite Group Databasebeta

← #6591607

Subgroups — #6591607

Order 512.

2041 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3328C212yesmaximal
2#36188C211yesmaximal, characteristic
3C4×D4×Q8C211yesmaximal, characteristic
4#33196C211yesmaximal, characteristic, complemented
5#32906C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28255C211yesmaximal, characteristic, complemented
8#27649C212yesmaximal, complemented
9#25323C212yesmaximal, complemented
10#25226C214yesmaximal, complemented
11#24818C212yesmaximal, complemented
12#24760C212yesmaximal, complemented
13#24540C214yesmaximal, complemented
14#22650C211yesmaximal, characteristic
15#22504C212yesmaximal
16C4×#462C211yesmaximal, characteristic, complemented
17#20647C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×D4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#3328—216no—
22C2×#487C2212yes—
23C2×#485C2214yes—
24C2×#475C2212yes—
25C2×#470C2214yes—
26C2×#462C2212yes—
27C2×#462C2211yescharacteristic
28C2×#462C2212yes—
29C2×#451C2214yes—
30C2×C4×Q16C2212yes—
31C2×#436C2212yes—
32C2×#423C2212yes—
33C2×#419C2214yes—
34#2868C2211yescharacteristic
35#2857C2212yes—
36#2568C2212yes—
37#2547—21no—
38#2547—21no—
39#2499C2214yes—
40#2496—22no—
41#2496—22no—
42#2365—21no—
43#2365—21no—
44#2365—21no—
45#2365C2211yescharacteristic
46#2360—22no—
47#2360C2212yes—
48Q8×C4⋊C4C2211yescharacteristic
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—
57#2285C2212yes—
58C42×Q8C2211yescharacteristic
59C42×D4—21no—
60C42×D4C2211yescharacteristic
61#2085C2214yes—
62#2074C2214yes—
63#2072C2214yes—
64#2066C2214yes—
65#2056C2214yes—
66#2056C2214yes—
67#1990C2212yes—
68#1988C2212yes—
69#1981C2211yescharacteristic
70#1948C2212yes—
71#1948C2212yes—
72#1947C2211yescharacteristic
73#1939C2212yes—
74#1928C2212yes—
75#1894C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1878C2212yes—
79#1872C2214yes—
80#1851C2214yes—
81#1842C2211yescharacteristic
82#1825C2214yes—
83#1805C2214yes—
84#1805C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1803C2211yescharacteristic
91#1803C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#1736C2214yes—
96C23×Q8C2312yes—
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C22×M16C2314yes—
100#556—28no—
101#556—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×#126—24no—
111C2×#124—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2312yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—28no—
119C2×C4×Q8—44no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4D411yescharacteristic
130C2×C4×D4D411yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×#117C2314yes—
133C22×C4⋊C4—22no—
134C22×C4⋊C4C2312yes—
135C22×C42—22no—
136C22×C42C2312yes—
137#487—28no—
138#485—216no—
139#475—28no—
140#470—216no—
141#462—28no—
142#462—24no—
143#462—44no—
144#451—216no—
145C4×Q16—28no—
146#436—28no—
147C2×#107C2312yes—
148C2×#106C2314yes—
149#423—44no—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2311yescharacteristic
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2314yes—
156#419—216no—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×C22⋊C8C2314yes—
167C2×C4×C8C2312yes—
168#400—24no—
169#397—24no—
170#397—24no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#391C2314yes—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2312yes—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—24no—
194#387—24no—
195#385—22no—
196#385—22no—
197#385—22no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2312yes—
204#384C2312yes—
205#382C2314yes—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
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×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2412yes—
264C2×Q16—24no—
265C2×Q16—24no—
266C2×Q16—24no—
267C2×Q16—28no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×M16—28no—
271C2×M16—44no—
272C2×M16—44no—
273C22×C8C2412yes—
274C22×C8C2414yes—
275C22×C8C2412yes—
276C4⋊Q8—28no—
277C4⋊Q8—24no—
278C4⋊Q8—22no—
279C4⋊Q8—24no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—44no—
283C4⋊Q8—44no—
284C4⋊Q8C2×D414yes—
285#126—416no—
286#124—48no—
287C22⋊Q8—24no—
288C22⋊Q8—216no—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C22⋊Q8C2×D418yes—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8—24no—
295C4×Q8—22no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—48no—
302C4×Q8—42no—
303C4×Q8—42no—
304C4×Q8—41no—
305C4×Q8—48no—
306C4×Q8—48no—
307C4×Q8C2×D414yes—
308C4×Q8#4114yes—
309C4×D4—28no—
310C4×D4—24no—
311C4×D4—42no—
312C4×D4—44no—
313C4×D4C2×D414yes—
314C4×D4#4114yes—
315#117—28no—
316#117—28no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—42no—
343C2×C4⋊C4—48no—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2414yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2411yescharacteristic
353C2×C22⋊C4—22no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—21no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—48no—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D412yes—
376C2×C42C2×D412yes—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#107—24no—
385#106—28no—
386C4⋊C8—24no—
387C4⋊C8—22no—
388C4⋊C8—22no—
389C4⋊C8—24no—
390C4⋊C8—42no—
391C4⋊C8—44no—
392#103—24no—
393#103—28no—
394#103—28no—
395#103—28no—
396#103—28no—
397#103—24no—
398#103—24no—
399#103—42no—
400#103—44no—
401C22⋊C8—28no—
402C4×C8—24no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—24no—
410#95—24no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—24no—
417#95—24no—
418#95C2×D412yes—
419#95C2×D412yes—
420#95C2×D412yes—
421C24C22×D412yes—
422C2×Q8—28no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—216no—
428C2×Q8—22no—
429C2×Q8—22no—
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—44no—
439C2×Q8—44no—
440C2×Q8—48no—
441C2×Q8—48no—
442C2×Q8—48no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8C22×D414yes—
447C2×Q8C22×D412yes—
448C2×Q8C2×#4112yes—
449C2×Q82−1+412yes—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4#12114yes—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—28no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—82no—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D414yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4114yes—
489C22×C4C2511yescharacteristic
490Q16—48no—
491Q16—88no—
492M16—88no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—28no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—81no—
537C4⋊C4—82no—
538C4⋊C4#12114yes—
539C4⋊C4C22×D412yes—
540C4⋊C4C2×#4112yes—
541C22⋊C4—28no—
542C22⋊C4—24no—
543C22⋊C4—42no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4#12118yes—
548C42—24no—
549C42—24no—
550C42—24no—
551C42—24no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—21no—
557C42—21no—
558C42—41no—
559C42—44no—
560C42—44no—
561C42—42no—
562C42—48no—
563C42—44no—
564C42—44no—
565C42—42no—
566C42—42no—
567C42—81no—
568C42#13714yes—
569C42C22×D412yes—
570C42C2×#4112yes—
571C42C2×#4112yes—
572C42H(5,𝔽2)12yes—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23C2×#12114yes—
580C23C23×D411yescharacteristic
581Q8—22no—
582Q8—22no—
583Q8—22no—
584Q8—24no—
585Q8—24no—
586Q8—416no—
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—42no—
593Q8—88no—
594Q8#54714yes—
595D4—28no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—416no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—84no—
647C2×C4—88no—
648C2×C4C2×#12114yes—
649C2×C4C2×#12112yes—
650C2×C4C2×#12112yes—
651C2×C4#54612yes—
652C2×C4#54612yes—
653C2×C4C2×#13712yes—
654C2×C4C2×#13712yes—
655C2×C4C2×#13712yes—
656C2×C4#57512yes—
657C2×C4#57812yes—
658C2×C4C23×D411yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×2−1+411yescharacteristic
663C2×C4#58511yescharacteristic
664C8—42no—
665C8—42no—
666C8—44no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22C22×#12111yescharacteristic
675C22C2×#54611yescharacteristic
676C22#348911yescharacteristic
677C22C22×#13711yescharacteristic
678C22C2×#57511yescharacteristic
679C22C2×#57811yescharacteristic
680C22#358611yescharacteristic
681C4—22no—
682C4—24no—
683C4—24no—
684C4—22no—
685C4—22no—
686C4—42no—
687C4—44no—
688C4—41no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—82no—
693C4—82no—
694C4—81no—
695C4—82no—
696C4—84no—
697C4#303714yes—
698C4#332814yescomplemented
699C4C2×#54711yescharacteristic
700C4#347511yescharacteristic
701C4#349611yescharacteristic
702C4#352011yescharacteristic
703C2—44no—
704C2#3615311yescharacteristic
705C2C2×#303711yescharacteristic
706C2#6043211yescharacteristic
707C2C2×#332811yescharacteristic
708C2#6116511yescharacteristic
709C2#6211311yescharacteristic
710C2#6214611yescharacteristic
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. #33196C2semidirect
5. #32906C2semidirect
6. #28717C2semidirect
7. #28255C2semidirect
8. #27649C2semidirect
9. #25323C2semidirect
10. #25226C2semidirect
11. #24818C2semidirect
12. #24760C2semidirect
13. #24540C2semidirect
16. C4×#462C2semidirect
17. #20647C2semidirect
698. C4#3328semidirect

Maximal subgroups

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

subgroupindex
C2×#33282
#361882
C4×D4×Q82
#331962
#329062
#287172
#282552
#276492
#253232
#252262
#248182
#247602
#245402
#226502
#225042
C4×#4622
#206472