Finite Group Databasebeta

← #6591606

Subgroups — #6591606

Order 512.

2127 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3329C212yesmaximal, complemented
2#36187C211yesmaximal, characteristic
3C4×D4×Q8C211yesmaximal, characteristic, complemented
4#33198C211yesmaximal, characteristic, complemented
5#32903C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28252C211yesmaximal, characteristic, complemented
8#27649C212yesmaximal, complemented
9#25319C212yesmaximal, complemented
10#25224C214yesmaximal, complemented
11#24814C212yesmaximal, complemented
12#24761C212yesmaximal, complemented
13#24541C214yesmaximal, complemented
14#22650C211yesmaximal, characteristic, complemented
15#22503C212yesmaximal, complemented
16C4×#460C211yesmaximal, characteristic, complemented
17#20647C212yesmaximal, complemented
18C2×#555C2212yes—
19C2×D4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#3329—216no—
22C2×#487C2212yes—
23C2×#485C2214yes—
24C2×#477C2212yes—
25C2×#468C2214yes—
26C2×#460C2211yescharacteristic
27C2×#460C2212yes—
28C2×#461C2212yes—
29C2×#448C2214yes—
30C2×C4×SD16C2212yes—
31C2×#436C2212yes—
32C2×#423C2212yes—
33C2×#419C2214yes—
34#2868C2211yescharacteristic
35#2856C2212yes—
36#2563C2212yes—
37#2537—21no—
38#2537—21no—
39#2500C2214yes—
40#2491—22no—
41#2491—22no—
42#2365—21no—
43#2365C2211yescharacteristic
44#2360—22no—
45#2360C2212yes—
46#2354—21no—
47#2354—21no—
48Q8×C4⋊C4C2211yescharacteristic
49D4×C4⋊C4—21no—
50D4×C4⋊C4C2211yescharacteristic
51Q8×C22⋊C4C2212yes—
52C4×C4⋊Q8—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×#127C2211yescharacteristic
55C4×C22⋊Q8—22no—
56C4×C22⋊Q8C2212yes—
57#2285C2212yes—
58C42×Q8C2211yescharacteristic
59C42×D4—21no—
60C42×D4C2211yescharacteristic
61#2085C2214yes—
62#2074C2214yes—
63#2072C2214yes—
64#2067C2214yes—
65#2055C2214yes—
66#2055C2214yes—
67#1990C2212yes—
68#1987C2212yes—
69#1982C2211yescharacteristic
70#1948C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1939C2212yes—
74#1926C2212yes—
75#1895C2214yes—
76#1880C2214yes—
77#1880C2214yes—
78#1880C2212yes—
79#1874C2214yes—
80#1851C2214yes—
81#1842C2211yescharacteristic
82#1825C2214yes—
83#1805C2214yes—
84#1805C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2211yescharacteristic
90#1803C2212yes—
91#1802C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#1736C2214yes—
96C23×Q8C2312yes—
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99C22×M16C2314yes—
100#555—28no—
101#555—28no—
102D4×Q8—24no—
103D4×Q8D418yescomplemented
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#127C2311yescharacteristic
110C2×#124—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2312yes—
113C2×#121—22no—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—28no—
120C2×C4×Q8—44no—
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×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4D411yescharacteristic
132C2×C4×D4D411yescharacteristic
133C2×C4×D4C2312yes—
134C2×C4×D4C2311yescharacteristic
135C2×#117—22no—
136C2×#117C2314yes—
137C22×C4⋊C4—22no—
138C22×C4⋊C4C2312yes—
139C22×C42—22no—
140C22×C42C2312yes—
141#487—28no—
142#485—216no—
143#477—28no—
144#468—216no—
145#461—28no—
146#460—24no—
147#460—44no—
148#448—216no—
149C4×SD16—28no—
150#436—28no—
151C2×#107C2314yes—
152C2×#106C2312yes—
153#423—44no—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2311yescharacteristic
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2314yes—
160#419—216no—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#102C2314yes—
170C2×C22⋊C8C2314yes—
171C2×C4×C8C2312yes—
172#394—24no—
173#394—24no—
174#393—24no—
175#391—22no—
176#391—22no—
177#391C2312yes—
178#391C2312yes—
179#390—21no—
180#390—21no—
181#390—22no—
182#390—22no—
183#390C2312yes—
184#390C2314yes—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—22no—
190#389—22no—
191#389C2312yes—
192#388—24no—
193#388—24no—
194#387—22no—
195#387—22no—
196#387—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384—21no—
201#384—21no—
202#384—21no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2312yes—
207#382C2314yes—
208#380—22no—
209#380—22no—
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—22no—
221C4×C4⋊C4—22no—
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×Q8C2411yescharacteristic
257C22×D4—24no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4C2×D411yescharacteristic
261C22×D4C2412yes—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4C2×D412yes—
266C23×C4C2×D412yes—
267C23×C4C2412yes—
268C2×SD16—24no—
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—28no—
272C2×SD16—44no—
273C2×SD16—44no—
274C2×M16—28no—
275C2×M16—44no—
276C2×M16—44no—
277C22×C8C2412yes—
278C22×C8C2414yes—
279C22×C8C2412yes—
280C4⋊Q8—28no—
281C4⋊Q8—24no—
282C4⋊Q8—22no—
283C4⋊Q8—24no—
284C4⋊Q8—82no—
285C4⋊Q8—82no—
286C4⋊Q8C2×D414yes—
287#127—22no—
288#127—22no—
289#124—416no—
290C22⋊Q8—24no—
291C22⋊Q8—216no—
292C22⋊Q8C2×D418yes—
293#121—28no—
294#121—28no—
295#121—48no—
296#121—84no—
297#121—84no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—24no—
301C4×Q8—22no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—48no—
306C4×Q8—42no—
307C4×Q8—42no—
308C4×Q8—41no—
309C4×Q8—48no—
310C4×Q8C2×D414yes—
311C4×Q8#4114yes—
312C4×D4—28no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—42no—
317C4×D4—44no—
318C4×D4—44no—
319C4×D4—44no—
320C4×D4C2×D414yes—
321C4×D4#4114yes—
322#117—28no—
323#117—28no—
324#117—48no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
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—22no—
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⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2414yes—
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C22⋊C4—22no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4C2×D412yes—
367C2×C22⋊C4C2×D412yes—
368C2×C22⋊C4C2×D412yes—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—24no—
372C2×C42—24no—
373C2×C42—24no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—48no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D412yes—
384C2×C42C2×D412yes—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2412yes—
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392#107—28no—
393#106—24no—
394C4⋊C8—24no—
395C4⋊C8—22no—
396C4⋊C8—22no—
397C4⋊C8—24no—
398C4⋊C8—42no—
399C4⋊C8—44no—
400#103—24no—
401#103—28no—
402#103—28no—
403#103—28no—
404#103—28no—
405#102—24no—
406#102—24no—
407#102—42no—
408#102—44no—
409C22⋊C8—28no—
410C4×C8—24no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—24no—
418#95—24no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95C2×D412yes—
423#95C2×D412yes—
424#95C2×D412yes—
425C24—22no—
426C24C22×D412yes—
427C2×Q8—28no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—216no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—48no—
440C2×Q8—48no—
441C2×Q8—48no—
442C2×Q8—44no—
443C2×Q8—82no—
444C2×Q8—82no—
445C2×Q8C22×D414yes—
446C2×Q8C22×D412yes—
447C2×Q8C2×#4112yes—
448C2×Q82−1+412yes—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—48no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—84no—
465C2×D4—84no—
466C2×D4#12114yes—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—28no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—82no—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D414yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C2×#4114yes—
505C22×C4C2511yescharacteristic
506SD16—48no—
507SD16—88no—
508M16—88no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—22no—
513C2×C8—24no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—28no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C4⋊C4—81no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4#12114yes—
551C4⋊C4C22×D412yes—
552C4⋊C4C2×#4112yes—
553C22⋊C4—28no—
554C22⋊C4—24no—
555C22⋊C4—42no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—48no—
559C22⋊C4—48no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—84no—
563C22⋊C4—84no—
564C22⋊C4#12118yes—
565C42—24no—
566C42—24no—
567C42—24no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—21no—
574C42—21no—
575C42—41no—
576C42—44no—
577C42—44no—
578C42—42no—
579C42—48no—
580C42—44no—
581C42—42no—
582C42—42no—
583C42—81no—
584C42#13714yes—
585C42C22×D412yes—
586C42C2×#4112yes—
587C42C2×#4112yes—
588C42H(5,𝔽2)12yes—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23C2×#12114yes—
603C23C23×D411yescharacteristic
604Q8—22no—
605Q8—22no—
606Q8—22no—
607Q8—24no—
608Q8—24no—
609Q8—416no—
610Q8—42no—
611Q8—42no—
612Q8—164no—
613Q8#54714yes—
614D4—28no—
615D4—42no—
616D4—42no—
617D4—42no—
618D4—42no—
619D4—84no—
620D4—84no—
621D4—168no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—21no—
644C2×C4—22no—
645C2×C4—21no—
646C2×C4—22no—
647C2×C4—21no—
648C2×C4—22no—
649C2×C4—416no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—48no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—44no—
669C2×C4—84no—
670C2×C4—88no—
671C2×C4—88no—
672C2×C4C2×#12114yes—
673C2×C4C2×#12112yes—
674C2×C4C2×#12112yes—
675C2×C4#54612yes—
676C2×C4#54612yes—
677C2×C4C2×#13712yes—
678C2×C4C2×#13712yes—
679C2×C4C2×#13712yes—
680C2×C4#57512yes—
681C2×C4#57712yes—
682C2×C4C23×D411yescharacteristic
683C2×C4C22×#4111yescharacteristic
684C2×C4C22×#4111yescharacteristic
685C2×C4C2×H(5,𝔽2)11yescharacteristic
686C2×C4C2×2−1+411yescharacteristic
687C2×C4#58511yescharacteristic
688C8—42no—
689C8—42no—
690C8—44no—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22C22×#12111yescharacteristic
706C22C2×#54611yescharacteristic
707C22#348911yescharacteristic
708C22C22×#13711yescharacteristic
709C22C2×#57511yescharacteristic
710C22C2×#57711yescharacteristic
711C22#358511yescharacteristic
712C4—22no—
713C4—24no—
714C4—24no—
715C4—22no—
716C4—22no—
717C4—42no—
718C4—44no—
719C4—41no—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—84no—
724C4—81no—
725C4—82no—
726C4—82no—
727C4#303714yes—
728C4#332914yescomplemented
729C4C2×#54711yescharacteristic
730C4#347511yescharacteristic
731C4#349611yescharacteristic
732C4#352011yescharacteristic
733C2—44no—
734C2—162no—
735C2#3615311yescharacteristic
736C2C2×#303711yescharacteristic
737C2#6043111yescharacteristic
738C2C2×#332911yescharacteristic
739C2#6116611yescharacteristic
740C2#6211411yescharacteristic
741C2#6214511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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
1. C2×#3329C2semidirect
3. C4×D4×Q8C2semidirect
4. #33198C2semidirect
5. #32903C2semidirect
6. #28717C2semidirect
7. #28252C2semidirect
8. #27649C2semidirect
9. #25319C2semidirect
10. #25224C2semidirect
11. #24814C2semidirect
12. #24761C2semidirect
13. #24541C2semidirect
14. #22650C2semidirect
15. #22503C2semidirect
16. C4×#460C2semidirect
17. #20647C2semidirect
103. D4×Q8D4semidirect
728. C4#3329semidirect

Maximal subgroups

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

subgroupindex
C2×#33292
#361872
C4×D4×Q82
#331982
#329032
#287172
#282522
#276492
#253192
#252242
#248142
#247612
#245412
#226502
#225032
C4×#4602
#206472