Finite Group Databasebeta

← #6373347

Subgroups — #6373347

Order 512.

2352 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic
2#34225C211yesmaximal, characteristic, complemented
3C4×#518C211yesmaximal, characteristic
4#21614C212yesmaximal, complemented
5C4×#476C212yesmaximal, complemented
6#21587C214yesmaximal, complemented
7#21504C212yesmaximal, complemented
8C4×#461C212yesmaximal, complemented
9#21459C214yesmaximal
10#21392C211yesmaximal, characteristic, complemented
11#21362C211yesmaximal, characteristic, complemented
12#21133C212yesmaximal, complemented
13#21130C212yesmaximal, complemented
14#20738C211yesmaximal, characteristic
15#20506C212yesmaximal, complemented
16#20504C212yesmaximal, complemented
17#20470C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#518C2212yes—
20C22×C4×D4C2211yescharacteristic
21#3101C4116yescomplemented
22C2×#483C2212yes—
23C2×#476C2211yescharacteristic
24C2×#476C2211yescharacteristic
25C2×#461C2211yescharacteristic
26C2×#461C2211yescharacteristic
27C2×#450C2212yes—
28C2×#440C2214yes—
29C2×C4×SD16C2214yes—
30C2×C4×SD16C2212yes—
31C2×C4×SD16C2212yes—
32C2×#432C2211yescharacteristic
33C2×#431C2211yescharacteristic
34#2370—22no—
35#2362—22no—
36#2356—21no—
37#2351—21no—
38#2347—21no—
39#2346—21no—
40#2323C2214yes—
41#2320C2212yescomplemented
42C4×C4⋊Q8C2212yescomplemented
43C4×C4⋊Q8C2212yescomplemented
44#2285C2211yescharacteristic
45#2276C2212yescomplemented
46#2275—21no—
47#2275—21no—
48#2275C2211yescharacteristic
49#2275C2211yescharacteristic
50#2275C2211yescharacteristic
51#2275C2211yescharacteristic
52#2274C2212yes—
53#2273—22no—
54C42×D4—22no—
55C42×D4C2212yes—
56C42×D4C2212yes—
57C4×#117C2211yescharacteristic
58C4×#117C2211yescharacteristic
59C2×C4×C22⋊C4—21no—
60C2×C4×C22⋊C4—21no—
61#1987C2212yescomplemented
62#1981C2212yescomplemented
63#1948C2212yescomplemented
64#1947C2212yescomplemented
65#1928C2214yescomplemented
66#1927C2214yescomplemented
67#1920C2214yescomplemented
68#1919C2214yescomplemented
69#1877C2214yes—
70#1877C2214yes—
71#1873C2214yes—
72#1864C2214yes—
73#1862C2214yes—
74#1854C2214yes—
75#1824C2212yes—
76#1802C2212yes—
77#1802C2212yes—
78#1786C2212yes—
79#1776C2214yescomplemented
80C4×#107C2212yescomplemented
81C4×#106C2214yescomplemented
82C4×#106C2212yescomplemented
83#1767C2212yescomplemented
84C4×C4⋊C8C2212yescomplemented
85#1763C2214yescomplemented
86#1763C2212yescomplemented
87#1763C2212yescomplemented
88#1762C2212yes—
89#1762C2212yes—
90#1762C2214yes—
91C4×#103C2214yescomplemented
92C4×#102C2214yes—
93#1750C2214yes—
94#1746C2212yescomplemented
95#1726C2212yescomplemented
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C22×SD16C2312yes—
100#533—24no—
101#533C2×C418yescomplemented
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2312yes—
107#518—28no—
108#518—28no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—24no—
118C2×C4×D4C2×C418yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×#117—21no—
128C2×#117—21no—
129C2×#117C2312yes—
130C2×#117C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C42—22no—
136C22×C42—22no—
137C22×C42C2311yescharacteristic
138#483C2×C4116yescomplemented
139#476C2×C418yescomplemented
140#476C2×C418yescomplemented
141#461C2×C418yescomplemented
142#461C2×C418yescomplemented
143#450C2×C4116yes—
144#440—216no—
145C4×SD16—216no—
146C4×SD16C2×C4116yescomplemented
147C4×SD16C2×C4116yescomplemented
148#432—24no—
149#431C2×C418yescomplemented
150C2×#107C2314yes—
151C2×#106C2314yes—
152C2×#106C2312yes—
153C2×#106C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2311yescharacteristic
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×C22⋊C8C2312yes—
166C2×C22⋊C8C2312yes—
167C2×#97C2312yes—
168C2×#97C2314yes—
169C2×C4×C8C2314yes—
170C2×C4×C8C2312yes—
171#391—22no—
172#391C2314yes—
173#389—24no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#388—24no—
179#388—24no—
180#387—24no—
181#387—24no—
182#384C2312yes—
183#383—22no—
184#382—21no—
185#382—22no—
186#380—21no—
187#380C2312yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—24no—
196C4×C4⋊C4D411yescharacteristic
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—21no—
209C4×C22⋊C4—21no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D411yescharacteristic
227C4×C22⋊C4D411yescharacteristic
228C4×C22⋊C4D412yes—
229#376—21no—
230#376—22no—
231#376C2311yescharacteristic
232#376C2311yescharacteristic
233C2×#95—21no—
234C2×#95—21no—
235C2×#95—21no—
236C2×#95—21no—
237C43—21no—
238C43C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#336C2314yes—
246#336C2314yes—
247C25—21no—
248C22×Q8—21no—
249C22×Q8C2412yes—
250C22×D4—24no—
251C22×D4—24no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4C22×C414yes—
255C22×D4C2411yescharacteristic
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—28no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—84no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2411yescharacteristic
271C2×SD16—28no—
272C2×SD16—28no—
273C2×SD16C22×C418yes—
274C2×SD16C22×C418yes—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8C22×C414yes—
281C4⋊Q8C22×C414yes—
282#125—28no—
283C22⋊Q8—28no—
284C22⋊Q8—24no—
285C22⋊Q8C22×C418yes—
286C4×Q8—28no—
287C4×Q8—24no—
288C4×Q8—28no—
289C4×Q8—28no—
290C4×Q8—84no—
291C4×Q8C22×C418yes—
292C4×D4—216no—
293C4×D4—216no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—28no—
299C4×D4—28no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4C22×C414yes—
307C4×D4C22×C414yes—
308C4×D4C22×C414yes—
309C4×D4C22×C414yes—
310C4×D4#4118yes—
311C4×D4#4118yes—
312#117—28no—
313#117—24no—
314#117—24no—
315#117—24no—
316#117—24no—
317#117—82no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4C22×C414yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C22⋊C4—216no—
339C2×C22⋊C4—216no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—24no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—84no—
367C2×C42—84no—
368C2×C42C22×C414yes—
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42#4114yes—
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2412yes—
380#107—28no—
381#106—28no—
382#106C22×C418yes—
383#106C22×C418yes—
384C4⋊C8—24no—
385C4⋊C8C22×C414yes—
386C4⋊C8C22×C414yes—
387#103—28no—
388#103—28no—
389#103C22×C418yes—
390#103C22×C418yes—
391#102—28no—
392#102—28no—
393#102C22×C418yes—
394#102C22×C418yes—
395C22⋊C8—24no—
396C22⋊C8C22×C418yes—
397#97—24no—
398#97—28no—
399C4×C8—28no—
400C4×C8C22×C418yes—
401#95—21no—
402#95—21no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—24no—
410#95—22no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95C2×D411yescharacteristic
415#95C2×D411yescharacteristic
416#95C2×D411yescharacteristic
417#95C2×D411yescharacteristic
418#95C2×D412yes—
419#95C2×D412yes—
420C24—28no—
421C24—21no—
422C24—21no—
423C24—24no—
424C24—82no—
425C24—81no—
426C24—81no—
427C24C22×D411yescharacteristic
428C2×Q8—24no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8C23×C414yes—
435C2×D4—28no—
436C2×D4—216no—
437C2×D4—28no—
438C2×D4—28no—
439C2×D4—24no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C23×C412yes—
445C2×D4C23×C412yes—
446C2×D4C2×#4114yes—
447C2×D4C2×#4114yes—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—24no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—816no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4C4×D414yes—
484C22×C4C4×D414yes—
485C22×C4C23×C412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C2×#4112yes—
496C22×C4C2511yescharacteristic
497SD16—216no—
498SD16—216no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8C23×C414yes—
505C2×C8C23×C414yes—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—84no—
525C4⋊C4C23×C414yes—
526C4⋊C4C23×C412yes—
527C4⋊C4C23×C412yes—
528C4⋊C4C2×#4114yes—
529C4⋊C4C2×#4114yes—
530C22⋊C4—28no—
531C22⋊C4—28no—
532C22⋊C4—28no—
533C22⋊C4—28no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—28no—
541C22⋊C4—28no—
542C22⋊C4—82no—
543C42—28no—
544C42—28no—
545C42—24no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—42no—
550C42—42no—
551C42—48no—
552C42—44no—
553C42—81no—
554C42—81no—
555C42—82no—
556C42—82no—
557C42—82no—
558C42—82no—
559C42—82no—
560C42—84no—
561C42C4×D414yes—
562C42C4×D414yes—
563C42C23×C412yes—
564C42C23×C412yes—
565C42C2×#4112yes—
566C42C2×#4112yes—
567C42C2×#4114yes—
568C42C2×#4114yes—
569C23—216no—
570C23—21no—
571C23—21no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—24no—
576C23—24no—
577C23—22no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—42no—
583C23—42no—
584C23—84no—
585C23—84no—
586C23—88no—
587C23—88no—
588C23C2×C4×D412yes—
589C23C23×D411yescharacteristic
590Q8—24no—
591Q8—24no—
592Q8—48no—
593Q8—44no—
594D4—216no—
595D4—22no—
596D4—22no—
597D4—24no—
598D4—24no—
599D4C4×#4114yes—
600D4C4×#4114yes—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—48no—
638C2×C4—48no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—88no—
643C2×C4—88no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—88no—
648C2×C4—88no—
649C2×C4C2×C4×D412yes—
650C2×C4C2×C4×D412yes—
651C2×C4C2×C4×D412yes—
652C2×C4C2×C4×D412yes—
653C2×C4C2×C4×D412yes—
654C2×C4#52514yes—
655C2×C4#52514yes—
656C2×C4#57714yes—
657C2×C4C24×C411yescharacteristic
658C2×C4C22×#4112yes—
659C2×C4C22×#4112yes—
660C2×C4C22×#4111yescharacteristic
661C8—24no—
662C8—24no—
663C8—44no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—24no—
668C22—24no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—44no—
675C22—42no—
676C22—44no—
677C22—44no—
678C22—42no—
679C22—816no—
680C22C22×C4×D411yescharacteristic
681C22C2×#52512yes—
682C22C2×#57712yes—
683C22#358511yescharacteristic
684C22#358611yescharacteristic
685C4—22no—
686C4—22no—
687C4—22no—
688C4—24no—
689C4—24no—
690C4—22no—
691C4—42no—
692C4—42no—
693C4—44no—
694C4—42no—
695C4—44no—
696C4—42no—
697C4—44no—
698C4—44no—
699C4—84no—
700C4#294914yes—
701C4C2×C4×#4111yescharacteristic
702C4C2×C4×#4111yescharacteristic
703C4#344412yes—
704C2—42no—
705C2—44no—
706C2—44no—
707C2C4×#52511yescharacteristic
708C2C2×#294911yescharacteristic
709C2#6017011yescharacteristic
710C2#6059112yes—
711C2#6059712yes—
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. #34225C2semidirect
4. #21614C2semidirect
5. C4×#476C2semidirect
6. #21587C2semidirect
7. #21504C2semidirect
8. C4×#461C2semidirect
10. #21392C2semidirect
11. #21362C2semidirect
12. #21133C2semidirect
13. #21130C2semidirect
15. #20506C2semidirect
16. #20504C2semidirect
17. #20470C2semidirect
21. #3101C4semidirect
41. #2320C22semidirect
42. C4×C4⋊Q8C22semidirect
43. C4×C4⋊Q8C22semidirect
45. #2276C22semidirect
61. #1987C22semidirect
62. #1981C22semidirect
63. #1948C22semidirect
64. #1947C22semidirect
65. #1928C22semidirect
66. #1927C22semidirect
67. #1920C22semidirect
68. #1919C22semidirect
79. #1776C22semidirect
80. C4×#107C22semidirect
81. C4×#106C22semidirect
82. C4×#106C22semidirect
83. #1767C22semidirect
84. C4×C4⋊C8C22semidirect
85. #1763C22semidirect
86. #1763C22semidirect
87. #1763C22semidirect
91. C4×#103C22semidirect
94. #1746C22semidirect
95. #1726C22semidirect
101. #533C2×C4semidirect
138. #483C2×C4semidirect
139. #476C2×C4semidirect
140. #476C2×C4semidirect
141. #461C2×C4semidirect
142. #461C2×C4semidirect
146. C4×SD16C2×C4semidirect
147. C4×SD16C2×C4semidirect
149. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
#342252
C4×#5182
#216142
C4×#4762
#215872
#215042
C4×#4612
#214592
#213922
#213622
#211332
#211302
#207382
#205062
#205042
#204702