Finite Group Databasebeta

← #6389761

Subgroups — #6389761

Order 512.

1914 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#34225C211yesmaximal, characteristic
3#34209C211yesmaximal, characteristic
4#23168C214yesmaximal, complemented
5#23166C214yesmaximal, complemented
6#22624C214yesmaximal
7#22622C214yesmaximal
8#22248C211yesmaximal, characteristic, complemented
9#22247C211yesmaximal, characteristic, complemented
10#22246C211yesmaximal, characteristic, complemented
11#22246C211yesmaximal, characteristic, complemented
12#21835C211yesmaximal, characteristic
13#21834C211yesmaximal, characteristic
14#21727C211yesmaximal, characteristic, complemented
15#21726C211yesmaximal, characteristic, complemented
16C4×#500C211yesmaximal, characteristic, complemented
17C4×#498C211yesmaximal, characteristic, complemented
18#20824C211yesmaximal, characteristic
19#20472C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#519—21no—
23#3165C4116yescomplemented
24C2×#501C2214yes—
25C2×#500C2211yescharacteristic
26C2×#500C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#428C2214yes—
31C2×C4×M16C2211yescharacteristic
32#2370C2212yes—
33#2370C2212yes—
34#2365—21no—
35#2365—21no—
36#2365C2212yes—
37#2365C2212yes—
38#2362—21no—
39#2362—21no—
40#2356—21no—
41#2351—21no—
42#2347C2212yes—
43#2346C2212yes—
44#2344—21no—
45#2344—21no—
46#2344C2212yes—
47#2344C2212yes—
48#2327—21no—
49#2327—21no—
50#2323—21no—
51#2323—21no—
52#2320—21no—
53#2310—21no—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62#2285C2211yescharacteristic
63#2285C2211yescharacteristic
64#2276—21no—
65C4×#117C2211yescharacteristic
66#1990C2211yescharacteristic
67#1990C2211yescharacteristic
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1989C2214yes—
75#1987C2214yes—
76#1944C2214yes—
77#1943C2214yes—
78#1942C2214yes—
79#1941C2214yes—
80#1921C2214yes—
81#1921C2214yes—
82#1920C2214yes—
83#1920C2214yes—
84#1919C2214yes—
85#1919C2214yes—
86#1918C2214yes—
87#1918C2214yes—
88#1845C2211yescharacteristic
89#1844C2211yescharacteristic
90#1834C2211yescharacteristic
91#1834C2211yescharacteristic
92#1825C2214yes—
93#1824C2214yes—
94#1809C2214yes—
95#1776C2214yes—
96#1776C2214yes—
97#1776C2214yes—
98C4×#107C2214yes—
99C4×#106C2214yes—
100#1771C2212yes—
101#1771C2212yes—
102#1753C2212yes—
103#1753C2212yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C42×C8C2211yescharacteristic
108C23×Q8—21no—
109C22×M16C2312yes—
110#533—24no—
111#533C2×C418yes—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—24no—
115C2×C4⋊Q8C2×C418yes—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121C2×#125C2312yes—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124#519—48no—
125C2×C4×Q8—22no—
126C2×C4×Q8—22no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×#117—21no—
138C2×#117—21no—
139C2×#117—22no—
140C2×#117—22no—
141C2×#117D411yescharacteristic
142C2×#117D411yescharacteristic
143C2×#117C2312yes—
144C22×C4⋊C4—21no—
145C22×C4⋊C4C2312yes—
146C22×C42C2311yescharacteristic
147#501C2×C4132yes—
148#500C2×C418yes—
149#500C2×C418yes—
150#498C2×C418yes—
151#498C2×C418yes—
152#431—24no—
153#428C2×C4132yes—
154C2×#107C2314yes—
155C2×#107C2314yes—
156C2×#107C2314yes—
157C2×#106C2314yes—
158C2×#106C2314yes—
159C2×#106C2314yes—
160C2×C4⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C4×M16C2×C418yes—
163C2×#97C2312yes—
164C2×#97C2312yes—
165C2×#97C2312yes—
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2312yes—
169C2×C4×C8C2312yes—
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172#391—22no—
173#391—22no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391—21no—
182#389—22no—
183#389—22no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#388—22no—
201#388—22no—
202#388—22no—
203#388—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#383—22no—
221#382—21no—
222#382—21no—
223#382—22no—
224#381—22no—
225#381—22no—
226#380—21no—
227#380—21no—
228#380C2312yes—
229#380C2312yes—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2312yes—
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2311yescharacteristic
256C4×C4⋊C4C2311yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4D411yescharacteristic
259C4×C22⋊C4C2311yescharacteristic
260C4×C22⋊C4C2311yescharacteristic
261#376—22no—
262#376D411yescharacteristic
263#376D411yescharacteristic
264C43C2311yescharacteristic
265C43C2311yescharacteristic
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#340C2314yes—
273#340C2314yes—
274#340C2314yes—
275#340C2314yes—
276#336C2312yes—
277#336C2312yes—
278#336C2312yes—
279#336C2312yes—
280C22×Q8—28no—
281C22×Q8—21no—
282C22×Q8—21no—
283C22×Q8—22no—
284C22×Q8—21no—
285C22×Q8—21no—
286C22×Q8—44no—
287C22×Q8—44no—
288C23×C4—22no—
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2411yescharacteristic
292C2×M16—24no—
293C2×M16—24no—
294C2×M16C22×C418yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—28no—
300C4⋊Q8—28no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—24no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8C22×C414yes—
307C4⋊Q8C22×C414yes—
308C4⋊Q8C22×C418yes—
309C4⋊Q8C22×C414yes—
310C4⋊Q8C22×C414yes—
311#125—24no—
312#125C22×C418yes—
313C22⋊Q8—28no—
314C22⋊Q8—28no—
315C4×Q8—28no—
316C4×Q8—28no—
317C4×Q8—44no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8—48no—
321C4×Q8—48no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8—44no—
325C4×Q8—82no—
326C4×Q8—82no—
327#117—24no—
328#117—24no—
329#117—42no—
330#117—48no—
331#117—48no—
332#117—44no—
333#117—82no—
334#117C22×C418yes—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4C22×C418yes—
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C22⋊C4C2×D411yescharacteristic
379C2×C22⋊C4C2×D411yescharacteristic
380C2×C22⋊C4C2×D411yescharacteristic
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42C22×C414yes—
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42#4114yes—
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410#107—28no—
411#107C22×C4116yes—
412#107C22×C4116yes—
413#106—28no—
414#106C22×C4116yes—
415#106C22×C4116yes—
416C4⋊C8—24no—
417C22⋊C8—24no—
418#97—24no—
419#97—24no—
420#97C22×C418yes—
421C4×C8—24no—
422C4×C8—24no—
423C4×C8—22no—
424C4×C8—22no—
425C4×C8C22×C414yes—
426C4×C8C22×C414yes—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454#95C2×D411yescharacteristic
455C24C22×D411yescharacteristic
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—28no—
459C2×Q8—24no—
460C2×Q8—24no—
461C2×Q8—44no—
462C2×Q8—44no—
463C2×Q8—432no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—28no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4C4×D414yes—
490C22×C4C4×D414yes—
491C22×C4C23×C412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C2×#4112yes—
506C22×C4C2511yescharacteristic
507M16—216no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8C23×C414yes—
517C2×C8C23×C414yes—
518C2×C82−1+414yes—
519C2×C82−1+414yes—
520C4⋊C4—28no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4C23×C414yes—
557C4⋊C4C23×C414yes—
558C4⋊C4C23×C414yes—
559C4⋊C4C23×C414yes—
560C22⋊C4—28no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—44no—
566C22⋊C4—82no—
567C42—28no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—44no—
580C42—44no—
581C42—44no—
582C42—44no—
583C42—44no—
584C42—44no—
585C42—44no—
586C42—82no—
587C42—82no—
588C42—82no—
589C42C4×D414yes—
590C42C4×D414yes—
591C42AGL(1,ℤ/8)14yes—
592C42C23×C412yes—
593C42C23×C412yes—
594C42C2×#4112yes—
595C42C2×#4112yes—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23C2×C4×D412yes—
603C23C23×D411yescharacteristic
604Q8—416no—
605Q8—48no—
606Q8—48no—
607Q8—48no—
608Q8—48no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—24no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—416no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4C2×C4×D412yes—
667C2×C4C2×C4×D412yes—
668C2×C4C2×C4×D412yes—
669C2×C4C2×C4×D412yes—
670C2×C4C2×C4×D412yes—
671C2×C4C2×AGL(1,ℤ/8)12yes—
672C2×C4C2×AGL(1,ℤ/8)12yes—
673C2×C4C2×AGL(1,ℤ/8)12yes—
674C2×C4#57512yes—
675C2×C4#57712yes—
676C2×C4#57812yes—
677C2×C4C24×C411yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C2×C4C2×2−1+412yes—
680C2×C4C2×2−1+412yes—
681C8—24no—
682C8—42no—
683C8—42no—
684C8#51918yescomplemented
685C22—22no—
686C22—22no—
687C22—22no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22C22×C4×D411yescharacteristic
693C22#349911yescharacteristic
694C22#349911yescharacteristic
695C22C22×AGL(1,ℤ/8)11yescharacteristic
696C22C2×#57511yescharacteristic
697C22C2×#57711yescharacteristic
698C22C2×#57811yescharacteristic
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—44no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—44no—
712C4—44no—
713C4—44no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4C4×AGL(1,ℤ/8)14yes—
719C4#294714yes—
720C4C2×#51912yes—
721C4#350912yes—
722C2—42no—
723C2#3420911yescharacteristic
724C2C2×C4×AGL(1,ℤ/8)11yescharacteristic
725C2C2×#294711yescharacteristic
726C2#6247711yescharacteristic
727C2#6247911yescharacteristic
728C2#6248211yescharacteristic
729C2#6248311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
4. #23168C2semidirect
5. #23166C2semidirect
8. #22248C2semidirect
9. #22247C2semidirect
10. #22246C2semidirect
11. #22246C2semidirect
14. #21727C2semidirect
15. #21726C2semidirect
16. C4×#500C2semidirect
17. C4×#498C2semidirect
23. #3165C4semidirect
684. C8#519semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#342252
#342092
#231682
#231662
#226242
#226222
#222482
#222472
#222462
#218352
#218342
#217272
#217262
C4×#5002
C4×#4982
#208242
#204722