Finite Group Databasebeta

← #6450141

Subgroups — #6450141

Order 512.

3430 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#553C211yesmaximal, characteristic
2C2×#3166C211yesmaximal, characteristic, complemented
3C2×#3077C211yesmaximal, characteristic, complemented
4#34946C211yesmaximal, characteristic, complemented
5#25714C214yesmaximal, complemented
6#25713C214yesmaximal, complemented
7#25707C218yesmaximal, complemented
8#23480C212yesmaximal, complemented
9#23480C212yesmaximal, complemented
10#21314C211yesmaximal, characteristic, complemented
11#21190C211yesmaximal, characteristic, complemented
12#20845C212yesmaximal, complemented
13#20844C212yesmaximal, complemented
14C4×#418C211yesmaximal, characteristic, complemented
15C2×#553—28no—
16C2×#553—22no—
17C2×#553—22no—
18C2×#553—28no—
19C2×#553C2214yes—
20C2×#553C2214yes—
21C2×#533C2211yescharacteristic
22C22×#126—21no—
23C22×#125C2211yescharacteristic
24C22×#123—21no—
25C22×#121—21no—
26C22×#121C2211yescharacteristic
27C22×#121C2211yescharacteristic
28C22×C4×D4—21no—
29C22×C4×D4C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#3166—28no—
34C2×#501C2212yescomplemented
35C2×#501C2212yescomplemented
36C2×#499C2214yescomplemented
37#3077—28no—
38C2×#478C2214yescomplemented
39C2×#476C2212yescomplemented
40C2×#474C2212yescomplemented
41C2×#428C2212yes—
42C2×#428C2212yes—
43C2×#428C2212yes—
44C2×#423C2211yescharacteristic
45C2×#418C2212yes—
46C2×#418C2212yes—
47C2×#407C2212yes—
48C2×#407C2212yes—
49C2×C4×M16C2211yescharacteristic
50#2563C2212yescomplemented
51#2562C2214yescomplemented
52#2561—21no—
53#2561—21no—
54#2547C2212yescomplemented
55#2503—21no—
56#2435—21no—
57#2398—22no—
58C4×#124—22no—
59#2280C2212yes—
60#2280C2212yes—
61C4×#117C2211yescharacteristic
62#2034C2218yescomplemented
63#2034C2218yescomplemented
64#2033C2218yescomplemented
65#2033C2218yescomplemented
66#1989C2214yescomplemented
67#1989C2214yescomplemented
68#1955C2214yescomplemented
69#1954C2214yescomplemented
70#1854C2218yescomplemented
71#1854C2218yescomplemented
72#1853C2218yescomplemented
73#1852C2218yescomplemented
74#1762C2214yescomplemented
75C4×#102C2214yescomplemented
76#1736C2212yes—
77#1736C2212yes—
78#1736C2212yes—
79C23×D4—21no—
80C23×D4—21no—
81C23×D4D411yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C22×M16C2312yes—
88C22×M16C2312yes—
89#553—232no—
90#553—232no—
91#553—216no—
92#553—216no—
93#553—216no—
94#553—216no—
95#533—24no—
96#533—24no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×#126—216no—
100C2×#126—24no—
101C2×#126—24no—
102C2×#125—21no—
103C2×#125—21no—
104C2×#125—24no—
105C2×#125D418yes—
106C2×#125C2312yes—
107C2×#125C2312yes—
108C2×#125C2312yes—
109C2×#124—22no—
110C2×#123—216no—
111C2×#123—24no—
112C2×#123—24no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×#121—216no—
116C2×#121—24no—
117C2×#121—24no—
118C2×#121—24no—
119C2×#121—24no—
120C2×#121—22no—
121C2×#121—22no—
122C2×#121—24no—
123C2×#121—24no—
124C2×#121C2314yes—
125C2×#121C2314yes—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×D4—216no—
129C2×C4×D4—24no—
130C2×C4×D4—24no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—24no—
134C2×C4×D4—24no—
135C2×C4×D4—24no—
136C2×C4×D4D418yes—
137C2×C4×D4D412yes—
138C2×C4×D4D412yes—
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×#117—22no—
144C2×#117—24no—
145C2×#117—24no—
146C2×#117C2314yes—
147C2×#117C2312yes—
148C2×#117C2312yes—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—22no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—24no—
154C22×C4⋊C4—24no—
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4C2314yes—
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C22⋊C4—21no—
161C22×C22⋊C4—21no—
162C22×C22⋊C4—21no—
163C22×C22⋊C4—21no—
164C22×C22⋊C4D411yescharacteristic
165C22×C22⋊C4D411yescharacteristic
166C22×C42—21no—
167C22×C42C2311yescharacteristic
168#501—28no—
169#501—28no—
170#499—216no—
171#478—216no—
172#476—28no—
173#474—28no—
174#428—28no—
175#428—28no—
176#428—28no—
177C2×#107C2314yes—
178C2×#107C2314yes—
179C2×#107C2314yes—
180C2×#106C2314yes—
181C2×#106C2314yes—
182C2×#106C2314yes—
183#423—24no—
184C2×C4⋊C8C2312yes—
185C2×C4⋊C8C2312yes—
186#418—28no—
187#418—28no—
188C2×#102C2314yes—
189C2×#102C2314yes—
190C2×#102C2318yes—
191#407—28no—
192#407—28no—
193C2×C22⋊C8C2318yes—
194C2×C22⋊C8C2318yes—
195C4×M16—24no—
196C2×#97C2312yes—
197C2×C4×C8C2312yes—
198#400—24no—
199#397—22no—
200#397—22no—
201#394—24no—
202#393—22no—
203#393—22no—
204#391—22no—
205#390—22no—
206#388—24no—
207#388—24no—
208#382—22no—
209#381—22no—
210#381—22no—
211#380C2314yes—
212#380C2314yes—
213#380C2314yes—
214#380C2314yes—
215#379—22no—
216#379—22no—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—24no—
221C4×C22⋊C4—24no—
222C4×C22⋊C4—21no—
223#376—21no—
224C43—21no—
225#340C2318yes—
226#340C2318yes—
227#340C2318yes—
228C25—21no—
229C25C2×D411yescharacteristic
230C25C2×D411yescharacteristic
231C22×Q8—21no—
232C22×Q8—21no—
233C22×D4—28no—
234C22×D4—24no—
235C22×D4—28no—
236C22×D4—24no—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—24no—
241C22×D4—24no—
242C22×D4—44no—
243C22×D4—48no—
244C22×D4—84no—
245C22×D4C2×D414yes—
246C22×D4C2×D412yes—
247C22×D4C2×D412yes—
248C22×D4C2412yes—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—28no—
253C23×C4—24no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—24no—
259C23×C4—24no—
260C23×C4—24no—
261C23×C4—24no—
262C23×C4—28no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—84no—
266C23×C4—84no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D412yes—
270C23×C4C2×D412yes—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2412yes—
274C23×C4C2412yes—
275C23×C4C2411yescharacteristic
276C2×M16—24no—
277C2×M16—24no—
278C2×M16—24no—
279C2×M16—24no—
280C2×M16—24no—
281C2×M16#4118yes—
282C22×C8C2414yes—
283C22×C8C2414yes—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288#126—232no—
289#126—232no—
290#125—28no—
291#125—28no—
292#125—24no—
293#125—24no—
294#125—24no—
295#125—24no—
296#125C2×D418yes—
297#125C2×D418yes—
298#124—48no—
299#123—232no—
300#123—232no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303#121—232no—
304#121—232no—
305#121—216no—
306#121—216no—
307#121—28no—
308#121—28no—
309#121—28no—
310#121—28no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×D4—232no—
314C4×D4—232no—
315C4×D4—28no—
316C4×D4—28no—
317C4×D4—28no—
318C4×D4—28no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4C2×D418yes—
326C4×D4C2×D418yes—
327#117—216no—
328#117—28no—
329#117—28no—
330#117—24no—
331#117—24no—
332#117—24no—
333#117—24no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—216no—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—28no—
351C2×C4⋊C4—28no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D414yes—
359C2×C4⋊C4C2×D414yes—
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2414yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C22⋊C4—28no—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—28no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D412yes—
386C2×C22⋊C4C2×D412yes—
387C2×C22⋊C4C2×D412yes—
388C2×C42—28no—
389C2×C42—24no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D414yes—
397C2×C42C2×D414yes—
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400#107—28no—
401#107—28no—
402#107—28no—
403#106—28no—
404#106—28no—
405#106—28no—
406C4⋊C8—24no—
407C4⋊C8—24no—
408#102—28no—
409#102—28no—
410#102—216no—
411C22⋊C8—216no—
412C22⋊C8—216no—
413#97—24no—
414C4×C8—24no—
415#95—22no—
416#95—22no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421C24—22no—
422C24—24no—
423C24—22no—
424C24—24no—
425C24—24no—
426C24—22no—
427C24—24no—
428C24—42no—
429C24—42no—
430C24—42no—
431C24—44no—
432C24—44no—
433C24—44no—
434C24C22×D412yes—
435C24C22×D412yes—
436C24C22×D411yescharacteristic
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—82no—
441C2×D4—216no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—216no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—28no—
449C2×D4—28no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—24no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—416no—
456C2×D4—432no—
457C2×D4—48no—
458C2×D4—48no—
459C2×D4—44no—
460C2×D4—48no—
461C2×D4—88no—
462C2×D4—816no—
463C2×D4C22×D414yes—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—216no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—28no—
488C22×C4—28no—
489C22×C4—24no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—84no—
504C22×C4—84no—
505C22×C4—88no—
506C22×C4—816no—
507C22×C4—816no—
508C22×C4#12014yes—
509C22×C4#12014yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C2×#4114yes—
520C22×C42−1+414yes—
521C22×C4C2511yescharacteristic
522M16—216no—
523M16—48no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8C2×#4118yes—
530C4⋊C4—216no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—28no—
534C4⋊C4—28no—
535C4⋊C4—28no—
536C4⋊C4—28no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4C22×D414yes—
554C4⋊C4C22×D414yes—
555C4⋊C4C22×D414yes—
556C22⋊C4—216no—
557C22⋊C4—216no—
558C22⋊C4—216no—
559C22⋊C4—28no—
560C22⋊C4—28no—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C22⋊C4—24no—
564C22⋊C4—24no—
565C22⋊C4—48no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—48no—
569C22⋊C4—84no—
570C22⋊C4—84no—
571C42—216no—
572C42—44no—
573C42—44no—
574C42—44no—
575C42—82no—
576C42#12014yes—
577C42#12014yes—
578C42C22×D412yes—
579C42C22×D412yes—
580C42C22×D412yes—
581C42C22×D412yes—
582C23—22no—
583C23—22no—
584C23—28no—
585C23—24no—
586C23—22no—
587C23—24no—
588C23—22no—
589C23—24no—
590C23—22no—
591C23—24no—
592C23—24no—
593C23—22no—
594C23—22no—
595C23—416no—
596C23—44no—
597C23—48no—
598C23—44no—
599C23—48no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—48no—
605C23—44no—
606C23—44no—
607C23—88no—
608C23—816no—
609C23C2×#12012yes—
610C23#54714yes—
611C23#57514yes—
612C23C23×D411yescharacteristic
613Q8—84no—
614Q8—84no—
615D4—24no—
616D4—24no—
617D4—48no—
618D4—48no—
619D4—432no—
620D4—48no—
621D4—44no—
622D4—816no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—48no—
637C2×C4—48no—
638C2×C4—48no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—48no—
649C2×C4—48no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—88no—
660C2×C4—88no—
661C2×C4—816no—
662C2×C4C2×#12012yes—
663C2×C4C2×#12012yes—
664C2×C4C2×#12012yes—
665C2×C4C2×#12012yes—
666C2×C4C2×#12012yes—
667C2×C4#53214yes—
668C2×C4#53614yes—
669C2×C4#54714yes—
670C2×C4#54818yes—
671C2×C4C23×D411yescharacteristic
672C2×C4C23×D411yescharacteristic
673C2×C4C22×#4112yes—
674C2×C4C2×2−1+412yes—
675C8—28no—
676C8—44no—
677C22—28no—
678C22—22no—
679C22—22no—
680C22—28no—
681C22—24no—
682C22—416no—
683C22—44no—
684C22—42no—
685C22—42no—
686C22—44no—
687C22—44no—
688C22—42no—
689C22—44no—
690C22—42no—
691C22—44no—
692C22—44no—
693C22—42no—
694C22—88no—
695C22—816no—
696C22—88no—
697C22#243214yes—
698C22C22×#12011yescharacteristic
699C22C2×#54712yes—
700C22C2×#57512yes—
701C22#358512yes—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—42no—
707C4—42no—
708C4—44no—
709C4—44no—
710C4—44no—
711C4—84no—
712C4—82no—
713C4—82no—
714C4#243214yes—
715C4C2×#53211yescharacteristic
716C4C2×#53611yescharacteristic
717C4C2×#54812yes—
718C2—24no—
719C2—44no—
720C2—82no—
721C2—84no—
722C2C2×#243211yescharacteristic
723C2#6035911yescharacteristic
724C2#6036711yescharacteristic
725C2#6125514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#553C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3166C2semidirect
3. C2×#3077C2semidirect
4. #34946C2semidirect
5. #25714C2semidirect
6. #25713C2semidirect
7. #25707C2semidirect
8. #23480C2semidirect
9. #23480C2semidirect
10. #21314C2semidirect
11. #21190C2semidirect
12. #20845C2semidirect
13. #20844C2semidirect
14. C4×#418C2semidirect
34. C2×#501C22semidirect
35. C2×#501C22semidirect
36. C2×#499C22semidirect
38. C2×#478C22semidirect
39. C2×#476C22semidirect
40. C2×#474C22semidirect
50. #2563C22semidirect
51. #2562C22semidirect
54. #2547C22semidirect
62. #2034C22semidirect
63. #2034C22semidirect
64. #2033C22semidirect
65. #2033C22semidirect
66. #1989C22semidirect
67. #1989C22semidirect
68. #1955C22semidirect
69. #1954C22semidirect
70. #1854C22semidirect
71. #1854C22semidirect
72. #1853C22semidirect
73. #1852C22semidirect
74. #1762C22semidirect
75. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5532
C2×#31662
C2×#30772
#349462
#257142
#257132
#257072
#234802
#213142
#211902
#208452
#208442
C4×#4182