Finite Group Databasebeta

← #6477488

Subgroups — #6477488

Order 512.

3130 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C2×#3195C211yesmaximal, characteristic, complemented
3C2×#3095C211yesmaximal, characteristic, complemented
4#35010C211yesmaximal, characteristic
5#33010C212yesmaximal, complemented
6#32984C212yesmaximal, complemented
7#29641C212yesmaximal, complemented
8#29599C212yesmaximal, complemented
9#25558C218yesmaximal, complemented
10#24108C214yesmaximal, complemented
11#23980C214yesmaximal, complemented
12#21765C211yesmaximal, characteristic, complemented
13#21742C211yesmaximal, characteristic
14#20702C211yesmaximal, characteristic, complemented
15C2×#556—28no—
16C2×#556—22no—
17C2×#556—22no—
18C2×#556—28no—
19C2×#556C2214yes—
20C2×#556C2214yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#126—21no—
23C22×#126—21no—
24C22×#124—21no—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C4×Q8—21no—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C22×#117C2211yescharacteristic
33#3195—28no—
34#3095—28no—
35C2×#484C2212yes—
36C2×#485C2214yescomplemented
37C2×#483C2212yes—
38C2×#483C2212yes—
39C2×#483C2212yescomplemented
40C2×#470C2214yescomplemented
41C2×#451C2212yes—
42C2×#451C2212yescomplemented
43C2×#428C2212yescomplemented
44C2×#419C2212yescomplemented
45C2×#407C2211yescharacteristic
46C2×#407C2211yescharacteristic
47#2855C2212yes—
48#2848C2212yes—
49#2627—21no—
50#2626—21no—
51#2579—22no—
52#2564—22no—
53#2487C2212yes—
54#2417C2214yes—
55#2409C2212yes—
56#2315—21no—
57#2308—21no—
58C2×#380C2211yescharacteristic
59#2058C2218yescomplemented
60#2035C2218yescomplemented
61#2024C2218yescomplemented
62#2001C2218yescomplemented
63#1894C2214yescomplemented
64#1878C2218yescomplemented
65#1873C2218yescomplemented
66#1872C2214yescomplemented
67#1870C2214yes—
68#1870C2214yes—
69#1863C2218yescomplemented
70#1854C2218yescomplemented
71#1825C2214yescomplemented
72#1824C2212yes—
73#1823C2212yes—
74#1823C2212yes—
75#1817C2212yes—
76#1787C2212yes—
77#1779C2212yes—
78#1736C2214yescomplemented
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C22×Q16C2314yes—
86C22×M16C2312yes—
87C22×M16C2312yes—
88#556—232no—
89#556—232no—
90#556—216no—
91#556—216no—
92#556—216no—
93#556—216no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8C2314yes—
98C2×#126—216no—
99C2×#126—216no—
100C2×#126—24no—
101C2×#126—24no—
102C2×#126—24no—
103C2×#126—24no—
104C2×#124—216no—
105C2×#124—24no—
106C2×#124—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8D418yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2314yes—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×C22⋊Q8C2314yes—
124C2×C4×Q8—216no—
125C2×C4×Q8—24no—
126C2×C4×Q8—24no—
127C2×C4×Q8—24no—
128C2×C4×Q8—24no—
129C2×C4×Q8—22no—
130C2×C4×Q8C2314yes—
131C2×C4×D4—24no—
132C2×C4×D4—24no—
133C2×C4×D4D412yes—
134C2×C4×D4D412yes—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×#117—24no—
138C2×#117—22no—
139C2×#117—21no—
140C2×#117—21no—
141C2×#117—24no—
142C2×#117—24no—
143C2×#117D418yes—
144C2×#117C2314yes—
145C2×#117C2312yes—
146C2×#117C2312yes—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4D411yescharacteristic
153C22×C4⋊C4C2311yescharacteristic
154C22×C4⋊C4C2311yescharacteristic
155C22×C4⋊C4C2311yescharacteristic
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2312yes—
158C22×C22⋊C4—21no—
159C22×C22⋊C4—21no—
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×C42—21no—
168C22×C42—21no—
169C22×C42C2311yescharacteristic
170#485—216no—
171#484—28no—
172#483—28no—
173#483—28no—
174#483—28no—
175#470—216no—
176#451—28no—
177#451—28no—
178#428—28no—
179C2×#107C2314yes—
180C2×#106C2314yes—
181C2×#106C2314yes—
182#419—28no—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2314yes—
187C2×#103C2318yes—
188#407—24no—
189#407—24no—
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2318yes—
195#400—22no—
196#399—22no—
197#398—24no—
198#396—24no—
199#395—22no—
200#393—22no—
201#391C2314yes—
202#389C2314yes—
203#387—24no—
204#386—24no—
205#386—22no—
206#386—22no—
207#385—24no—
208#385—22no—
209#385—22no—
210#385—22no—
211#385—22no—
212#381—22no—
213#380—21no—
214#380—21no—
215#380—24no—
216#380C2312yes—
217#380C2314yes—
218#380C2312yes—
219#379—22no—
220C2×#95—21no—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25C2×D411yescharacteristic
226C25C2×D411yescharacteristic
227C22×Q8—28no—
228C22×Q8—22no—
229C22×Q8—21no—
230C22×Q8—24no—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—44no—
235C22×Q8—44no—
236C22×Q8—44no—
237C22×Q8—44no—
238C22×Q8C2×D414yes—
239C22×Q8C2412yes—
240C22×Q8C2412yes—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—42no—
244C22×D4C2×D412yes—
245C22×D4C2×D412yes—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D414yes—
265C23×C4C2×D414yes—
266C23×C4C2412yes—
267C23×C4C2411yescharacteristic
268C23×C4C2411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×Q16—28no—
271C2×Q16—48no—
272C2×M16—24no—
273C2×M16—24no—
274C2×M16—44no—
275C2×M16—44no—
276C22×C8C2414yes—
277C22×C8C2414yes—
278C4⋊Q8—216no—
279C4⋊Q8—28no—
280C4⋊Q8—28no—
281#126—232no—
282#126—232no—
283#126—232no—
284#126—232no—
285#124—232no—
286#124—232no—
287C22⋊Q8—216no—
288C22⋊Q8—28no—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—28no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—44no—
299C22⋊Q8—44no—
300C22⋊Q8—48no—
301C22⋊Q8C2×D418yes—
302C22⋊Q8C2×D418yes—
303C4×Q8—232no—
304C4×Q8—232no—
305C4×Q8—216no—
306C4×Q8—28no—
307C4×Q8—28no—
308C4×D4—28no—
309C4×D4—28no—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—24no—
314#117—216no—
315#117—28no—
316#117—28no—
317#117—24no—
318#117—24no—
319#117—28no—
320#117—28no—
321#117C2×D418yes—
322#117C2×D418yes—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2414yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C22⋊C4—28no—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—44no—
378C2×C22⋊C4—44no—
379C2×C22⋊C4C2×D412yes—
380C2×C22⋊C4C2×D412yes—
381C2×C22⋊C4C2×D412yes—
382C2×C22⋊C4C2×D412yes—
383C2×C42—28no—
384C2×C42—28no—
385C2×C42—28no—
386C2×C42—24no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—24no—
392C2×C42C2×D412yes—
393C2×C42C2412yes—
394#107—28no—
395#106—28no—
396#106—28no—
397#103—28no—
398#103—28no—
399#103—28no—
400#103—28no—
401#103—48no—
402C22⋊C8—24no—
403C22⋊C8—24no—
404C22⋊C8—24no—
405C22⋊C8—24no—
406C22⋊C8—48no—
407#95—22no—
408#95—22no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95—24no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24#12014yes—
423C24C22×D412yes—
424C24C22×D411yescharacteristic
425C24C22×D411yescharacteristic
426C24C22×D411yescharacteristic
427C2×Q8—216no—
428C2×Q8—24no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—28no—
433C2×Q8—28no—
434C2×Q8—48no—
435C2×Q8—44no—
436C2×Q8—432no—
437C2×Q8—48no—
438C2×Q8—48no—
439C2×Q8—48no—
440C2×Q8—42no—
441C2×Q8—48no—
442C2×Q8—82no—
443C2×Q8—84no—
444C2×Q8C22×D414yes—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—416no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—28no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—44no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4#12014yes—
498C22×C4#12014yes—
499C22×C4#12014yes—
500C22×C4#13714yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C2511yescharacteristic
514Q16—88no—
515M16—48no—
516M16—84no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—28no—
520C2×C8—28no—
521C4⋊C4—216no—
522C4⋊C4—216no—
523C4⋊C4—216no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—28no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—84no—
541C4⋊C4—82no—
542C4⋊C4—84no—
543C4⋊C4C22×D414yes—
544C4⋊C4C22×D414yes—
545C4⋊C4C22×D414yes—
546C22⋊C4—216no—
547C22⋊C4—216no—
548C22⋊C4—28no—
549C22⋊C4—28no—
550C22⋊C4—28no—
551C22⋊C4—28no—
552C22⋊C4—24no—
553C22⋊C4—24no—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—84no—
561C22⋊C4—84no—
562C42—216no—
563C42—216no—
564C42—28no—
565C42—28no—
566C42—24no—
567C42—22no—
568C42—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—28no—
580C23—24no—
581C23—22no—
582C23—24no—
583C23—22no—
584C23—22no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23C2×#12012yes—
595C23C2×#12012yes—
596C23C2×#12012yes—
597C23C2×#13712yes—
598C23#57512yes—
599C23C23×D411yescharacteristic
600Q8—28no—
601Q8—48no—
602Q8—432no—
603Q8—44no—
604Q8—44no—
605Q8—44no—
606Q8—84no—
607D4—48no—
608D4—48no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—28no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—48no—
633C2×C4—44no—
634C2×C4—48no—
635C2×C4—44no—
636C2×C4—48no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—48no—
640C2×C4—48no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—164no—
657C2×C4C2×#12012yes—
658C2×C4C2×#12012yes—
659C2×C4C2×#12012yes—
660C2×C4C2×#13714yes—
661C2×C4C23×D411yescharacteristic
662C2×C4C23×D411yescharacteristic
663C2×C4#58514yes—
664C8—44no—
665C8—44no—
666C22—22no—
667C22—24no—
668C22—24no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—44no—
682C22—44no—
683C22—42no—
684C22—42no—
685C22—88no—
686C22—88no—
687C22#300714yes—
688C22C22×#12011yescharacteristic
689C22#348512yes—
690C22C22×#13711yescharacteristic
691C22C2×#57511yescharacteristic
692C22#358612yes—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—24no—
697C4—42no—
698C4—44no—
699C4—44no—
700C4—44no—
701C4—42no—
702C4—44no—
703C4—82no—
704C4—84no—
705C4—82no—
706C4#344912yes—
707C4#348512yes—
708C2—24no—
709C2—42no—
710C2—42no—
711C2—82no—
712C2#3508811yescharacteristic
713C2C2×#300711yescharacteristic
714C2#6036811yescharacteristic
715C2#6171212yes—
716C2#6172312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
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×#3195C2semidirect
3. C2×#3095C2semidirect
5. #33010C2semidirect
6. #32984C2semidirect
7. #29641C2semidirect
8. #29599C2semidirect
9. #25558C2semidirect
10. #24108C2semidirect
11. #23980C2semidirect
12. #21765C2semidirect
14. #20702C2semidirect
36. C2×#485C22semidirect
39. C2×#483C22semidirect
40. C2×#470C22semidirect
42. C2×#451C22semidirect
43. C2×#428C22semidirect
44. C2×#419C22semidirect
59. #2058C22semidirect
60. #2035C22semidirect
61. #2024C22semidirect
62. #2001C22semidirect
63. #1894C22semidirect
64. #1878C22semidirect
65. #1873C22semidirect
66. #1872C22semidirect
69. #1863C22semidirect
70. #1854C22semidirect
71. #1825C22semidirect
78. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5562
C2×#31952
C2×#30952
#350102
#330102
#329842
#296412
#295992
#255582
#241082
#239802
#217652
#217422
#207022