Finite Group Databasebeta

← #6477316

Subgroups — #6477316

Order 512.

3360 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C2×#3195C211yesmaximal, characteristic, complemented
3C2×#3193C211yesmaximal, characteristic
4C2×#2855C211yesmaximal, characteristic, complemented
5#33010C212yesmaximal, complemented
6#33007C212yesmaximal, complemented
7#29643C212yesmaximal, complemented
8#29626C212yesmaximal, complemented
9#25505C218yesmaximal, complemented
10#24066C214yesmaximal, complemented
11#23937C214yesmaximal, complemented
12#21765C211yesmaximal, characteristic, complemented
13#21764C211yesmaximal, characteristic
14#20682C211yesmaximal, characteristic, complemented
15C2×#529—22no—
16C2×#529—22no—
17C2×#529—28no—
18C2×#529—28no—
19C2×#529C2214yes—
20C2×#529C2214yes—
21C22×#126—21no—
22C22×#125C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C22×#123—21no—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C22×#121C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4—21no—
30C22×C4×D4C2211yescharacteristic
31C22×#117—21no—
32C23×C42C2211yescharacteristic
33#3195—28no—
34#3193—28no—
35C2×#484C2212yes—
36C2×#483C2212yes—
37C2×#483C2212yescomplemented
38C2×#480C2212yes—
39C2×#470C2214yescomplemented
40C2×#468C2214yes—
41C2×#451C2212yes—
42C2×#451C2212yescomplemented
43C2×#448C2212yes—
44C2×#450C2212yes—
45C2×#407C2211yescharacteristic
46C2×#407C2211yescharacteristic
47#2855—21no—
48#2855—21no—
49#2855—22no—
50#2855C2212yes—
51#2855C2212yes—
52#2855C2214yescomplemented
53C2×#391C2212yescomplemented
54C2×#389C2212yescomplemented
55C2×#387—22no—
56C2×#385—21no—
57C2×#381—21no—
58C2×#380C2211yescharacteristic
59#2059C2218yescomplemented
60#2042C2218yescomplemented
61#2026C2218yescomplemented
62#2015C2218yescomplemented
63#1895C2214yes—
64#1894C2214yescomplemented
65#1878C2218yescomplemented
66#1877C2218yescomplemented
67#1874C2214yes—
68#1872C2214yescomplemented
69#1864C2218yescomplemented
70#1863C2218yescomplemented
71#1824C2212yes—
72#1823C2212yes—
73#1817C2212yes—
74#1817C2212yes—
75#1817C2214yescomplemented
76#1787C2212yes—
77#1786C2212yes—
78#1728C2214yescomplemented
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4—21no—
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C24×C4D411yescharacteristic
85C24×C4D411yescharacteristic
86C24×C4C2311yescharacteristic
87C22×Q16C2314yes—
88C22×SD16C2314yes—
89C22×M16C2312yes—
90C22×M16C2312yes—
91#529—232no—
92#529—232no—
93#529—216no—
94#529—216no—
95#529—216no—
96#529—216no—
97C2×#126—216no—
98C2×#126—24no—
99C2×#126—24no—
100C2×#125—24no—
101C2×#125—24no—
102C2×#125—22no—
103C2×#125C2314yes—
104C2×#124—24no—
105C2×#124—24no—
106C2×#124—22no—
107C2×#124C2314yes—
108C2×#123—216no—
109C2×#123—24no—
110C2×#123—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8D418yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×#121—24no—
124C2×#121—21no—
125C2×#121—21no—
126C2×#121D418yes—
127C2×#121C2312yes—
128C2×#121C2312yes—
129C2×C4×Q8—24no—
130C2×C4×Q8—24no—
131C2×C4×Q8—22no—
132C2×C4×Q8C2314yes—
133C2×C4×D4—216no—
134C2×C4×D4—24no—
135C2×C4×D4—24no—
136C2×C4×D4—24no—
137C2×C4×D4—24no—
138C2×C4×D4—22no—
139C2×C4×D4C2314yes—
140C2×#117—216no—
141C2×#117—24no—
142C2×#117—24no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—21no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—22no—
149C22×C4⋊C4C2311yescharacteristic
150C22×C4⋊C4C2311yescharacteristic
151C22×C4⋊C4C2311yescharacteristic
152C22×C4⋊C4C2312yes—
153C22×C22⋊C4—21no—
154C22×C22⋊C4—21no—
155C22×C22⋊C4—21no—
156C22×C22⋊C4—21no—
157C22×C22⋊C4—21no—
158C22×C22⋊C4—21no—
159C22×C22⋊C4—21no—
160C22×C42—24no—
161C22×C42—24no—
162C22×C42—21no—
163C22×C42D411yescharacteristic
164C22×C42D411yescharacteristic
165C22×C42D412yes—
166C22×C42D412yes—
167C22×C42C2311yescharacteristic
168C22×C42C2311yescharacteristic
169C22×C42C2312yes—
170C22×C42C2312yes—
171#484—28no—
172#483—28no—
173#483—28no—
174#480—28no—
175#470—216no—
176#468—216no—
177#451—28no—
178#451—28no—
179#450—28no—
180#448—28no—
181C2×#107C2314yes—
182C2×#106C2314yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2318yes—
187C2×#102C2314yes—
188C2×#102C2318yes—
189#407—24no—
190#407—24no—
191C2×C22⋊C8C2312yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2312yes—
195#391—22no—
196#391—22no—
197#391C2314yes—
198#391C2314yes—
199#389—22no—
200#389—22no—
201#389C2314yes—
202#389C2314yes—
203#387—24no—
204#387—24no—
205#387—24no—
206#387—24no—
207#385—22no—
208#385—22no—
209#385—24no—
210#381—22no—
211#381—22no—
212#381—24no—
213#380—21no—
214#380—21no—
215#380C2312yes—
216#380C2312yes—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95—22no—
221#340C2318yes—
222#340C2318yes—
223#336C2318yes—
224#336C2318yes—
225C25—21no—
226C25C2×D411yescharacteristic
227C22×Q8—22no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—44no—
233C22×Q8—44no—
234C22×Q8C2×D414yes—
235C22×Q8C2412yes—
236C22×Q8C2412yes—
237C22×D4—28no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—44no—
243C22×D4—48no—
244C22×D4C2×D414yes—
245C22×D4C2412yes—
246C23×C4—28no—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—44no—
258C23×C4—42no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D412yes—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D414yes—
274C23×C4C2×D414yes—
275C23×C4C2411yescharacteristic
276C23×C4C2411yescharacteristic
277C23×C4C2411yescharacteristic
278C2×Q16—28no—
279C2×Q16—48no—
280C2×SD16—28no—
281C2×SD16—48no—
282C2×M16—24no—
283C2×M16—24no—
284C2×M16—44no—
285C2×M16—44no—
286C22×C8C2414yes—
287C22×C8C2414yes—
288#126—232no—
289#126—232no—
290#125—216no—
291#125—28no—
292#125—28no—
293#124—216no—
294#124—28no—
295#124—28no—
296#123—232no—
297#123—232no—
298C22⋊Q8—28no—
299C22⋊Q8—28no—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—24no—
305C22⋊Q8—24no—
306C22⋊Q8—44no—
307C22⋊Q8—44no—
308C22⋊Q8C2×D418yes—
309C22⋊Q8C2×D418yes—
310#121—28no—
311#121—28no—
312#121—24no—
313#121—24no—
314#121C2×D418yes—
315#121C2×D418yes—
316C4×Q8—216no—
317C4×Q8—28no—
318C4×Q8—28no—
319C4×D4—232no—
320C4×D4—232no—
321C4×D4—216no—
322C4×D4—28no—
323C4×D4—28no—
324#117—232no—
325#117—232no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—48no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2414yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—28no—
357C2×C22⋊C4—28no—
358C2×C22⋊C4—28no—
359C2×C22⋊C4—28no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—28no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4—44no—
374C2×C42—28no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42—28no—
378C2×C42—28no—
379C2×C42—24no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—24no—
385C2×C42—24no—
386C2×C42—24no—
387C2×C42—24no—
388C2×C42C2×D412yes—
389C2×C42C2×D412yes—
390C2×C42C2×D412yes—
391C2×C42C2×D412yes—
392C2×C42C2×D412yes—
393C2×C42C2×D412yes—
394C2×C42C2412yes—
395C2×C42C2412yes—
396#107—28no—
397#106—28no—
398#103—28no—
399#103—28no—
400#103—28no—
401#103—48no—
402#102—28no—
403#102—48no—
404C22⋊C8—24no—
405C22⋊C8—24no—
406C22⋊C8—24no—
407C22⋊C8—24no—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416C24—22no—
417C24—22no—
418C24—24no—
419C24—24no—
420C24—22no—
421C24—42no—
422C24—44no—
423C24—44no—
424C24—42no—
425C24—42no—
426C24#12014yes—
427C24C22×D411yescharacteristic
428C24C22×D411yescharacteristic
429C24C22×D411yescharacteristic
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—28no—
435C2×Q8—28no—
436C2×Q8—48no—
437C2×Q8—44no—
438C2×Q8—48no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—84no—
442C2×Q8—84no—
443C2×Q8C22×D414yes—
444C2×D4—28no—
445C2×D4—28no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—28no—
449C2×D4—48no—
450C2×D4—44no—
451C2×D4—48no—
452C2×D4—432no—
453C2×D4—48no—
454C2×D4—48no—
455C2×D4C22×D414yes—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—216no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—416no—
496C22×C4—432no—
497C22×C4—48no—
498C22×C4—416no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—48no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4#12014yes—
512C22×C4#12014yes—
513C22×C4#12014yes—
514C22×C4#13714yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C2511yescharacteristic
529Q16—88no—
530SD16—88no—
531M16—84no—
532M16—84no—
533C2×C8—24no—
534C2×C8—24no—
535C2×C8—28no—
536C2×C8—28no—
537C4⋊C4—216no—
538C4⋊C4—216no—
539C4⋊C4—216no—
540C4⋊C4—28no—
541C4⋊C4—28no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—28no—
549C4⋊C4—44no—
550C4⋊C4—48no—
551C4⋊C4—48no—
552C4⋊C4—42no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4C22×D414yes—
556C4⋊C4C22×D414yes—
557C22⋊C4—216no—
558C22⋊C4—216no—
559C22⋊C4—216no—
560C22⋊C4—28no—
561C22⋊C4—28no—
562C22⋊C4—28no—
563C22⋊C4—28no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C42—216no—
569C42—24no—
570C42—24no—
571C42—24no—
572C42—24no—
573C42—24no—
574C42—24no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C23—22no—
580C23—22no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—22no—
585C23—28no—
586C23—28no—
587C23—21no—
588C23—21no—
589C23—22no—
590C23—24no—
591C23—44no—
592C23—48no—
593C23—48no—
594C23—48no—
595C23—48no—
596C23—44no—
597C23—48no—
598C23—42no—
599C23—42no—
600C23—44no—
601C23—88no—
602C23—88no—
603C23C2×#12012yes—
604C23C2×#12012yes—
605C23C2×#12012yes—
606C23C2×#13712yes—
607C23#57514yes—
608C23#57512yes—
609C23C23×D411yescharacteristic
610Q8—28no—
611Q8—48no—
612Q8—44no—
613Q8—44no—
614Q8—44no—
615Q8—84no—
616D4—28no—
617D4—48no—
618D4—416no—
619D4—416no—
620D4—44no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—28no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—48no—
649C2×C4—48no—
650C2×C4—432no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—48no—
654C2×C4—48no—
655C2×C4—48no—
656C2×C4—44no—
657C2×C4—48no—
658C2×C4—44no—
659C2×C4—48no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—88no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4C2×#12012yes—
674C2×C4C2×#12012yes—
675C2×C4C2×#12012yes—
676C2×C4C2×#13714yes—
677C2×C4C23×D411yescharacteristic
678C2×C4C23×D411yescharacteristic
679C2×C4#58514yes—
680C8—44no—
681C8—44no—
682C22—22no—
683C22—22no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—416no—
694C22—42no—
695C22—42no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—816no—
700C22—84no—
701C22—84no—
702C22#300514yes—
703C22#300714yes—
704C22C22×#12011yescharacteristic
705C22#344912yes—
706C22C22×#13711yescharacteristic
707C22C2×#57512yes—
708C22C2×#57511yescharacteristic
709C4—22no—
710C4—22no—
711C4—22no—
712C4—24no—
713C4—44no—
714C4—42no—
715C4—42no—
716C4—44no—
717C4—42no—
718C4—44no—
719C4—82no—
720C4—84no—
721C4—82no—
722C4#348512yes—
723C4#348512yes—
724C2—24no—
725C2—24no—
726C2—42no—
727C2—84no—
728C2#3508811yescharacteristic
729C2C2×#300511yescharacteristic
730C2C2×#300711yescharacteristic
731C2#6095012yes—
732C2#6096412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3195C2semidirect
4. C2×#2855C2semidirect
5. #33010C2semidirect
6. #33007C2semidirect
7. #29643C2semidirect
8. #29626C2semidirect
9. #25505C2semidirect
10. #24066C2semidirect
11. #23937C2semidirect
12. #21765C2semidirect
14. #20682C2semidirect
37. C2×#483C22semidirect
39. C2×#470C22semidirect
42. C2×#451C22semidirect
52. #2855C22semidirect
53. C2×#391C22semidirect
54. C2×#389C22semidirect
59. #2059C22semidirect
60. #2042C22semidirect
61. #2026C22semidirect
62. #2015C22semidirect
64. #1894C22semidirect
65. #1878C22semidirect
66. #1877C22semidirect
68. #1872C22semidirect
69. #1864C22semidirect
70. #1863C22semidirect
75. #1817C22semidirect
78. #1728C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#31952
C2×#31932
C2×#28552
#330102
#330072
#296432
#296262
#255052
#240662
#239372
#217652
#217642
#206822