Finite Group Databasebeta

← #6436522

Subgroups — #6436522

Order 512.

3261 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic, complemented
2C2×#3108C212yesmaximal, complemented
3#36621C211yesmaximal, characteristic
4#29695C214yesmaximal, complemented
5#29668C214yesmaximal, complemented
6#29641C211yesmaximal, characteristic, complemented
7#29638C211yesmaximal, characteristic, complemented
8#29627C211yesmaximal, characteristic, complemented
9#29624C211yesmaximal, characteristic, complemented
10#29599C212yesmaximal, complemented
11#29595C212yesmaximal
12#23571C214yesmaximal, complemented
13#23568C214yesmaximal, complemented
14#20735C212yesmaximal, complemented
15#20714C211yesmaximal, characteristic, complemented
16C2×#533—22no—
17C2×#533—22no—
18C2×#533—216no—
19C2×#533C2214yes—
20C2×#533C2214yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×#529C2211yescharacteristic
24C22×#125—21no—
25C22×C22⋊Q8—21no—
26C22×C22⋊Q8C2212yes—
27C22×C4×Q8—21no—
28C22×C4×Q8C2212yes—
29C22×#117—21no—
30C23×C42C2211yescharacteristic
31#3108—216no—
32C2×#483C2212yes—
33C2×#480C2212yes—
34C2×#479C2214yescomplemented
35C2×#464C2214yescomplemented
36C2×#451C2212yes—
37C2×#448C2212yes—
38C2×C4×Q16C2214yescomplemented
39C2×C4×SD16C2214yescomplemented
40C2×#431C2212yes—
41#2855C2211yescharacteristic
42#2854C2211yescharacteristic
43#2853—21no—
44#2852—22no—
45#2851C2212yes—
46#2850C2212yes—
47#2849—21no—
48#2848C2211yescharacteristic
49#2847C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51C4×#123—21no—
52C4×#121C2212yes—
53C4×#120—21no—
54#2290—22no—
55#2283C2211yescharacteristic
56#2058C2214yescomplemented
57#2057C2214yescomplemented
58#2043C2214yescomplemented
59#2043C2214yescomplemented
60#2036C2218yes—
61#2035C2218yes—
62#2024C2214yescomplemented
63#2023C2214yescomplemented
64#2016C2214yescomplemented
65#2013C2214yescomplemented
66#2001C2218yes—
67#2000C2218yes—
68#1823C2212yes—
69#1823C2212yes—
70#1817C2211yescharacteristic
71#1817C2211yescharacteristic
72#1805C2218yescomplemented
73#1797C2214yes—
74#1787C2212yes—
75#1787C2212yes—
76#1779C2212yes—
77C4×#103C2218yescomplemented
78C4×C22⋊C8C2214yes—
79C23×Q8—21no—
80C23×Q8C2312yes—
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C22×Q16C2314yes—
85C22×SD16C2314yes—
86#533—232no—
87#533—232no—
88#533—232no—
89#533—232no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8C2314yes—
95C2×C4⋊Q8C2314yes—
96#529—28no—
97C2×#126—21no—
98C2×#125—216no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125C2311yescharacteristic
102C2×#124C2311yescharacteristic
103C2×#123—21no—
104C2×#123—21no—
105C2×C22⋊Q8—216no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—28no—
111C2×C22⋊Q8C2311yescharacteristic
112C2×C22⋊Q8C2318yes—
113C2×C22⋊Q8C2312yes—
114C2×#121C2311yescharacteristic
115C2×#121C2312yes—
116C2×#120—21no—
117C2×C4×Q8—216no—
118C2×C4×Q8—24no—
119C2×C4×Q8—24no—
120C2×C4×Q8—28no—
121C2×C4×Q8—22no—
122C2×C4×Q8—22no—
123C2×C4×Q8D4116yescomplemented
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2314yes—
126C2×C4×Q8C2314yes—
127C2×C4×D4—21no—
128C2×C4×D4—22no—
129C2×C4×D4C2311yescharacteristic
130C2×#117—216no—
131C2×#117—24no—
132C2×#117—24no—
133C2×#117—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—22no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4C2312yes—
140C22×C22⋊C4—22no—
141C22×C42—22no—
142C22×C42—28no—
143C22×C42D412yes—
144C22×C42D411yescharacteristic
145C22×C42D411yescharacteristic
146C22×C42D412yes—
147C22×C42C2312yes—
148C22×C42C2311yescharacteristic
149C22×C42C2312yes—
150C22×C42C2311yescharacteristic
151#483—28no—
152#480—28no—
153#479—216no—
154#464—216no—
155#451—28no—
156#448—28no—
157C4×Q16—216no—
158C4×SD16—216no—
159#431—28no—
160C2×#107C2314yes—
161C2×#106C2314yes—
162C2×C4⋊C8C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2318yes—
167C2×#103C2318yes—
168C2×#102C2314yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171C2×C22⋊C8C2318yes—
172C2×C4×C8C2314yes—
173#391C2312yes—
174#390C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#386—22no—
186#386—21no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—21no—
191#385—22no—
192#385—21no—
193#385—22no—
194#385—22no—
195#385—22no—
196#385—21no—
197#383—21no—
198#382C2312yes—
199#381—21no—
200#381—21no—
201#380C2311yescharacteristic
202#380C2311yescharacteristic
203#380C2312yes—
204#380C2312yes—
205#379—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—21no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211#376—21no—
212#340C2318yes—
213#340C2318yes—
214#336C2314yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C22×Q8—28no—
218C22×Q8—24no—
219C22×Q8—22no—
220C22×Q8—24no—
221C22×Q8—48no—
222C22×Q8—44no—
223C22×Q8—44no—
224C22×Q8C2×D418yes—
225C22×Q8C2411yescharacteristic
226C22×Q8C2414yes—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4C2411yescharacteristic
230C23×C4—24no—
231C23×C4—28no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—21no—
238C23×C4—44no—
239C23×C4—42no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C2×Q16—28no—
253C2×Q16—48no—
254C2×SD16—28no—
255C2×SD16—48no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—216no—
259C4⋊Q8—216no—
260C4⋊Q8—216no—
261C4⋊Q8—216no—
262#126—44no—
263#125—232no—
264#125—232no—
265#125—24no—
266#124—24no—
267#123—44no—
268#123—44no—
269C22⋊Q8—232no—
270C22⋊Q8—232no—
271C22⋊Q8—232no—
272C22⋊Q8—216no—
273C22⋊Q8—216no—
274C22⋊Q8—24no—
275C22⋊Q8—44no—
276#121—24no—
277#121—44no—
278#120—44no—
279C4×Q8—232no—
280C4×Q8—232no—
281C4×Q8—216no—
282C4×Q8—216no—
283C4×Q8—28no—
284C4×Q8—28no—
285C4×Q8—24no—
286C4×Q8C2×D4116yes—
287C4×Q8C2×D4116yes—
288C4×D4—24no—
289C4×D4—44no—
290C4×D4—48no—
291#117—232no—
292#117—232no—
293#117—44no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—216no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—216no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4C2×D418yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C22⋊C4—216no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C42—216no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—28no—
342C2×C42—28no—
343C2×C42—28no—
344C2×C42—28no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D418yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—28no—
359#106—28no—
360C4⋊C8—28no—
361#103—28no—
362#103—28no—
363#103—28no—
364#103—48no—
365#103—48no—
366#102—28no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369C22⋊C8—48no—
370C4×C8—28no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382C24—22no—
383C24—24no—
384C24—22no—
385C24—22no—
386C24—21no—
387C24—42no—
388C24C22×D412yes—
389C24C22×D411yescharacteristic
390C2×Q8—216no—
391C2×Q8—28no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—24no—
397C2×Q8—416no—
398C2×Q8—48no—
399C2×Q8—432no—
400C2×Q8—48no—
401C2×Q8—416no—
402C2×Q8—48no—
403C2×Q8C22×D418yes—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—42no—
412C22×C4—232no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4#12014yes—
457C22×C4#12014yes—
458C22×C4#13514yes—
459C22×C4#13718yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D414yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C2511yescharacteristic
470Q16—416no—
471SD16—416no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—28no—
475C2×C8—28no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—232no—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4C22×D418yes—
496C22⋊C4—232no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C42—24no—
511C42—24no—
512C42—24no—
513C42—24no—
514C42—24no—
515C42—216no—
516C42—216no—
517C42—24no—
518C42—42no—
519C42—44no—
520C42#12014yes—
521C42#12014yes—
522C42C22×D414yes—
523C42C22×D414yes—
524C23—22no—
525C23—22no—
526C23—28no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—48no—
534C23—48no—
535C23—44no—
536C23—44no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23C2×#12012yes—
543C23C2×#13512yes—
544C23C2×#13714yes—
545C23#57512yes—
546C23C23×D411yescharacteristic
547Q8—28no—
548Q8—28no—
549Q8—416no—
550Q8—432no—
551Q8—48no—
552Q8—44no—
553D4—44no—
554D4—88no—
555D4—84no—
556C2×C4—216no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—216no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—416no—
583C2×C4—48no—
584C2×C4—416no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—84no—
602C2×C4C2×#12012yes—
603C2×C4C2×#12012yes—
604C2×C4C2×#12012yes—
605C2×C4C2×#12012yes—
606C2×C4C2×#12012yes—
607C2×C4C2×#13514yes—
608C2×C4C2×#13714yes—
609C2×C4C2×#13714yes—
610C2×C4C23×D412yes—
611C2×C4#58514yes—
612C8—44no—
613C8—44no—
614C22—24no—
615C22—24no—
616C22—22no—
617C22—22no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—416no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—82no—
627C22—82no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22#300214yes—
632C22C22×#12011yescharacteristic
633C22#344912yes—
634C22C22×#13511yescharacteristic
635C22C22×#13712yes—
636C22C2×#57511yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—24no—
641C4—22no—
642C4—24no—
643C4—42no—
644C4—48no—
645C4—48no—
646C4—82no—
647C4—82no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4#300414yes—
652C4#344914yes—
653C2—24no—
654C2—44no—
655C2—161no—
656C2#3463211yescharacteristic
657C2C2×#300211yescharacteristic
658C2C2×#300411yescharacteristic
659C2#6094712yes—
660C2#6095012yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#533C2semidirect
2. C2×#3108C2semidirect
4. #29695C2semidirect
5. #29668C2semidirect
6. #29641C2semidirect
7. #29638C2semidirect
8. #29627C2semidirect
9. #29624C2semidirect
10. #29599C2semidirect
12. #23571C2semidirect
13. #23568C2semidirect
14. #20735C2semidirect
15. #20714C2semidirect
34. C2×#479C22semidirect
35. C2×#464C22semidirect
38. C2×C4×Q16C22semidirect
39. C2×C4×SD16C22semidirect
56. #2058C22semidirect
57. #2057C22semidirect
58. #2043C22semidirect
59. #2043C22semidirect
62. #2024C22semidirect
63. #2023C22semidirect
64. #2016C22semidirect
65. #2013C22semidirect
72. #1805C22semidirect
77. C4×#103C22semidirect
123. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#31082
#366212
#296952
#296682
#296412
#296382
#296272
#296242
#295992
#295952
#235712
#235682
#207352
#207142