Finite Group Databasebeta

← #6477306

Subgroups — #6477306

Order 512.

3275 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic, complemented
2C2×#3190C212yesmaximal
3#36626C211yesmaximal, characteristic
4#33008C212yesmaximal, complemented
5#33003C212yesmaximal, complemented
6#29641C211yesmaximal, characteristic, complemented
7#29638C211yesmaximal, characteristic, complemented
8#29627C211yesmaximal, characteristic, complemented
9#29624C211yesmaximal, characteristic, complemented
10#25498C214yesmaximal, complemented
11#25497C214yesmaximal, complemented
12#24065C214yesmaximal, complemented
13#23935C214yesmaximal, complemented
14#21762C212yesmaximal, complemented
15C2×#1817C211yesmaximal, characteristic, complemented
16C2×#533—22no—
17C2×#533—22no—
18C2×#533—216no—
19C2×#533C2214yes—
20C2×#533C2214yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125—21no—
24C2×#525C2211yescharacteristic
25C22×C22⋊Q8—21no—
26C22×C22⋊Q8C2212yes—
27C22×C4×Q8—21no—
28C22×C4×Q8C2212yes—
29C22×#117—21no—
30C23×C42C2211yescharacteristic
31#3190—216no—
32C2×#485C2214yes—
33C2×#483C2212yes—
34C2×#480C2212yes—
35C2×#467C2214yes—
36C2×#465C2214yes—
37C2×#452C2214yes—
38C2×#451C2212yes—
39C2×#448C2212yes—
40C22×C22⋊C8C2212yes—
41#2855C2212yes—
42#2855C2211yescharacteristic
43#2855C2211yescharacteristic
44#2854C2212yes—
45#2854C2211yescharacteristic
46#2854C2211yescharacteristic
47#2853—22no—
48#2849—21no—
49#2849—21no—
50C4×#123—21no—
51C4×#123—21no—
52C4×#123—22no—
53#2297C2212yes—
54#2296C2212yes—
55#2280C2211yescharacteristic
56#2058C2214yescomplemented
57#2057C2214yescomplemented
58#2043C2214yescomplemented
59#2043C2214yescomplemented
60#2024C2214yescomplemented
61#2023C2214yescomplemented
62#2016C2214yescomplemented
63#2013C2214yescomplemented
64#1892C2218yes—
65#1880C2218yescomplemented
66#1878C2218yescomplemented
67#1873C2218yes—
68#1865C2218yescomplemented
69#1863C2218yescomplemented
70#1823C2212yes—
71#1823C2212yes—
72#1817C2214yes—
73#1817C2211yescharacteristic
74#1817C2211yescharacteristic
75#1817C2212yes—
76#1787C2212yes—
77#1787C2212yes—
78C2×#336C2214yes—
79C22×#41C2311yescharacteristic
80C23×Q8—21no—
81C23×Q8C2312yes—
82C24×C4D411yescharacteristic
83C24×C4D411yescharacteristic
84C24×C4C2311yescharacteristic
85C22×Q16C2314yes—
86C22×SD16C2314yes—
87C23×C8C2314yes—
88#533—232no—
89#533—232no—
90#533—232no—
91#533—232no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2314yes—
97C2×C4⋊Q8C2314yes—
98C2×#125—216no—
99C2×#125—24no—
100C2×#125—24no—
101#525—28no—
102C2×#123—21no—
103C2×#123—21no—
104C2×#123—22no—
105C2×#123—21no—
106C2×C22⋊Q8—216no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—28no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8D4116yes—
113C2×C22⋊Q8C2311yescharacteristic
114C2×C22⋊Q8C2311yescharacteristic
115C2×C22⋊Q8C2314yes—
116C2×C22⋊Q8C2314yes—
117C2×#121C2311yescharacteristic
118C2×#121C2311yescharacteristic
119C2×#120—21no—
120C2×C4×Q8—216no—
121C2×C4×Q8—28no—
122C2×C4×Q8—28no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8—24no—
126C2×C4×Q8C2318yes—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×#117—216no—
130C2×#117—24no—
131C2×#117—24no—
132C2×#117C2311yescharacteristic
133C2×#117C2312yes—
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—28no—
142C22×C42—22no—
143C22×C42D412yes—
144C22×C42D411yescharacteristic
145C22×C42D411yescharacteristic
146C22×C42D412yes—
147C22×C42C2311yescharacteristic
148C22×C42C2311yescharacteristic
149C22×C42C2312yes—
150C22×C42C2312yes—
151#485—216no—
152#483—28no—
153#480—28no—
154#467—216no—
155#465—216no—
156#452—216no—
157#451—28no—
158#448—28no—
159C2×#107C2314yes—
160C2×#106C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2318yes—
165C2×#103C2318yes—
166C2×#102C2314yes—
167C2×C22⋊C8—28no—
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2314yes—
171#391C2312yes—
172#391C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#386—22no—
188#386—22no—
189#386—21no—
190#386—21no—
191#386—22no—
192#385—22no—
193#385—22no—
194#385—21no—
195#385—21no—
196#385—22no—
197#381—21no—
198#381—21no—
199#381—22no—
200#381—21no—
201#380C2311yescharacteristic
202#380C2311yescharacteristic
203#380C2312yes—
204#379—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—21no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211#340C2318yes—
212#340C2318yes—
213#336C2314yes—
214#336C2314yes—
215#336C2318yes—
216C25C2×D411yescharacteristic
217C2×#41—24no—
218C2×#41—44no—
219C22×Q8—28no—
220C22×Q8—24no—
221C22×Q8—22no—
222C22×Q8—24no—
223C22×Q8—48no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8C2×D418yes—
227C22×Q8C2411yescharacteristic
228C22×Q8C2414yes—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2411yescharacteristic
233C23×C4—24no—
234C23×C4—28no—
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—24no—
239C23×C4—21no—
240C23×C4—44no—
241C23×C4—42no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D418yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×Q16—28no—
258C2×Q16—48no—
259C2×SD16—28no—
260C2×SD16—48no—
261C22×C8—28no—
262C22×C8—48no—
263C22×C8C2414yes—
264C22×C8C2414yes—
265C4⋊Q8—216no—
266C4⋊Q8—216no—
267C4⋊Q8—216no—
268C4⋊Q8—216no—
269#125—232no—
270#125—232no—
271#123—44no—
272#123—44no—
273#123—44no—
274#123—48no—
275C22⋊Q8—232no—
276C22⋊Q8—232no—
277C22⋊Q8—216no—
278C22⋊Q8—216no—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8C2×D4116yes—
284C22⋊Q8C2×D4116yes—
285#121—24no—
286#121—24no—
287#120—44no—
288C4×Q8—232no—
289C4×Q8—232no—
290C4×Q8—232no—
291C4×Q8—216no—
292C4×Q8—216no—
293C4×D4—44no—
294C4×D4—44no—
295#117—232no—
296#117—232no—
297#117—24no—
298#117—44no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—216no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—216no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4C2×D418yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C22⋊C4—216no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C42—216no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—28no—
344C2×C42—28no—
345C2×C42—28no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—28no—
349C2×C42—28no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360#107—28no—
361#106—28no—
362#103—28no—
363#103—28no—
364#103—28no—
365#103—48no—
366#103—48no—
367#102—28no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C22⋊C8—28no—
371#95—22no—
372#95—21no—
373#95—21no—
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—21no—
386C24—42no—
387C24#12014yes—
388C24C22×D411yescharacteristic
389C24C22×D411yescharacteristic
390C24C22×D411yescharacteristic
391#41—88no—
392C2×Q8—216no—
393C2×Q8—28no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—28no—
397C2×Q8—28no—
398C2×Q8—24no—
399C2×Q8—416no—
400C2×Q8—48no—
401C2×Q8—432no—
402C2×Q8—48no—
403C2×Q8—416no—
404C2×Q8—48no—
405C2×Q8C22×D418yes—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—42no—
415C22×C4—232no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—416no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—416no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4#12014yes—
459C22×C4#12014yes—
460C22×C4#12014yes—
461C22×C4#13514yes—
462C22×C4#13718yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C2511yescharacteristic
474Q16—88no—
475SD16—88no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—28no—
479C2×C8—28no—
480C2×C8—88no—
481C4⋊C4—216no—
482C4⋊C4—216no—
483C4⋊C4—232no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—216no—
487C4⋊C4—28no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4C22×D418yes—
500C22⋊C4—216no—
501C22⋊C4—216no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C42—232no—
516C42—24no—
517C42—24no—
518C42—24no—
519C42—24no—
520C42—24no—
521C42—24no—
522C42—24no—
523C42—24no—
524C42—44no—
525C42—42no—
526C23—22no—
527C23—22no—
528C23—21no—
529C23—22no—
530C23—21no—
531C23—22no—
532C23—28no—
533C23—28no—
534C23—21no—
535C23—22no—
536C23—21no—
537C23—44no—
538C23—48no—
539C23—48no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23C2×#12012yes—
546C23C2×#12012yes—
547C23C2×#12012yes—
548C23C2×#13512yes—
549C23C2×#13714yes—
550C23#57512yes—
551C23C23×D411yescharacteristic
552Q8—28no—
553Q8—28no—
554Q8—416no—
555Q8—432no—
556Q8—48no—
557Q8—44no—
558D4—44no—
559D4—84no—
560D4—84no—
561D4—84no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—216no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—216no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—416no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—416no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—84no—
609C2×C4C2×#12012yes—
610C2×C4C2×#12012yes—
611C2×C4C2×#12012yes—
612C2×C4C2×#13514yes—
613C2×C4C2×#13714yes—
614C2×C4C2×#13714yes—
615C2×C4C23×D412yes—
616C2×C4#58514yes—
617C8—44no—
618C8—44no—
619C22—22no—
620C22—22no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—416no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—82no—
635C22—82no—
636C22—81no—
637C22—81no—
638C22#300214yes—
639C22#300414yes—
640C22C22×#12011yescharacteristic
641C22#344912yes—
642C22C22×#13511yescharacteristic
643C22C22×#13712yes—
644C22C2×#57511yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—24no—
649C4—24no—
650C4—24no—
651C4—48no—
652C4—42no—
653C4—48no—
654C4—81no—
655C4—82no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4#348514yes—
660C2—24no—
661C2—24no—
662C2—42no—
663C2—161no—
664C2#3508811yescharacteristic
665C2C2×#300211yescharacteristic
666C2C2×#300411yescharacteristic
667C2#6094712yes—
668C2#6096312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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
1. C22×#533C2semidirect
4. #33008C2semidirect
5. #33003C2semidirect
6. #29641C2semidirect
7. #29638C2semidirect
8. #29627C2semidirect
9. #29624C2semidirect
10. #25498C2semidirect
11. #25497C2semidirect
12. #24065C2semidirect
13. #23935C2semidirect
14. #21762C2semidirect
15. C2×#1817C2semidirect
56. #2058C22semidirect
57. #2057C22semidirect
58. #2043C22semidirect
59. #2043C22semidirect
60. #2024C22semidirect
61. #2023C22semidirect
62. #2016C22semidirect
63. #2013C22semidirect
65. #1880C22semidirect
66. #1878C22semidirect
68. #1865C22semidirect
69. #1863C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#31902
#366262
#330082
#330032
#296412
#296382
#296272
#296242
#254982
#254972
#240652
#239352
#217622
C2×#18172