Finite Group Databasebeta

← #6434668

Subgroups — #6434668

Order 512.

3243 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic, complemented
2C2×#3091C212yesmaximal
3#36626C211yesmaximal, characteristic
4#29645C214yesmaximal, complemented
5#29644C214yesmaximal, complemented
6#29643C212yesmaximal, complemented
7#29642C211yesmaximal, characteristic, complemented
8#29641C211yesmaximal, characteristic, complemented
9#29640C212yesmaximal, complemented
10#29639C211yesmaximal, characteristic, complemented
11#29638C211yesmaximal, characteristic, complemented
12#23551C214yesmaximal, complemented
13#23451C214yesmaximal, complemented
14#20699C212yesmaximal, 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×#125C2212yes—
24C2×#525C2211yescharacteristic
25C22×C22⋊Q8—21no—
26C22×C22⋊Q8—21no—
27C22×C4×Q8—21no—
28C22×C4×Q8—21no—
29C22×#117C2212yes—
30C23×C42C2211yescharacteristic
31#3091—216no—
32C2×#484C2212yes—
33C2×#481C2212yes—
34C2×#485C2214yes—
35C2×#482C2214yes—
36C2×#483C2212yes—
37C2×#480C2212yes—
38C2×#427C2214yes—
39C2×#416C2214yes—
40C22×C22⋊C8C2212yes—
41#2855C2211yescharacteristic
42#2855C2211yescharacteristic
43#2855C2212yes—
44#2854C2211yescharacteristic
45#2854C2211yescharacteristic
46#2854C2212yes—
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#2059C2218yescomplemented
57#2059C2218yescomplemented
58#2058C2214yescomplemented
59#2058C2214yescomplemented
60#2057C2214yescomplemented
61#2057C2214yescomplemented
62#2026C2218yescomplemented
63#2025C2218yescomplemented
64#2024C2214yescomplemented
65#2024C2214yescomplemented
66#2023C2214yescomplemented
67#2023C2214yescomplemented
68#1825C2218yes—
69#1824C2212yes—
70#1824C2212yes—
71#1823C2212yes—
72#1823C2212yes—
73#1817C2211yescharacteristic
74#1817C2211yescharacteristic
75#1817C2212yes—
76#1817C2214yes—
77#1735C2218yes—
78C2×#336C2214yes—
79C22×#41C2311yescharacteristic
80C23×Q8—21no—
81C23×Q8—21no—
82C24×C4D411yescharacteristic
83C24×C4D411yescharacteristic
84C24×C4C2311yescharacteristic
85C23×C8C2314yes—
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—
96C2×#125—28no—
97C2×#125—28no—
98C2×#125—24no—
99C2×#125C2318yes—
100#525—28no—
101C2×#123—21no—
102C2×#123—22no—
103C2×#123—21no—
104C2×#123—21no—
105C2×C22⋊Q8—216no—
106C2×C22⋊Q8—216no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8C2311yescharacteristic
112C2×C22⋊Q8C2311yescharacteristic
113C2×#121C2311yescharacteristic
114C2×#121C2311yescharacteristic
115C2×#120—21no—
116C2×C4×Q8—216no—
117C2×C4×Q8—216no—
118C2×C4×Q8—24no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8—24no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×#117—28no—
125C2×#117—22no—
126C2×#117—22no—
127C2×#117D4116yes—
128C2×#117C2311yescharacteristic
129C2×#117C2312yes—
130C2×#117C2314yes—
131C2×#117C2314yes—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C22⋊C4—22no—
139C22×C42—28no—
140C22×C42—22no—
141C22×C42D412yes—
142C22×C42D411yescharacteristic
143C22×C42D411yescharacteristic
144C22×C42D412yes—
145C22×C42C2311yescharacteristic
146C22×C42C2311yescharacteristic
147C22×C42C2312yes—
148C22×C42C2312yes—
149#485—216no—
150#484—28no—
151#483—28no—
152#482—216no—
153#481—28no—
154#480—28no—
155#427—216no—
156C2×#107C2314yes—
157C2×#107C2314yes—
158C2×#106C2314yes—
159C2×#106C2314yes—
160#416—216no—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×C22⋊C8—28no—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2314yes—
169#391C2312yes—
170#391C2312yes—
171#390C2312yes—
172#390C2312yes—
173#389C2312yes—
174#389C2312yes—
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—22no—
187#386—21no—
188#386—22no—
189#386—21no—
190#385—22no—
191#385—22no—
192#385—21no—
193#385—22no—
194#385—21no—
195#381—21no—
196#381—22no—
197#381—21no—
198#381—21no—
199#380C2311yescharacteristic
200#380C2311yescharacteristic
201#380C2312yes—
202#379—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#336C2314yes—
214#336C2314yes—
215#336C2318yes—
216C25C2×D411yescharacteristic
217C2×#41—24no—
218C2×#41—28no—
219C22×Q8—28no—
220C22×Q8—28no—
221C22×Q8—24no—
222C22×Q8—24no—
223C22×Q8—44no—
224C22×Q8—48no—
225C22×Q8—44no—
226C22×Q8C2411yescharacteristic
227C22×D4—21no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4C2411yescharacteristic
231C23×C4—28no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—24no—
237C23×C4—21no—
238C23×C4—44no—
239C23×C4—42no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D418yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C22×C8—28no—
256C22×C8—216no—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—216no—
260C4⋊Q8—216no—
261C4⋊Q8—216no—
262C4⋊Q8—216no—
263#125—232no—
264#125—216no—
265#125—216no—
266#123—44no—
267#123—48no—
268#123—44no—
269#123—44no—
270C22⋊Q8—232no—
271C22⋊Q8—232no—
272C22⋊Q8—232no—
273C22⋊Q8—232no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276#121—24no—
277#121—24no—
278#120—44no—
279C4×Q8—232no—
280C4×Q8—232no—
281C4×Q8—232no—
282C4×Q8—232no—
283C4×D4—44no—
284C4×D4—44no—
285#117—216no—
286#117—216no—
287#117—28no—
288#117—28no—
289#117—24no—
290#117—28no—
291#117C2×D4116yes—
292#117C2×D4116yes—
293C2×C4⋊C4—216no—
294C2×C4⋊C4—216no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4C2×D418yes—
312C2×C4⋊C4C2×D418yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2414yes—
319C2×C22⋊C4—216no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
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—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C42—216no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—28no—
339C2×C42—28no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—28no—
344C2×C42—28no—
345C2×C42—24no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2412yes—
355C2×C42C2412yes—
356#107—28no—
357#107—28no—
358#106—28no—
359#106—28no—
360#103—28no—
361#103—28no—
362#102—28no—
363#102—28no—
364C22⋊C8—24no—
365C22⋊C8—24no—
366C22⋊C8—28no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—21no—
382C24—42no—
383C24#12014yes—
384C24C22×D411yescharacteristic
385C24C22×D411yescharacteristic
386C24C22×D411yescharacteristic
387#41—416no—
388C2×Q8—216no—
389C2×Q8—216no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—432no—
393C2×Q8—432no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—48no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—232no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4#12014yes—
451C22×C4#12014yes—
452C22×C4#12014yes—
453C22×C4#13514yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D414yes—
463C22×C4C22×D411yescharacteristic
464C22×C42−1+418yes—
465C22×C4C2511yescharacteristic
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—28no—
469C2×C8—28no—
470C2×C8—416no—
471C4⋊C4—232no—
472C4⋊C4—232no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4C22×D418yes—
490C4⋊C4C22×D418yes—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C42—24no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—24no—
512C42—216no—
513C42—216no—
514C42—24no—
515C42—24no—
516C42—42no—
517C42—44no—
518C23—22no—
519C23—21no—
520C23—22no—
521C23—22no—
522C23—21no—
523C23—28no—
524C23—22no—
525C23—28no—
526C23—21no—
527C23—22no—
528C23—21no—
529C23—44no—
530C23—48no—
531C23—48no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23C2×#12012yes—
538C23C2×#12012yes—
539C23C2×#12012yes—
540C23#53614yes—
541C23C2×#13512yes—
542C23#57512yes—
543C23C23×D411yescharacteristic
544Q8—432no—
545Q8—432no—
546Q8—44no—
547D4—44no—
548D4—84no—
549D4—84no—
550D4—84no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—416no—
578C2×C4—44no—
579C2×C4—416no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—84no—
597C2×C4C2×#12012yes—
598C2×C4C2×#12012yes—
599C2×C4C2×#12012yes—
600C2×C4#53614yes—
601C2×C4#536116yes—
602C2×C4C2×#13514yes—
603C2×C4#57514yes—
604C2×C4C23×D412yes—
605C2×C4C2×2−1+414yes—
606C8—44no—
607C8—44no—
608C22—22no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—416no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—82no—
624C22—82no—
625C22—81no—
626C22—81no—
627C22#261414yes—
628C22#300214yes—
629C22C22×#12011yescharacteristic
630C22C2×#53612yes—
631C22C22×#13511yescharacteristic
632C22C2×#57512yes—
633C22C2×#57511yescharacteristic
634C4—22no—
635C4—22no—
636C4—24no—
637C4—24no—
638C4—42no—
639C4—48no—
640C4—42no—
641C4—48no—
642C4—81no—
643C4—82no—
644C4—82no—
645C4—81no—
646C4—81no—
647C4#261414yes—
648C4C2×#53614yes—
649C2—24no—
650C2—24no—
651C2—42no—
652C2—161no—
653C2C2×#261411yescharacteristic
654C2C2×#300211yescharacteristic
655C2#6035911yescharacteristic
656C2#6084612yes—
657C2#6085412yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#525C22
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
4. #29645C2semidirect
5. #29644C2semidirect
6. #29643C2semidirect
7. #29642C2semidirect
8. #29641C2semidirect
9. #29640C2semidirect
10. #29639C2semidirect
11. #29638C2semidirect
12. #23551C2semidirect
13. #23451C2semidirect
14. #20699C2semidirect
15. C2×#1817C2semidirect
56. #2059C22semidirect
57. #2059C22semidirect
58. #2058C22semidirect
59. #2058C22semidirect
60. #2057C22semidirect
61. #2057C22semidirect
62. #2026C22semidirect
63. #2025C22semidirect
64. #2024C22semidirect
65. #2024C22semidirect
66. #2023C22semidirect
67. #2023C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#30912
#366262
#296452
#296442
#296432
#296422
#296412
#296402
#296392
#296382
#235512
#234512
#206992
C2×#18172