Finite Group Databasebeta

← #6491250

Subgroups — #6491250

Order 512.

2935 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2C2×#3261C212yesmaximal, complemented
3#35310C211yesmaximal, characteristic
4#32960C212yesmaximal, complemented
5#32906C212yesmaximal, complemented
6#29701C212yesmaximal, complemented
7#29674C212yesmaximal, complemented
8#25512C214yesmaximal, complemented
9#24386C212yesmaximal
10#24380C212yesmaximal
11#24307C212yesmaximal, complemented
12#24212C212yesmaximal, complemented
13#24165C214yesmaximal, complemented
14#21898C212yesmaximal, complemented
15#20628C211yesmaximal, characteristic
16C2×#541—216no—
17C2×#541—22no—
18C2×#541—22no—
19C2×#541C2212yes—
20C2×#541C2212yes—
21C2×#541C2212yes—
22C2×#541C2211yescharacteristic
23C2×#541C2212yes—
24C22×C4⋊Q8C2211yescharacteristic
25C22×#126—22no—
26C22×#125—22no—
27C22×#125C2211yescharacteristic
28C22×C22⋊Q8C2212yes—
29C22×C4×Q8C2212yes—
30C22×#117C2211yescharacteristic
31#3261—216no—
32C2×#479C2214yescomplemented
33C2×#479C2214yescomplemented
34C2×#470C2214yes—
35C2×#467C2214yes—
36C2×#462C2214yescomplemented
37C2×#462C2214yescomplemented
38C2×#451C2212yes—
39C2×#451C2212yes—
40C2×#424C2212yes—
41C2×#407C2211yescharacteristic
42C22×C22⋊C8C2211yescharacteristic
43#2857C2211yescharacteristic
44#2851C2211yescharacteristic
45#2848C2212yes—
46#2629C2211yescharacteristic
47#2625C2212yes—
48#2621C2211yescharacteristic
49#2480—22no—
50#2478—22no—
51#2475—22no—
52#2473—22no—
53#2323C2212yes—
54#2314C2211yescharacteristic
55C4×C22⋊Q8C2212yes—
56#2059C2214yescomplemented
57#2043C2214yescomplemented
58#2036C2214yes—
59#2035C2214yes—
60#2026C2214yescomplemented
61#2016C2214yescomplemented
62#2001C2214yes—
63#2001C2214yes—
64#1989C2214yescomplemented
65#1981C2214yescomplemented
66#1894C2214yes—
67#1892C2214yes—
68#1878C2214yescomplemented
69#1878C2214yescomplemented
70#1872C2212yes—
71#1872C2212yes—
72#1865C2214yescomplemented
73#1863C2214yescomplemented
74#1853C2212yes—
75#1852C2212yes—
76#1805C2214yescomplemented
77#1803C2214yescomplemented
78#1798C2212yes—
79#1798C2212yes—
80#1797C2212yes—
81#1797C2212yes—
82#1787C2212yes—
83#1787C2212yes—
84#1729C2212yes—
85C23×Q8C2312yes—
86C24×C4C2311yescharacteristic
87C22×Q16C2314yes—
88C22×Q16C2314yes—
89C22×M16C2312yes—
90C23×C8C2312yes—
91#541—232no—
92#541—232no—
93#541—28no—
94#541—28no—
95#541—216no—
96#541—216no—
97#541—28no—
98#541—28no—
99#541D4116yes—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#126—232no—
106C2×#126—24no—
107C2×#126—24no—
108C2×#126—24no—
109C2×#126—24no—
110C2×#126—22no—
111C2×#125—232no—
112C2×#125—28no—
113C2×#125—24no—
114C2×#125—24no—
115C2×#125—24no—
116C2×#125—24no—
117C2×#125—22no—
118C2×#125—22no—
119C2×#125C2311yescharacteristic
120C2×#125C2312yes—
121C2×#125C2312yes—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8—216no—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2312yes—
127C2×C22⋊Q8C2314yes—
128C2×C22⋊Q8C2314yes—
129C2×C22⋊Q8C2312yes—
130C2×C4×Q8—216no—
131C2×C4×Q8—22no—
132C2×C4×Q8—22no—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2314yes—
135C2×C4×Q8C2314yes—
136C2×#117—28no—
137C2×#117D412yes—
138C2×#117D412yes—
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C22×C4⋊C4—22no—
144C22×C4⋊C4—22no—
145C22×C4⋊C4—22no—
146C22×C4⋊C4—22no—
147C22×C4⋊C4D412yes—
148C22×C4⋊C4C2312yes—
149C22×C22⋊C4—22no—
150C22×C22⋊C4D412yes—
151C22×C42—22no—
152C22×C42C2311yescharacteristic
153C22×C42C2311yescharacteristic
154#479—216no—
155#479—216no—
156#470—216no—
157#467—216no—
158#462—216no—
159#462—216no—
160#451—44no—
161#451—44no—
162C2×#107C2314yes—
163C2×#106C2314yes—
164#424—28no—
165C2×C4⋊C8C2314yes—
166C2×C4⋊C8C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×#103C2314yes—
175#407—42no—
176C2×C22⋊C8—42no—
177C2×C22⋊C8C2312yes—
178C2×C22⋊C8C2312yes—
179C2×C22⋊C8C2312yes—
180C2×C22⋊C8C2312yes—
181#400—22no—
182#398—22no—
183#398—22no—
184#398—22no—
185#397—22no—
186#397—22no—
187#397—22no—
188#396—22no—
189#391C2312yes—
190#391C2312yes—
191#389—22no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#388—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#385—22no—
203#385—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#384C2311yescharacteristic
208#382C2311yescharacteristic
209#380—22no—
210#380C2312yes—
211#380C2312yes—
212#380C2312yes—
213#379—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25C2×D411yescharacteristic
224C22×Q8—28no—
225C22×Q8—22no—
226C22×Q8—24no—
227C22×Q8—24no—
228C22×Q8—44no—
229C22×Q8C2412yes—
230C22×Q8C2414yes—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—24no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—42no—
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×Q16—28no—
246C2×Q16—28no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×Q16—44no—
251C2×M16—24no—
252C2×M16—42no—
253C2×M16—42no—
254C22×C8—24no—
255C22×C8—42no—
256C22×C8—42no—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—216no—
260C4⋊Q8—28no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8C2×D418yes—
264#126—232no—
265#126—232no—
266#126—232no—
267#126—232no—
268#126—48no—
269#125—232no—
270#125—232no—
271#125—232no—
272#125—216no—
273#125—232no—
274#125—28no—
275#125—24no—
276#125—24no—
277#125—48no—
278#125C2×D418yes—
279C22⋊Q8—216no—
280C22⋊Q8—216no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—44no—
288C22⋊Q8C2×D4116yes—
289C4×Q8—216no—
290C4×Q8—216no—
291C4×Q8—28no—
292C4×Q8—28no—
293C4×Q8—28no—
294C4×Q8—28no—
295C4×Q8C2×D4116yes—
296#117—24no—
297#117—28no—
298#117—28no—
299#117—24no—
300#117—24no—
301#117C2×D418yes—
302C2×C4⋊C4—216no—
303C2×C4⋊C4—216no—
304C2×C4⋊C4—216no—
305C2×C4⋊C4—216no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2414yes—
334C2×C22⋊C4—216no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—216no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—28no—
362#106—28no—
363C4⋊C8—28no—
364C4⋊C8—28no—
365#103—28no—
366#103—28no—
367#103—28no—
368#103—28no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375C22⋊C8—42no—
376C22⋊C8—42no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388C24—22no—
389C24—22no—
390C24—24no—
391C24—41no—
392C24—42no—
393C24C22×D411yescharacteristic
394C24C22×D411yescharacteristic
395C24C22×D411yescharacteristic
396C2×Q8—28no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—416no—
407C2×Q8—416no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8C22×D418yes—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×D411yescharacteristic
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C22×C4C2511yescharacteristic
460Q16—88no—
461Q16—88no—
462M16—84no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—84no—
470C4⋊C4—28no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—216no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4#12018yes—
497C4⋊C4C22×D418yes—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4#12018yes—
510C42—216no—
511C42—216no—
512C42—24no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—44no—
517C42—82no—
518C42C22×D414yes—
519C42C22×D414yes—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—22no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—44no—
533C23—48no—
534C23—44no—
535C23—84no—
536C23—82no—
537C23C2×#12012yes—
538C23#53712yes—
539C23#57814yes—
540C23C23×D411yescharacteristic
541Q8—28no—
542Q8—28no—
543Q8—416no—
544Q8—44no—
545Q8—44no—
546Q8—44no—
547Q8—44no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4C2×#12014yes—
597C2×C4C2×#12014yes—
598C2×C4C2×#12012yes—
599C2×C4#53712yes—
600C2×C4C23×D412yes—
601C2×C4C2×H(5,𝔽2)11yescharacteristic
602C2×C4C2×2−1+411yescharacteristic
603C2×C4#58512yes—
604C8—44no—
605C8—44no—
606C22—24no—
607C22—22no—
608C22—24no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—42no—
613C22—41no—
614C22—41no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—41no—
619C22—41no—
620C22—88no—
621C22#326114yescomplemented
622C22C22×#12011yescharacteristic
623C22C2×#53711yescharacteristic
624C22#344911yescharacteristic
625C22C2×#57812yes—
626C22#358612yes—
627C4—22no—
628C4—22no—
629C4—24no—
630C4—24no—
631C4—41no—
632C4—48no—
633C4—42no—
634C4—42no—
635C4—48no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4#348812yes—
641C4#348912yes—
642C2—24no—
643C2—42no—
644C2—81no—
645C2#3525111yescharacteristic
646C2#6037512yes—
647C2C2×#326111yescharacteristic
648C2#6092111yescharacteristic
649C2#6098312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3261C2semidirect
4. #32960C2semidirect
5. #32906C2semidirect
6. #29701C2semidirect
7. #29674C2semidirect
8. #25512C2semidirect
11. #24307C2semidirect
12. #24212C2semidirect
13. #24165C2semidirect
14. #21898C2semidirect
32. C2×#479C22semidirect
33. C2×#479C22semidirect
36. C2×#462C22semidirect
37. C2×#462C22semidirect
56. #2059C22semidirect
57. #2043C22semidirect
60. #2026C22semidirect
61. #2016C22semidirect
64. #1989C22semidirect
65. #1981C22semidirect
68. #1878C22semidirect
69. #1878C22semidirect
72. #1865C22semidirect
73. #1863C22semidirect
76. #1805C22semidirect
77. #1803C22semidirect
621. C22#3261semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#32612
#353102
#329602
#329062
#297012
#296742
#255122
#243862
#243802
#243072
#242122
#241652
#218982
#206282