Finite Group Databasebeta

← #6472775

Subgroups — #6472775

Order 512.

3358 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3#32983C214yesmaximal, complemented
4#29895C214yesmaximal, complemented
5#27603C214yesmaximal, complemented
6#24552C218yesmaximal, complemented
7#24006C214yesmaximal, complemented
8#21940C212yesmaximal, complemented
9#21759C212yesmaximal, complemented
10C2×#1779C211yesmaximal, characteristic
11C25×C4C2211yescharacteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C23×C4⋊C4C2212yes—
15C23×C22⋊C4—21no—
16C2×#484C2218yescomplemented
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2214yes—
19#2848—21no—
20#2848—22no—
21#2848—21no—
22#2848C2212yes—
23#2848C2212yes—
24#2848C2214yes—
25C2×#385—22no—
26C2×#385—21no—
27C2×#380—22no—
28C2×#380—21no—
29C2×#380—22no—
30C2×#380—21no—
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2214yes—
35C2×#380C2212yes—
36C2×#380C2214yes—
37C2×#379—21no—
38C22×#95—21no—
39#2076C22116yescomplemented
40#2036C22116yescomplemented
41#1892C2218yescomplemented
42#1872C2218yescomplemented
43#1872C2218yescomplemented
44#1852C2218yescomplemented
45#1852C2218yescomplemented
46#1852C2218yescomplemented
47#1824C2214yescomplemented
48#1824C2218yescomplemented
49#1808C2218yescomplemented
50#1787C2214yescomplemented
51#1779C2214yes—
52#1779C2212yes—
53#1779C2212yes—
54C26D411yescharacteristic
55C23×Q8C2311yescharacteristic
56C24×C4—28no—
57C24×C4—22no—
58C24×C4—48no—
59C24×C4D411yescharacteristic
60C24×C4D411yescharacteristic
61C24×C4D412yes—
62C24×C4D411yescharacteristic
63C24×C4D412yes—
64C24×C4C2311yescharacteristic
65C24×C4C2312yes—
66C24×C4C2312yes—
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C23×C8C2314yes—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—48no—
76C2×C22⋊Q8C2314yes—
77C2×C22⋊Q8C2312yes—
78C2×C22⋊Q8C2312yes—
79C2×C22⋊Q8C2314yes—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—48no—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—22no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—22no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—48no—
100#484—416no—
101C2×#107C23116yescomplemented
102C2×#103C23116yescomplemented
103C2×C22⋊C8—28no—
104C2×C22⋊C8—48no—
105C2×C22⋊C8C2318yes—
106C2×C22⋊C8C2314yes—
107C2×C22⋊C8C2314yes—
108C2×C22⋊C8C2318yes—
109#385—24no—
110#385—22no—
111#385—24no—
112#385—24no—
113#385—24no—
114#385—22no—
115#385—24no—
116#380—22no—
117#380—22no—
118#380—21no—
119#380—21no—
120#380—24no—
121#380—24no—
122#380—24no—
123#380—24no—
124#380—24no—
125#380—22no—
126#380—22no—
127#380C2312yes—
128#380C2314yes—
129#380C2314yes—
130#380C2318yes—
131#380C2314yes—
132#380C2314yes—
133#380C2312yes—
134#379—22no—
135#379—24no—
136#379—22no—
137C2×#95—24no—
138C2×#95—24no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—21no—
143#340C23116yescomplemented
144#340C23116yescomplemented
145C25—24no—
146C25—24no—
147C25—44no—
148C25—44no—
149C25Q1618yes—
150C25C2×D411yescharacteristic
151C25C2×D412yes—
152C25C2×D412yes—
153C25C2×D411yescharacteristic
154C25C2×D411yescharacteristic
155C22×Q8—21no—
156C22×Q8—24no—
157C22×Q8—44no—
158C22×Q8C2412yes—
159C23×C4—24no—
160C23×C4—28no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—28no—
164C23×C4—28no—
165C23×C4—24no—
166C23×C4—28no—
167C23×C4—28no—
168C23×C4—24no—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—464no—
189C23×C4—48no—
190C23×C4—48no—
191C23×C4—48no—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2×D412yes—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D412yes—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4#4118yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C22×C8—28no—
212C22×C8—48no—
213C22×C8C2418yes—
214C22⋊Q8—28no—
215C22⋊Q8—28no—
216C22⋊Q8—216no—
217C22⋊Q8—48no—
218C22⋊Q8—48no—
219C22⋊Q8—48no—
220C22⋊Q8—44no—
221C22⋊Q8—44no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—216no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—48no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C2418yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2414yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—416no—
258C2×C22⋊C4—416no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—48no—
263#107—416no—
264#103—416no—
265C22⋊C8—28no—
266C22⋊C8—28no—
267C22⋊C8—216no—
268C22⋊C8—48no—
269#95—28no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—22no—
274#95—22no—
275#95—24no—
276C24—22no—
277C24—24no—
278C24—22no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—22no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—22no—
288C24—24no—
289C24—24no—
290C24—232no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—432no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—416no—
301C24—432no—
302C24—416no—
303C24#12314yes—
304C24C2×Q1614yes—
305C24C2×Q1618yes—
306C24C2×Q1618yes—
307C24C2×Q1614yes—
308C24C2×Q1614yes—
309C24AGL(1,ℤ/8)14yes—
310C24C22×D412yes—
311C24C22×D411yescharacteristic
312C24C22×D411yescharacteristic
313C24C22×D411yescharacteristic
314C24C22×D412yes—
315C2×Q8—24no—
316C2×Q8—28no—
317C2×Q8—44no—
318C2×Q8—42no—
319C2×Q8—88no—
320C22×C4—232no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—22no—
352C22×C4—416no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—432no—
366C22×C4—432no—
367C22×C4—416no—
368C22×C4—432no—
369C22×C4—432no—
370C22×C4—416no—
371C22×C4—416no—
372C22×C4—416no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—832no—
390C22×C4#12314yes—
391C22×C4#12314yes—
392C22×C4#12314yes—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C2×#4114yes—
399C22×C4C2×#4114yes—
400C22×C4C2×#4114yes—
401C22×C4H(5,𝔽2)18yes—
402C22×C4C2511yescharacteristic
403C2×C8—28no—
404C2×C8—216no—
405C2×C8—88no—
406C4⋊C4—24no—
407C4⋊C4—216no—
408C4⋊C4—28no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—44no—
412C4⋊C4—48no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—416no—
420C22⋊C4—48no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—48no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C23—28no—
430C23—216no—
431C23—28no—
432C23—28no—
433C23—22no—
434C23—216no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—216no—
445C23—216no—
446C23—28no—
447C23—22no—
448C23—216no—
449C23—216no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—44no—
456C23—44no—
457C23—48no—
458C23—48no—
459C23—48no—
460C23—48no—
461C23—48no—
462C23—48no—
463C23—48no—
464C23—48no—
465C23—48no—
466C23—48no—
467C23—48no—
468C23—48no—
469C23—48no—
470C23—44no—
471C23—4128no—
472C23—48no—
473C23—44no—
474C23#484116yescomplemented
475C23C2×#12312yes—
476C23C2×#12312yes—
477C23C2×#12312yes—
478C23#53414yes—
479C23C22×Q1614yes—
480C23C22×Q1612yes—
481C23C22×Q1612yes—
482C23C22×Q1612yes—
483C23C22×Q1614yes—
484C23C2×AGL(1,ℤ/8)12yes—
485C23C2×AGL(1,ℤ/8)12yes—
486C23C2×AGL(1,ℤ/8)12yes—
487C23#57514yes—
488C23C23×D411yescharacteristic
489Q8—48no—
490Q8—84no—
491C2×C4—216no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—216no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—44no—
516C2×C4—48no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—48no—
527C2×C4—88no—
528C2×C4—84no—
529C2×C4—816no—
530C2×C4—816no—
531C2×C4—816no—
532C2×C4—84no—
533C2×C4C2×#12312yes—
534C2×C4C2×#12312yes—
535C2×C4C2×#12312yes—
536C2×C4#53414yes—
537C2×C4C22×#4112yes—
538C2×C4C2×H(5,𝔽2)14yes—
539C8—48no—
540C22—232no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—22no—
555C22—22no—
556C22—24no—
557C22—22no—
558C22—22no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—416no—
563C22—432no—
564C22—432no—
565C22—416no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22C2×#48414yes—
571C22C2×#48414yes—
572C22C2×#48414yes—
573C22#318918yes—
574C22C22×#12311yescharacteristic
575C22C2×#53412yes—
576C22C23×Q1611yescharacteristic
577C22C22×AGL(1,ℤ/8)11yescharacteristic
578C22C2×#57512yes—
579C4—22no—
580C4—24no—
581C4—22no—
582C4—22no—
583C4—42no—
584C4—84no—
585C4—88no—
586C4—82no—
587C4—82no—
588C4#349014yes—
589C2—24no—
590C2—28no—
591C2—24no—
592C2—24no—
593C2—44no—
594C2#3499211yescharacteristic
595C2C22×#48411yescharacteristic
596C2#6051111yescharacteristic
597C2C2×#318912yes—
598C2#6075012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32983C2semidirect
4. #29895C2semidirect
5. #27603C2semidirect
6. #24552C2semidirect
7. #24006C2semidirect
8. #21940C2semidirect
9. #21759C2semidirect
16. C2×#484C22semidirect
39. #2076C22semidirect
40. #2036C22semidirect
41. #1892C22semidirect
42. #1872C22semidirect
43. #1872C22semidirect
44. #1852C22semidirect
45. #1852C22semidirect
46. #1852C22semidirect
47. #1824C22semidirect
48. #1824C22semidirect
49. #1808C22semidirect
50. #1787C22semidirect
101. C2×#107C23semidirect
102. C2×#103C23semidirect
143. #340C23semidirect
144. #340C23semidirect
474. C23#484semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C22×#3802
#329832
#298952
#276032
#245522
#240062
#219402
#217592
C2×#17792