Finite Group Databasebeta

← #6500817

Subgroups — #6500817

Order 512.

2198 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2414C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic, complemented
3#27614C212yesmaximal, complemented
4#27601C212yesmaximal, complemented
5#27585C212yesmaximal, complemented
6#26214C218yesmaximal, complemented
7#26142C218yesmaximal, complemented
8#24002C212yesmaximal, complemented
9#21764C212yesmaximal, complemented
10#21742C211yesmaximal, characteristic, complemented
11#20846C212yesmaximal, complemented
12C22×#121C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C2×#518—21no—
15C22×C4×D4C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C2×#480C2214yescomplemented
18C2×#428C2214yescomplemented
19C2×#407C2212yes—
20#2414—24no—
21#2414C2218yes—
22C2×#394—21no—
23C2×#392—21no—
24#2349—24no—
25D4×C4⋊C4C2214yescomplemented
26D4×C4⋊C4C2214yescomplemented
27#2316—21no—
28#2315—21no—
29#2315—21no—
30C2×#390—21no—
31C2×#390C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#388—21no—
34#2285C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38#2100C2214yescomplemented
39#2094C2214yescomplemented
40#2083C2214yescomplemented
41#2077C2218yescomplemented
42#2071C2214yescomplemented
43#2033C2218yescomplemented
44#1926C2218yescomplemented
45#1921C2218yescomplemented
46#1895C2218yescomplemented
47#1874C2214yescomplemented
48#1874C2214yescomplemented
49#1874C2214yescomplemented
50#1870C2218yescomplemented
51#1854C2218yescomplemented
52#1853C2214yes—
53#1853C2214yescomplemented
54#1852C2214yes—
55#1824C2214yes—
56#1823C2214yescomplemented
57#1817C2212yes—
58#1786C2214yes—
59#1779C2212yes—
60#1736C2214yescomplemented
61C23×D4—21no—
62C23×D4—21no—
63C23×D4D411yescharacteristic
64C23×D4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C22×M16C2314yes—
68C2×#121—22no—
69C2×#121—22no—
70C2×#121—28no—
71C2×#121—44no—
72C2×#121C2314yes—
73C2×#121C2314yes—
74#518—48no—
75C2×C4×D4—28no—
76C2×C4×D4—24no—
77C2×C4×D4—24no—
78C2×C4×D4D414yes—
79C2×C4×D4C2314yes—
80C2×#117—22no—
81C2×#117—22no—
82C2×#117—44no—
83C2×#117C2314yes—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4D411yescharacteristic
102C22×C22⋊C4D411yescharacteristic
103C22×C42—21no—
104C22×C42C2311yescharacteristic
105#480—48no—
106#428—48no—
107C2×#107C2318yes—
108C2×#106C2318yes—
109C2×#102C2318yes—
110C2×#102C2318yes—
111#407—28no—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×C22⋊C8C2314yes—
116#394—28no—
117#392—28no—
118#390—28no—
119#390—22no—
120#390C2314yes—
121#389—24no—
122#389—24no—
123#389—22no—
124#389C2314yes—
125#388—28no—
126#385—24no—
127#385—24no—
128#385—24no—
129#385—24no—
130#384C2312yes—
131#384C2312yes—
132#382—22no—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380—22no—
139#380C2314yes—
140#380C2314yes—
141#380C2314yes—
142#380C2314yes—
143C4×C4⋊C4—22no—
144C4×C4⋊C4C2314yes—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—21no—
148C2×#95—21no—
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#336C2314yes—
154#336C2314yes—
155C25—21no—
156C25C2×D411yescharacteristic
157C25C2×D411yescharacteristic
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—21no—
164C22×D4—24no—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—82no—
171C22×D4—82no—
172C22×D4—82no—
173C22×D4C2×D414yes—
174C22×D4C2412yes—
175C23×C4—21no—
176C23×C4—24no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4#4118yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C2×M16—28no—
198C2×M16—44no—
199C2×M16—44no—
200C22×C8C2418yes—
201#121—28no—
202#121—28no—
203#121—28no—
204#121—44no—
205#121—48no—
206#121—48no—
207C4×D4—216no—
208C4×D4—28no—
209C4×D4—28no—
210C4×D4—416no—
211C4×D4—416no—
212#117—28no—
213#117—44no—
214#117—48no—
215#117—48no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2414yes—
239C2×C4⋊C4C2414yes—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2412yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—82no—
261C2×C22⋊C4—82no—
262C2×C22⋊C4C2×D414yes—
263C2×C22⋊C4C2×D414yes—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—44no—
268C2×C42C2×D412yes—
269C2×C42C2412yes—
270#107—216no—
271#106—48no—
272#102—48no—
273#102—48no—
274C22⋊C8—28no—
275C22⋊C8—28no—
276C22⋊C8—44no—
277C22⋊C8—44no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24#12314yes—
305C24AGL(1,ℤ/8)14yes—
306C24C22×D412yes—
307C24C22×D412yes—
308C24C22×D411yescharacteristic
309C2×D4—232no—
310C2×D4—28no—
311C2×D4—28no—
312C2×D4—28no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—48no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—42no—
322C2×D4—88no—
323C2×D4—84no—
324C2×D4—84no—
325C2×D4—84no—
326C2×D4—816no—
327C2×D4—816no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—416no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
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—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—88no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4#12314yes—
379C22×C4#12318yes—
380C22×C4#13714yes—
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C2×#4114yes—
387C22×C4C2×#4114yes—
388C22×C4C2×#4114yes—
389C22×C4C2511yescharacteristic
390M16—416no—
391C2×C8—28no—
392C2×C8—28no—
393C2×C8—28no—
394C4⋊C4—24no—
395C4⋊C4—28no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—28no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—84no—
412C4⋊C4H(5,𝔽2)14yes—
413C22⋊C4—28no—
414C22⋊C4—28no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—48no—
422C22⋊C4—84no—
423C22⋊C4—84no—
424C42—24no—
425C42—24no—
426C42—48no—
427C42—48no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—28no—
445C23—28no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—84no—
466C23—84no—
467C23—84no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23C2×#12314yes—
473C23C2×#12312yes—
474C23C2×AGL(1,ℤ/8)14yes—
475C23C2×AGL(1,ℤ/8)12yes—
476C23C2×#13712yes—
477C23C23×D411yescharacteristic
478D4—232no—
479D4—48no—
480D4—84no—
481D4—816no—
482D4—816no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—48no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—164no—
524C2×C4—164no—
525C2×C4—164no—
526C2×C4—164no—
527C2×C4C2×#12312yes—
528C2×C4C2×#12314yes—
529C2×C4C2×#13714yes—
530C2×C4C22×#4112yes—
531C2×C4C2×H(5,𝔽2)12yes—
532C2×C4#58512yes—
533C8—48no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—88no—
555C22—88no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—168no—
564C22#309214yes—
565C22C22×#12311yescharacteristic
566C22#348411yescharacteristic
567C22#348511yescharacteristic
568C22C22×AGL(1,ℤ/8)11yescharacteristic
569C22C22×#13711yescharacteristic
570C22#358511yescharacteristic
571C22#358511yescharacteristic
572C4—22no—
573C4—24no—
574C4—24no—
575C4—42no—
576C4—42no—
577C4—44no—
578C4—84no—
579C4—82no—
580C4—84no—
581C4—82no—
582C4#350514yes—
583C2—24no—
584C2—44no—
585C2—44no—
586C2—162no—
587C2#3535111yescharacteristic
588C2C2×#309211yescharacteristic
589C2#6051811yescharacteristic
590C2#6156911yescharacteristic
591C2#6158811yescharacteristic
592C2#6170711yescharacteristic
593C2#6172111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2414C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34181C2semidirect
3. #27614C2semidirect
4. #27601C2semidirect
5. #27585C2semidirect
6. #26214C2semidirect
7. #26142C2semidirect
8. #24002C2semidirect
9. #21764C2semidirect
10. #21742C2semidirect
11. #20846C2semidirect
17. C2×#480C22semidirect
18. C2×#428C22semidirect
25. D4×C4⋊C4C22semidirect
26. D4×C4⋊C4C22semidirect
38. #2100C22semidirect
39. #2094C22semidirect
40. #2083C22semidirect
41. #2077C22semidirect
42. #2071C22semidirect
43. #2033C22semidirect
44. #1926C22semidirect
45. #1921C22semidirect
46. #1895C22semidirect
47. #1874C22semidirect
48. #1874C22semidirect
49. #1874C22semidirect
50. #1870C22semidirect
51. #1854C22semidirect
53. #1853C22semidirect
56. #1823C22semidirect
60. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24142
#341812
#276142
#276012
#275852
#262142
#261422
#240022
#217642
#217422
#208462