Finite Group Databasebeta

← #6519037

Subgroups — #6519037

Order 512.

2232 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2535C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic, complemented
3#27603C212yesmaximal, complemented
4#27601C212yesmaximal, complemented
5#25022C218yesmaximal, complemented
6#24977C218yesmaximal, complemented
7#24005C212yesmaximal, complemented
8#24002C212yesmaximal, complemented
9#21986C212yesmaximal, complemented
10#21790C211yesmaximal, characteristic, complemented
11#20738C212yesmaximal, complemented
12C22×#121C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C2×#518—21no—
15C22×C4×D4C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#481C2214yescomplemented
18C2×#480C2214yescomplemented
19C2×#432C2212yes—
20#2535—24no—
21#2535C2218yes—
22C2×#397—21no—
23C2×#392—21no—
24#2349—24no—
25D4×C4⋊C4C2214yescomplemented
26D4×C4⋊C4C2214yescomplemented
27#2316—21no—
28#2315—21no—
29#2315—21no—
30C2×#386—21no—
31C2×#386—21no—
32#2285C2212yes—
33C2×#382C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2212yes—
38#2077C2218yescomplemented
39#2076C2218yescomplemented
40#2034C2218yescomplemented
41#2033C2218yescomplemented
42#1955C2218yescomplemented
43#1949C2218yescomplemented
44#1874C2214yescomplemented
45#1874C2214yescomplemented
46#1853C2214yes—
47#1853C2214yescomplemented
48#1852C2214yes—
49#1852C2214yescomplemented
50#1847C2214yescomplemented
51#1836C2214yescomplemented
52#1824C2214yes—
53#1824C2214yescomplemented
54#1823C2214yescomplemented
55#1802C2214yescomplemented
56#1802C2214yescomplemented
57#1786C2214yes—
58#1779C2212yes—
59#1779C2212yes—
60#1762C2218yescomplemented
61C23×D4—21no—
62C23×D4—21no—
63C23×D4D411yescharacteristic
64C23×D4C2311yescharacteristic
65C24×C4—21no—
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C2×#121—22no—
69C2×#121—22no—
70C2×#121—44no—
71C2×#121—44no—
72C2×#121C2314yes—
73C2×#121C2314yes—
74#518—48no—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4—22no—
78C2×C4×D4—28no—
79C2×C4×D4—28no—
80C2×C4×D4D414yes—
81C2×C4×D4C2314yes—
82C2×C4×D4C2314yes—
83C2×#117—22no—
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—22no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4D411yescharacteristic
100C22×C22⋊C4D411yescharacteristic
101C22×C42C2311yescharacteristic
102#481—48no—
103#480—48no—
104#432—28no—
105C2×#107C2318yes—
106C2×#106C2318yes—
107C2×C4⋊C8C2314yes—
108C2×#102C2318yes—
109C2×#102C2318yes—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×#97C2314yes—
116#397—28no—
117#392—28no—
118#389—24no—
119#389—24no—
120#386—28no—
121#386—28no—
122#385—24no—
123#385—24no—
124#385—24no—
125#385—24no—
126#384C2312yes—
127#384C2312yes—
128#382—22no—
129#382—22no—
130#382C2314yes—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380C2314yes—
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—
149C2×#95—21no—
150C2×#95—21no—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155C25—21no—
156C25C2×D411yescharacteristic
157C25C2×D411yescharacteristic
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—21no—
165C22×D4—24no—
166C22×D4—42no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—82no—
172C22×D4—82no—
173C22×D4C2×D414yes—
174C22×D4C2412yes—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—82no—
191C23×C4—82no—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2×D411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C22×C8C2418yes—
202#121—28no—
203#121—28no—
204#121—44no—
205#121—44no—
206#121—48no—
207#121—48no—
208C4×D4—216no—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—28no—
212C4×D4—28no—
213C4×D4—416no—
214C4×D4—416no—
215C4×D4—48no—
216#117—48no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2414yes—
238C2×C4⋊C4C2414yes—
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2412yes—
242C2×C22⋊C4—24no—
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—48no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4C2×D414yes—
260C2×C22⋊C4C2×D414yes—
261C2×C42—24no—
262C2×C42C2×D412yes—
263C2×C42#4118yes—
264C2×C42C2412yes—
265#107—48no—
266#106—48no—
267C4⋊C8—28no—
268#102—216no—
269#102—48no—
270C22⋊C8—28no—
271C22⋊C8—44no—
272C22⋊C8—44no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275#97—28no—
276#95—24no—
277#95—24no—
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—24no—
289C24—22no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24AGL(1,ℤ/8)14yes—
305C24#13714yes—
306C24C22×D412yes—
307C24C22×D412yes—
308C24C22×D411yescharacteristic
309C2×D4—232no—
310C2×D4—28no—
311C2×D4—28no—
312C2×D4—24no—
313C2×D4—28no—
314C2×D4—24no—
315C2×D4—22no—
316C2×D4—24no—
317C2×D4—416no—
318C2×D4—42no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—42no—
324C2×D4—84no—
325C2×D4—84no—
326C2×D4—816no—
327C2×D4—816no—
328C2×D4H(5,𝔽2)14yes—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
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—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—48no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—816no—
385C22×C4#12314yes—
386C22×C4#12314yes—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C2×#4114yes—
393C22×C4C2511yescharacteristic
394C2×C8—28no—
395C2×C8—28no—
396C2×C8—28no—
397C4⋊C4—28no—
398C4⋊C4—24no—
399C4⋊C4—28no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—28no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C22⋊C4—28no—
415C22⋊C4—28no—
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—48no—
425C42#12318yes—
426C42C2×#4114yes—
427C42C2×#4114yes—
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—24no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—48no—
452C23—48no—
453C23—48no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23C2×#12312yes—
473C23C2×AGL(1,ℤ/8)14yes—
474C23C2×AGL(1,ℤ/8)12yes—
475C23C2×#13714yes—
476C23C2×#13712yes—
477C23C23×D411yescharacteristic
478D4—232no—
479D4—28no—
480D4—48no—
481D4—44no—
482D4—816no—
483D4—816no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—28no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—816no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—164no—
525C2×C4—164no—
526C2×C4—164no—
527C2×C4—164no—
528C2×C4C2×#12312yes—
529C2×C4C2×#12314yes—
530C2×C4C2×#12314yes—
531C2×C4C22×#4112yes—
532C2×C4C2×H(5,𝔽2)12yes—
533C2×C4#58512yes—
534C8—48no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—88no—
559C22—88no—
560C22—816no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22#309214yes—
566C22C22×#12311yescharacteristic
567C22#348411yescharacteristic
568C22#348511yescharacteristic
569C22C22×AGL(1,ℤ/8)11yescharacteristic
570C22C22×#13711yescharacteristic
571C22#358511yescharacteristic
572C22#358511yescharacteristic
573C4—24no—
574C4—24no—
575C4—22no—
576C4—42no—
577C4—42no—
578C4—84no—
579C4—82no—
580C4—82no—
581C4—84no—
582C4—82no—
583C4#349414yes—
584C2—24no—
585C2—44no—
586C2—44no—
587C2—84no—
588C2#3554011yescharacteristic
589C2C2×#309211yescharacteristic
590C2#6051811yescharacteristic
591C2#6156911yescharacteristic
592C2#6157011yescharacteristic
593C2#6170711yescharacteristic
594C2#6170811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2535C2
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. #27603C2semidirect
4. #27601C2semidirect
5. #25022C2semidirect
6. #24977C2semidirect
7. #24005C2semidirect
8. #24002C2semidirect
9. #21986C2semidirect
10. #21790C2semidirect
11. #20738C2semidirect
17. C2×#481C22semidirect
18. C2×#480C22semidirect
25. D4×C4⋊C4C22semidirect
26. D4×C4⋊C4C22semidirect
38. #2077C22semidirect
39. #2076C22semidirect
40. #2034C22semidirect
41. #2033C22semidirect
42. #1955C22semidirect
43. #1949C22semidirect
44. #1874C22semidirect
45. #1874C22semidirect
47. #1853C22semidirect
49. #1852C22semidirect
50. #1847C22semidirect
51. #1836C22semidirect
53. #1824C22semidirect
54. #1823C22semidirect
55. #1802C22semidirect
56. #1802C22semidirect
60. #1762C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25352
#341812
#276032
#276012
#250222
#249772
#240052
#240022
#219862
#217902
#207382