Finite Group Databasebeta

← #6519038

Subgroups — #6519038

Order 512.

2016 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#27601C212yesmaximal, complemented
4#27600C212yesmaximal, complemented
5#25024C218yesmaximal, complemented
6#24979C218yesmaximal, complemented
7#24006C212yesmaximal, complemented
8#24003C212yesmaximal, complemented
9#21986C212yesmaximal, complemented
10#21790C211yesmaximal, characteristic
11#20740C212yesmaximal, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C2×#518—21no—
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#484C2214yescomplemented
18C2×#483C2214yescomplemented
19C2×#432C2212yes—
20#2536—24no—
21#2536C2218yes—
22C2×#398—21no—
23C2×#394—21no—
24#2349—24no—
25D4×C4⋊C4C2214yes—
26D4×C4⋊C4C2214yes—
27#2316—21no—
28#2315—21no—
29#2315—21no—
30C2×#388—21no—
31C2×#388—21no—
32#2285C2212yes—
33C2×#382C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2212yes—
38#2077C2218yescomplemented
39#2075C2218yescomplemented
40#2036C2218yescomplemented
41#2035C2218yescomplemented
42#1955C2218yes—
43#1949C2218yescomplemented
44#1872C2214yescomplemented
45#1872C2214yescomplemented
46#1853C2214yes—
47#1853C2214yescomplemented
48#1852C2214yes—
49#1852C2214yescomplemented
50#1847C2214yescomplemented
51#1836C2214yescomplemented
52#1824C2214yescomplemented
53#1823C2214yes—
54#1823C2214yescomplemented
55#1803C2214yescomplemented
56#1803C2214yescomplemented
57#1787C2214yes—
58#1779C2212yes—
59#1779C2212yes—
60#1763C2218yescomplemented
61C23×Q8C2311yescharacteristic
62C23×D4D411yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—44no—
68C2×C22⋊Q8—44no—
69C2×C22⋊Q8C2314yes—
70C2×C22⋊Q8C2314yes—
71#518—48no—
72C2×C4×Q8—22no—
73C2×C4×Q8—28no—
74C2×C4×Q8C2314yes—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4—28no—
78C2×C4×D4D414yes—
79C2×C4×D4C2314yes—
80C2×#117—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
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⋊C4D411yescharacteristic
99C22×C22⋊C4D411yescharacteristic
100C22×C42—21no—
101C22×C42C2311yescharacteristic
102#484—48no—
103#483—48no—
104#432—28no—
105C2×#107C2318yes—
106C2×#106C2318yes—
107C2×C4⋊C8C2314yes—
108C2×#103C2318yes—
109C2×#103C2318yes—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115C2×#97C2314yes—
116#398—28no—
117#394—28no—
118#389—24no—
119#389—24no—
120#388—28no—
121#388—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—
155C25C2×D411yescharacteristic
156C25C2×D411yescharacteristic
157C22×Q8—24no—
158C22×Q8—21no—
159C22×Q8—24no—
160C22×Q8—42no—
161C22×Q8C2412yes—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4C2×D414yes—
166C23×C4—21no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C22×C8C2418yes—
190C22⋊Q8—28no—
191C22⋊Q8—28no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C22⋊Q8—48no—
195C22⋊Q8—48no—
196C4×Q8—28no—
197C4×Q8—28no—
198C4×Q8—48no—
199C4×D4—216no—
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—416no—
203C4×D4—416no—
204#117—48no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2414yes—
232C2×C4⋊C4C2414yes—
233C2×C4⋊C4C2412yes—
234C2×C4⋊C4C2412yes—
235C2×C4⋊C4C2412yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4C2×D414yes—
252C2×C22⋊C4C2×D414yes—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—44no—
256C2×C42C2×D412yes—
257C2×C42#4118yes—
258C2×C42C2412yes—
259#107—48no—
260#106—48no—
261C4⋊C8—28no—
262#103—216no—
263#103—48no—
264C22⋊C8—28no—
265C22⋊C8—44no—
266C22⋊C8—44no—
267C22⋊C8—44no—
268C22⋊C8—44no—
269#97—28no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24AGL(1,ℤ/8)14yes—
291C24#13714yes—
292C24C22×D412yes—
293C24C22×D412yes—
294C24C22×D411yescharacteristic
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—416no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8H(5,𝔽2)14yes—
303C2×D4—232no—
304C2×D4—28no—
305C2×D4—28no—
306C2×D4—28no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—88no—
354C22×C4—88no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4#12314yes—
364C22×C4#12314yes—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D412yes—
370C22×C4C2×#4114yes—
371C22×C4C2511yescharacteristic
372C2×C8—28no—
373C2×C8—28no—
374C2×C8—28no—
375C4⋊C4—28no—
376C4⋊C4—24no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—28no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—48no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C22⋊C4—28no—
396C22⋊C4—28no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C42—48no—
405C42—48no—
406C42#12318yes—
407C42C2×#4114yes—
408C42C2×#4114yes—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—28no—
426C23—28no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23C2×#12312yes—
440C23C2×AGL(1,ℤ/8)14yes—
441C23C2×AGL(1,ℤ/8)12yes—
442C23C2×#13714yes—
443C23C2×#13712yes—
444C23C23×D411yescharacteristic
445Q8—28no—
446Q8—48no—
447Q8—44no—
448D4—232no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—28no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—28no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—48no—
469C2×C4—48no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—164no—
493C2×C4—164no—
494C2×C4—164no—
495C2×C4—164no—
496C2×C4C2×#12314yes—
497C2×C4C2×#12312yes—
498C2×C4C2×#12314yes—
499C2×C4C22×#4112yes—
500C2×C4C2×H(5,𝔽2)12yes—
501C2×C4#58512yes—
502C8—48no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—88no—
524C22—88no—
525C22#309214yes—
526C22C22×#12311yescharacteristic
527C22#348411yescharacteristic
528C22#348511yescharacteristic
529C22C22×AGL(1,ℤ/8)11yescharacteristic
530C22C22×#13711yescharacteristic
531C22#358611yescharacteristic
532C22#358611yescharacteristic
533C4—24no—
534C4—24no—
535C4—22no—
536C4—42no—
537C4—42no—
538C4—44no—
539C4—84no—
540C4—82no—
541C4—82no—
542C4—84no—
543C4—82no—
544C4#349414yes—
545C2—24no—
546C2—44no—
547C2—44no—
548C2#3554011yescharacteristic
549C2C2×#309211yescharacteristic
550C2#6051911yescharacteristic
551C2#6157211yescharacteristic
552C2#6157311yescharacteristic
553C2#6170411yescharacteristic
554C2#6170511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
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
3. #27601C2semidirect
4. #27600C2semidirect
5. #25024C2semidirect
6. #24979C2semidirect
7. #24006C2semidirect
8. #24003C2semidirect
9. #21986C2semidirect
11. #20740C2semidirect
17. C2×#484C22semidirect
18. C2×#483C22semidirect
38. #2077C22semidirect
39. #2075C22semidirect
40. #2036C22semidirect
41. #2035C22semidirect
43. #1949C22semidirect
44. #1872C22semidirect
45. #1872C22semidirect
47. #1853C22semidirect
49. #1852C22semidirect
50. #1847C22semidirect
51. #1836C22semidirect
52. #1824C22semidirect
54. #1823C22semidirect
55. #1803C22semidirect
56. #1803C22semidirect
60. #1763C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
#341812
#276012
#276002
#250242
#249792
#240062
#240032
#219862
#217902
#207402