Finite Group Databasebeta

← #6434311

Subgroups — #6434311

Order 512.

1784 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35341C211yesmaximal, characteristic, complemented
2#35266C211yesmaximal, characteristic, complemented
3#30355C214yesmaximal, complemented
4#30349C214yesmaximal, complemented
5#30346C214yesmaximal, complemented
6#30340C214yesmaximal, complemented
7#23616C212yesmaximal, complemented
8#23553C211yesmaximal, characteristic, complemented
9#23544C212yesmaximal, complemented
10#23504C211yesmaximal, characteristic, complemented
11C2×#2025C211yesmaximal, characteristic, complemented
12#23428C211yesmaximal, characteristic
13#21235C212yesmaximal, complemented
14#21092C212yesmaximal, complemented
15#20430C211yesmaximal, characteristic, complemented
16C22×AGL(1,ℤ/8)C2212yes—
17C2×#532C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C2×#428C2212yes—
20C2×#424C2212yes—
21C2×#417C2212yes—
22C2×#416C2212yes—
23C2×#416C2212yes—
24C22×#102C2212yes—
25C2×#405C2212yes—
26#2861C2214yescomplemented
27#2860C2214yes—
28#2644C2214yes—
29#2642C2212yescomplemented
30#2641C2212yescomplemented
31#2591—24no—
32#2562—22no—
33#2562—22no—
34#2513—22no—
35#2512—22no—
36#2480—24no—
37#2317C2211yescharacteristic
38#2316C2211yescharacteristic
39#2311C2212yes—
40#2311C2212yes—
41#2309C2211yescharacteristic
42#2309C2211yescharacteristic
43#2296C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2068C2214yescomplemented
47#2067C2214yescomplemented
48#2066C2214yescomplemented
49#2065C2214yescomplemented
50#2059C2214yescomplemented
51#2059C2214yescomplemented
52#2055C2214yes—
53#2055C2214yes—
54#2054C2214yes—
55#2054C2214yes—
56#2043C2212yescomplemented
57#2042C2212yescomplemented
58#2042C2212yescomplemented
59#2041C2212yescomplemented
60#2025C2212yes—
61#2025C2212yes—
62#2025C2214yes—
63#2013C2212yes—
64#2012C2214yes—
65#2011C2212yes—
66#1973C2214yes—
67#1972C2214yes—
68#1970C2214yes—
69#1968C2214yes—
70#1936C2214yescomplemented
71#1935C2214yescomplemented
72#1934C2214yescomplemented
73#1933C2214yescomplemented
74#1768C2214yescomplemented
75#1767C2212yescomplemented
76C4×C4⋊C8C2212yescomplemented
77#1736C2212yes—
78C2×#340C2212yes—
79#1728C2211yescharacteristic
80C2×#336C2211yescharacteristic
81C22×#41C2312yes—
82C23×D4C2311yescharacteristic
83C2×AGL(1,ℤ/8)—28no—
84C2×AGL(1,ℤ/8)—44no—
85C2×AGL(1,ℤ/8)—44no—
86C22×SD16C2314yes—
87C22×D8C2314yes—
88C22×M16C2312yes—
89C23×C8C2312yes—
90#532—44no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#126—24no—
94C2×#125—22no—
95C2×#125—44no—
96C2×#125C2312yes—
97C2×#125C2312yes—
98C2×#124C2312yes—
99C2×#121—22no—
100C2×C4×D4—22no—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C4⋊C4—22no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—22no—
109C22×C42C2311yescharacteristic
110#428—44no—
111C2×#107C2314yes—
112C2×#106C2314yes—
113#424—28no—
114C2×C4⋊C8C2314yes—
115C2×C4⋊C8C2314yes—
116#417—28no—
117#416—44no—
118#416—44no—
119C2×#103C2314yes—
120C2×#103C2312yes—
121C2×#103C2312yes—
122C2×#103C2312yes—
123C2×#103C2312yes—
124C2×#102—44no—
125C2×#102C2314yes—
126C2×#102C2312yes—
127C2×#102C2312yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2312yes—
131C2×#102C2312yes—
132#405—44no—
133C2×#97C2314yes—
134C2×C4×C8C2314yes—
135#400—24no—
136#400—24no—
137#397—24no—
138#397—24no—
139#397—24no—
140#396—24no—
141#394—24no—
142#394—24no—
143#391C2314yes—
144#391C2314yes—
145#390—21no—
146#390—22no—
147#390—21no—
148#390—22no—
149#390C2314yes—
150#390C2314yes—
151#390C2314yes—
152#390C2314yes—
153#390C2312yes—
154#390C2312yes—
155#389—22no—
156#389C2312yes—
157#389C2312yes—
158#389C2314yes—
159#389C2314yes—
160#388—24no—
161#388—24no—
162#388—24no—
163#388—24no—
164#388—24no—
165#388—24no—
166#388—24no—
167#386—24no—
168#386—24no—
169#385—22no—
170#385—22no—
171#384—21no—
172#382—21no—
173#380—22no—
174#380—22no—
175#380—22no—
176#380—22no—
177#380—22no—
178#379—22no—
179#379—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4D412yes—
182C4×C4⋊C4D412yes—
183C4×C22⋊C4D414yes—
184#376C2312yes—
185C43C2312yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2312yes—
191#336C2312yes—
192#336C2311yescharacteristic
193#336C2311yescharacteristic
194#336C2312yes—
195C25—21no—
196C2×#41—24no—
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—24no—
201C2×#41—24no—
202C22×Q8C2412yes—
203C22×D4—22no—
204C22×D4—21no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4C2412yes—
211C22×D4C2412yes—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4C2×D412yes—
216C23×C4C2411yescharacteristic
217AGL(1,ℤ/8)—432no—
218C2×SD16—28no—
219C2×SD16—44no—
220C2×SD16—44no—
221C2×D8—28no—
222C2×D8—44no—
223C2×D8—44no—
224C2×M16—24no—
225C2×M16—24no—
226C2×M16#4118yes—
227C22×C8—24no—
228C22×C8—24no—
229C22×C8—24no—
230C22×C8C2412yes—
231C22×C8C2412yes—
232C22×C8C2414yes—
233C4⋊Q8—42no—
234#127—42no—
235#126—88no—
236#125—44no—
237#125—44no—
238#125—48no—
239#124—42no—
240#124—42no—
241#121—48no—
242C4×D4—48no—
243#117—24no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—24no—
248#117—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—84no—
284C2×C22⋊C4C2×D412yes—
285C2×C22⋊C4C2×D412yes—
286C2×C42—24no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42C2×D412yes—
290C2×C42C2×D414yes—
291C2×C42C2×D414yes—
292C2×C42C2412yes—
293C2×C42C2412yes—
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296#107—44no—
297#106—44no—
298C4⋊C8—28no—
299C4⋊C8—28no—
300#103—28no—
301#103—42no—
302#103—42no—
303#103—42no—
304#103—42no—
305#102—28no—
306#102—44no—
307#102—44no—
308#102—42no—
309#102—42no—
310#102—42no—
311#102—42no—
312#97—44no—
313C4×C8—44no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95C2×D414yes—
323#95C2×D414yes—
324C24—22no—
325C24—22no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—82no—
330C24—82no—
331C24C22×D411yescharacteristic
332#41—48no—
333#41—48no—
334#41—48no—
335#41—48no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—82no—
362C2×D4—82no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4C22×D412yes—
380C22×C4C22×D414yes—
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×D412yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C2×#4114yes—
386C22×C4H(5,𝔽2)14yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C42−1+412yes—
389C22×C4C2511yescharacteristic
390SD16—416no—
391D8—416no—
392M16—216no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—48no—
402C2×C8C2×#4118yes—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—82no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—48no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C42—24no—
439C42—24no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—44no—
445C42—44no—
446C42—42no—
447C42—42no—
448C23—22no—
449C23—22no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—88no—
463C23—84no—
464C23—162no—
465C23—162no—
466C23—164no—
467C23#53212yes—
468C23#53512yes—
469C23#53712yes—
470C23C23×D411yescharacteristic
471Q8—42no—
472Q8—42no—
473Q8—42no—
474Q8—42no—
475D4—42no—
476D4—44no—
477D4—42no—
478D4—44no—
479D4—42no—
480D4—42no—
481D4—88no—
482D4—88no—
483D4—84no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—84no—
512C2×C4—88no—
513C2×C4—88no—
514C2×C4—162no—
515C2×C4#53212yes—
516C2×C4#53512yes—
517C2×C4#53712yes—
518C2×C4#53818yes—
519C2×C4#54218yes—
520C2×C4C22×#4112yes—
521C2×C4C2×H(5,𝔽2)11yescharacteristic
522C2×C4C2×H(5,𝔽2)12yes—
523C2×C4C2×2−1+411yescharacteristic
524C8—28no—
525C8—42no—
526C8—42no—
527C22—24no—
528C22—22no—
529C22—22no—
530C22—24no—
531C22—82no—
532C22—82no—
533C22—84no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—84no—
543C22—168no—
544C22#266214yes—
545C22C2×#53211yescharacteristic
546C22C2×#53511yescharacteristic
547C22C2×#53711yescharacteristic
548C22#358412yes—
549C22#358512yes—
550C4—44no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4—82no—
559C4—84no—
560C4#266214yes—
561C4C2×#53812yes—
562C4C2×#54212yes—
563C2—24no—
564C2—162no—
565C2—162no—
566C2C2×#266211yescharacteristic
567C2#6072411yescharacteristic
568C2#6072511yescharacteristic
569C2#6081512yes—
570C2#6093312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23428C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35341C2semidirect
2. #35266C2semidirect
3. #30355C2semidirect
4. #30349C2semidirect
5. #30346C2semidirect
6. #30340C2semidirect
7. #23616C2semidirect
8. #23553C2semidirect
9. #23544C2semidirect
10. #23504C2semidirect
11. C2×#2025C2semidirect
13. #21235C2semidirect
14. #21092C2semidirect
15. #20430C2semidirect
26. #2861C22semidirect
29. #2642C22semidirect
30. #2641C22semidirect
46. #2068C22semidirect
47. #2067C22semidirect
48. #2066C22semidirect
49. #2065C22semidirect
50. #2059C22semidirect
51. #2059C22semidirect
56. #2043C22semidirect
57. #2042C22semidirect
58. #2042C22semidirect
59. #2041C22semidirect
70. #1936C22semidirect
71. #1935C22semidirect
72. #1934C22semidirect
73. #1933C22semidirect
74. #1768C22semidirect
75. #1767C22semidirect
76. C4×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#353412
#352662
#303552
#303492
#303462
#303402
#236162
#235532
#235442
#235042
C2×#20252
#234282
#212352
#210922
#204302