Finite Group Databasebeta

← #6481828

Subgroups — #6481828

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3123C212yesmaximal
2#37387C211yesmaximal, characteristic, complemented
3#34255C211yesmaximal, characteristic
4#33117C212yesmaximal, complemented
5#33114C212yesmaximal, complemented
6#29723C212yesmaximal
7#29714C212yesmaximal
8#24690C214yesmaximal, complemented
9#24686C214yesmaximal, complemented
10#24646C214yesmaximal, complemented
11#24630C214yesmaximal, complemented
12#21957C211yesmaximal, characteristic, complemented
13#21873C212yesmaximal, complemented
14C2×#541C2212yes—
15C2×#518C2211yescharacteristic
16#3123—216no—
17C2×#485C2214yes—
18C2×#478C2214yes—
19C2×#461C2214yes—
20C2×#450C2214yes—
21C2×#440C2214yes—
22C2×C4×SD16C2214yes—
23C2×#433C2212yes—
24#2857C2212yes—
25#2726—22no—
26#2726—22no—
27#2722—24no—
28#2621C2212yes—
29#2565C2214yescomplemented
30#2546C2214yescomplemented
31#2357C2214yes—
32#2354C2214yes—
33#2342—24no—
34#2319—22no—
35#2317C2211yescharacteristic
36#2309C2211yescharacteristic
37#2296C2211yescharacteristic
38#2293—22no—
39C2×#390C2211yescharacteristic
40#2274C2211yescharacteristic
41C4×#117C2211yescharacteristic
42#2073C2212yes—
43#2073C2212yes—
44#2071C2214yes—
45#2055C2214yes—
46#2055C2214yes—
47#2055C2214yes—
48#2055C2214yes—
49#2013C2214yes—
50#2013C2212yes—
51#2013C2212yes—
52#1992C2212yescomplemented
53#1991C2212yescomplemented
54#1981C2212yescomplemented
55#1981C2214yescomplemented
56#1981C2212yescomplemented
57#1960C2212yescomplemented
58#1956C2212yescomplemented
59#1949C2214yescomplemented
60#1948C2212yescomplemented
61#1947C2212yescomplemented
62#1940C2214yescomplemented
63#1939C2214yescomplemented
64#1928C2214yescomplemented
65#1928C2214yescomplemented
66#1927C2214yescomplemented
67#1927C2214yescomplemented
68#1873C2214yes—
69#1872C2214yes—
70#1805C2214yescomplemented
71#1803C2214yescomplemented
72#1798C2212yes—
73#1797C2212yes—
74C23×D4C2311yescharacteristic
75C22×SD16C2314yes—
76C22×SD16C2314yes—
77#541—28no—
78#541—28no—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2312yes—
82C2×#126—24no—
83C2×#125—24no—
84C2×#125C2312yes—
85C2×C22⋊Q8C2314yes—
86#518—24no—
87#518D418yes—
88C2×C4×Q8C2314yes—
89C2×C4×D4—22no—
90C2×C4×D4C2314yes—
91C2×#117—24no—
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4—22no—
98C22×C22⋊C4—22no—
99C22×C42C2311yescharacteristic
100#485—216no—
101#478—216no—
102#461—216no—
103#450—216no—
104#440—216no—
105C4×SD16—216no—
106#433—28no—
107C2×#107C2314yes—
108C2×#106C2314yes—
109C2×C4⋊C8C2314yes—
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2312yes—
112C2×C4⋊C8C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×C22⋊C8C2314yes—
122C2×#97C2312yes—
123C2×C4×C8C2312yes—
124#398—24no—
125#398—24no—
126#398—24no—
127#398—24no—
128#398—24no—
129#397—24no—
130#391C2314yes—
131#391C2314yes—
132#390—22no—
133#390—21no—
134#390—21no—
135#390C2312yes—
136#390C2312yes—
137#390C2314yes—
138#389—22no—
139#389—24no—
140#389—24no—
141#389—22no—
142#389—22no—
143#389D412yes—
144#389D412yes—
145#388—24no—
146#388—24no—
147#388—24no—
148#388—24no—
149#387—24no—
150#387—24no—
151#385—22no—
152#385D414yes—
153#384C2312yes—
154#384C2312yes—
155#382C2312yes—
156#382C2312yes—
157#381—24no—
158#380—22no—
159#380—22no—
160#380—24no—
161#379—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4C2312yes—
164C4×C4⋊C4C2314yes—
165C4×C4⋊C4C2312yes—
166C4×C4⋊C4C2314yes—
167C4×C22⋊C4—22no—
168C4×C22⋊C4—22no—
169C4×C22⋊C4—24no—
170#376—21no—
171C43—21no—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#336C2314yes—
177#336C2312yes—
178#336C2312yes—
179C25—21no—
180C22×Q8C2414yes—
181C22×D4—21no—
182C22×D4—22no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4C2×D414yes—
186C22×D4C2412yes—
187C23×C4—24no—
188C23×C4—21no—
189C23×C4C2×D412yes—
190C23×C4C2411yescharacteristic
191C2×SD16—28no—
192C2×SD16—28no—
193C2×SD16—44no—
194C2×SD16—44no—
195C2×SD16—44no—
196C2×SD16—44no—
197C22×C8C2414yes—
198C22×C8C2414yes—
199C4⋊Q8—24no—
200C4⋊Q8—24no—
201C4⋊Q8—42no—
202C4⋊Q8—42no—
203C4⋊Q8—44no—
204#126—416no—
205#125—24no—
206#125—24no—
207#125—416no—
208C22⋊Q8—28no—
209C22⋊Q8—28no—
210C4×Q8—28no—
211C4×Q8—28no—
212C4×D4—216no—
213C4×D4—28no—
214C4×D4C2×D4116yes—
215#117—22no—
216#117—42no—
217#117—42no—
218#117—88no—
219#117C2×D414yes—
220#117C2×D414yes—
221#117#4114yes—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2411yescharacteristic
240C2×C4⋊C4C2411yescharacteristic
241C2×C4⋊C4C2414yes—
242C2×C4⋊C4C2412yes—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—216no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—84no—
253C2×C22⋊C4C2×D412yes—
254C2×C22⋊C4C2×D412yes—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—42no—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2412yes—
265C2×C42C2412yes—
266C2×C42C2412yes—
267#107—28no—
268#106—28no—
269C4⋊C8—28no—
270C4⋊C8—42no—
271C4⋊C8—42no—
272C4⋊C8—44no—
273#103—28no—
274#103—28no—
275#103—44no—
276#103—44no—
277#103—44no—
278#103—44no—
279#102—28no—
280#102—28no—
281C22⋊C8—28no—
282#97—24no—
283C4×C8—24no—
284#95—22no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95C2×D414yes—
294#95C2×D414yes—
295C24—22no—
296C24—24no—
297C24—42no—
298C24—42no—
299C24—82no—
300C24—82no—
301C24C22×D411yescharacteristic
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—48no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—48no—
316C2×D4C22×D414yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—216no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—82no—
333C22×C4C22×D414yes—
334C22×C4C22×D412yes—
335C22×C4C22×D414yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D412yes—
338C22×C4C22×D412yes—
339C22×C4C2×#4112yes—
340C22×C4C2511yescharacteristic
341SD16—416no—
342SD16—416no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C4⋊C4—28no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—48no—
361C4⋊C4—48no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—44no—
365C4⋊C4—84no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—84no—
369C4⋊C4#12118yes—
370C4⋊C4C22×D414yes—
371C22⋊C4—24no—
372C22⋊C4—216no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—48no—
376C22⋊C4—48no—
377C22⋊C4—84no—
378C22⋊C4—84no—
379C22⋊C4—84no—
380C22⋊C4#12118yes—
381C42—24no—
382C42—22no—
383C42—22no—
384C42—24no—
385C42—41no—
386C42—41no—
387C42—42no—
388C42—48no—
389C42—44no—
390C42—82no—
391C42—84no—
392C42C22×D414yes—
393C42C2×#4114yes—
394C23—22no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—24no—
399C23—48no—
400C23—44no—
401C23—42no—
402C23—42no—
403C23—44no—
404C23—88no—
405C23—164no—
406C23—162no—
407C23—162no—
408C23C2×#12112yes—
409C23C23×D411yescharacteristic
410Q8—44no—
411Q8—44no—
412Q8—44no—
413Q8—44no—
414D4—28no—
415D4—48no—
416D4—44no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—84no—
447C2×C4—816no—
448C2×C4—162no—
449C2×C4C2×#12112yes—
450C2×C4C2×#12118yes—
451C2×C4#57714yes—
452C2×C4C23×D411yescharacteristic
453C2×C4C22×#4111yescharacteristic
454C2×C4#58512yes—
455C2×C4#58512yes—
456C8—44no—
457C8—44no—
458C22—24no—
459C22—22no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—48no—
464C22—44no—
465C22—84no—
466C22—82no—
467C22—82no—
468C22—168no—
469C22C22×#12111yescharacteristic
470C22#348511yescharacteristic
471C22#348711yescharacteristic
472C22C2×#57712yes—
473C22#358612yes—
474C4—28no—
475C4—24no—
476C4—22no—
477C4—44no—
478C4—44no—
479C4—42no—
480C4—84no—
481C4—84no—
482C4—81no—
483C4—82no—
484C4—81no—
485C4#344912yes—
486C4#349312yes—
487C2—42no—
488C2—84no—
489C2#3515011yescharacteristic
490C2#6042912yes—
491C2#6175312yes—
492C2#6193311yescharacteristic
493C2#6194311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #37387C2semidirect
4. #33117C2semidirect
5. #33114C2semidirect
8. #24690C2semidirect
9. #24686C2semidirect
10. #24646C2semidirect
11. #24630C2semidirect
12. #21957C2semidirect
13. #21873C2semidirect
29. #2565C22semidirect
30. #2546C22semidirect
52. #1992C22semidirect
53. #1991C22semidirect
54. #1981C22semidirect
55. #1981C22semidirect
56. #1981C22semidirect
57. #1960C22semidirect
58. #1956C22semidirect
59. #1949C22semidirect
60. #1948C22semidirect
61. #1947C22semidirect
62. #1940C22semidirect
63. #1939C22semidirect
64. #1928C22semidirect
65. #1928C22semidirect
66. #1927C22semidirect
67. #1927C22semidirect
70. #1805C22semidirect
71. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31232
#373872
#342552
#331172
#331142
#297232
#297142
#246902
#246862
#246462
#246302
#219572
#218732