Finite Group Databasebeta

← #59000

Subgroups — #59000

Order 256.

1680 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C212yesmaximal, complemented
2C2×#525C212yesmaximal, complemented
3#2679C212yesmaximal, complemented
4#2668C212yesmaximal, complemented
5#2542C211yesmaximal, characteristic, complemented
6#2455C212yesmaximal, complemented
7#2432C212yesmaximal, complemented
8#2427C212yesmaximal, complemented
9#2422C214yesmaximal, complemented
10#2418C214yesmaximal, complemented
11#2415C212yesmaximal, complemented
12#2408C211yesmaximal, characteristic, complemented
13#2363C212yesmaximal, complemented
14#2358C212yesmaximal, complemented
15C4×#120C211yesmaximal, characteristic, complemented
16C22×#41C2212yes—
17C23×D4C2211yescharacteristic, complemented
18C24×C4C2211yescharacteristic, complemented
19#546—216no—
20C2×C4⋊Q8C2211yescharacteristic, complemented
21C2×#124C2212yescomplemented
22#525—216no—
23C2×#123C2214yescomplemented
24C2×#123C2214yescomplemented
25C2×C22⋊Q8C2212yescomplemented
26C2×C22⋊Q8C2212yescomplemented
27C2×C22⋊Q8C2212yescomplemented
28C2×#121C2212yes—
29C2×#121C2214yes—
30C2×#121C2212yes—
31C2×#120C2212yes—
32C2×#120C2214yes—
33C2×#120C2211yescharacteristic
34C2×#120C2214yescomplemented
35C2×C4×D4C2212yes—
36C2×C4×D4C2211yescharacteristic
37C2×C4×D4C2214yescomplemented
38C2×C4×D4C2212yescomplemented
39C2×C4×D4C2214yescomplemented
40C2×#117C2212yescomplemented
41C22×C22⋊C4C2214yescomplemented
42#399C2212yescomplemented
43#399C2212yescomplemented
44#397C2214yescomplemented
45#397C2214yescomplemented
46#394C2214yes—
47#394C2212yescomplemented
48#394C2214yescomplemented
49#394C2214yescomplemented
50#394C2214yescomplemented
51#392C2212yes—
52#391C2212yescomplemented
53#390C2212yes—
54#390C2212yescomplemented
55#390C2214yescomplemented
56#389C2212yescomplemented
57#389C2214yescomplemented
58#388C2214yescomplemented
59#388C2214yescomplemented
60#386C2211yescharacteristic
61#386C2212yes—
62#386C2214yescomplemented
63#386C2214yescomplemented
64#385C2214yescomplemented
65#385C2214yescomplemented
66#385C2212yescomplemented
67#385C2211yescharacteristic, complemented
68#384C2212yescomplemented
69#380C2211yescharacteristic, complemented
70#380C2214yescomplemented
71#379C2212yescomplemented
72#379C2211yescharacteristic, complemented
73C4×C4⋊C4C2211yescharacteristic, complemented
74C4×C22⋊C4C2214yescomplemented
75C4×C22⋊C4C2212yescomplemented
76C4×C22⋊C4C2211yescharacteristic, complemented
77C25C2312yes—
78C25C2311yescharacteristic
79C2×#41—28no—
80C2×#41—28no—
81C2×#41—44no—
82C22×Q8C2312yes—
83C22×D4—28no—
84C22×D4—42no—
85C22×D4C2312yes—
86C22×D4C2314yes—
87C22×D4C2314yes—
88C22×D4C2312yes—
89C22×D4C2312yes—
90C22×D4C2314yes—
91C22×D4C2311yescharacteristic
92C23×C4—28no—
93C23×C4—42no—
94C23×C4C2311yescharacteristic
95C23×C4C2311yescharacteristic
96C23×C4C2314yes—
97C23×C4C2311yescharacteristic
98C23×C4C2312yes—
99C23×C4C2312yes—
100C23×C4C2314yes—
101C4⋊Q8—42no—
102#124—28no—
103#123—216no—
104#123—216no—
105C22⋊Q8—28no—
106C22⋊Q8—28no—
107C22⋊Q8—28no—
108#121—28no—
109#121—216no—
110#121D4116yescomplemented
111#120—216no—
112#120—216no—
113#120—24no—
114#120—28no—
115C4×D4—216no—
116C4×D4—28no—
117C4×D4—28no—
118C4×D4—216no—
119C4×D4—42no—
120#117—28no—
121C2×C4⋊C4C2312yes—
122C2×C4⋊C4C2312yes—
123C2×C4⋊C4C2314yes—
124C2×C4⋊C4C2314yes—
125C2×C4⋊C4C2314yes—
126C2×C4⋊C4C2311yescharacteristic
127C2×C4⋊C4C2312yes—
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2314yes—
130C2×C4⋊C4C2312yes—
131C2×C22⋊C4—216no—
132C2×C22⋊C4C2312yes—
133C2×C22⋊C4C2314yes—
134C2×C22⋊C4C2314yes—
135C2×C22⋊C4C2314yes—
136C2×C22⋊C4C2312yes—
137C2×C22⋊C4C2314yes—
138C2×C22⋊C4C2314yes—
139C2×C22⋊C4C2314yes—
140C2×C22⋊C4C2314yes—
141C2×C22⋊C4C2314yes—
142C2×C22⋊C4C2314yes—
143C2×C22⋊C4C2312yes—
144C2×C22⋊C4C2311yescharacteristic
145C2×C22⋊C4C2312yes—
146C2×C22⋊C4C2311yescharacteristic
147C2×C22⋊C4C2311yescharacteristic
148C2×C22⋊C4C2311yescharacteristic
149C2×C22⋊C4C2312yes—
150C2×C22⋊C4C2314yes—
151C2×C22⋊C4C2314yes—
152C2×C42C2314yes—
153C2×C42C2312yes—
154C2×C42C2311yescharacteristic
155C2×C42C2312yes—
156#95C2312yes—
157#95C2312yes—
158#95C2312yes—
159#95C2314yes—
160#95C2312yes—
161#95C2314yes—
162#95C2312yes—
163#95C2314yes—
164C24—24no—
165C24—22no—
166C24—28no—
167C24—24no—
168C24—41no—
169C24—42no—
170C24—41no—
171C24—42no—
172C24—44no—
173C24—42no—
174C24C2412yes—
175C24C2414yes—
176C24C2411yescharacteristic
177C24C2411yescharacteristic
178#41—432no—
179C2×Q8—24no—
180C2×Q8—24no—
181C2×Q8—24no—
182C2×D4—24no—
183C2×D4—28no—
184C2×D4—28no—
185C2×D4—28no—
186C2×D4—28no—
187C2×D4—24no—
188C2×D4—28no—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—41no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—42no—
198C2×D4—44no—
199C2×D4—416no—
200C2×D4C2×D418yes—
201C2×D4C2×D4116yes—
202C2×D4#4114yescomplemented
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—28no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—28no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—416no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4C2×D418yes—
222C22×C4C2412yes—
223C22×C4C2411yescharacteristic
224C22×C4C2412yes—
225C22×C4C2414yes—
226C22×C4C2411yescharacteristic
227C22×C4C2414yes—
228C22×C4C2411yescharacteristic
229C22×C4C2414yes—
230C22×C4C2412yes—
231C22×C4C2411yescharacteristic
232C22×C4C2411yescharacteristic
233C4⋊C4—24no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—28no—
238C4⋊C4—44no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4C2×D418yes—
243C22⋊C4—28no—
244C22⋊C4—28no—
245C22⋊C4—24no—
246C22⋊C4—28no—
247C22⋊C4—28no—
248C22⋊C4—24no—
249C22⋊C4—22no—
250C22⋊C4—24no—
251C22⋊C4—28no—
252C22⋊C4—28no—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—24no—
256C22⋊C4—28no—
257C22⋊C4—44no—
258C22⋊C4—42no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4C2×D4116yes—
263C42—28no—
264C42—24no—
265C42—22no—
266C42—42no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—22no—
272C23—24no—
273C23—21no—
274C23—21no—
275C23—22no—
276C23—24no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—24no—
281C23—24no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—21no—
287C23—44no—
288C23—48no—
289C23—48no—
290C23—416no—
291C23—48no—
292C23—44no—
293C23—44no—
294C23—88no—
295C23—84no—
296C23—84no—
297C23C22×D418yes—
298C23C22×D414yes—
299C23C2×#4112yes—
300C23C2×#4112yes—
301C23C2511yescharacteristic
302Q8—48no—
303D4—216no—
304D4—416no—
305D4—44no—
306D4—416no—
307D4—48no—
308D4—48no—
309D4—416no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—216no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—44no—
336C2×C4—48no—
337C2×C4—416no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—88no—
341C2×C4C22×D418yes—
342C2×C4C22×D414yes—
343C2×C4C22×D414yes—
344C2×C4C2×#4112yes—
345C2×C4H(5,𝔽2)12yes—
346C22—24no—
347C22—28no—
348C22—24no—
349C22—24no—
350C22—24no—
351C22—28no—
352C22—24no—
353C22—28no—
354C22—22no—
355C22—22no—
356C22—416no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—44no—
361C22—44no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—44no—
367C22—41no—
368C22—42no—
369C22—44no—
370C22—41no—
371C22—816no—
372C22C23×D412yes—
373C22C22×#4111yescharacteristic
374C22C2×H(5,𝔽2)11yescharacteristic
375C22#58512yes—
376C22#58511yescharacteristic
377C4—24no—
378C4—28no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—44no—
383C4—42no—
384C4—44no—
385C4—42no—
386C4—41no—
387C4D4214yes—
388C2—44no—
389C2—42no—
390C2—48no—
391C2—42no—
392C2C2×D4211yescharacteristic
393C2D4×#4112yes—
394C2#348512yes—
395C2#350511yescharacteristic
396C2#354011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#546C2semidirect
2. C2×#525C2semidirect
3. #2679C2semidirect
4. #2668C2semidirect
5. #2542C2semidirect
6. #2455C2semidirect
7. #2432C2semidirect
8. #2427C2semidirect
9. #2422C2semidirect
10. #2418C2semidirect
11. #2415C2semidirect
12. #2408C2semidirect
13. #2363C2semidirect
14. #2358C2semidirect
15. C4×#120C2semidirect
17. C23×D4C22semidirect
18. C24×C4C22semidirect
20. C2×C4⋊Q8C22semidirect
21. C2×#124C22semidirect
23. C2×#123C22semidirect
24. C2×#123C22semidirect
25. C2×C22⋊Q8C22semidirect
26. C2×C22⋊Q8C22semidirect
27. C2×C22⋊Q8C22semidirect
34. C2×#120C22semidirect
37. C2×C4×D4C22semidirect
38. C2×C4×D4C22semidirect
39. C2×C4×D4C22semidirect
40. C2×#117C22semidirect
41. C22×C22⋊C4C22semidirect
42. #399C22semidirect
43. #399C22semidirect
44. #397C22semidirect
45. #397C22semidirect
47. #394C22semidirect
48. #394C22semidirect
49. #394C22semidirect
50. #394C22semidirect
52. #391C22semidirect
54. #390C22semidirect
55. #390C22semidirect
56. #389C22semidirect
57. #389C22semidirect
58. #388C22semidirect
59. #388C22semidirect
62. #386C22semidirect
63. #386C22semidirect
64. #385C22semidirect
65. #385C22semidirect
66. #385C22semidirect
67. #385C22semidirect
68. #384C22semidirect
69. #380C22semidirect
70. #380C22semidirect
71. #379C22semidirect
72. #379C22semidirect
73. C4×C4⋊C4C22semidirect
74. C4×C22⋊C4C22semidirect
75. C4×C22⋊C4C22semidirect
76. C4×C22⋊C4C22semidirect
110. #121D4semidirect
202. C2×D4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#5252
#26792
#26682
#25422
#24552
#24322
#24272
#24222
#24182
#24152
#24082
#23632
#23582
C4×#1202