Finite Group Databasebeta

← #52472

Subgroups — #52472

Order 256.

1117 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2842C211yesmaximal, characteristic, complemented
2#2829C213yesmaximal, complemented
3#2825C211yesmaximal, characteristic, complemented
4#2813C213yesmaximal, complemented
5#2801C211yesmaximal, characteristic, complemented
6#2780C213yesmaximal, complemented
7#2678C213yesmaximal, complemented
8#2675C211yesmaximal, characteristic, complemented
9#2656C211yesmaximal, characteristic, complemented
10#2643C211yesmaximal, characteristic, complemented
11#2443C213yesmaximal, complemented
12#2424C213yesmaximal, complemented
13#2416C213yesmaximal, complemented
14#2410C213yesmaximal, complemented
15C2×#396C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C2×#123C2213yescomplemented
18C2×#121C2213yes—
19C2×#121C2213yes—
20C2×#121C2213yescomplemented
21C2×#121C2213yescomplemented
22C2×#121C2213yescomplemented
23C2×#120C2213yes—
24C2×#120C2213yes—
25C2×C4×D4C2213yescomplemented
26C22×C4⋊C4C2213yescomplemented
27C22×C22⋊C4C2213yes—
28#401C2211yescharacteristic, complemented
29#400C2213yescomplemented
30#399C2213yescomplemented
31#399C2211yescharacteristic, complemented
32#399C2213yescomplemented
33#398C2213yescomplemented
34#397C2213yescomplemented
35#397C2213yescomplemented
36#396C2211yescharacteristic
37#396C2213yes—
38#396C2213yes—
39#396C2213yescomplemented
40#396C2213yescomplemented
41#396C2211yescharacteristic, complemented
42#396C2211yescharacteristic, complemented
43#396C2213yescomplemented
44#394C2213yes—
45#394C2213yes—
46#394C2213yescomplemented
47#394C2213yescomplemented
48#394C2213yescomplemented
49#394C2213yescomplemented
50#394C2213yescomplemented
51#392C2213yes—
52#392C2213yes—
53#392C2213yes—
54#392C2211yescharacteristic, complemented
55#392C2213yescomplemented
56#390C2213yes—
57#390C2213yescomplemented
58#389C2213yescomplemented
59#388C2213yescomplemented
60#388C2213yescomplemented
61#387C2213yescomplemented
62#386C2213yes—
63#386C2213yes—
64#386C2213yes—
65#386C2213yes—
66#386C2213yescomplemented
67#385C2213yes—
68#385C2213yescomplemented
69#385C2213yescomplemented
70#380C2213yescomplemented
71#379C2213yes—
72C2×#95C2211yescharacteristic, complemented
73C25C2311yescharacteristic
74C25C2311yescharacteristic
75C22×D4—46no—
76C22×D4C2313yes—
77C22×D4C2311yescharacteristic
78C22×D4C2311yescharacteristic
79C22×D4C2313yes—
80C22×D4C2313yes—
81C22×D4C2313yes—
82C22×D4C2313yes—
83C22×D4C2313yes—
84C22×D4C2311yescharacteristic
85C22×D4C2311yescharacteristic
86C23×C4C2313yes—
87C23×C4C2313yes—
88C23×C4C2313yes—
89C23×C4C2311yescharacteristic
90#123—46no—
91#121—46no—
92#121—46no—
93#121—46no—
94#121—46no—
95#121—46no—
96#120—46no—
97#120—46no—
98C4×D4—46no—
99C2×C4⋊C4—46no—
100C2×C4⋊C4C2313yes—
101C2×C4⋊C4C2313yes—
102C2×C4⋊C4C2313yes—
103C2×C4⋊C4C2313yes—
104C2×C4⋊C4C2313yes—
105C2×C4⋊C4C2313yes—
106C2×C4⋊C4C2313yes—
107C2×C4⋊C4C2313yes—
108C2×C4⋊C4C2313yes—
109C2×C4⋊C4C2313yes—
110C2×C22⋊C4—46no—
111C2×C22⋊C4C2313yes—
112C2×C22⋊C4C2313yes—
113C2×C22⋊C4C2313yes—
114C2×C22⋊C4C2313yes—
115C2×C22⋊C4C2313yes—
116C2×C22⋊C4C2313yes—
117C2×C22⋊C4C2313yes—
118C2×C22⋊C4C2313yes—
119C2×C22⋊C4C2313yes—
120C2×C22⋊C4C2313yes—
121C2×C22⋊C4C2313yes—
122C2×C22⋊C4C2313yes—
123C2×C22⋊C4C2313yes—
124C2×C22⋊C4C2313yes—
125C2×C22⋊C4C2313yes—
126C2×C22⋊C4C2313yes—
127C2×C22⋊C4C2313yes—
128C2×C22⋊C4C2313yes—
129C2×C22⋊C4C2313yes—
130C2×C22⋊C4C2313yes—
131C2×C42C2313yes—
132#95C2311yescharacteristic
133#95C2313yes—
134#95C2313yes—
135#95C2313yes—
136#95C2313yes—
137#95C2313yes—
138#95C2313yes—
139#95C2313yes—
140#95C2313yes—
141#95C2313yes—
142C24—22no—
143C24—26no—
144C24—22no—
145C24—43no—
146C24—43no—
147C24—43no—
148C24C2411yescharacteristic
149C24C2411yescharacteristic
150C24C2411yescharacteristic
151C24C2413yes—
152C24C2411yescharacteristic
153C24C2411yescharacteristic
154C2×D4—26no—
155C2×D4—26no—
156C2×D4—26no—
157C2×D4—26no—
158C2×D4—26no—
159C2×D4—26no—
160C2×D4—26no—
161C2×D4—43no—
162C2×D4—43no—
163C2×D4—43no—
164C2×D4—43no—
165C2×D4—43no—
166C2×D4—43no—
167C2×D4—43no—
168C2×D4—43no—
169C2×D4—43no—
170C2×D4—43no—
171C2×D4—43no—
172C2×D4—43no—
173C2×D4—43no—
174C2×D4—43no—
175C2×D4—43no—
176C2×D4—416no—
177C22×C4—26no—
178C22×C4—26no—
179C22×C4—26no—
180C22×C4—26no—
181C22×C4—43no—
182C22×C4—43no—
183C22×C4—43no—
184C22×C4—43no—
185C22×C4—43no—
186C22×C4—43no—
187C22×C4C2413yes—
188C22×C4C2411yescharacteristic
189C22×C4C2411yescharacteristic
190C22×C4C2413yes—
191C22×C4C2413yes—
192C22×C4C2413yes—
193C22×C4C2413yes—
194C22×C4C2413yes—
195C22×C4C2413yes—
196C4⋊C4—26no—
197C4⋊C4—26no—
198C4⋊C4—43no—
199C4⋊C4—43no—
200C4⋊C4—43no—
201C4⋊C4—43no—
202C4⋊C4—43no—
203C4⋊C4—43no—
204C4⋊C4—43no—
205C4⋊C4—43no—
206C22⋊C4—26no—
207C22⋊C4—26no—
208C22⋊C4—26no—
209C22⋊C4—26no—
210C22⋊C4—26no—
211C22⋊C4—43no—
212C22⋊C4—43no—
213C22⋊C4—43no—
214C22⋊C4—43no—
215C22⋊C4—43no—
216C22⋊C4—43no—
217C22⋊C4—43no—
218C22⋊C4—43no—
219C22⋊C4—43no—
220C22⋊C4—43no—
221C22⋊C4—43no—
222C22⋊C4—43no—
223C22⋊C4—43no—
224C22⋊C4—43no—
225C22⋊C4—43no—
226C42—43no—
227C23—23no—
228C23—21no—
229C23—23no—
230C23—23no—
231C23—23no—
232C23—23no—
233C23—23no—
234C23—23no—
235C23—23no—
236C23—23no—
237C23—23no—
238C23—23no—
239C23—23no—
240C23—23no—
241C23—23no—
242C23—23no—
243C23—23no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—412no—
248C23—44no—
249C23—44no—
250C23—86no—
251C23—86no—
252C23—86no—
253C23H(5,𝔽2)16yes—
254C232−1+412yes—
255C23C2511yescharacteristic
256D4—44no—
257D4—44no—
258D4—412no—
259D4—44no—
260D4—44no—
261D4—86no—
262D4—86no—
263D4—86no—
264D4—86no—
265D4—86no—
266C2×C4—23no—
267C2×C4—23no—
268C2×C4—23no—
269C2×C4—23no—
270C2×C4—23no—
271C2×C4—23no—
272C2×C4—23no—
273C2×C4—23no—
274C2×C4—23no—
275C2×C4—23no—
276C2×C4—23no—
277C2×C4—23no—
278C2×C4—23no—
279C2×C4—23no—
280C2×C4—23no—
281C2×C4—23no—
282C2×C4—23no—
283C2×C4—23no—
284C2×C4—23no—
285C2×C4—23no—
286C2×C4—23no—
287C2×C4—23no—
288C2×C4—412no—
289C2×C4—44no—
290C2×C4—86no—
291C2×C4—86no—
292C2×C4H(5,𝔽2)16yes—
293C22—26no—
294C22—26no—
295C22—26no—
296C22—41no—
297C22—43no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—43no—
302C22—43no—
303C22—43no—
304C22—43no—
305C22—43no—
306C22—43no—
307C22—43no—
308C22—43no—
309C22—43no—
310C22—43no—
311C22—43no—
312C22—43no—
313C22—43no—
314C22—88no—
315C22—88no—
316C22#56014yes—
317C22C2×H(5,𝔽2)13yes—
318C22C2×H(5,𝔽2)13yes—
319C22C2×2−1+411yescharacteristic
320C4—26no—
321C4—41no—
322C4—41no—
323C4—43no—
324C4—43no—
325C4—43no—
326C4—43no—
327C4—43no—
328C4—43no—
329C2—24no—
330C2—42no—
331C2—83no—
332C2—81no—
333C2—81no—
334C2—81no—
335C2C2×#56011yescharacteristic
336C2#353613yes—
337C2#353713yes—
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. #2842C2semidirect
2. #2829C2semidirect
3. #2825C2semidirect
4. #2813C2semidirect
5. #2801C2semidirect
6. #2780C2semidirect
7. #2678C2semidirect
8. #2675C2semidirect
9. #2656C2semidirect
10. #2643C2semidirect
11. #2443C2semidirect
12. #2424C2semidirect
13. #2416C2semidirect
14. #2410C2semidirect
15. C2×#396C2semidirect
17. C2×#123C22semidirect
20. C2×#121C22semidirect
21. C2×#121C22semidirect
22. C2×#121C22semidirect
25. C2×C4×D4C22semidirect
26. C22×C4⋊C4C22semidirect
28. #401C22semidirect
29. #400C22semidirect
30. #399C22semidirect
31. #399C22semidirect
32. #399C22semidirect
33. #398C22semidirect
34. #397C22semidirect
35. #397C22semidirect
39. #396C22semidirect
40. #396C22semidirect
41. #396C22semidirect
42. #396C22semidirect
43. #396C22semidirect
46. #394C22semidirect
47. #394C22semidirect
48. #394C22semidirect
49. #394C22semidirect
50. #394C22semidirect
54. #392C22semidirect
55. #392C22semidirect
57. #390C22semidirect
58. #389C22semidirect
59. #388C22semidirect
60. #388C22semidirect
61. #387C22semidirect
66. #386C22semidirect
68. #385C22semidirect
69. #385C22semidirect
70. #380C22semidirect
72. C2×#95C22semidirect

Maximal subgroups

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

subgroupindex
#28422
#28292
#28252
#28132
#28012
#27802
#26782
#26752
#26562
#26432
#24432
#24242
#24162
#24102
C2×#3962