Finite Group Databasebeta

← #19461

Subgroups — #19461

Order 256.

348 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3207C211yesmaximal, characteristic, complemented
2#3100C211yesmaximal, characteristic, complemented
3#3036C211yesmaximal, characteristic, complemented
4#2974C211yesmaximal, characteristic, complemented
5#1701C211yesmaximal, characteristic, complemented
6#1699C211yesmaximal, characteristic, complemented
7#1687C211yesmaximal, characteristic, complemented
8#1685C211yesmaximal, characteristic, complemented
9#1681C211yesmaximal, characteristic, complemented
10#1679C211yesmaximal, characteristic, complemented
11#1667C211yesmaximal, characteristic, complemented
12#1665C211yesmaximal, characteristic, complemented
13#1574C211yesmaximal, characteristic, complemented
14#1557C211yesmaximal, characteristic, complemented
15#1553C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#532C2211yescharacteristic
18#532C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20#493C2211yescharacteristic, complemented
21#492C2211yescharacteristic, complemented
22#487C2211yescharacteristic, complemented
23#486C2211yescharacteristic, complemented
24#480—21no—
25#474C2211yescharacteristic
26#474C2211yescharacteristic
27#463—21no—
28#463—21no—
29#463—21no—
30#463—21no—
31#461C2211yescharacteristic
32#461C2211yescharacteristic
33#459C2211yescharacteristic
34#459C2211yescharacteristic
35#459C2211yescharacteristic
36#459C2211yescharacteristic
37#449—21no—
38#449—21no—
39#447—21no—
40C4×D8—21no—
41C4×D8—21no—
42#436C2211yescharacteristic
43#436C2211yescharacteristic
44#435—21no—
45#435—21no—
46C8×D4C2211yescharacteristic
47C8×D4C2211yescharacteristic
48#432C2211yescharacteristic
49#431C2211yescharacteristic
50#431C2211yescharacteristic
51#423C2211yescharacteristic
52#417—21no—
53#407—21no—
54C2×C22⋊C8—21no—
55#335C2211yescharacteristic, complemented
56#335C2211yescharacteristic, complemented
57#334C2211yescharacteristic, complemented
58#334C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#332C2211yescharacteristic, complemented
62#332C2211yescharacteristic, complemented
63#331C2211yescharacteristic, complemented
64#331C2211yescharacteristic, complemented
65#331C2211yescharacteristic, complemented
66#331C2211yescharacteristic, complemented
67C2×#41—21no—
68C2×#41—21no—
69C22×D4D411yescharacteristic
70C23×C4—21no—
71AGL(1,ℤ/8)—44no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8—21no—
81C2×D8—21no—
82C2×M16—21no—
83C2×M16—21no—
84C2×M16—21no—
85C22×C8—21no—
86C22×C8—21no—
87C4⋊Q8C2311yescharacteristic
88C4⋊Q8C2311yescharacteristic
89#127C2311yescharacteristic
90#127C2311yescharacteristic
91#124—21no—
92#124—21no—
93#121—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4D411yescharacteristic
104C4×D4D411yescharacteristic
105C4×D4C2311yescharacteristic
106C4×D4C2311yescharacteristic
107C2×C4⋊C4D411yescharacteristic
108C2×C22⋊C4—21no—
109C2×C42C2311yescharacteristic
110#107—21no—
111#107—21no—
112C4⋊C8—21no—
113C4⋊C8D411yescharacteristic
114C4⋊C8D411yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#102—21no—
126#102—21no—
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133C22⋊C8—21no—
134C22⋊C8—21no—
135C22⋊C8—21no—
136C22⋊C8—21no—
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8D411yescharacteristic
140C22⋊C8D411yescharacteristic
141#97—21no—
142C4×C8—21no—
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145C24—21no—
146#41—42no—
147#41—42no—
148#41—81no—
149#41—81no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×D4—22no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4C2×D411yescharacteristic
164C2×D4C2×D411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4#4111yescharacteristic
177SD16—42no—
178SD16—42no—
179SD16—81no—
180SD16—81no—
181D8—42no—
182D8—42no—
183D8—42no—
184D8—42no—
185D8—81no—
186D8—81no—
187M16—42no—
188M16—81no—
189M16—81no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8C2×D411yescharacteristic
199C2×C8C2×D411yescharacteristic
200C2×C8C2×D411yescharacteristic
201C2×C8C2×D411yescharacteristic
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C42C2×D411yescharacteristic
218C42C2×D411yescharacteristic
219C42#4111yescharacteristic
220C42C2411yescharacteristic
221C23—21no—
222C23—21no—
223C23—42no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23—81no—
228C23#12111yescharacteristic
229Q8—41no—
230Q8—41no—
231Q8—41no—
232Q8—41no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—42no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—81no—
246D4—81no—
247D4—82no—
248D4—82no—
249D4AGL(1,ℤ/8)12yes—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4#12111yescharacteristic
269C2×C4#12111yescharacteristic
270C2×C4#12111yescharacteristic
271C2×C4C22×D411yescharacteristic
272C2×C4C22×D411yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C8—21no—
275C8—21no—
276C8—21no—
277C8—21no—
278C8—41no—
279C8—41no—
280C8—41no—
281C8—41no—
282C8—41no—
283C8—41no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—21no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—84no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22C2×#12111yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—41no—
306C4—41no—
307C4—81no—
308C4—81no—
309C4C2×AGL(1,ℤ/8)11yescharacteristic
310C4#57511yescharacteristic
311C4#57611yescharacteristic
312C4#57611yescharacteristic
313C2—41no—
314C2—42no—
315C2—81no—
316C2—161no—
317C2—161no—
318C2#304111yescharacteristic
319C2#305511yescharacteristic
320C2#306711yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3207C2semidirect
2. #3100C2semidirect
3. #3036C2semidirect
4. #2974C2semidirect
5. #1701C2semidirect
6. #1699C2semidirect
7. #1687C2semidirect
8. #1685C2semidirect
9. #1681C2semidirect
10. #1679C2semidirect
11. #1667C2semidirect
12. #1665C2semidirect
13. #1574C2semidirect
14. #1557C2semidirect
15. #1553C2semidirect
20. #493C22semidirect
21. #492C22semidirect
22. #487C22semidirect
23. #486C22semidirect
55. #335C22semidirect
56. #335C22semidirect
57. #334C22semidirect
58. #334C22semidirect
59. #332C22semidirect
60. #332C22semidirect
61. #332C22semidirect
62. #332C22semidirect
63. #331C22semidirect
64. #331C22semidirect
65. #331C22semidirect
66. #331C22semidirect

Maximal subgroups

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

subgroupindex
#32072
#31002
#30362
#29742
#17012
#16992
#16872
#16852
#16812
#16792
#16672
#16652
#15742
#15572
#15532