Finite Group Databasebeta

← #33877

Subgroups — #33877

Order 256.

305 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3336C211yesmaximal, characteristic, complemented
2D4×D8C211yesmaximal, characteristic
3#2244C211yesmaximal, characteristic, complemented
4#2234C211yesmaximal, characteristic, complemented
5#2232C211yesmaximal, characteristic, complemented
6#2228C211yesmaximal, characteristic, complemented
7#2220C211yesmaximal, characteristic, complemented
8#2217C211yesmaximal, characteristic, complemented
9#2207C211yesmaximal, characteristic, complemented
10#2187C211yesmaximal, characteristic, complemented
11#2187C211yesmaximal, characteristic, complemented
12#2175C211yesmaximal, characteristic, complemented
13#2168C211yesmaximal, characteristic, complemented
14#2140C211yesmaximal, characteristic, complemented
15C2×#357C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18#552—21no—
19D42—21no—
20C2×SD32C2211yescharacteristic
21C2×SD32C2211yescharacteristic, complemented
22C2×SD32C2211yescharacteristic, complemented
23C2×M32C2211yescharacteristic, complemented
24C22×C16C2211yescharacteristic, complemented
25#499C2211yescharacteristic, complemented
26#493C2211yescharacteristic
27#485—21no—
28#483—21no—
29#479—21no—
30#467C2211yescharacteristic, complemented
31#467C2211yescharacteristic, complemented
32#466C2211yescharacteristic
33#466C2211yescharacteristic
34#459—21no—
35#447—21no—
36#447—21no—
37C4×Q16C2211yescharacteristic, complemented
38C4×D8C2211yescharacteristic
39C8×D4C2211yescharacteristic
40#427C2211yescharacteristic, complemented
41C2×#107C2211yescharacteristic, complemented
42#367C2211yescharacteristic, complemented
43#366C2211yescharacteristic, complemented
44#366C2211yescharacteristic, complemented
45C4⋊C16C2211yescharacteristic, complemented
46#358C2211yescharacteristic, complemented
47#357C2211yescharacteristic
48#357C2211yescharacteristic
49#357C2211yescharacteristic
50#357C2211yescharacteristic, complemented
51#357C2211yescharacteristic, complemented
52#357C2211yescharacteristic, complemented
53#357C2211yescharacteristic, complemented
54#357C2211yescharacteristic, complemented
55#357C2211yescharacteristic, complemented
56C22⋊C16C2211yescharacteristic, complemented
57C22⋊C16C2211yescharacteristic, complemented
58#346C2211yescharacteristic, complemented
59C22×D4—21no—
60C22×D4—21no—
61C22×D4—41no—
62C2×Q16C2311yescharacteristic
63C2×D8—22no—
64C2×D8—22no—
65C2×D8—21no—
66C2×D8—21no—
67C2×D8—21no—
68C2×D8—41no—
69C2×D8—41no—
70C2×D8C2311yescharacteristic
71C2×D8C2311yescharacteristic
72C2×D8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#127—21no—
76#126—41no—
77#125—21no—
78#125—41no—
79#123—41no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82#121—21no—
83#121—21no—
84#121—41no—
85#120—41no—
86#120—41no—
87C4×Q8—21no—
88C4×D4—21no—
89C4×D4D411yescharacteristic
90#117—21no—
91#117—41no—
92C2×C4⋊C4—21no—
93SD32—22no—
94SD32—41no—
95SD32—41no—
96M32—41no—
97C2×C16—41no—
98C2×C16C2311yescharacteristic
99C2×C16C2311yescharacteristic
100C2×C16C2311yescharacteristic
101C2×C16C2311yescharacteristic
102#107—21no—
103#107C2311yescharacteristic
104#107C2311yescharacteristic
105#107C2311yescharacteristic
106#107C2311yescharacteristic
107#106—21no—
108#106—21no—
109C4⋊C8D411yescharacteristic
110#103—21no—
111#103—21no—
112#103—21no—
113#102—21no—
114#102—21no—
115#102—21no—
116C22⋊C8D411yescharacteristic
117C22⋊C8D411yescharacteristic
118C4×C8C2311yescharacteristic
119C24—41no—
120C24—41no—
121C2×Q8—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—81no—
134C2×D4—81no—
135C2×D4—84no—
136C2×D4D811yescharacteristic
137C22×C4—41no—
138C22×C4—41no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141Q16—21no—
142Q16—21no—
143D8—21no—
144D8—21no—
145D8—21no—
146D8—21no—
147D8—21no—
148D8—42no—
149D8—42no—
150D8—81no—
151D8#4112yes—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8C2×D411yescharacteristic
157C2×C8C2×D411yescharacteristic
158C2×C8C2×D411yescharacteristic
159C2×C8C2411yescharacteristic
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4D811yescharacteristic
171C22⋊C4—41no—
172C22⋊C4—41no—
173C22⋊C4—41no—
174C22⋊C4—41no—
175C22⋊C4—41no—
176C22⋊C4—41no—
177C22⋊C4D811yescharacteristic
178C22⋊C4D811yescharacteristic
179C42—41no—
180C42—41no—
181C42C2×D411yescharacteristic
182C16—21no—
183C16—21no—
184C16—21no—
185C16—21no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—82no—
190C23—82no—
191C23—82no—
192C23—82no—
193C23—161no—
194C23—161no—
195C23C2×D811yescharacteristic
196C23C2×D811yescharacteristic
197Q8—41no—
198Q8—41no—
199D4—22no—
200D4—22no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—41no—
205D4—41no—
206D4—82no—
207D4—82no—
208D4—82no—
209D4—82no—
210D4—82no—
211D4—161no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4C2×D811yescharacteristic
224C2×C4C2×D811yescharacteristic
225C2×C4C2×D811yescharacteristic
226C2×C4C2×D811yescharacteristic
227C2×C4C22×D411yescharacteristic
228C8—21no—
229C8—21no—
230C8—21no—
231C8C2×#4111yescharacteristic
232C8H(5,𝔽2)11yescharacteristic
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—164no—
248C22—164no—
249C22#50912yes—
250C22C22×D811yescharacteristic
251C4—21no—
252C4—21no—
253C4—21no—
254C4—21no—
255C4—81no—
256C4—81no—
257C4—81no—
258C4—81no—
259C4—81no—
260C4#54611yescharacteristic
261C4#57711yescharacteristic
262C2—22no—
263C2—41no—
264C2—82no—
265C2—161no—
266C2—161no—
267C2#329411yescharacteristic
268C2C2×#50911yescharacteristic
269C2#341611yescharacteristic
subgroupG/H
center, socleC22C22×D8
commutator, Frattini, cosocleC2×C8C24
prime radicalD4×D8C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3336C2semidirect
3. #2244C2semidirect
4. #2234C2semidirect
5. #2232C2semidirect
6. #2228C2semidirect
7. #2220C2semidirect
8. #2217C2semidirect
9. #2207C2semidirect
10. #2187C2semidirect
11. #2187C2semidirect
12. #2175C2semidirect
13. #2168C2semidirect
14. #2140C2semidirect
15. C2×#357C2semidirect
21. C2×SD32C22semidirect
22. C2×SD32C22semidirect
23. C2×M32C22semidirect
24. C22×C16C22semidirect
25. #499C22semidirect
30. #467C22semidirect
31. #467C22semidirect
37. C4×Q16C22semidirect
40. #427C22semidirect
41. C2×#107C22semidirect
42. #367C22semidirect
43. #366C22semidirect
44. #366C22semidirect
45. C4⋊C16C22semidirect
46. #358C22semidirect
50. #357C22semidirect
51. #357C22semidirect
52. #357C22semidirect
53. #357C22semidirect
54. #357C22semidirect
55. #357C22semidirect
56. C22⋊C16C22semidirect
57. C22⋊C16C22semidirect
58. #346C22semidirect

Maximal subgroups

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

subgroupindex
#33362
D4×D82
#22442
#22342
#22322
#22282
#22202
#22172
#22072
#21872
#21752
#21682
#21402
C2×#3572