Finite Group Databasebeta

← #33873

Subgroups — #33873

Order 256.

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

normal subgroups only

This group has 301 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#3334C211yesmaximal, characteristic, complemented
2#3280C211yesmaximal, characteristic, complemented
3#2243C211yesmaximal, characteristic, complemented
4#2235C211yesmaximal, characteristic, complemented
5#2233C211yesmaximal, characteristic, complemented
6#2226C211yesmaximal, characteristic, complemented
7#2218C211yesmaximal, characteristic, complemented
8#2217C211yesmaximal, characteristic, complemented
9#2210C211yesmaximal, characteristic, complemented
10#2186C211yesmaximal, characteristic, complemented
11#2184C211yesmaximal, characteristic, complemented
12#2177C211yesmaximal, characteristic, complemented
13#2168C211yesmaximal, characteristic, complemented
14#2139C211yesmaximal, characteristic, complemented
15C2×#357C211yesmaximal, characteristic, complemented
16C2×#135C2211yescharacteristic
17C22×D8C2211yescharacteristic
18#547—21no—
19#546—21no—
20C2×SD32C2211yescharacteristic, complemented
21C2×D16C2211yescharacteristic
22C2×D16C2211yescharacteristic
23C2×M32C2211yescharacteristic, complemented
24C22×C16C2211yescharacteristic
25#500C2211yescharacteristic, complemented
26#495C2211yescharacteristic
27#480—21no—
28#480—21no—
29#474—21no—
30#467C2211yescharacteristic
31#466C2211yescharacteristic
32#466C2211yescharacteristic
33#466C2211yescharacteristic, complemented
34#463—21no—
35#450—21no—
36#449—21no—
37C4×D8C2211yescharacteristic
38C4×D8C2211yescharacteristic, complemented
39C8×D4C2211yescharacteristic
40C2×#107C2211yescharacteristic
41C2×#107C2211yescharacteristic, complemented
42#367C2211yescharacteristic, complemented
43#367C2211yescharacteristic, complemented
44#367C2211yescharacteristic, complemented
45C4⋊C16C2211yescharacteristic, complemented
46#358C2211yescharacteristic, complemented
47#358C2211yescharacteristic, complemented
48#358C2211yescharacteristic, complemented
49#357C2211yescharacteristic
50#357C2211yescharacteristic
51#357C2211yescharacteristic
52#357C2211yescharacteristic
53#357C2211yescharacteristic
54#357C2211yescharacteristic
55#357C2211yescharacteristic, complemented
56C22⋊C16C2211yescharacteristic
57C22⋊C16C2211yescharacteristic, complemented
58#346C2211yescharacteristic, complemented
59C2×#41—21no—
60C2×#41—41no—
61C22×D4—21no—
62#135—22no—
63#135—41no—
64C2×Q16C2311yescharacteristic
65C2×SD16—21no—
66C2×SD16—21no—
67C2×D8—22no—
68C2×D8—21no—
69C2×D8—41no—
70C2×D8C2311yescharacteristic
71C2×D8C2311yescharacteristic
72C2×D8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—21no—
76#124—21no—
77#123—41no—
78#123—41no—
79#123—41no—
80C22⋊Q8—21no—
81C22⋊Q8—41no—
82#121—21no—
83#121—21no—
84#121—21no—
85#121—41no—
86#120—41no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4D411yescharacteristic
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C22⋊C4—41no—
93SD32—41no—
94D16—22no—
95D16—41no—
96M32—41no—
97C2×C16—41no—
98C2×C16C2311yescharacteristic
99C2×C16C2311yescharacteristic
100C2×C16C2311yescharacteristic
101C2×C16C2311yescharacteristic
102#107—21no—
103#107—21no—
104#107—21no—
105#107C2311yescharacteristic
106#107C2311yescharacteristic
107#107C2311yescharacteristic
108#107C2311yescharacteristic
109C4⋊C8D411yescharacteristic
110#103—21no—
111#102—21no—
112#102—21no—
113#102—21no—
114#102—21no—
115#102—21no—
116C22⋊C8D411yescharacteristic
117C22⋊C8D411yescharacteristic
118C4×C8C2311yescharacteristic
119C24—41no—
120#41—42no—
121#41—81no—
122#41—84no—
123C2×Q8—21no—
124C2×Q8—41no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—42no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4—81no—
135C2×D4D811yescharacteristic
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143Q16—21no—
144Q16—21no—
145SD16—42no—
146SD16—81no—
147D8—21no—
148D8—21no—
149D8—21no—
150D8—21no—
151D8—21no—
152D8—42no—
153D8#4112yescomplemented
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8C2×D411yescharacteristic
159C2×C8C2×D411yescharacteristic
160C2×C8C2×D411yescharacteristic
161C2×C8C2411yescharacteristic
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
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⋊C4—41no—
178C22⋊C4—41no—
179C22⋊C4D811yescharacteristic
180C22⋊C4D811yescharacteristic
181C42C2×D411yescharacteristic
182C16—21no—
183C16—21no—
184C16—21no—
185C16—21no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—82no—
190C23—82no—
191C23—161no—
192C23C2×D811yescharacteristic
193C23C2×D811yescharacteristic
194Q8—41no—
195Q8—41no—
196Q8—82no—
197D4—22no—
198D4—22no—
199D4—41no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—82no—
205D4—82no—
206D4—82no—
207D4—82no—
208D4—161no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—161no—
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#50912yes—
249C22C22×D811yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—81no—
255C4—81no—
256C4—81no—
257C4—81no—
258C4—81no—
259C4#54611yescharacteristic
260C4#57711yescharacteristic
261C2—22no—
262C2—41no—
263C2—82no—
264C2—161no—
265C2—161no—
266C2#329411yescharacteristic
267C2C2×#50911yescharacteristic
268C2#341611yescharacteristic
subgroupG/H
center, socleC22C22×D8
commutator, Frattini, cosocleC2×C8C24
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. #3334C2semidirect
2. #3280C2semidirect
3. #2243C2semidirect
4. #2235C2semidirect
5. #2233C2semidirect
6. #2226C2semidirect
7. #2218C2semidirect
8. #2217C2semidirect
9. #2210C2semidirect
10. #2186C2semidirect
11. #2184C2semidirect
12. #2177C2semidirect
13. #2168C2semidirect
14. #2139C2semidirect
15. C2×#357C2semidirect
20. C2×SD32C22semidirect
23. C2×M32C22semidirect
25. #500C22semidirect
33. #466C22semidirect
38. C4×D8C22semidirect
41. C2×#107C22semidirect
42. #367C22semidirect
43. #367C22semidirect
44. #367C22semidirect
45. C4⋊C16C22semidirect
46. #358C22semidirect
47. #358C22semidirect
48. #358C22semidirect
55. #357C22semidirect
57. C22⋊C16C22semidirect
58. #346C22semidirect
153. D8#41semidirect

Maximal subgroups

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

subgroupindex
#33342
#32802
#22432
#22352
#22332
#22262
#22182
#22172
#22102
#21862
#21842
#21772
#21682
#21392
C2×#3572