Finite Group Databasebeta

← #33523

Subgroups — #33523

Order 256.

569 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#576C211yesmaximal, characteristic
2C2×#510C211yesmaximal, characteristic, complemented
3C22×SD32C211yesmaximal, characteristic, complemented
4#3059C211yesmaximal, characteristic, complemented
5#2189C214yesmaximal, complemented
6#2187C214yesmaximal, complemented
7#2140C211yesmaximal, characteristic, complemented
8#2138C211yesmaximal, characteristic, complemented
9#2113C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)—21no—
11#576—24no—
12#576—24no—
13C2×AGL(1,ℤ/8)—21no—
14C2×AGL(1,ℤ/8)—22no—
15C2×#135C2212yes—
16C2×#135C2211yescharacteristic
17C22×Q16C2211yescharacteristic, complemented
18C22×D8C2212yes—
19C22×D8C2211yescharacteristic
20C2×#131C2211yescharacteristic
21#536—21no—
22#510—24no—
23C2×Q32C2212yescomplemented
24C2×SD32—24no—
25C2×SD32C2214yescomplemented
26C2×SD32C2212yescomplemented
27C2×M32C2211yescharacteristic, complemented
28C22×C16C2211yescharacteristic, complemented
29#470—22no—
30#467C2214yescomplemented
31#427C2211yescharacteristic, complemented
32#419—21no—
33#358C2212yescomplemented
34#357C2212yescomplemented
35#357C2214yescomplemented
36C22⋊C16C2214yescomplemented
37H(5,𝔽2)—28no—
38H(5,𝔽2)—44no—
39C2×#41—22no—
40C2×#41—21no—
41C2×#41—41no—
42C2×#41D411yescharacteristic
43C22×Q8—21no—
44C22×D4—22no—
45C22×D4—21no—
46C22×D4—42no—
47C22×D4—41no—
48AGL(1,ℤ/8)—28no—
49AGL(1,ℤ/8)—216no—
50#135—24no—
51#135—24no—
52#135—22no—
53#135D414yes—
54C2×Q16—22no—
55C2×Q16—22no—
56C2×Q16—21no—
57C2×Q16C2312yes—
58C2×Q16C2311yescharacteristic
59C2×SD16—22no—
60C2×SD16—21no—
61C2×D8—24no—
62C2×D8—24no—
63C2×D8—22no—
64C2×D8—21no—
65C2×D8—22no—
66C2×D8D414yes—
67C2×D8C2311yescharacteristic
68C2×D8C2312yes—
69#131—22no—
70#131—22no—
71C2×M16D411yescharacteristic
72C2×M16D412yes—
73C22×C8C2312yes—
74C22×C8C2311yescharacteristic
75C4⋊Q8—41no—
76#124—41no—
77#123—42no—
78C22⋊Q8—24no—
79#117—21no—
80Q32—24no—
81SD32—28no—
82SD32—24no—
83M32—22no—
84C2×C16—22no—
85C2×C16C2312yes—
86C2×C16C2312yes—
87#107C2312yes—
88#106—21no—
89#103—24no—
90C24—42no—
91C24—41no—
92#41—22no—
93#41—22no—
94#41—28no—
95#41—24no—
96#41—44no—
97#41—48no—
98#41—42no—
99C2×Q8—21no—
100C2×Q8—22no—
101C2×Q8—41no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8D811yescharacteristic
105C2×D4—28no—
106C2×D4—24no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—44no—
110C2×D4—416no—
111C2×D4—48no—
112C2×D4—42no—
113C2×D4—41no—
114C2×D4—42no—
115C2×D4—41no—
116C2×D4—42no—
117C2×D4D812yes—
118C2×D4D811yescharacteristic
119C22×C4—42no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124Q16—21no—
125Q16—22no—
126Q16—22no—
127Q16—42no—
128Q16C2×D412yes—
129SD16—28no—
130SD16—24no—
131D8—28no—
132D8—24no—
133D8—22no—
134D8—21no—
135D8C2×D412yes—
136D8C2×D414yes—
137M16—22no—
138M16—24no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8C2×D412yes—
142C2×C8C2×D411yescharacteristic
143C2×C8C2×D412yes—
144C2×C8C2×D412yes—
145C2×C8C2411yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—44no—
148C22⋊C4—41no—
149C22⋊C4—44no—
150C42—41no—
151C16—22no—
152C16—22no—
153C23—48no—
154C23—42no—
155C23—41no—
156C23—44no—
157C23—162no—
158C23—164no—
159C23C2×D812yes—
160C23C2×D811yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—42no—
165Q8—42no—
166Q8—82no—
167D4—24no—
168D4—22no—
169D4—24no—
170D4—22no—
171D4—48no—
172D4—48no—
173D4—44no—
174D4—41no—
175D4—42no—
176D4—44no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—48no—
180C2×C4—41no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—82no—
185C2×C4#12012yes—
186C2×C4C2×D812yes—
187C2×C4C2×D811yescharacteristic
188C2×C4AGL(1,ℤ/8)12yes—
189C2×C4C22×D411yescharacteristic
190C8—22no—
191C8#12012yes—
192C8C22×D411yescharacteristic
193C8C22×D411yescharacteristic
194C22—22no—
195C22—24no—
196C22—22no—
197C22—44no—
198C22—42no—
199C22—84no—
200C22—82no—
201C22—164no—
202C22—168no—
203C22#44712yes—
204C22C22×D811yescharacteristic
205C4—22no—
206C4—42no—
207C4—82no—
208C4—82no—
209C4#44712yes—
210C4C2×#12011yescharacteristic
211C4C2×AGL(1,ℤ/8)11yescharacteristic
212C2—22no—
213C2—42no—
214C2—84no—
215C2—82no—
216C2C2×#44711yescharacteristic
217C2#341612yes—
subgroupG/H
center, socleC22C22×D8
commutator, Frattini, cosocleC2×C8C24
prime radicalC2×#576C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#510C2semidirect
3. C22×SD32C2semidirect
4. #3059C2semidirect
5. #2189C2semidirect
6. #2187C2semidirect
7. #2140C2semidirect
8. #2138C2semidirect
9. #2113C2semidirect
17. C22×Q16C22semidirect
23. C2×Q32C22semidirect
25. C2×SD32C22semidirect
26. C2×SD32C22semidirect
27. C2×M32C22semidirect
28. C22×C16C22semidirect
30. #467C22semidirect
31. #427C22semidirect
33. #358C22semidirect
34. #357C22semidirect
35. #357C22semidirect
36. C22⋊C16C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5762
C2×#5102
C22×SD322
#30592
#21892
#21872
#21402
#21382
#21132