Finite Group Databasebeta

← #10160

Subgroups — #10160

Order 256.

289 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 61 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#3205C211yesmaximal, characteristic
2#3141C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#1965C211yesmaximal, characteristic, complemented
5#1548C211yesmaximal, characteristic, complemented
6#1492C211yesmaximal, characteristic, complemented
7#1278C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#127C2211yescharacteristic
11#493—21no—
12#489—22no—
13#487—21no—
14#486—22no—
15#486—21no—
16#486—21no—
17#486—22no—
18#459—22no—
19#447—22no—
20#431C2211yescharacteristic
21#423C2211yescharacteristic
22#417—22no—
23C2×#102—22no—
24C2×#100—22no—
25C4×M16C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27#333—22no—
28D4⋊C8—22no—
29C2×#41—21no—
30C22×Q8—21no—
31C22×D4—21no—
32C22×D4—21no—
33C22×D4D412yes—
34C2×D8—24no—
35C2×M16—21no—
36C2×M16D412yes—
37C22×C8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8D412yes—
42C4⋊Q8D411yescharacteristic
43#127—21no—
44#127—21no—
45#127—21no—
46#127—21no—
47#127D411yescharacteristic
48#124—21no—
49#121—21no—
50#121—21no—
51C4×D4—21no—
52C4×D4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C42C2311yescharacteristic
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8D412yes—
59#103—24no—
60#102—28no—
61#102—24no—
62#102—24no—
63#100—44no—
64C22⋊C8—21no—
65#97—21no—
66C4×C8—21no—
67C4×C8—21no—
68C4×C8—21no—
69C4×C8D412yes—
70C24—22no—
71#41—42no—
72#41—42no—
73C2×Q8—21no—
74C2×Q8—22no—
75C2×Q8—41no—
76C2×Q8—82no—
77C2×D4—21no—
78C2×D4—22no—
79C2×D4—22no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—44no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—42no—
89C2×D4—42no—
90C2×D4—82no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4#4111yescharacteristic
96D8—84no—
97M16—22no—
98M16—22no—
99M16—42no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—42no—
104C2×C8—41no—
105C4⋊C4—22no—
106C4⋊C4—21no—
107C4⋊C4—22no—
108C4⋊C4—21no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C42C2×D411yescharacteristic
114C42C2×D411yescharacteristic
115C42C2×D411yescharacteristic
116C42C2×D411yescharacteristic
117C23—41no—
118C23—42no—
119C23—44no—
120C23—82no—
121C23—82no—
122C23#12111yescharacteristic
123Q8—42no—
124Q8—82no—
125Q8—82no—
126D4—44no—
127D4—42no—
128D4—42no—
129D4—44no—
130D4—42no—
131D4—82no—
132D4—82no—
133D4—82no—
134D4—82no—
135D4—81no—
136D4—81no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—82no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4#12011yescharacteristic
145C2×C4#12011yescharacteristic
146C2×C4#12011yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4#12711yescharacteristic
150C2×C4#13512yes—
151C2×C4AGL(1,ℤ/8)12yes—
152C8—42no—
153C8—42no—
154C8—42no—
155C8—81no—
156C22—21no—
157C22—21no—
158C22—82no—
159C22—81no—
160C22—81no—
161C22—82no—
162C22—81no—
163C22—82no—
164C22—164no—
165C22#39211yescharacteristic
166C22#46312yes—
167C4—21no—
168C4—21no—
169C4—81no—
170C4—82no—
171C4#44912yes—
172C4#45512yes—
173C4#47212yes—
174C4#49511yescharacteristic
175C4#49611yescharacteristic
176C2—22no—
177C2—161no—
178C2—162no—
179C2#200411yescharacteristic
180C2#201211yescharacteristic
181C2#201811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3205C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3141C2semidirect
3. C2×#486C2semidirect
4. #1965C2semidirect
5. #1548C2semidirect
6. #1492C2semidirect
7. #1278C2semidirect

Maximal subgroups

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

subgroupindex
#32052
#31412
C2×#4862
#19652
#15482
#14922
#12782