Finite Group Databasebeta

← #10169

Subgroups — #10169

Order 256.

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

normal subgroups only

This group has 278 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#3208C211yesmaximal, characteristic
2#3139C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic
4#1965C211yesmaximal, characteristic
5#1546C211yesmaximal, characteristic, complemented
6#1493C211yesmaximal, characteristic, complemented
7#1278C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#127C2211yescharacteristic
11#492—21no—
12#491—22no—
13#487—21no—
14#487—21no—
15#486—22no—
16#486—21no—
17#486—22no—
18#460—22no—
19#450—22no—
20#431C2211yescharacteristic
21#423C2211yescharacteristic
22#419—22no—
23C2×#102—22no—
24C2×#100—22no—
25C4×M16C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27#332—22no—
28Q8⋊C8—22no—
29C22×Q8—21no—
30C22×D4—21no—
31C22×D4—21no—
32C22×D4D412yes—
33C2×SD16—24no—
34C2×M16—21no—
35C2×M16D412yes—
36C22×C8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8D412yes—
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43#127—21no—
44#127—21no—
45#127—21no—
46#127—21no—
47#125—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52#117—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C42C2311yescharacteristic
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8D412yes—
59#103—24no—
60#103—24no—
61#102—28no—
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—
71C2×Q8—22no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8—41no—
75C2×Q8—42no—
76C2×Q8—42no—
77C2×D4—22no—
78C2×D4—22no—
79C2×D4—44no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4—41no—
84C2×D4—42no—
85C2×D4—42no—
86C2×D4—82no—
87C22×C4—41no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4#4111yescharacteristic
91SD16—84no—
92M16—22no—
93M16—22no—
94M16—42no—
95C2×C8—22no—
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—42no—
99C2×C8—41no—
100C4⋊C4—22no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—22no—
105C4⋊C4—21no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C22⋊C4—41no—
111C42—41no—
112C42C2×D411yescharacteristic
113C42C2×D411yescharacteristic
114C42C2×D411yescharacteristic
115C42C2×D411yescharacteristic
116C23—42no—
117C23—44no—
118C23—82no—
119C23—82no—
120C23#12111yescharacteristic
121Q8—44no—
122Q8—42no—
123Q8—42no—
124Q8—81no—
125Q8—81no—
126D4—44no—
127D4—44no—
128D4—82no—
129D4—82no—
130D4—82no—
131D4—81no—
132D4—81no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—82no—
139C2×C4#12011yescharacteristic
140C2×C4#12011yescharacteristic
141C2×C4#12011yescharacteristic
142C2×C4#12111yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12711yescharacteristic
145C2×C4#13512yes—
146C2×C4AGL(1,ℤ/8)12yes—
147C8—42no—
148C8—42no—
149C8—42no—
150C8—81no—
151C22—21no—
152C22—21no—
153C22—82no—
154C22—82no—
155C22—82no—
156C22—164no—
157C22#39211yescharacteristic
158C22#46312yes—
159C4—21no—
160C4—21no—
161C4—81no—
162C4—82no—
163C4—81no—
164C4#44912yes—
165C4#45512yes—
166C4#47112yes—
167C4#49511yescharacteristic
168C4#49611yescharacteristic
169C2—22no—
170C2—162no—
171C2#200411yescharacteristic
172C2#201211yescharacteristic
173C2#201911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
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. #3139C2semidirect
5. #1546C2semidirect
6. #1493C2semidirect
7. #1278C2semidirect

Maximal subgroups

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

subgroupindex
#32082
#31392
C2×#4862
#19652
#15462
#14932
#12782