Finite Group Databasebeta

← #10170

Subgroups — #10170

Order 256.

287 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 65 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#3206C211yesmaximal, characteristic
2#3152C211yesmaximal, characteristic
3C2×#487C211yesmaximal, characteristic, complemented
4#1997C211yesmaximal, characteristic, complemented
5#1546C211yesmaximal, characteristic, complemented
6#1492C211yesmaximal, characteristic
7#1278C211yesmaximal, characteristic, complemented
8C2×#137—22no—
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12#497—22no—
13#494—21no—
14#492—21no—
15#492—21no—
16#487—22no—
17#487—21no—
18#487—22no—
19#461—22no—
20#448—22no—
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23C2×#103—22no—
24C4×M16C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#343—22no—
27#332—22no—
28D4⋊C8—22no—
29C2×#41—21no—
30C22×Q8—21no—
31C22×Q8D412yes—
32#137—44no—
33#137—44no—
34C2×Q16—24no—
35C2×SD16—24no—
36C2×SD16—24no—
37C2×M16—21no—
38C2×M16D412yes—
39C22×C8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8—21no—
45C4⋊Q8—21no—
46C4⋊Q8—21no—
47C4⋊Q8D412yes—
48C4⋊Q8D411yescharacteristic
49#127D411yescharacteristic
50#124—21no—
51#121—21no—
52#121—21no—
53C4×D4—21no—
54C4×D4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C42C2311yescharacteristic
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8D412yes—
63#103—28no—
64#103—24no—
65C22⋊C8—21no—
66#97—21no—
67C4×C8—21no—
68C4×C8—21no—
69C4×C8—21no—
70C4×C8D412yes—
71#41—42no—
72#41—42no—
73C2×Q8—22no—
74C2×Q8—22no—
75C2×Q8—21no—
76C2×Q8—22no—
77C2×Q8—44no—
78C2×Q8—41no—
79C2×Q8—42no—
80C2×Q8—42no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—41no—
85C22×C4—22no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4#4111yescharacteristic
91Q16—48no—
92SD16—48no—
93SD16—84no—
94M16—22no—
95M16—42no—
96M16SD1614yescomplemented
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—42no—
101C2×C8—41no—
102C4⋊C4—22no—
103C4⋊C4—21no—
104C4⋊C4—22no—
105C4⋊C4—21no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C42C2×D411yescharacteristic
115C42C2×D411yescharacteristic
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C23—41no—
119C23#12111yescharacteristic
120Q8—44no—
121Q8—44no—
122Q8—42no—
123Q8—44no—
124Q8—81no—
125Q8—81no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—82no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—82no—
136C2×C4—81no—
137C2×C4—81no—
138C2×C4—82no—
139C2×C4#12011yescharacteristic
140C2×C4#12011yescharacteristic
141C2×C4#12011yescharacteristic
142C2×C4#12111yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12711yescharacteristic
145C2×C4C2×SD1612yes—
146C2×C4AGL(1,ℤ/8)12yes—
147C8—24no—
148C8—42no—
149C8—42no—
150C8—81no—
151C22—21no—
152C22—21no—
153C22—81no—
154C22—81no—
155C22—81no—
156C22#39211yescharacteristic
157C22#46012yes—
158C4—21no—
159C4—21no—
160C4—82no—
161C4—81no—
162C4—82no—
163C4#45012yes—
164C4#45512yes—
165C4#47312yes—
166C4#49211yescharacteristic
167C4#49611yescharacteristic
168C2—22no—
169C2—161no—
170C2#200411yescharacteristic
171C2#201411yescharacteristic
172C2#202011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
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
3. C2×#487C2semidirect
4. #1997C2semidirect
5. #1546C2semidirect
7. #1278C2semidirect
96. M16SD16semidirect

Maximal subgroups

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

subgroupindex
#32062
#31522
C2×#4872
#19972
#15462
#14922
#12782