Finite Group Databasebeta

← #10570

Subgroups — #10570

Order 256.

302 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 69 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#2659C211yesmaximal, characteristic
2#1853C212yesmaximal
3#1444C211yesmaximal, characteristic
4#1294C212yesmaximal
5#1280C211yesmaximal, characteristic
6C2×C22⋊Q8—21no—
7C22×C4⋊C4—21no—
8C22×C4⋊C4C2211yescharacteristic
9C2×#106—22no—
10C2×C22⋊C8C2212yes—
11#398—22no—
12#395—22no—
13#380—21no—
14#380C2212yes—
15C2×#95—21no—
16C2×#95C2211yescharacteristic
17C2×#95C2211yescharacteristic
18#340—22no—
19C22×Q8—21no—
20C23×C4—21no—
21C23×C4—21no—
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—22no—
26C22⋊Q8—42no—
27C22⋊Q8—42no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—42no—
34C2×C4⋊C4—42no—
35C2×C4⋊C4—42no—
36C2×C4⋊C4—42no—
37C2×C4⋊C4D412yes—
38C2×C4⋊C4D412yes—
39C2×C22⋊C4—22no—
40#106—44no—
41C22⋊C8—22no—
42C22⋊C8Q814yescomplemented
43#95—22no—
44#95—22no—
45#95—22no—
46#95—21no—
47#95—42no—
48#95Q812yescomplemented
49C24C2×D411yescharacteristic
50C2×Q8—42no—
51C2×Q8—81no—
52C2×Q8—81no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—22no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—22no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4SD1612yes—
80C22×C4SD1612yes—
81C22×C4C2×D412yes—
82C22×C4C2×Q812yes—
83C22×C4C2×Q811yescharacteristic
84C22×C4#4111yescharacteristic
85C2×C8—24no—
86C2×C8—42no—
87C2×C8—42no—
88C4⋊C4—24no—
89C4⋊C4—42no—
90C4⋊C4—42no—
91C4⋊C4—42no—
92C4⋊C4—42no—
93C4⋊C4—44no—
94C4⋊C4—48no—
95C4⋊C4—81no—
96C4⋊C4—81no—
97C22⋊C4—42no—
98C22⋊C4—42no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—21no—
103C23#12011yescharacteristic
104C23C22⋊Q811yescharacteristic
105C23C4⋊Q811yescharacteristic
106C23C2×SD1611yescharacteristic
107C23C2×SD1611yescharacteristic
108C23AGL(1,ℤ/8)11yescharacteristic
109C23#13711yescharacteristic
110Q8—84no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—24no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4C22⋊Q812yes—
147C2×C4C4⋊Q812yes—
148C8—44no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22#39511yescharacteristic
162C22#44811yescharacteristic
163C22#45011yescharacteristic
164C22#45311yescharacteristic
165C22#45611yescharacteristic
166C22B(4,𝔽2)11yescharacteristic
167C22#45811yescharacteristic
168C22#47511yescharacteristic
169C22#47611yescharacteristic
170C22#49811yescharacteristic
171C22#50111yescharacteristic
172C4—22no—
173C4—22no—
174C4—44no—
175C4—42no—
176C4—42no—
177C4—82no—
178C4—82no—
179C4—82no—
180C2—21no—
181C2—21no—
182C2—21no—
183C2—21no—
184C2#159611yescharacteristic
185C2#160411yescharacteristic
186C2#161111yescharacteristic
187C2#161311yescharacteristic
188C2#205811yescharacteristic
189C2#206011yescharacteristic
190C2#206211yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
42. C22⋊C8Q8semidirect
48. #95Q8semidirect

Maximal subgroups

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

subgroupindex
#26592
#18532
#14442
#12942
#12802