Finite Group Databasebeta

← #7784

Subgroups — #7784

Order 256.

218 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 51 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#2806C211yesmaximal, characteristic
2#2057C211yesmaximal, characteristic
3#2024C211yesmaximal, characteristic
4#2016C211yesmaximal, characteristic
5#1272C211yesmaximal, characteristic
6#1272C211yesmaximal, characteristic
7#1271C211yesmaximal, characteristic
8C22×Q16—21no—
9C2×C4⋊Q8C2211yescharacteristic
10C2×#107—21no—
11C2×#103—21no—
12C2×#103—21no—
13#398—21no—
14#398—21no—
15#395—21no—
16#391—21no—
17#391C2211yescharacteristic
18#391C2211yescharacteristic
19#389—21no—
20#389C2211yescharacteristic
21#387—21no—
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8—21no—
28C22×Q8D411yescharacteristic
29C22×Q8D411yescharacteristic
30C2×Q16—42no—
31C2×Q16—42no—
32C2×Q16—42no—
33C22×C8—21no—
34C22×C8—21no—
35C22×C8—21no—
36C4⋊Q8—42no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C4⋊C4D411yescharacteristic
47C2×C4⋊C4D411yescharacteristic
48C2×C42—21no—
49C2×C42C2311yescharacteristic
50#107—42no—
51#103—42no—
52#103—42no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61C2×Q8—21no—
62C2×Q8—22no—
63C2×Q8—21no—
64C2×Q8—21no—
65C2×Q8—41no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×Q8—41no—
69C2×Q8—41no—
70C2×Q8—41no—
71C2×Q8—41no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×Q8Q1612yescomplemented
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—41no—
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85Q16—48no—
86C2×C8—22no—
87C2×C8—22no—
88C2×C8—41no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8—41no—
92C2×C8—41no—
93C2×C8—41no—
94C2×C8—41no—
95C4⋊C4—21no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—81no—
106C4⋊C4—81no—
107C4⋊C4—81no—
108C4⋊C4—81no—
109C4⋊C4Q1612yescomplemented
110C42—41no—
111C42—81no—
112C23#12011yescharacteristic
113Q8—42no—
114Q8—42no—
115Q8—42no—
116Q8—42no—
117Q8—81no—
118Q8—81no—
119Q8—81no—
120Q8—81no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4C2×Q1611yescharacteristic
149C2×C4C2×Q1611yescharacteristic
150C2×C4#13511yescharacteristic
151C2×C4AGL(1,ℤ/8)11yescharacteristic
152C2×C4#13711yescharacteristic
153C2×C4#13711yescharacteristic
154C8—42no—
155C8—42no—
156C8—81no—
157C22#44911yescharacteristic
158C22#45111yescharacteristic
159C22#45111yescharacteristic
160C22#45311yescharacteristic
161C22#45511yescharacteristic
162C22#45611yescharacteristic
163C22B(4,𝔽2)11yescharacteristic
164C4—21no—
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—42no—
171C4—42no—
172C4—81no—
173C4—81no—
174C4—82no—
175C4—81no—
176C4—81no—
177C2#160011yescharacteristic
178C2#160211yescharacteristic
179C2#160511yescharacteristic
180C2#163311yescharacteristic
181C2#164411yescharacteristic
182C2#164811yescharacteristic
183C2#166111yescharacteristic
subgroupG/H
center, socle, prime radicalC23#120
commutator, Frattini, cosocleC2×C42C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
74. C2×Q8Q16semidirect
109. C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#28062
#20572
#20242
#20162
#12722
#12712