Finite Group Databasebeta

← #7783

Subgroups — #7783

Order 256.

212 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 49 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#2805C211yesmaximal, characteristic
2#2058C211yesmaximal, characteristic
3#2057C211yesmaximal, characteristic
4#2016C211yesmaximal, characteristic
5#1272C211yesmaximal, characteristic
6#1271C211yesmaximal, characteristic
7#1271C211yesmaximal, characteristic
8C22×Q16—21no—
9C2×C4⋊Q8C2211yescharacteristic
10C2×#107—21no—
11C2×#106—21no—
12C2×#103—21no—
13#400—21no—
14#400—21no—
15#395—21no—
16#391C2211yescharacteristic
17#389—21no—
18#389C2211yescharacteristic
19#389C2211yescharacteristic
20#387—21no—
21#387—21no—
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8—21no—
28C22×Q8D411yescharacteristic
29C2×Q16—42no—
30C2×Q16—42no—
31C2×Q16—42no—
32C22×C8—21no—
33C22×C8—21no—
34C22×C8—21no—
35C4⋊Q8—42no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C4⋊C4D411yescharacteristic
47C2×C42—21no—
48C2×C42C2311yescharacteristic
49#107—42no—
50#106—42no—
51#103—42no—
52#95—21no—
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—22no—
62C2×Q8—21no—
63C2×Q8—21no—
64C2×Q8—41no—
65C2×Q8—41no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×Q8—41no—
69C2×Q8—41no—
70C2×Q8—41no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—41no—
78C22×C4C2×D411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81Q16—48no—
82C2×C8—22no—
83C2×C8—22no—
84C2×C8—22no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8—41no—
88C2×C8—41no—
89C2×C8—41no—
90C2×C8—41no—
91C4⋊C4—21no—
92C4⋊C4—22no—
93C4⋊C4—22no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—41no—
97C4⋊C4—41no—
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⋊C4Q1612yescomplemented
108C42—41no—
109C42—81no—
110C23#12011yescharacteristic
111Q8—42no—
112Q8—42no—
113Q8—42no—
114Q8—81no—
115Q8—81no—
116Q8—81no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—22no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
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—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4C2×Q1611yescharacteristic
145C2×C4#13511yescharacteristic
146C2×C4#13511yescharacteristic
147C2×C4AGL(1,ℤ/8)11yescharacteristic
148C2×C4#13711yescharacteristic
149C2×C4#13711yescharacteristic
150C8—42no—
151C8—42no—
152C8—42no—
153C22#44911yescharacteristic
154C22#45111yescharacteristic
155C22#45211yescharacteristic
156C22#45311yescharacteristic
157C22#45611yescharacteristic
158C22#45611yescharacteristic
159C22B(4,𝔽2)11yescharacteristic
160C4—21no—
161C4—21no—
162C4—21no—
163C4—21no—
164C4—21no—
165C4—21no—
166C4—42no—
167C4—81no—
168C4—81no—
169C4—82no—
170C4—81no—
171C4—81no—
172C4—81no—
173C2#160011yescharacteristic
174C2#160311yescharacteristic
175C2#160511yescharacteristic
176C2#163211yescharacteristic
177C2#164811yescharacteristic
178C2#165211yescharacteristic
179C2#166011yescharacteristic
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
107. C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#28052
#20582
#20572
#20162
#12722
#12712