Finite Group Databasebeta

← #12244

Subgroups — #12244

Order 256.

199 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 37 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#2722C211yesmaximal, characteristic
2#2062C211yesmaximal, characteristic
3#1608C211yesmaximal, characteristic
4#1608C211yesmaximal, characteristic
5#1344C211yesmaximal, characteristic
6#1344C211yesmaximal, characteristic
7#1326C211yesmaximal, characteristic
8#543C2211yescharacteristic
9C2×#126—21no—
10#483—21no—
11#483—21no—
12#398—21no—
13#398—21no—
14#397C2211yescharacteristic
15#397C2211yescharacteristic
16#389—21no—
17#387—21no—
18#385—21no—
19#385C2211yescharacteristic
20#380—21no—
21#342—21no—
22#342C2211yescharacteristic
23#328—21no—
24#328—21no—
25#323C2211yescharacteristic
26#323C2211yescharacteristic
27C23×C4D411yescharacteristic
28#126—42no—
29#126—42no—
30C22⋊Q8—21no—
31C22⋊Q8—21no—
32C22⋊Q8—41no—
33C22⋊Q8—41no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—41no—
41C2×C4⋊C4D411yescharacteristic
42C2×C4⋊C4D411yescharacteristic
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4D411yescharacteristic
48C2×C22⋊C4C2311yescharacteristic
49C2×C42—21no—
50#106—41no—
51#106—41no—
52#103—41no—
53#103—41no—
54C22⋊C8—21no—
55C22⋊C8—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—41no—
62#95D411yescharacteristic
63#95D411yescharacteristic
64C24C2×D411yescharacteristic
65C2×Q8—21no—
66C2×Q8—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—42no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4C2×D411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C2×C8—41no—
81C2×C8—41no—
82C4⋊C4—21no—
83C4⋊C4—21no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C4⋊C4—42no—
90C4⋊C4—42no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—81no—
94C4⋊C4—82no—
95C22⋊C4—21no—
96C22⋊C4—22no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—41no—
105C22⋊C4SD1612yes—
106C42—42no—
107C23—21no—
108C23—21no—
109C23—41no—
110C23—41no—
111C23#12011yescharacteristic
112C23C2×SD1611yescharacteristic
113C23AGL(1,ℤ/8)11yescharacteristic
114Q8—81no—
115Q8—81no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—42no—
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—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4—81no—
147C8—81no—
148C8—81no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—41no—
154C22—41no—
155C22—41no—
156C22—41no—
157C22—81no—
158C22#45011yescharacteristic
159C22#45611yescharacteristic
160C22B(4,𝔽2)11yescharacteristic
161C4—42no—
162C4—41no—
163C4—42no—
164C4—41no—
165C4—81no—
166C4—81no—
167C4—82no—
168C4—81no—
169C4—81no—
170C4—81no—
171C4—81no—
172C2—21no—
173C2—21no—
174C2—41no—
175C2—41no—
176C2#160411yescharacteristic
177C2#219711yescharacteristic
178C2#220111yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radicalC24C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#27222
#20622
#16082
#13442
#13262