Finite Group Databasebeta

← #7788

Subgroups — #7788

Order 256.

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

normal subgroups only

This group has 209 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#2724C211yesmaximal, characteristic
2#2099C211yesmaximal, characteristic, complemented
3#2083C211yesmaximal, characteristic, complemented
4#2071C211yesmaximal, characteristic
5#1271C211yesmaximal, characteristic, complemented
6#1271C211yesmaximal, characteristic, complemented
7#1270C211yesmaximal, characteristic
8C2×#123—21no—
9C2×#107—21no—
10C2×#102—21no—
11#398—21no—
12#397—21no—
13#397—21no—
14#390C2211yescharacteristic
15#389C2211yescharacteristic
16#389C2211yescharacteristic
17#385—21no—
18#385—21no—
19#384C2211yescharacteristic
20#340—21no—
21#340—21no—
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×D4D411yescharacteristic
28C23×C4—21no—
29C22×C8—21no—
30C22×C8—21no—
31C22×C8—21no—
32#123—42no—
33#123—42no—
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⋊C4D411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C42C2311yescharacteristic
51#107—42no—
52#102—42no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58C24—21no—
59C2×D4—21no—
60C2×D4—41no—
61C2×D4—41no—
62C2×D4D812yes—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—42no—
69C22×C4—41no—
70C22×C4—81no—
71C22×C4—81no—
72C22×C4C2×D411yescharacteristic
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C2×C8—22no—
76C2×C8—41no—
77C2×C8—41no—
78C2×C8—41no—
79C2×C8—41no—
80C2×C8—41no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—41no—
84C4⋊C4—21no—
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—21no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—42no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—81no—
101C22⋊C4—42no—
102C22⋊C4—42no—
103C22⋊C4—42no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C42—41no—
109C23—22no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23#12011yescharacteristic
117D4—42no—
118D4—81no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
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—41no—
139C2×C4—41no—
140C2×C4—84no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4C2×D811yescharacteristic
145C2×C4#13511yescharacteristic
146C2×C4#13511yescharacteristic
147C2×C4AGL(1,ℤ/8)11yescharacteristic
148C2×C4AGL(1,ℤ/8)11yescharacteristic
149C2×C4#13711yescharacteristic
150C8—42no—
151C8—81no—
152C8—81no—
153C22—42no—
154C22—42no—
155C22—42no—
156C22—81no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22#44711yescharacteristic
161C22#44911yescharacteristic
162C22#45211yescharacteristic
163C22#45411yescharacteristic
164C22#45511yescharacteristic
165C22#45611yescharacteristic
166C22#45811yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—81no—
174C4—81no—
175C4—81no—
176C4—81no—
177C4—81no—
178C4—161no—
179C2—82no—
180C2#160711yescharacteristic
181C2#160911yescharacteristic
182C2#161511yescharacteristic
183C2#162511yescharacteristic
184C2#164011yescharacteristic
185C2#164711yescharacteristic
186C2#166411yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
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
2. #2099C2semidirect
3. #2083C2semidirect
5. #1271C2semidirect
6. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#27242
#20992
#20832
#20712
#12712
#12702