Finite Group Databasebeta

← #9446

Subgroups — #9446

Order 256.

258 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 59 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#2921C211yesmaximal, characteristic, complemented
2#2500C211yesmaximal, characteristic
3#1820C211yesmaximal, characteristic, complemented
4#1510C212yesmaximal
5#1284C212yesmaximal, complemented
6C2×#121C2211yescharacteristic
7C2×#117C2211yescharacteristic
8#433—24no—
9#423—21no—
10#407C2212yes—
11C4×M16—21no—
12#394—21no—
13#394—21no—
14#390C411yescharacteristic, complemented
15#390C411yescharacteristic, complemented
16#388—21no—
17#388—21no—
18#384C411yescharacteristic, complemented
19#382C411yescharacteristic, complemented
20#381C2212yes—
21C4×C22⋊C4C2211yescharacteristic
22#342—21no—
23#342—21no—
24#327—24no—
25C22×D4—21no—
26C22×D4C2×C411yescharacteristic
27C23×C4C2311yescharacteristic
28C2×M16—22no—
29#121—22no—
30#121—22no—
31#117—22no—
32#117—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4D411yescharacteristic
46C2×C22⋊C4D411yescharacteristic
47C2×C42C2×C411yescharacteristic
48C2×C42C2×C411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42D411yescharacteristic
51C4⋊C8—24no—
52C22⋊C8—22no—
53C22⋊C8D414yes—
54#97—22no—
55C4×C8—22no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—22no—
60#95—22no—
61C24—21no—
62C24C2×D411yescharacteristic
63C2×D4—22no—
64C2×D4—41no—
65C2×D4—41no—
66C2×D4—42no—
67C2×D4—81no—
68C2×D4—81no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—42no—
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22×C411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D412yes—
82C22×C4#4112yes—
83M16—42no—
84M16—42no—
85C2×C8—24no—
86C4⋊C4—22no—
87C4⋊C4—22no—
88C4⋊C4—22no—
89C4⋊C4—81no—
90C4⋊C4—81no—
91C22⋊C4—22no—
92C22⋊C4—22no—
93C22⋊C4—42no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C22⋊C4—81no—
101C22⋊C4—81no—
102C42—21no—
103C42—21no—
104C42—21no—
105C42—21no—
106C42—41no—
107C42—41no—
108C23—21no—
109C23—21no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23C2×C22⋊C411yescharacteristic
120C23#12112yes—
121D4—82no—
122D4—84no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—42no—
133C2×C4—42no—
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—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4C4×D412yes—
150C2×C4#12011yescharacteristic
151C2×C4#12311yescharacteristic
152C8—44no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—22no—
157C22—22no—
158C22—81no—
159C22—81no—
160C22—81no—
161C22—81no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22#38611yescharacteristic
168C22#41011yescharacteristic
169C22#41011yescharacteristic
170C4—21no—
171C4—22no—
172C4—21no—
173C4—42no—
174C4—41no—
175C4—41no—
176C4—41no—
177C4—41no—
178C4—81no—
179C4—81no—
180C4—162no—
181C2—21no—
182C2—21no—
183C2—42no—
184C2—161no—
185C2#188411yescharacteristic
186C2#189911yescharacteristic
187C2#191011yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2921C2semidirect
3. #1820C2semidirect
5. #1284C2semidirect
14. #390C4semidirect
15. #390C4semidirect
18. #384C4semidirect
19. #382C4semidirect

Maximal subgroups

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

subgroupindex
#29212
#25002
#18202
#15102
#12842