Finite Group Databasebeta

← #9494

Subgroups — #9494

Order 256.

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

normal subgroups only

This group has 257 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#2598C211yesmaximal, characteristic
3#1490C212yesmaximal, complemented
4#1464C211yesmaximal, characteristic, complemented
5#1288C212yesmaximal, complemented
6C2×C4×D4C2211yescharacteristic
7C2×#117C2211yescharacteristic
8#433—24no—
9#432C2212yes—
10#423—21no—
11#407—21no—
12C4×M16C2212yes—
13#400C411yescharacteristic
14#399C411yescharacteristic
15#394C411yescharacteristic
16#392C411yescharacteristic
17#388—21no—
18#386—21no—
19#386—21no—
20#385—21no—
21#383C2211yescharacteristic
22D4⋊C8—24no—
23#324—21no—
24#323—21no—
25C22×D4—21no—
26C22×D4C2×C411yescharacteristic
27C23×C4—21no—
28C23×C4D411yescharacteristic
29C2×M16—22no—
30C4×D4—22no—
31C4×D4—22no—
32#117—22no—
33#117—22no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4D411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4C2×C411yescharacteristic
44C2×C22⋊C4D411yescharacteristic
45C2×C42D411yescharacteristic
46C2×C42C2311yescharacteristic
47C4⋊C8—24no—
48C4⋊C8—22no—
49C22⋊C8—24no—
50C22⋊C8—22no—
51#97—22no—
52#97—22no—
53C4×C8D414yescomplemented
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95C2×C411yescharacteristic
59#95C2×C411yescharacteristic
60C24—21no—
61C24C22⋊C411yescharacteristic
62C2×D4—22no—
63C2×D4—42no—
64C2×D4—42no—
65C2×D4—42no—
66C2×D4—81no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—42no—
75C22×C4—81no—
76C22×C4—81no—
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83M16—42no—
84M16—42no—
85C2×C8—24no—
86C2×C8—42no—
87C4⋊C4—22no—
88C4⋊C4—22no—
89C4⋊C4—22no—
90C4⋊C4—81no—
91C22⋊C4—22no—
92C22⋊C4—22no—
93C22⋊C4—42no—
94C22⋊C4—42no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—81no—
98C22⋊C4—81no—
99C42—21no—
100C42—21no—
101C42C2×D412yes—
102C42#4112yes—
103C23—22no—
104C23—21no—
105C23—21no—
106C23—21no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23C2×C22⋊C411yescharacteristic
116D4—24no—
117D4—42no—
118D4—84no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—42no—
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—84no—
142C2×C4C4×D412yes—
143C2×C4#12011yescharacteristic
144C2×C4#12112yes—
145C2×C4#12311yescharacteristic
146C8—44no—
147C8—82no—
148C22—21no—
149C22—21no—
150C22—42no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22—41no—
155C22—41no—
156C22—41no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22—81no—
161C22—81no—
162C22#38611yescharacteristic
163C22#42111yescharacteristic
164C22#42111yescharacteristic
165C4—22no—
166C4—21no—
167C4—21no—
168C4—22no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—81no—
173C4—81no—
174C4—81no—
175C2—21no—
176C2—21no—
177C2—81no—
178C2—82no—
179C2#189111yescharacteristic
180C2#189711yescharacteristic
181C2#191011yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#392C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2921C2semidirect
3. #1490C2semidirect
4. #1464C2semidirect
5. #1288C2semidirect
53. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
#29212
#25982
#14902
#14642
#12882