Finite Group Databasebeta

← #9205

Subgroups — #9205

Order 256.

345 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 67 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#2617C211yesmaximal, characteristic
2#2312C211yesmaximal, characteristic, complemented
3#1519C212yesmaximal, complemented
4C2×#323C211yesmaximal, characteristic, complemented
5#1289C212yesmaximal, complemented
6C2×#121C2211yescharacteristic
7C2×#120—21no—
8C22×C22⋊C4C2211yescharacteristic
9C2×C22⋊C8C2212yes—
10#397C412yescomplemented
11#394—22no—
12#392C412yescomplemented
13#388—24no—
14#386—21no—
15#380C2211yescharacteristic
16#380C2212yes—
17#379—21no—
18C4×C22⋊C4—21no—
19#327—24no—
20#323—22no—
21C25D411yescharacteristic
22C22×D4—21no—
23C22×D4—22no—
24C22×D4C2×C411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4C2311yescharacteristic
27C22×C8—22no—
28#121—22no—
29#121—22no—
30#120—44no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4C2×C412yes—
44C2×C22⋊C4D412yes—
45C2×C42—22no—
46C22⋊C8—22no—
47C22⋊C8—22no—
48#95—22no—
49#95—22no—
50#95C2×C412yes—
51#95D414yescomplemented
52C24—21no—
53C24—21no—
54C24—22no—
55C24—22no—
56C24—21no—
57C24—81no—
58C24—81no—
59C24C22⋊C412yes—
60C24C2×D411yescharacteristic
61C2×D4—22no—
62C2×D4—44no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4—42no—
66C2×D4—42no—
67C2×D4—82no—
68C2×D4—82no—
69C2×D4—81no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4—24no—
75C22×C4—81no—
76C22×C4—81no—
77C22×C4C22⋊C412yes—
78C22×C4C22×C411yescharacteristic
79C22×C4C2×D412yes—
80C22×C4C2×D411yescharacteristic
81C22×C4#4112yes—
82C2×C8—42no—
83C2×C8—82no—
84C4⋊C4—22no—
85C4⋊C4—82no—
86C4⋊C4—81no—
87C22⋊C4—24no—
88C22⋊C4—24no—
89C22⋊C4—44no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C22⋊C4—42no—
93C22⋊C4—42no—
94C22⋊C4—82no—
95C22⋊C4—82no—
96C22⋊C4—82no—
97C42—82no—
98C23—22no—
99C23—22no—
100C23—22no—
101C23—41no—
102C23—41no—
103C23—42no—
104C23—42no—
105C23—42no—
106C23—42no—
107C23—42no—
108C23—42no—
109C23—42no—
110C23—42no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—81no—
115C23—81no—
116C23—84no—
117C23—84no—
118C23C2×C22⋊C411yescharacteristic
119C23#12011yescharacteristic
120C23#12311yescharacteristic
121C23#13512yes—
122C23AGL(1,ℤ/8)12yes—
123D4—82no—
124D4—88no—
125D4—82no—
126D4—82no—
127C2×C4—22no—
128C2×C4—41no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—44no—
135C2×C4—44no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—44no—
143C2×C4—81no—
144C2×C4C4×D412yes—
145C2×C4#12112yes—
146C8—82no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—42no—
154C22—42no—
155C22—44no—
156C22—42no—
157C22—42no—
158C22—42no—
159C22—82no—
160C22—84no—
161C22—84no—
162C22—82no—
163C22—82no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22#38611yescharacteristic
168C22#41611yescharacteristic
169C22#41811yescharacteristic
170C22#44912yes—
171C22#48212yes—
172C4—22no—
173C4—22no—
174C4—42no—
175C4—84no—
176C4—82no—
177C4—81no—
178C2—21no—
179C2—22no—
180C2—21no—
181C2—22no—
182C2—82no—
183C2—161no—
184C2#187511yescharacteristic
185C2#191111yescharacteristic
186C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#2617C2
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
2. #2312C2semidirect
3. #1519C2semidirect
4. C2×#323C2semidirect
5. #1289C2semidirect
10. #397C4semidirect
12. #392C4semidirect
51. #95D4semidirect

Maximal subgroups

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

subgroupindex
#26172
#23122
#15192
C2×#3232
#12892