Finite Group Databasebeta

← #8370

Subgroups — #8370

Order 256.

404 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#559C211yesmaximal, characteristic
2C4×#101C211yesmaximal, characteristic, complemented
3#1527C212yesmaximal, complemented
4#1490C212yesmaximal, complemented
5#1467C211yesmaximal, characteristic, complemented
6#559—24no—
7#559—24no—
8C2×#127C411yescharacteristic
9C2×#124—21no—
10C2×#124C411yescharacteristic
11C2×#124C2211yescharacteristic
12C2×#123—21no—
13C2×#123—21no—
14C2×C4×Q8C411yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4—21no—
17C2×C4×D4C411yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#432C2212yes—
20C2×#101—21no—
21C4×M16C2212yes—
22#343—21no—
23#329—24no—
24D4⋊C8—24no—
25#324—21no—
26#324—21no—
27C22×Q8C2×C411yescharacteristic
28C22×Q8D411yescharacteristic
29C22×D4—21no—
30C22×D4—21no—
31C22×D4C2×C411yescharacteristic
32C22×D4C2×C411yescharacteristic
33C23×C4—21no—
34C23×C4—21no—
35C2×M16—22no—
36#127—22no—
37#127—22no—
38#124—28no—
39#124—22no—
40#124—22no—
41#124—22no—
42#124D414yes—
43#123—28no—
44#123—28no—
45C4×Q8—22no—
46C4×Q8—22no—
47C4×Q8—22no—
48C4×Q8—22no—
49C4×D4—28no—
50C4×D4—22no—
51C4×D4—22no—
52C4×D4—22no—
53C4×D4D414yes—
54C2×C4⋊C4—21no—
55C2×C4⋊C4C2×C411yescharacteristic
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C42—21no—
66C2×C42C2×C411yescharacteristic
67C2×C42C2×C411yescharacteristic
68C2×C42C2311yescharacteristic
69C4⋊C8—22no—
70#101—24no—
71C22⋊C8—22no—
72#97—22no—
73#97—22no—
74C4×C8—22no—
75C24—21no—
76C24—21no—
77C2×Q8—22no—
78C2×Q8—21no—
79C2×Q8—22no—
80C2×Q8—41no—
81C2×Q8—42no—
82C2×Q8C22⋊C412yes—
83C2×D4—24no—
84C2×D4—24no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—42no—
88C2×D4—42no—
89C2×D4—42no—
90C2×D4—42no—
91C2×D4—82no—
92C2×D4C22⋊C412yes—
93C2×D4C22⋊C412yes—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—24no—
97C22×C4—24no—
98C22×C4—82no—
99C22×C4—82no—
100C22×C4C22⋊C411yescharacteristic
101C22×C4C22⋊C411yescharacteristic
102C22×C4C22⋊C411yescharacteristic
103C22×C4C22⋊C411yescharacteristic
104C22×C4C22×C411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107M16—24no—
108M16—42no—
109C2×C8—42no—
110C2×C8—42no—
111C4⋊C4—24no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4C22⋊C412yes—
117C22⋊C4—24no—
118C22⋊C4—24no—
119C22⋊C4—24no—
120C22⋊C4—24no—
121C22⋊C4—24no—
122C22⋊C4—24no—
123C42—24no—
124C42—21no—
125C42—21no—
126C42—21no—
127C42—21no—
128C42C2×D412yes—
129C42C2×D412yes—
130C23—22no—
131C23—22no—
132C23—42no—
133C23—42no—
134C23—42no—
135C23—42no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23C2×C22⋊C411yescharacteristic
141Q8—44no—
142Q8—42no—
143Q8—42no—
144D4—24no—
145D4—42no—
146D4—42no—
147D4—48no—
148D4—42no—
149D4—84no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—84no—
168C2×C4—84no—
169C2×C4#10412yes—
170C2×C4C2×C22⋊C411yescharacteristic
171C2×C4C2×C22⋊C411yescharacteristic
172C2×C4#12012yes—
173C2×C4#12012yes—
174C8—44no—
175C8—82no—
176C22—21no—
177C22—21no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—82no—
185C22—82no—
186C22—82no—
187C22—82no—
188C22#37911yescharacteristic
189C22C2×#10411yescharacteristic
190C22#42111yescharacteristic
191C4—22no—
192C4—21no—
193C4—22no—
194C4—22no—
195C4—42no—
196C4—41no—
197C4—41no—
198C4—42no—
199C4—41no—
200C4#41312yes—
201C2—21no—
202C2—21no—
203C2—82no—
204C2—82no—
205C2#178511yescharacteristic
206C2#178911yescharacteristic
207C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#127C4
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
2. C4×#101C2semidirect
3. #1527C2semidirect
4. #1490C2semidirect
5. #1467C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5592
C4×#1012
#15272
#14902
#14672