Finite Group Databasebeta

← #8373

Subgroups — #8373

Order 256.

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

normal subgroups only

This group has 407 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×#553C211yesmaximal, characteristic
2C4×#100C211yesmaximal, characteristic, complemented
3#1528C212yesmaximal, complemented
4#1490C212yesmaximal, complemented
5#1466C211yesmaximal, characteristic, complemented
6#553—24no—
7#553—24no—
8C2×#126—21no—
9C2×#125C2211yescharacteristic
10C2×#123—21no—
11C2×#121—21no—
12C2×#121C411yescharacteristic
13C2×#121C411yescharacteristic
14C2×C4×D4—21no—
15C2×C4×D4C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17C2×#117C411yescharacteristic
18C22×C4⋊C4C411yescharacteristic
19#432C2212yes—
20C2×#100—21no—
21C4×M16C2212yes—
22#343—21no—
23#330—24no—
24D4⋊C8—24no—
25#323—21no—
26#323—21no—
27C22×D4—21no—
28C22×D4—21no—
29C22×D4C2×C411yescharacteristic
30C22×D4D411yescharacteristic
31C23×C4—21no—
32C23×C4—21no—
33C23×C4C2×C411yescharacteristic
34C23×C4C2×C411yescharacteristic
35C2×M16—22no—
36#126—28no—
37#125—22no—
38#125D414yes—
39#123—28no—
40#121—28no—
41#121—22no—
42#121—22no—
43#121—22no—
44#121—22no—
45C4×D4—28no—
46C4×D4—22no—
47C4×D4—22no—
48C4×D4—22no—
49C4×D4D414yes—
50#117—22no—
51#117—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4C2×C411yescharacteristic
58C2×C4⋊C4C2×C411yescharacteristic
59C2×C4⋊C4C2×C411yescharacteristic
60C2×C4⋊C4D411yescharacteristic
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4D411yescharacteristic
66C2×C22⋊C4D411yescharacteristic
67C2×C42—21no—
68C2×C42C2311yescharacteristic
69C4⋊C8—22no—
70#100—24no—
71C22⋊C8—22no—
72#97—22no—
73#97—22no—
74C4×C8—22no—
75C24—21no—
76C24C22⋊C411yescharacteristic
77C24C22⋊C411yescharacteristic
78C2×D4—24no—
79C2×D4—22no—
80C2×D4—24no—
81C2×D4—21no—
82C2×D4—22no—
83C2×D4—42no—
84C2×D4—41no—
85C2×D4—42no—
86C2×D4—42no—
87C2×D4—82no—
88C2×D4C22⋊C412yes—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—24no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—24no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—82no—
101C22×C4—82no—
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—21no—
114C4⋊C4—21no—
115C4⋊C4—24no—
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4C22⋊C412yes—
119C4⋊C4C22⋊C412yes—
120C4⋊C4C22⋊C412yes—
121C22⋊C4—24no—
122C22⋊C4—24no—
123C22⋊C4—24no—
124C22⋊C4—24no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C42—24no—
128C42C2×D412yes—
129C42C2×D412yes—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—22no—
134C23—21no—
135C23—21no—
136C23—41no—
137C23—41no—
138C23—42no—
139C23—42no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23#10412yes—
145C23C2×C22⋊C411yescharacteristic
146D4—24no—
147D4—44no—
148D4—42no—
149D4—48no—
150D4—84no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—84no—
171C2×C4—84no—
172C2×C4C2×C22⋊C411yescharacteristic
173C2×C4C2×C22⋊C411yescharacteristic
174C2×C4#12012yes—
175C2×C4#12012yes—
176C8—44no—
177C8—82no—
178C22—21no—
179C22—21no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22—41no—
185C22—42no—
186C22—41no—
187C22—41no—
188C22—42no—
189C22—41no—
190C22—42no—
191C22—81no—
192C22—81no—
193C22—82no—
194C22—82no—
195C22#37911yescharacteristic
196C22C2×#10411yescharacteristic
197C22#42111yescharacteristic
198C4—22no—
199C4—21no—
200C4—22no—
201C4—42no—
202C4—42no—
203C4—41no—
204C4—41no—
205C4—42no—
206C4#41312yes—
207C2—21no—
208C2—21no—
209C2—42no—
210C2—81no—
211C2—82no—
212C2#178511yescharacteristic
213C2#178911yescharacteristic
214C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#553C2
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×#100C2semidirect
3. #1528C2semidirect
4. #1490C2semidirect
5. #1466C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5532
C4×#1002
#15282
#14902
#14662