Finite Group Databasebeta

← #9679

Subgroups — #9679

Order 256.

343 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 99 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#2857C211yesmaximal, characteristic
2C4×C22⋊C8C211yesmaximal, characteristic
3C2×#328C214yesmaximal
4C4⋊#95C211yesmaximal, characteristic
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×C22⋊Q8C2212yes—
7C22×C4⋊C4—21no—
8C22×C42C2211yescharacteristic
9C2×C22⋊C8C2212yes—
10C2×C4×C8—21no—
11#391C412yescomplemented
12#389—21no—
13#389—21no—
14#384C411yescharacteristic, complemented
15#380—21no—
16#380—21no—
17C2×#95—21no—
18C2×#95C2212yes—
19#336—21no—
20#328—28no—
21C22×Q8C2×C412yes—
22C23×C4—22no—
23C23×C4D411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4C2311yescharacteristic
26C22×C8—22no—
27C4⋊Q8—42no—
28C22⋊Q8—24no—
29C22⋊Q8—24no—
30C2×C4⋊C4—24no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—42no—
37C2×C4⋊C4—42no—
38C2×C4⋊C4C2×C412yes—
39C2×C22⋊C4—22no—
40C2×C42—42no—
41C2×C42C2×C411yescharacteristic
42C2×C42C2×C411yescharacteristic
43C2×C42D411yescharacteristic
44C2×C42D411yescharacteristic
45C22⋊C8—22no—
46C22⋊C8D414yes—
47C4×C8—42no—
48#95—22no—
49#95—22no—
50#95—24no—
51#95—42no—
52C24C2×D411yescharacteristic
53C2×Q8—24no—
54C2×Q8—42no—
55C2×Q8—42no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—24no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—41no—
62C22×C4—41no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4SD1612yes—
76C22×C4SD1612yes—
77C22×C4Q1612yes—
78C22×C4Q1612yes—
79C22×C4C22×C411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D412yes—
82C22×C4#4112yes—
83C2×C8—24no—
84C2×C8—42no—
85C2×C8—42no—
86C4⋊C4—24no—
87C4⋊C4—42no—
88C4⋊C4—48no—
89C4⋊C4—42no—
90C4⋊C4—42no—
91C4⋊C4—42no—
92C4⋊C4—44no—
93C22⋊C4—42no—
94C22⋊C4—42no—
95C42—22no—
96C42—22no—
97C42—41no—
98C42—41no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—21no—
103C23C2×C22⋊C411yescharacteristic
104C23#12411yescharacteristic
105C23#12711yescharacteristic
106C23C2×SD1611yescharacteristic
107C23C2×SD1611yescharacteristic
108C23C2×Q1611yescharacteristic
109C23C2×Q1611yescharacteristic
110Q8—84no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—24no—
122C2×C4—24no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—44no—
128C2×C4—44no—
129C2×C4—44no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—82no—
137C2×C4—82no—
138C2×C4—82no—
139C2×C4—82no—
140C2×C4#9912yes—
141C2×C4#10312yes—
142C2×C4#10312yes—
143C2×C4#10312yes—
144C2×C4#10312yes—
145C2×C4#10412yes—
146C2×C4C4×D412yes—
147C2×C4#12112yes—
148C8—44no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22#39011yescharacteristic
162C22C2×#9911yescharacteristic
163C22#41011yescharacteristic
164C22C2×#10311yescharacteristic
165C22C2×#10311yescharacteristic
166C22C2×#10411yescharacteristic
167C22#42111yescharacteristic
168C22#48711yescharacteristic
169C22#48711yescharacteristic
170C22#49211yescharacteristic
171C22#49411yescharacteristic
172C4—22no—
173C4—22no—
174C4—22no—
175C4—22no—
176C4—22no—
177C4—84no—
178C4—82no—
179C4—82no—
180C4#32814yescomplemented
181C2—21no—
182C2—21no—
183C2—21no—
184C2—21no—
185C2C2×#32811yescharacteristic
186C2#150311yescharacteristic
187C2#151811yescharacteristic
188C2#152011yescharacteristic
189C2#195611yescharacteristic
190C2#196411yescharacteristic
191C2#198011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊Q8C4semidirect
11. #391C4semidirect
14. #384C4semidirect
180. C4#328semidirect

Maximal subgroups

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

subgroupindex
#28572
C4×C22⋊C82
C2×#3282
C4⋊#952