Finite Group Databasebeta

← #7617

Subgroups — #7617

Order 256.

275 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C4×C22⋊Q8C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic
3#1728C211yesmaximal, characteristic
4#1272C212yesmaximal, complemented
5#1271C212yesmaximal, complemented
6C22×M16—21no—
7C2×C22⋊Q8C411yescharacteristic
8C2×C4×Q8C411yescharacteristic
9C22×C42C2211yescharacteristic
10C2×C4⋊C8—21no—
11C2×C22⋊C8—21no—
12#391C2211yescharacteristic
13#389C2211yescharacteristic
14#387—21no—
15#385—21no—
16#380C411yescharacteristic
17C4×C4⋊C4—21no—
18C4×C4⋊C4C411yescharacteristic
19C4×C22⋊C4—21no—
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C22×Q8C2×C411yescharacteristic
24C23×C4C2×C411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4D411yescharacteristic
27C2×M16—24no—
28C2×M16—42no—
29C2×M16—42no—
30C22×C8—22no—
31C22×C8—22no—
32C22⋊Q8—22no—
33C22⋊Q8—22no—
34C4×Q8—42no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C42—21no—
44C2×C42—21no—
45C2×C42—42no—
46C2×C42C2×C411yescharacteristic
47C2×C42D411yescharacteristic
48C2×C42D411yescharacteristic
49C2×C42C2311yescharacteristic
50C4⋊C8—42no—
51C22⋊C8—42no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57C24C22⋊C411yescharacteristic
58C2×Q8—21no—
59C2×Q8—41no—
60C2×Q8—41no—
61C2×Q8M1612yescomplemented
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4SD1612yes—
80C22×C4Q1612yes—
81C22×C4C22×C411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84M16—48no—
85C2×C8—24no—
86C2×C8—42no—
87C2×C8—42no—
88C2×C8—42no—
89C2×C8—42no—
90C2×C8—42no—
91C4⋊C4—24no—
92C4⋊C4—21no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—42no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4M1612yescomplemented
99C22⋊C4—24no—
100C22⋊C4—42no—
101C42—22no—
102C42—22no—
103C42—41no—
104C42—41no—
105C42—42no—
106C42—42no—
107C23—21no—
108C23—21no—
109C23—21no—
110C23—21no—
111C23—21no—
112C23—21no—
113C23#10312yes—
114C23C2×C22⋊C411yescharacteristic
115Q8—42no—
116Q8—42no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—24no—
124C2×C4—24no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4#9912yes—
148C2×C4#10312yes—
149C2×C4#10412yes—
150C2×C4C2×M1611yescharacteristic
151C2×C4C2×M1611yescharacteristic
152C2×C4C2×SD1611yescharacteristic
153C2×C4C2×Q1611yescharacteristic
154C2×C4AGL(1,ℤ/8)11yescharacteristic
155C2×C4#13711yescharacteristic
156C8—44no—
157C8—82no—
158C22—22no—
159C22—22no—
160C22—22no—
161C22—41no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#40711yescharacteristic
166C22C2×#9911yescharacteristic
167C22#41311yescharacteristic
168C22C2×#10311yescharacteristic
169C22#41711yescharacteristic
170C22C2×#10411yescharacteristic
171C22#42111yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—41no—
179C4—42no—
180C4—42no—
181C4—41no—
182C4—81no—
183C4—81no—
184C4#32814yes—
185C2—42no—
186C2#146511yescharacteristic
187C2#148711yescharacteristic
188C2#148911yescharacteristic
189C2C2×#32811yescharacteristic
190C2#151311yescharacteristic
191C2#152911yescharacteristic
192C2#155011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1272C2semidirect
5. #1271C2semidirect
61. C2×Q8M16semidirect
98. C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊Q82
#17972
#17282
#12722
#12712