Finite Group Databasebeta

← #7634

Subgroups — #7634

Order 256.

278 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 69 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#1749C211yesmaximal, characteristic
3#1728C211yesmaximal, characteristic
4#1272C212yesmaximal, complemented
5#1271C212yesmaximal, complemented
6C22×M16—21no—
7C2×C22⋊Q8C411yescharacteristic
8C2×C4×Q8C411yescharacteristic
9C22×C42C2211yescharacteristic
10C2×C22⋊C8—21no—
11C2×#97—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
50C22⋊C8—24no—
51#97—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—22no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—22no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83M16—48no—
84C2×C8—24no—
85C2×C8—24no—
86C2×C8—42no—
87C2×C8—42no—
88C2×C8—42no—
89C2×C8—42no—
90C4⋊C4—24no—
91C4⋊C4—21no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—42no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4M1612yescomplemented
98C22⋊C4—24no—
99C22⋊C4—42no—
100C42—22no—
101C42—22no—
102C42—41no—
103C42—41no—
104C42—42no—
105C42—42no—
106C23—21no—
107C23—21no—
108C23—21no—
109C23—21no—
110C23—21no—
111C23—21no—
112C23#10412yes—
113C23C2×C22⋊C411yescharacteristic
114Q8—42no—
115Q8—42no—
116C2×C4—24no—
117C2×C4—24no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—44no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—41no—
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#9912yes—
147C2×C4#10412yes—
148C2×C4C2×M1611yescharacteristic
149C2×C4C2×M1611yescharacteristic
150C2×C4AGL(1,ℤ/8)11yescharacteristic
151C2×C4#13711yescharacteristic
152C2×C4#13712yes—
153C8—44no—
154C8—44no—
155C22—22no—
156C22—22no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#40711yescharacteristic
163C22C2×#9911yescharacteristic
164C22#41311yescharacteristic
165C22#41711yescharacteristic
166C22#41911yescharacteristic
167C22C2×#10411yescharacteristic
168C22C2×#10411yescharacteristic
169C4—21no—
170C4—21no—
171C4—21no—
172C4—22no—
173C4—21no—
174C4—22no—
175C4—22no—
176C4—41no—
177C4—42no—
178C4—42no—
179C4—81no—
180C4—81no—
181C2—42no—
182C2#146511yescharacteristic
183C2#148911yescharacteristic
184C2#149111yescharacteristic
185C2#150411yescharacteristic
186C2#150811yescharacteristic
187C2#152411yescharacteristic
188C2#155911yescharacteristic
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
97. 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
#17492
#17282
#12722
#12712