Finite Group Databasebeta

← #12692

Subgroups — #12692

Order 256.

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

normal subgroups only

This group has 488 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×#536C211yesmaximal, characteristic
2C2×#352C214yesmaximal, complemented
3#1785C211yesmaximal, characteristic
4#1782C211yesmaximal, characteristic
5C22×#41C2211yescharacteristic
6C23×Q8C411yescharacteristic, complemented
7#536—28no—
8C2×C4⋊Q8—21no—
9C2×#124—21no—
10C2×#123C412yescomplemented
11C2×#123C2212yes—
12C2×C22⋊Q8—21no—
13C2×C22⋊Q8—21no—
14C2×#117C411yescharacteristic, complemented
15C2×#101—21no—
16C2×#100C2212yes—
17C2×#99—21no—
18C2×#99C2212yes—
19#407—21no—
20#381—21no—
21#352—28no—
22C2×#41—24no—
23C2×#41—42no—
24C2×#41—42no—
25C22×Q8—21no—
26C22×Q8—24no—
27C22×Q8—21no—
28C22×Q8—42no—
29C22×Q8—42no—
30C22×Q8D411yescharacteristic
31C22×D4C2×C411yescharacteristic
32C22×D4D411yescharacteristic
33C22×D4C2311yescharacteristic
34C23×C4C2×C411yescharacteristic
35C23×C4C2×C412yes—
36C23×C4D411yescharacteristic
37C23×C4D411yescharacteristic
38C2×M16—22no—
39C4⋊Q8—28no—
40#124—28no—
41#123—24no—
42#123—24no—
43#123—24no—
44#123D418yes—
45C22⋊Q8—28no—
46C22⋊Q8—28no—
47#117—24no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C22⋊C4—22no—
52C2×C22⋊C4—22no—
53C2×C22⋊C4—22no—
54C2×C22⋊C4C2×C412yes—
55C2×C42—21no—
56#101—42no—
57#100—24no—
58#99—24no—
59#99—42no—
60C22⋊C8—42no—
61#95—42no—
62C24C22⋊C411yescharacteristic
63C24C22×C411yescharacteristic
64C24C2×D411yescharacteristic
65#41—416no—
66C2×Q8—24no—
67C2×Q8—24no—
68C2×Q8—22no—
69C2×Q8—22no—
70C2×Q8—41no—
71C2×Q8—416no—
72C2×Q8—42no—
73C2×Q8—42no—
74C2×Q8—42no—
75C2×Q8—42no—
76C2×D4—22no—
77C2×D4—22no—
78C2×D4—22no—
79C2×D4—41no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4—41no—
84C2×D4C2×D414yes—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4C22⋊C411yescharacteristic
101C22×C4C22⋊C411yescharacteristic
102C22×C4C22⋊C414yes—
103C22×C4C22⋊C411yescharacteristic
104C22×C4C2×D411yescharacteristic
105M16—42no—
106M16—42no—
107C2×C8—42no—
108C4⋊C4—24no—
109C4⋊C4—28no—
110C4⋊C4—28no—
111C22⋊C4—28no—
112C22⋊C4—22no—
113C22⋊C4—28no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C22⋊C4C22⋊C414yes—
117C42—24no—
118C23—21no—
119C23—21no—
120C23—21no—
121C23—21no—
122C23—21no—
123C23—21no—
124C23—21no—
125C23—42no—
126C23—42no—
127C23—42no—
128C23C2×C22⋊C411yescharacteristic
129C23C2×C22⋊C412yes—
130C23#12012yes—
131Q8—44no—
132Q8—48no—
133Q8—48no—
134D4—44no—
135D4—44no—
136D4—44no—
137C2×C4—28no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—44no—
150C2×C4—44no—
151C2×C4—44no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—44no—
156C2×C4—82no—
157C2×C4—84no—
158C2×C4#9912yes—
159C2×C4#9912yes—
160C2×C4#12012yes—
161C8—82no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—42no—
166C22—42no—
167C22—42no—
168C22—42no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—41no—
173C22—81no—
174C22—81no—
175C22#37911yescharacteristic
176C22C2×#9911yescharacteristic
177C22C2×#9911yescharacteristic
178C4—22no—
179C4—22no—
180C4—22no—
181C4—44no—
182C4—41no—
183C4—44no—
184C4—162no—
185C2—21no—
186C2—21no—
187C2—42no—
188C2—42no—
189C2—81no—
190C2#178111yescharacteristic
191C2#212312yes—
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC22×D4C23
prime radicalC22×#41C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#352C2semidirect
6. C23×Q8C4semidirect
10. C2×#123C4semidirect
14. C2×#117C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
C2×#3522
#17852
#17822