Finite Group Databasebeta

← #8525

Subgroups — #8525

Order 256.

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

normal subgroups only

This group has 347 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×#544C211yesmaximal, characteristic
2#1556C212yesmaximal, complemented
3#1534C212yesmaximal, complemented
4#1464C211yesmaximal, characteristic
5#1464C211yesmaximal, characteristic
6#544—24no—
7#544—24no—
8C2×C4⋊Q8—21no—
9C2×C4⋊Q8C411yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×#125—21no—
12C2×#125C411yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8—21no—
16C2×#117C411yescharacteristic
17C2×#117C2211yescharacteristic
18C22×C4⋊C4C411yescharacteristic
19#432C2212yes—
20#432C2212yes—
21#332—24no—
22#330—24no—
23#324—21no—
24#324—21no—
25#323—21no—
26#323—21no—
27C22×Q8—21no—
28C22×Q8—21no—
29C23×C4—21no—
30C23×C4C2×C411yescharacteristic
31C4⋊Q8—28no—
32C4⋊Q8—22no—
33C4⋊Q8—22no—
34C4⋊Q8—22no—
35C4⋊Q8D414yes—
36#125—28no—
37#125—22no—
38#125—22no—
39#125—22no—
40#125D414yes—
41C22⋊Q8—28no—
42C22⋊Q8—28no—
43#117—22no—
44#117—22no—
45#117—22no—
46#117—22no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4C2×C411yescharacteristic
57C2×C4⋊C4C2×C411yescharacteristic
58C2×C4⋊C4C2×C411yescharacteristic
59C2×C4⋊C4C2×C411yescharacteristic
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4D411yescharacteristic
62C2×C22⋊C4—21no—
63C2×C22⋊C4D411yescharacteristic
64C2×C22⋊C4D411yescharacteristic
65C2×C42—21no—
66C2×C42C2×C411yescharacteristic
67C2×C42C2311yescharacteristic
68C4⋊C8—22no—
69C4⋊C8—22no—
70C22⋊C8—22no—
71C22⋊C8—22no—
72#97—22no—
73#97—22no—
74C24C22⋊C411yescharacteristic
75C2×Q8—24no—
76C2×Q8—24no—
77C2×Q8—82no—
78C2×Q8—82no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—24no—
88C22×C4—82no—
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22⋊C411yescharacteristic
92C22×C4C22×C411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C2×C8—42no—
96C2×C8—42no—
97C4⋊C4—24no—
98C4⋊C4—24no—
99C4⋊C4—24no—
100C4⋊C4—24no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—24no—
106C4⋊C4—24no—
107C4⋊C4—24no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4C22⋊C412yes—
111C4⋊C4C22⋊C412yes—
112C4⋊C4C22⋊C412yes—
113C4⋊C4C22⋊C412yes—
114C22⋊C4—24no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C42—24no—
118C42—21no—
119C42—21no—
120C42C2×D412yes—
121C42C2×D412yes—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#10112yes—
127C23C2×C22⋊C411yescharacteristic
128Q8—84no—
129Q8—84no—
130C2×C4—24no—
131C2×C4—24no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—84no—
151C2×C4#10112yes—
152C2×C4C2×C22⋊C411yescharacteristic
153C2×C4C2×C22⋊C411yescharacteristic
154C2×C4#12012yes—
155C2×C4#12012yes—
156C8—82no—
157C8—82no—
158C22—21no—
159C22—21no—
160C22—22no—
161C22—42no—
162C22—42no—
163C22—41no—
164C22—41no—
165C22#37911yescharacteristic
166C22C2×#10111yescharacteristic
167C22C2×#10111yescharacteristic
168C4—22no—
169C4—22no—
170C4—22no—
171C4—21no—
172C4—21no—
173C4—42no—
174C4—42no—
175C4—42no—
176C4—41no—
177C4—41no—
178C4—42no—
179C2—21no—
180C2—21no—
181C2—42no—
182C2#178411yescharacteristic
183C2#179511yescharacteristic
184C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
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. #1556C2semidirect
3. #1534C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
#15562
#15342
#14642