Finite Group Databasebeta

← #8471

Subgroups — #8471

Order 256.

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

normal subgroups only

This group has 462 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
1#3499C211yesmaximal, characteristic
2#1997C212yesmaximal, complemented
3#1560C211yesmaximal, characteristic, complemented
4#1525C211yesmaximal, characteristic, complemented
5#1489C212yesmaximal, complemented
6C2×2−1+4C412yes—
7D4×Q8—22no—
8#547—22no—
9#547—24no—
10#536—22no—
11#536C412yes—
12#532—21no—
13#532C2211yescharacteristic
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8C2211yescharacteristic
16C4×#41C2211yescharacteristic
17#423C2212yes—
18C4×M16C2212yes—
19#343—24no—
20#333—21no—
21#331—21no—
22#329—22no—
23Q8⋊C8—22no—
24D4⋊C8—22no—
252−1+4—44no—
262−1+4—44no—
27C2×#41—22no—
28C2×#41—22no—
29C2×#41—24no—
30C2×#41C2×C411yescharacteristic
31C2×#41C2×C412yes—
32C22×Q8—22no—
33C22×Q8—22no—
34C22×Q8C2×C412yes—
35C2×M16—22no—
36C2×M16—22no—
37C4⋊Q8—24no—
38C4⋊Q8—22no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—22no—
42C4⋊Q8D411yescharacteristic
43#127D411yescharacteristic
44#124—22no—
45#124D412yes—
46#123—28no—
47#123—24no—
48C22⋊Q8—28no—
49C22⋊Q8—24no—
50#121—24no—
51#121—22no—
52C4×Q8D412yes—
53C4×D4—24no—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4D412yes—
57#117—21no—
58#117C2×C411yescharacteristic
59C2×C4⋊C4—24no—
60C2×C4⋊C4D414yes—
61C2×C42—21no—
62C2×C42C2311yescharacteristic
63C4⋊C8—22no—
64C4⋊C8—22no—
65#97—22no—
66C4×C8—22no—
67#41—22no—
68#41—22no—
69#41—48no—
70#41—42no—
71#41—42no—
72#41—416no—
73#41—44no—
74#41—44no—
75C2×Q8—24no—
76C2×Q8—22no—
77C2×Q8—22no—
78C2×Q8—42no—
79C2×Q8—42no—
80C2×Q8—48no—
81C2×Q8—44no—
82C2×Q8—44no—
83C2×Q8C22⋊C411yescharacteristic
84C2×Q8C22⋊C412yes—
85C2×D4—21no—
86C2×D4—24no—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4C22⋊C411yescharacteristic
90C2×D4C22⋊C412yes—
91C22×C4—21no—
92C22×C4—24no—
93C22×C4—24no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4C22⋊C414yes—
97C22×C4C22×C411yescharacteristic
98C22×C4C2×D412yes—
99M16—24no—
100M16—42no—
101M16—44no—
102C2×C8—42no—
103C2×C8—42no—
104C4⋊C4—28no—
105C4⋊C4—22no—
106C4⋊C4—24no—
107C4⋊C4—24no—
108C4⋊C4—22no—
109C4⋊C4C22⋊C412yes—
110C22⋊C4—28no—
111C22⋊C4—21no—
112C22⋊C4—22no—
113C22⋊C4—24no—
114C42—21no—
115C42—22no—
116C42C2×D412yes—
117C42C2×D411yescharacteristic
118C42C2×D411yescharacteristic
119C23—21no—
120C23—22no—
121C23C2×C22⋊C411yescharacteristic
122Q8—21no—
123Q8—42no—
124Q8—44no—
125Q8—48no—
126Q8—42no—
127Q8—42no—
128Q8#10012yes—
129D4—24no—
130D4—21no—
131D4—42no—
132D4—44no—
133D4—48no—
134D4—42no—
135D4—42no—
136D4#10012yes—
137C2×C4—24no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—48no—
147C2×C4—48no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4C2×C22⋊C411yescharacteristic
154C2×C4C2×C22⋊C411yescharacteristic
155C2×C4#12012yes—
156C2×C4#12012yes—
157C8—44no—
158C8—82no—
159C22—22no—
160C22—21no—
161C22—22no—
162C22—42no—
163C22—44no—
164C22—42no—
165C22#37911yescharacteristic
166C4—22no—
167C4—21no—
168C4—22no—
169C4—42no—
170C4—42no—
171C4—44no—
172C4C2×#10011yescharacteristic
173C4C2×#10011yescharacteristic
174C4#45412yes—
175C4#45612yes—
176C2—41no—
177C2—42no—
178C2—82no—
179C2#178311yescharacteristic
180C2#179312yes—
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#3499C2
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. #1997C2semidirect
3. #1560C2semidirect
4. #1525C2semidirect
5. #1489C2semidirect

Maximal subgroups

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

subgroupindex
#34992
#19972
#15602
#15252
#14892