Finite Group Databasebeta

← #8455

Subgroups — #8455

Order 256.

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

normal subgroups only

This group has 519 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#3486C211yesmaximal, characteristic
2#1784C211yesmaximal, characteristic, complemented
3#1504C214yesmaximal, complemented
4#1446C211yesmaximal, characteristic, complemented
5C2×2−1+4C411yescharacteristic, complemented
6C23×Q8C2211yescharacteristic
7D4×Q8—24no—
8#548—24no—
9#536—22no—
10#536C412yescomplemented
11#525—22no—
12#525C2212yes—
13#519C411yescharacteristic, complemented
14C2×#101—22no—
15#407C2212yes—
16C2×#95—22no—
17C2×#95C2212yes—
18#328—24no—
19#327—24no—
202−1+4—44no—
21C2×#41—22no—
22C2×#41—22no—
23C2×#41C2×C412yes—
24C22×Q8—22no—
25C22×Q8—24no—
26C22×Q8—44no—
27C22×Q8C2×C411yescharacteristic
28C22×Q8C2×C411yescharacteristic
29C22×Q8D412yes—
30C23×C4—22no—
31C23×C4D412yes—
32C23×C4C2311yescharacteristic
33C2×M16—22no—
34C4⋊Q8—24no—
35C4⋊Q8—22no—
36#124—24no—
37#124—22no—
38#123—24no—
39#123—22no—
40C22⋊Q8—24no—
41C22⋊Q8—28no—
42C22⋊Q8—24no—
43C22⋊Q8D414yes—
44#121—24no—
45#121D414yes—
46#120—22no—
47C4×Q8—24no—
48C4×D4—28no—
49C4×D4—24no—
50#117—22no—
51#117C2×C412yes—
52#101—44no—
53C22⋊C8—24no—
54#95—24no—
55#95—44no—
56C24C2×D411yescharacteristic
57#41—44no—
58#41—44no—
59#41—48no—
60#41—44no—
61C2×Q8—22no—
62C2×Q8—24no—
63C2×Q8—24no—
64C2×Q8—22no—
65C2×Q8—22no—
66C2×Q8—24no—
67C2×Q8—42no—
68C2×Q8—42no—
69C2×Q8—416no—
70C2×Q8—48no—
71C2×Q8C22⋊C412yes—
72C2×D4—22no—
73C2×D4—22no—
74C2×D4—24no—
75C2×D4C22⋊C412yes—
76C22×C4—24no—
77C22×C4—24no—
78C22×C4—24no—
79C22×C4—22no—
80C22×C4—24no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—44no—
84C22×C4—42no—
85C22×C4—44no—
86C22×C4—42no—
87C22×C4C22⋊C412yes—
88C22×C4C22⋊C412yes—
89C22×C4C22×C411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D414yes—
92M16—44no—
93C2×C8—42no—
94C4⋊C4—24no—
95C4⋊C4—28no—
96C4⋊C4—24no—
97C4⋊C4C22⋊C414yes—
98C22⋊C4—28no—
99C22⋊C4—22no—
100C22⋊C4—24no—
101C22⋊C4—24no—
102C42—24no—
103C42—22no—
104C23—22no—
105C23—22no—
106C23—22no—
107C23—22no—
108C23C2×C22⋊C411yescharacteristic
109C23#12012yes—
110Q8—28no—
111Q8—48no—
112Q8—42no—
113Q8—44no—
114Q8—44no—
115D4—42no—
116D4—44no—
117D4—44no—
118D4—44no—
119D4—44no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—44no—
130C2×C4—48no—
131C2×C4—48no—
132C2×C4—44no—
133C2×C4—48no—
134C2×C4—44no—
135C2×C4—84no—
136C2×C4—82no—
137C2×C4—82no—
138C2×C4#9914yes—
139C2×C4C2×C22⋊C412yes—
140C2×C4#12012yes—
141C8—82no—
142C22—24no—
143C22—22no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—22no—
148C22—44no—
149C22—42no—
150C22#37911yescharacteristic
151C22C2×#9912yes—
152C22#45412yes—
153C22#45612yes—
154C4—22no—
155C4—24no—
156C4—22no—
157C4—44no—
158C4—42no—
159C4—42no—
160C4—84no—
161C2—22no—
162C2—22no—
163C2—22no—
164C2—82no—
165C2#178111yescharacteristic
166C2#179312yes—
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#3486C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1784C2semidirect
3. #1504C2semidirect
4. #1446C2semidirect
5. C2×2−1+4C4semidirect
10. #536C4semidirect
13. #519C4semidirect

Maximal subgroups

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

subgroupindex
#34862
#17842
#15042
#14462