Finite Group Databasebeta

← #9911

Subgroups — #9911

Order 256.

296 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 85 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#2857C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic
3#1434C211yesmaximal, characteristic
4#1294C214yesmaximal
5C2×C4⋊Q8C411yescharacteristic, complemented
6C2×C22⋊Q8—21no—
7C22×C42C2211yescharacteristic
8C2×C4⋊C8—21no—
9C2×C22⋊C8C2212yes—
10#391—21no—
11#389—21no—
12#389C412yescomplemented
13#384C411yescharacteristic, complemented
14#380—21no—
15#380C2212yes—
16C2×#95—21no—
17C2×#95—21no—
18C2×#95C2212yes—
19#336—21no—
20C22×Q8—21no—
21C23×C4—22no—
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—22no—
26C4⋊Q8—42no—
27C22⋊Q8—44no—
28C2×C4⋊C4—21no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4C2×C412yes—
33C2×C4⋊C4C2×C412yes—
34C2×C22⋊C4—22no—
35C2×C42—42no—
36C2×C42C2×C411yescharacteristic
37C2×C42C2×C411yescharacteristic
38C2×C42D411yescharacteristic
39C2×C42D411yescharacteristic
40C4⋊C8—42no—
41C22⋊C8—24no—
42#95—22no—
43#95—22no—
44#95—22no—
45#95—42no—
46#95—42no—
47#95Q814yescomplemented
48C24C2×D411yescharacteristic
49C2×Q8—42no—
50C2×Q8—42no—
51C2×Q8—42no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—24no—
57C22×C4—41no—
58C22×C4—41no—
59C22×C4—41no—
60C22×C4—41no—
61C22×C4—41no—
62C22×C4—42no—
63C22×C4—42no—
64C22×C4—42no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22⋊C411yescharacteristic
71C22×C4C22⋊C411yescharacteristic
72C22×C4SD1612yes—
73C22×C4Q1612yes—
74C22×C4C22×C411yescharacteristic
75C22×C4C2×D411yescharacteristic
76C22×C4C2×Q812yes—
77C22×C4#4112yes—
78C2×C8—42no—
79C2×C8—42no—
80C2×C8—42no—
81C4⋊C4—42no—
82C4⋊C4—42no—
83C4⋊C4—42no—
84C4⋊C4—42no—
85C4⋊C4—44no—
86C4⋊C4—42no—
87C22⋊C4—44no—
88C42—22no—
89C42—22no—
90C42—41no—
91C42—41no—
92C23—21no—
93C23—21no—
94C23—21no—
95C23—21no—
96C23C2×C22⋊C411yescharacteristic
97C23C22⋊Q812yes—
98C23C2×SD1611yescharacteristic
99C23C2×Q1611yescharacteristic
100C23AGL(1,ℤ/8)12yes—
101Q8—84no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—44no—
120C2×C4—44no—
121C2×C4—44no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—44no—
129C2×C4—82no—
130C2×C4—82no—
131C2×C4—82no—
132C2×C4—82no—
133C2×C4#9912yes—
134C2×C4#10312yes—
135C2×C4#10312yes—
136C2×C4C4×Q812yes—
137C2×C4#12411yescharacteristic
138C2×C4C4⋊Q811yescharacteristic
139C8—82no—
140C22—22no—
141C22—22no—
142C22—22no—
143C22—22no—
144C22—21no—
145C22—21no—
146C22—21no—
147C22—21no—
148C22#39111yescharacteristic
149C22C2×#9911yescharacteristic
150C22#41011yescharacteristic
151C22C2×#10311yescharacteristic
152C22#41811yescharacteristic
153C22#42111yescharacteristic
154C22#42111yescharacteristic
155C22#47612yes—
156C22#47712yes—
157C4—21no—
158C4—22no—
159C4—21no—
160C4—22no—
161C4—22no—
162C4—22no—
163C4—41no—
164C4—84no—
165C4—82no—
166C4—82no—
167C2—22no—
168C2—22no—
169C2#151011yescharacteristic
170C2#151411yescharacteristic
171C2#151811yescharacteristic
172C2#152011yescharacteristic
173C2#198711yescharacteristic
174C2#199511yescharacteristic
175C2#199611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊Q8C4semidirect
12. #389C4semidirect
13. #384C4semidirect
47. #95Q8semidirect

Maximal subgroups

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

subgroupindex
#28572
#17972
#14342
#12942