Finite Group Databasebeta

← #22246

Subgroups — #22246

Order 256.

395 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 183 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×#498C211yesmaximal, characteristic
2C4×C4⋊Q8C211yesmaximal, characteristic
3C4×C4⋊Q8C211yesmaximal, characteristic
4#1990C211yesmaximal, characteristic
5#1990C211yesmaximal, characteristic
6#1920C214yesmaximal
7#1919C214yesmaximal
8#1834C211yesmaximal, characteristic
9#1744C211yesmaximal, characteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4×Q8—21no—
13C2×C4×Q8—21no—
14#498C418yescomplemented
15C2×#106C2214yes—
16C2×C4×C8C2211yescharacteristic
17C2×C4×C8C2211yescharacteristic
18#391—21no—
19#391—21no—
20#389—21no—
21#389—21no—
22#389C2212yes—
23#389C2212yes—
24#384C2211yescharacteristic
25#384C2211yescharacteristic
26C4×C4⋊C4—21no—
27C4×C4⋊C4—21no—
28C4×C4⋊C4C2212yes—
29C4×C4⋊C4C2212yes—
30C43C2211yescharacteristic
31#340C2214yes—
32#340C2214yes—
33#340C2214yes—
34#336C2212yes—
35#336C2212yes—
36C22×Q8—21no—
37C22×Q8—21no—
38C22×C8C2312yes—
39C22×C8C2312yes—
40C4⋊Q8—22no—
41C4⋊Q8—22no—
42C4⋊Q8C2×C414yes—
43C4⋊Q8C2×C414yes—
44C4×Q8—44no—
45C4×Q8—44no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4C2312yes—
55C2×C4⋊C4C2312yes—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2312yes—
58C2×C42—22no—
59C2×C42—22no—
60C2×C42D411yescharacteristic
61C2×C42D411yescharacteristic
62C2×C42D411yescharacteristic
63C2×C42D411yescharacteristic
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66C2×C42C2311yescharacteristic
67#106C2×C4116yes—
68C4×C8—22no—
69C4×C8C2×C414yes—
70#95—22no—
71#95—22no—
72#95—22no—
73#95—22no—
74C2×Q8—24no—
75C2×Q8—24no—
76C2×Q8—42no—
77C2×Q8—42no—
78C2×Q8—42no—
79C2×Q8—42no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2411yescharacteristic
91C2×C8—22no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—22no—
96C2×C8C22×C414yes—
97C4⋊C4—24no—
98C4⋊C4—24no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—24no—
102C4⋊C4—24no—
103C4⋊C4—44no—
104C4⋊C4—44no—
105C4⋊C4—42no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—42no—
109C4⋊C4C22×C414yes—
110C4⋊C4C22×C414yes—
111C42—22no—
112C42—22no—
113C42—22no—
114C42—22no—
115C42—22no—
116C42—44no—
117C42—44no—
118C42SD1614yes—
119C42C22×C412yes—
120C42#4112yes—
121C23C22×D411yescharacteristic
122Q8—48no—
123Q8—48no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—24no—
129C2×C4—24no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4C4×D412yes—
147C2×C4C4×D412yes—
148C2×C4C2×SD1612yes—
149C2×C4C2×SD1612yes—
150C2×C4C2×SD1612yes—
151C2×C4#13512yes—
152C2×C4AGL(1,ℤ/8)12yes—
153C2×C4#13712yes—
154C2×C4C23×C411yescharacteristic
155C2×C4C2×#4111yescharacteristic
156C2×C42−1+412yes—
157C2×C42−1+412yes—
158C8—24no—
159C8—42no—
160C22C2×C4×D411yescharacteristic
161C22#53611yescharacteristic
162C22#53611yescharacteristic
163C22C22×SD1611yescharacteristic
164C22C2×#13511yescharacteristic
165C22C2×AGL(1,ℤ/8)11yescharacteristic
166C22C2×#13711yescharacteristic
167C4—22no—
168C4—22no—
169C4—44no—
170C4—44no—
171C4—41no—
172C4—41no—
173C4—82no—
174C4—82no—
175C4C4×SD1614yes—
176C4#44014yes—
177C4#51912yes—
178C4#54112yes—
179C2#232711yescharacteristic
180C2C2×C4×SD1611yescharacteristic
181C2C2×#44011yescharacteristic
182C2#322711yescharacteristic
183C2#322811yescharacteristic
184C2#322911yescharacteristic
185C2#323111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #498C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4982
C4×C4⋊Q82
#19902
#19202
#19192
#18342
#17442