Finite Group Databasebeta

← #23747

Subgroups — #23747

Order 256.

429 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 163 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×#389C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic
3#2101C214yesmaximal, complemented
4#2100C212yesmaximal, complemented
5#2099C212yesmaximal, complemented
6#1736C212yesmaximal
7#1736C212yesmaximal
8#1728C211yesmaximal, characteristic
9C22×M16C2212yes—
10C2×#117C2212yes—
11C22×C4⋊C4—21no—
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C42C2211yescharacteristic
15#389—21no—
16#389—21no—
17#389C2212yes—
18#389C2212yes—
19#384C2211yescharacteristic
20#384C2211yescharacteristic
21#382C2212yes—
22#380—21no—
23#380—21no—
24C4×C4⋊C4—21no—
25C4×C4⋊C4—21no—
26C2×#95—21no—
27#340C2214yes—
28#340C2214yes—
29#340C2214yes—
30#340C2214yes—
31#336C2212yes—
32#336C2212yes—
33C23×C4—21no—
34C23×C4—21no—
35C23×C4D411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4C2311yescharacteristic
38C2×M16—24no—
39C2×M16—24no—
40C2×M16—24no—
41C22×C8C2314yes—
42#117—28no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—24no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—44no—
54C2×C4⋊C4Q814yes—
55C2×C4⋊C4C2312yes—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C22⋊C4—22no—
60C2×C42—24no—
61C2×C42—22no—
62C2×C42D411yescharacteristic
63C2×C42D411yescharacteristic
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66#95—22no—
67#95—22no—
68#95—22no—
69#95—22no—
70C24C2×D411yescharacteristic
71C22×C4—24no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×Q812yes—
93C22×C4#4112yes—
94C22×C4#4114yes—
95C22×C4#4114yes—
96C22×C4C2411yescharacteristic
97M16—48no—
98C2×C8—24no—
99C2×C8—24no—
100C2×C8—24no—
101C4⋊C4—24no—
102C4⋊C4—22no—
103C4⋊C4—24no—
104C4⋊C4—24no—
105C4⋊C4—24no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—44no—
109C4⋊C4—42no—
110C4⋊C4—42no—
111C4⋊C4—44no—
112C4⋊C4—42no—
113C4⋊C4C2×Q814yes—
114C22⋊C4—44no—
115C42—22no—
116C42—22no—
117C42—22no—
118C42—22no—
119C42—44no—
120C23—21no—
121C23—21no—
122C23—21no—
123C23—21no—
124C23C22⋊Q812yes—
125C23#12312yes—
126C23#12412yes—
127C23C22×D411yescharacteristic
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—24no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—44no—
140C2×C4—44no—
141C2×C4—48no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—84no—
154C2×C4C22⋊Q812yes—
155C2×C4#12312yes—
156C2×C4#12412yes—
157C2×C4#12518yes—
158C2×C4#12618yes—
159C2×C4AGL(1,ℤ/8)12yes—
160C2×C4#13712yes—
161C2×C4C22×Q811yescharacteristic
162C2×C4C2×#4112yes—
163C2×C4C2×#4112yes—
164C2×C4C2×#4111yescharacteristic
165C8—44no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22#40014yes—
173C22C2×C22⋊Q811yescharacteristic
174C22C2×#12311yescharacteristic
175C22C2×#12411yescharacteristic
176C22C2×AGL(1,ℤ/8)11yescharacteristic
177C22C2×#13711yescharacteristic
178C22#57511yescharacteristic
179C22#57511yescharacteristic
180C4—22no—
181C4—22no—
182C4—44no—
183C4—41no—
184C4—41no—
185C4—82no—
186C4—82no—
187C4—82no—
188C4#40014yes—
189C4C2×#12512yes—
190C4C2×#12612yes—
191C2—24no—
192C2C2×#40011yescharacteristic
193C2#307711yescharacteristic
194C2#307811yescharacteristic
195C2#309411yescharacteristic
196C2#309511yescharacteristic
197C2#314111yescharacteristic
198C2#314211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2101C2semidirect
4. #2100C2semidirect
5. #2099C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3892
#22852
#21012
#21002
#20992
#17362
#17282