Finite Group Databasebeta

← #29301

Subgroups — #29301

Order 256.

421 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 143 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
2C2×#389C211yesmaximal, characteristic
3#2083C212yesmaximal, complemented
4#2058C212yesmaximal, complemented
5#1943C214yesmaximal, complemented
6#1853C212yesmaximal
7#1852C212yesmaximal
8#1749C211yesmaximal, characteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C22⋊Q8—21no—
11C22×C4⋊C4—21no—
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C42C2211yescharacteristic
15C2×#107C2214yes—
16C2×#106C2214yes—
17C2×C22⋊C8C2212yes—
18C2×#97C2212yes—
19#391—21no—
20#389—21no—
21#389—21no—
22#389—21no—
23#389C2212yes—
24#389C2212yes—
25#389C2212yes—
26#384C2211yescharacteristic
27#380—21no—
28#380C2212yes—
29C2×#95—21no—
30#340C2214yes—
31#340C2214yes—
32#336C2212yes—
33C22×Q8—21no—
34C23×C4—21no—
35C23×C4—21no—
36C23×C4D411yescharacteristic
37C23×C4D411yescharacteristic
38C23×C4C2311yescharacteristic
39C22×C8C2314yes—
40C4⋊Q8—42no—
41C22⋊Q8—44no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—24no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—42no—
52C2×C4⋊C4—44no—
53C2×C4⋊C4C2312yes—
54C2×C4⋊C4C2312yes—
55C2×C4⋊C4C2312yes—
56C2×C4⋊C4C2312yes—
57C2×C22⋊C4—22no—
58C2×C42—42no—
59C2×C42D411yescharacteristic
60C2×C42D411yescharacteristic
61C2×C42C2311yescharacteristic
62C2×C42C2311yescharacteristic
63#107—28no—
64#106—44no—
65C22⋊C8Q818yescomplemented
66#97—42no—
67#95—22no—
68#95—22no—
69#95—22no—
70#95—22no—
71C24C2×D411yescharacteristic
72C2×Q8—42no—
73C2×Q8—42no—
74C2×Q8—42no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—24no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×Q814yes—
99C22×C4C2411yescharacteristic
100C2×C8—24no—
101C2×C8—24no—
102C2×C8C2×Q818yes—
103C4⋊C4—24no—
104C4⋊C4—22no—
105C4⋊C4—24no—
106C4⋊C4—24no—
107C4⋊C4—24no—
108C4⋊C4—42no—
109C4⋊C4—44no—
110C4⋊C4—42no—
111C4⋊C4—44no—
112C4⋊C4—42no—
113C4⋊C4—42no—
114C4⋊C4—44no—
115C4⋊C4#4114yes—
116C22⋊C4—44no—
117C42—22no—
118C42—22no—
119C42—41no—
120C42—41no—
121C23—21no—
122C23—21no—
123C23—21no—
124C23—21no—
125C23C4⋊Q812yes—
126C23AGL(1,ℤ/8)12yes—
127C23#13712yes—
128C23C22×D411yescharacteristic
129Q8—84no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—28no—
137C2×C4—24no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—44no—
145C2×C4—44no—
146C2×C4—48no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—84no—
159C2×C4C4⋊Q812yes—
160C2×C4AGL(1,ℤ/8)12yes—
161C2×C4#13712yes—
162C2×C4C22×Q812yes—
163C2×C4C2×#4112yes—
164C2×C4H(5,𝔽2)12yes—
165C8—28no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22#50114yes—
173C22#52511yescharacteristic
174C22C2×C4⋊Q811yescharacteristic
175C22#53411yescharacteristic
176C22C2×AGL(1,ℤ/8)11yescharacteristic
177C22C2×AGL(1,ℤ/8)11yescharacteristic
178C22C2×#13711yescharacteristic
179C22C2×#13711yescharacteristic
180C4—22no—
181C4—22no—
182C4—22no—
183C4—44no—
184C4—41no—
185C4—41no—
186C4—82no—
187C4—82no—
188C4—82no—
189C4#55414yes—
190C2—24no—
191C2#258311yescharacteristic
192C2#310911yescharacteristic
193C2#311311yescharacteristic
194C2C2×#50111yescharacteristic
195C2#316511yescharacteristic
196C2#319311yescharacteristic
197C2#319311yescharacteristic
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. #2083C2semidirect
4. #2058C2semidirect
5. #1943C2semidirect
65. C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#28572
C2×#3892
#20832
#20582
#19432
#18532
#18522
#17492