Finite Group Databasebeta

← #32771

Subgroups — #32771

Order 256.

374 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 117 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#2857C211yesmaximal, characteristic
3#2096C212yesmaximal, complemented
4#2084C212yesmaximal, complemented
5#2043C212yesmaximal, complemented
6#2035C214yesmaximal
7#2024C212yesmaximal, complemented
8#1817C211yesmaximal, characteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C22⋊Q8—21no—
12C2×C22⋊Q8C2212yes—
13C22×C42C2211yescharacteristic
14C2×#106C2214yes—
15C2×#103C2214yes—
16C2×#103C2214yes—
17C2×C22⋊C8C2212yes—
18#391—21no—
19#391C2212yes—
20#389—21no—
21#389—21no—
22#389—21no—
23#389C2212yes—
24#384C2211yescharacteristic
25#384C2211yescharacteristic
26#380—21no—
27#380—21no—
28#380—21no—
29#380C2212yes—
30#340C2214yes—
31#336C2212yes—
32#336C2212yes—
33C22×Q8—21no—
34C22×Q8C2312yes—
35C23×C4D411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4C2311yescharacteristic
38C22×C8C2314yes—
39C4⋊Q8—42no—
40C4⋊Q8—42no—
41C22⋊Q8—44no—
42C22⋊Q8—44no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4C2312yes—
53C2×C4⋊C4C2312yes—
54C2×C4⋊C4C2312yes—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—22no—
57C2×C42—42no—
58C2×C42D411yescharacteristic
59C2×C42D411yescharacteristic
60C2×C42C2311yescharacteristic
61C2×C42C2311yescharacteristic
62#106—44no—
63#103—44no—
64#103—44no—
65C22⋊C8—42no—
66#95—22no—
67#95—22no—
68#95—22no—
69#95—22no—
70C24C2×D411yescharacteristic
71C2×Q8—24no—
72C2×Q8—24no—
73C2×Q8—24no—
74C2×Q8—42no—
75C2×Q8—42no—
76C2×Q8—42no—
77C22×C4—22no—
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—41no—
89C22×C4SD1614yes—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2411yescharacteristic
96C2×C8—24no—
97C2×C8—24no—
98C2×C8—24no—
99C4⋊C4—22no—
100C4⋊C4—24no—
101C4⋊C4—22no—
102C4⋊C4—24no—
103C4⋊C4—42no—
104C4⋊C4—44no—
105C4⋊C4—42no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—42no—
109C4⋊C4—44no—
110C4⋊C4—42no—
111C4⋊C4—42no—
112C22⋊C4—44no—
113C22⋊C4—44no—
114C42—22no—
115C42—22no—
116C42—41no—
117C42—41no—
118C23—21no—
119C23—21no—
120C23—21no—
121C23—21no—
122C23—21no—
123C23—21no—
124C23C2×SD1612yes—
125C23C22×D411yescharacteristic
126Q8—48no—
127Q8—84no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
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—44no—
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—42no—
154C2×C4C2×SD1612yes—
155C2×C4C2×SD1612yes—
156C2×C4#13712yes—
157C2×C4#13712yes—
158C2×C4H(5,𝔽2)12yes—
159C2×C4H(5,𝔽2)12yes—
160C2×C42−1+412yes—
161C8—44no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22#53411yescharacteristic
170C22#53511yescharacteristic
171C22#53611yescharacteristic
172C22C22×SD1611yescharacteristic
173C22C2×#13711yescharacteristic
174C22C2×#13711yescharacteristic
175C22#57511yescharacteristic
176C4—22no—
177C4—22no—
178C4—22no—
179C4—22no—
180C4—41no—
181C4—82no—
182C4—82no—
183C4—82no—
184C4—82no—
185C4#56214yes—
186C2—42no—
187C2#281611yescharacteristic
188C2#318711yescharacteristic
189C2#319511yescharacteristic
190C2#320611yescharacteristic
191C2#321611yescharacteristic
192C2#322911yescharacteristic
193C2#323711yescharacteristic
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. #2096C2semidirect
4. #2084C2semidirect
5. #2043C2semidirect
7. #2024C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#20962
#20842
#20432
#20352
#20242
#18172