Finite Group Databasebeta

← #29922

Subgroups — #29922

Order 256.

386 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 129 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#2285C211yesmaximal, characteristic
3#2096C212yesmaximal, complemented
4#2072C212yesmaximal, complemented
5#2035C214yesmaximal
6#1981C212yesmaximal, complemented
7#1948C212yesmaximal, complemented
8#1797C211yesmaximal, characteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C22⋊Q8C2212yes—
11C2×#117—21no—
12C22×C42C2211yescharacteristic
13C2×#106C2214yes—
14C2×C4⋊C8C2212yes—
15C2×#103C2214yes—
16C2×#103C2214yes—
17C2×C22⋊C8C2212yes—
18#391C2212yes—
19#389—21no—
20#389—21no—
21#384C2211yescharacteristic
22#384C2211yescharacteristic
23#384C2211yescharacteristic
24#382—21no—
25#380—21no—
26#380—21no—
27#380—21no—
28#380C2212yes—
29C4×C4⋊C4—21no—
30C4×C4⋊C4C2212yes—
31#340C2214yes—
32#336C2212yes—
33C22×Q8C2312yes—
34C23×C4D411yescharacteristic
35C23×C4D411yescharacteristic
36C23×C4C2311yescharacteristic
37C22×C8C2314yes—
38C4⋊Q8—24no—
39C22⋊Q8—44no—
40#117—44no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4C2312yes—
51C2×C4⋊C4C2312yes—
52C2×C4⋊C4C2312yes—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—22no—
55C2×C42—24no—
56C2×C42—22no—
57C2×C42D411yescharacteristic
58C2×C42D411yescharacteristic
59C2×C42C2311yescharacteristic
60C2×C42C2311yescharacteristic
61#106—44no—
62C4⋊C8—24no—
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—22no—
73C2×Q8—24no—
74C2×Q8—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4—21no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—41no—
88C22×C4SD1614yes—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2411yescharacteristic
95C2×C8—24no—
96C2×C8—24no—
97C2×C8—24no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—24no—
102C4⋊C4—22no—
103C4⋊C4—42no—
104C4⋊C4—42no—
105C4⋊C4—44no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C4⋊C4—42no—
111C4⋊C4—42no—
112C22⋊C4—44no—
113C22⋊C4—44no—
114C42—22no—
115C42—22no—
116C42—22no—
117C42—44no—
118C42#4114yes—
119C23—21no—
120C23—21no—
121C23—21no—
122C23—21no—
123C23—21no—
124C23—21no—
125C23C2×SD1612yes—
126C23C22×D411yescharacteristic
127Q8—44no—
128Q8—44no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—28no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—44no—
140C2×C4—42no—
141C2×C4—42no—
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×C4#12312yes—
155C2×C4#12312yes—
156C2×C4C2×SD1612yes—
157C2×C4C2×SD1612yes—
158C2×C4#13712yes—
159C2×C4#13712yes—
160C2×C4C2×#4112yes—
161C2×C4H(5,𝔽2)11yescharacteristic
162C2×C4H(5,𝔽2)12yes—
163C2×C42−1+411yescharacteristic
164C8—44no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22C2×#12311yescharacteristic
173C22#53511yescharacteristic
174C22#53711yescharacteristic
175C22C22×SD1611yescharacteristic
176C22C2×#13711yescharacteristic
177C22C2×#13711yescharacteristic
178C22#57511yescharacteristic
179C4—22no—
180C4—22no—
181C4—22no—
182C4—41no—
183C4—82no—
184C4—82no—
185C4—82no—
186C4—82no—
187C4#48314yes—
188C4#54012yes—
189C4#54212yes—
190C2—42no—
191C2#262411yescharacteristic
192C2C2×#48311yescharacteristic
193C2#309511yescharacteristic
194C2#320611yescharacteristic
195C2#321611yescharacteristic
196C2#324611yescharacteristic
197C2#325511yescharacteristic
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. #2072C2semidirect
6. #1981C2semidirect
7. #1948C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#22852
#20962
#20722
#20352
#19812
#19482
#17972