Finite Group Databasebeta

← #29111

Subgroups — #29111

Order 256.

453 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 127 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×#391C211yesmaximal, characteristic
3#2072C212yesmaximal, complemented
4#2024C212yesmaximal, complemented
5#1935C214yesmaximal, complemented
6#1823C212yesmaximal
7#1787C212yesmaximal
8#1749C211yesmaximal, characteristic
9C23×Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C22⋊Q8—21no—
12C22×C4⋊C4C2211yescharacteristic
13C22×C42C2211yescharacteristic
14C2×#103C2214yes—
15C2×#103C2214yes—
16C2×C22⋊C8C2212yes—
17C2×#97C2212yes—
18#391—21no—
19#391—21no—
20#391—21no—
21#391C2212yes—
22#391C2212yes—
23#389—21no—
24#389C2212yes—
25#384C2211yescharacteristic
26#380—21no—
27#380C2212yes—
28C2×#95—21no—
29C2×#95—21no—
30#340C2214yes—
31#340C2214yes—
32#336C2212yes—
33C22×Q8—24no—
34C22×Q8—21no—
35C22×Q8—21no—
36C22×Q8—24no—
37C22×Q8—42no—
38C22×Q8C2312yes—
39C23×C4—21no—
40C23×C4—21no—
41C23×C4D411yescharacteristic
42C23×C4D411yescharacteristic
43C23×C4C2311yescharacteristic
44C22×C8C2314yes—
45C4⋊Q8—42no—
46C22⋊Q8—44no—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4C2312yes—
54C2×C4⋊C4C2312yes—
55C2×C4⋊C4C2312yes—
56C2×C22⋊C4—22no—
57C2×C42—42no—
58C2×C42D411yescharacteristic
59C2×C42D411yescharacteristic
60C2×C42C2311yescharacteristic
61C2×C42C2311yescharacteristic
62#103—28no—
63#103—44no—
64C22⋊C8—24no—
65#97—42no—
66#95—22no—
67#95—22no—
68#95—22no—
69#95—22no—
70#95—22no—
71#95—22no—
72C24C2×D411yescharacteristic
73C2×Q8—24no—
74C2×Q8—22no—
75C2×Q8—24no—
76C2×Q8—24no—
77C2×Q8—416no—
78C2×Q8—42no—
79C2×Q8—42no—
80C2×Q8—42no—
81C2×Q8—42no—
82C2×Q8—42no—
83C2×Q8#4114yes—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—24no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4#4114yes—
108C22×C4C2411yescharacteristic
109C2×C8—24no—
110C2×C8—24no—
111C2×C8—24no—
112C4⋊C4—24no—
113C4⋊C4—24no—
114C4⋊C4—24no—
115C4⋊C4—42no—
116C4⋊C4—44no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C22⋊C4—44no—
121C42—22no—
122C42—22no—
123C42—41no—
124C42—41no—
125C23—21no—
126C23—21no—
127C23—21no—
128C23—21no—
129C23#12412yes—
130C23AGL(1,ℤ/8)12yes—
131C23#13712yes—
132C23C22×D411yescharacteristic
133Q8—28no—
134Q8—48no—
135Q8—44no—
136Q8—84no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—28no—
144C2×C4—24no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—44no—
152C2×C4—44no—
153C2×C4—48no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—84no—
166C2×C4#12412yes—
167C2×C4#13712yes—
168C2×C4#13712yes—
169C2×C4C2×#4112yes—
170C2×C4C2×#4112yes—
171C2×C4H(5,𝔽2)12yes—
172C8—44no—
173C22—22no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—22no—
179C22#48914yes—
180C22#52511yescharacteristic
181C22C2×#12411yescharacteristic
182C22#53411yescharacteristic
183C22C2×AGL(1,ℤ/8)11yescharacteristic
184C22C2×#13711yescharacteristic
185C22C2×#13711yescharacteristic
186C22C2×#13711yescharacteristic
187C4—22no—
188C4—22no—
189C4—22no—
190C4—44no—
191C4—41no—
192C4—41no—
193C4—82no—
194C4—82no—
195C4—82no—
196C4#55114yes—
197C2—24no—
198C2#256111yescharacteristic
199C2#311211yescharacteristic
200C2#311311yescharacteristic
201C2C2×#48911yescharacteristic
202C2#313911yescharacteristic
203C2#319311yescharacteristic
204C2#319511yescharacteristic
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. #2072C2semidirect
4. #2024C2semidirect
5. #1935C2semidirect

Maximal subgroups

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

subgroupindex
#28572
C2×#3912
#20722
#20242
#19352
#18232
#17872
#17492