Finite Group Databasebeta

← #29757

Subgroups — #29757

Order 256.

442 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 137 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×#484C212yesmaximal
2#2857C211yesmaximal, characteristic
3C2×#389C211yesmaximal, characteristic
4#2083C212yesmaximal, complemented
5#2068C214yesmaximal, complemented
6#2043C212yesmaximal, complemented
7#1872C212yesmaximal
8#1797C211yesmaximal, characteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C22⋊Q8C2212yes—
11C22×C4⋊C4—21no—
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C42C2211yescharacteristic
15#484—28no—
16C2×#107C2214yes—
17C2×C4⋊C8C2212yes—
18C2×#103C2214yes—
19C2×#103C2214yes—
20C2×C22⋊C8C2212yes—
21#391C2212yes—
22#389—21no—
23#389—21no—
24#389—21no—
25#389—21no—
26#389C2212yes—
27#389C2212yes—
28#384C2211yescharacteristic
29#380—21no—
30#380—21no—
31C2×#95—21no—
32#340C2214yes—
33#336C2212yes—
34C22×Q8C2312yes—
35C23×C4—21no—
36C23×C4—21no—
37C23×C4D411yescharacteristic
38C23×C4D411yescharacteristic
39C23×C4C2311yescharacteristic
40C22×C8C2314yes—
41C4⋊Q8—42no—
42C22⋊Q8—24no—
43C22⋊Q8—24no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4—44no—
56C2×C4⋊C4D414yes—
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C22⋊C4—22no—
61C2×C42—42no—
62C2×C42D411yescharacteristic
63C2×C42D411yescharacteristic
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66#107—28no—
67C4⋊C8—42no—
68#103—28no—
69#103—44no—
70C22⋊C8—24no—
71#95—22no—
72#95—22no—
73#95—22no—
74#95—22no—
75C24C2×D411yescharacteristic
76C2×Q8—22no—
77C2×Q8—22no—
78C2×Q8—24no—
79C2×Q8—24no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—24no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4Q1614yes—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D412yes—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4#4112yes—
105C22×C4C2411yescharacteristic
106C2×C8—24no—
107C2×C8—24no—
108C2×C8—24no—
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—24no—
114C4⋊C4—42no—
115C4⋊C4—42no—
116C4⋊C4—44no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—44no—
120C4⋊C4—42no—
121C4⋊C4—44no—
122C4⋊C4C2×D414yes—
123C22⋊C4—44no—
124C42—22no—
125C42—22no—
126C42—41no—
127C42—41no—
128C23—21no—
129C23—21no—
130C23—21no—
131C23—21no—
132C23#12112yes—
133C23C2×Q1612yes—
134C23#13712yes—
135C23C22×D411yescharacteristic
136Q8—44no—
137Q8—44no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—44no—
153C2×C4—44no—
154C2×C4—48no—
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—42no—
166C2×C4—84no—
167C2×C4#12112yes—
168C2×C4C2×Q1612yes—
169C2×C4C2×Q1612yes—
170C2×C4AGL(1,ℤ/8)12yes—
171C2×C4C22×D411yescharacteristic
172C2×C4C2×#4111yescharacteristic
173C2×C4H(5,𝔽2)12yes—
174C2×C42−1+412yes—
175C8—44no—
176C22—22no—
177C22—22no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—22no—
182C22#46214yes—
183C22C2×#12111yescharacteristic
184C22#53411yescharacteristic
185C22#53611yescharacteristic
186C22C22×Q1611yescharacteristic
187C22C2×AGL(1,ℤ/8)11yescharacteristic
188C22C2×#13711yescharacteristic
189C22#57511yescharacteristic
190C4—22no—
191C4—22no—
192C4—22no—
193C4—22no—
194C4—41no—
195C4—44no—
196C4—82no—
197C4—82no—
198C4—82no—
199C4#53712yes—
200C4#53912yes—
201C2—24no—
202C2#261811yescharacteristic
203C2C2×#46211yescharacteristic
204C2#303611yescharacteristic
205C2#318911yescharacteristic
206C2#319311yescharacteristic
207C2#323011yescharacteristic
208C2#323611yescharacteristic
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
4. #2083C2semidirect
5. #2068C2semidirect
6. #2043C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
#28572
C2×#3892
#20832
#20682
#20432
#18722
#17972