Finite Group Databasebeta

← #23657

Subgroups — #23657

Order 256.

473 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 179 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×#428C212yesmaximal
2C2×#389C211yesmaximal, characteristic
3#2285C211yesmaximal, characteristic
4#2085C214yesmaximal, complemented
5#2084C212yesmaximal, complemented
6#2083C212yesmaximal, complemented
7#1736C212yesmaximal
8#1728C211yesmaximal, characteristic
9C22×M16C2212yes—
10C2×#117C2212yes—
11C22×C4⋊C4—21no—
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C42C2211yescharacteristic
15#428—28no—
16C2×#107C2214yes—
17C2×#106C2214yes—
18#389—21no—
19#389—21no—
20#389C2212yes—
21#389C2212yes—
22#384C2211yescharacteristic
23#384C2211yescharacteristic
24#382C2212yes—
25#380—21no—
26#380—21no—
27C4×C4⋊C4—21no—
28C4×C4⋊C4—21no—
29C2×#95—21no—
30#340C2214yes—
31#340C2214yes—
32#336C2212yes—
33#336C2212yes—
34C23×C4—21no—
35C23×C4—21no—
36C23×C4D411yescharacteristic
37C23×C4D411yescharacteristic
38C23×C4C2311yescharacteristic
39C2×M16—24no—
40C2×M16—24no—
41C2×M16Q818yes—
42C22×C8C2314yes—
43#117—24no—
44#117—24no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—44no—
56C2×C4⋊C4D414yes—
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C22⋊C4—22no—
62C2×C42—24no—
63C2×C42—22no—
64C2×C42D411yescharacteristic
65C2×C42D411yescharacteristic
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68#107—28no—
69#106—28no—
70#95—22no—
71#95—22no—
72#95—22no—
73#95—22no—
74C24C2×D411yescharacteristic
75C22×C4—24no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D412yes—
97C22×C4C2×Q814yes—
98C22×C4#4112yes—
99C22×C4#4114yes—
100C22×C4C2411yescharacteristic
101M16—216no—
102C2×C8—24no—
103C2×C8—24no—
104C2×C8C2×Q818yes—
105C4⋊C4—24no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—24no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—42no—
113C4⋊C4—42no—
114C4⋊C4—44no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—44no—
118C4⋊C4—42no—
119C4⋊C4C2×D414yes—
120C22⋊C4—44no—
121C42—22no—
122C42—22no—
123C42—22no—
124C42—22no—
125C42—44no—
126C23—21no—
127C23—21no—
128C23—21no—
129C23—21no—
130C23#12112yes—
131C23#12312yes—
132C23C4⋊Q812yes—
133C23C22×D411yescharacteristic
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—24no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—44no—
146C2×C4—44no—
147C2×C4—48no—
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—42no—
159C2×C4—84no—
160C2×C4#12112yes—
161C2×C4C22⋊Q818yes—
162C2×C4#12312yes—
163C2×C4#12518yes—
164C2×C4C4⋊Q812yes—
165C2×C4AGL(1,ℤ/8)12yes—
166C2×C4#13712yes—
167C2×C4C22×D411yescharacteristic
168C2×C4C22×Q812yes—
169C2×C4C2×#4112yes—
170C2×C4C2×#4111yescharacteristic
171C8—28no—
172C22—22no—
173C22—22no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22#39814yes—
179C22C2×#12111yescharacteristic
180C22C2×#12311yescharacteristic
181C22C2×C4⋊Q811yescharacteristic
182C22C2×AGL(1,ℤ/8)11yescharacteristic
183C22C2×#13711yescharacteristic
184C22#57511yescharacteristic
185C22#57511yescharacteristic
186C4—22no—
187C4—22no—
188C4—41no—
189C4—41no—
190C4—44no—
191C4—82no—
192C4—82no—
193C4—82no—
194C4#39814yes—
195C4C2×C22⋊Q812yes—
196C4C2×#12512yes—
197C2—24no—
198C2C2×#39811yescharacteristic
199C2#303611yescharacteristic
200C2#303711yescharacteristic
201C2#309411yescharacteristic
202C2#309511yescharacteristic
203C2#316511yescharacteristic
204C2#316611yescharacteristic
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. #2085C2semidirect
5. #2084C2semidirect
6. #2083C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4282
C2×#3892
#22852
#20852
#20842
#20832
#17362
#17282