Finite Group Databasebeta

← #33148

Subgroups — #33148

Order 256.

446 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 153 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×C4×C4⋊C4C211yesmaximal, characteristic
3#1987C212yesmaximal, complemented
4#1948C212yesmaximal, complemented
5#1920C214yesmaximal, complemented
6#1853C212yesmaximal
7#1823C212yesmaximal
8#1797C211yesmaximal, characteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C22⋊Q8—21no—
11C22×C4⋊C4C2211yescharacteristic
12C22×C4⋊C4C2211yescharacteristic
13C22×C42—21no—
14C22×C42C2211yescharacteristic
15C2×#106C2214yes—
16C2×C4⋊C8C2212yes—
17C2×C22⋊C8C2212yes—
18#391—21no—
19#389—21no—
20#389C2212yes—
21#384C2211yescharacteristic
22#380—21no—
23#380C2212yes—
24C4×C4⋊C4—21no—
25C4×C4⋊C4—21no—
26C4×C4⋊C4C2212yes—
27C4×C4⋊C4C2212yes—
28C2×#95—21no—
29#340C2214yes—
30#340C2214yes—
31#340C2214yes—
32#336C2212yes—
33C22×Q8—21no—
34C23×C4—21no—
35C23×C4—21no—
36C23×C4D411yescharacteristic
37C23×C4D411yescharacteristic
38C23×C4C2311yescharacteristic
39C22×C8C2314yes—
40C4⋊Q8—24no—
41C22⋊Q8—44no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—24no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4Q814yescomplemented
51C2×C4⋊C4C2312yes—
52C2×C4⋊C4C2312yes—
53C2×C4⋊C4C2312yes—
54C2×C4⋊C4C2312yes—
55C2×C22⋊C4—22no—
56C2×C42—24no—
57C2×C42—22no—
58C2×C42—22no—
59C2×C42—44no—
60C2×C42D411yescharacteristic
61C2×C42D411yescharacteristic
62C2×C42C2311yescharacteristic
63C2×C42C2311yescharacteristic
64#106—28no—
65C4⋊C8—24no—
66C22⋊C8—24no—
67#95—22no—
68#95—22no—
69#95—22no—
70#95—22no—
71C24C2×D411yescharacteristic
72C2×Q8—42no—
73C2×Q8—42no—
74C2×Q8—42no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—24no—
80C22×C4—24no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—41no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4SD1614yes—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×Q812yes—
99C22×C4#4112yes—
100C22×C4C2411yescharacteristic
101C2×C8—24no—
102C2×C8—24no—
103C2×C8—24no—
104C4⋊C4—24no—
105C4⋊C4—24no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—24no—
109C4⋊C4—24no—
110C4⋊C4—42no—
111C4⋊C4—44no—
112C4⋊C4—42no—
113C4⋊C4—42no—
114C4⋊C4C2×Q814yes—
115C4⋊C4#4114yes—
116C22⋊C4—44no—
117C42—22no—
118C42—22no—
119C42—22no—
120C42—44no—
121C42—44no—
122C42#4114yes—
123C23—21no—
124C23—21no—
125C23—21no—
126C23—21no—
127C23C22⋊Q812yes—
128C23C2×SD1612yes—
129C23AGL(1,ℤ/8)12yes—
130C23C22×D411yescharacteristic
131Q8—84no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—28no—
141C2×C4—24no—
142C2×C4—21no—
143C2×C4—24no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—44no—
147C2×C4—48no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—48no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4C22⋊Q812yes—
160C2×C4#12312yes—
161C2×C4#12312yes—
162C2×C4C2×SD1612yes—
163C2×C4C2×SD1612yes—
164C2×C4#13712yes—
165C2×C4C22×Q811yescharacteristic
166C2×C4C2×#4111yescharacteristic
167C2×C4C2×#4112yes—
168C2×C4C2×#4112yes—
169C8—44no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—22no—
174C22—22no—
175C22—22no—
176C22#47614yes—
177C22C2×C22⋊Q811yescharacteristic
178C22C2×#12311yescharacteristic
179C22#52511yescharacteristic
180C22C22×SD1611yescharacteristic
181C22C2×AGL(1,ℤ/8)11yescharacteristic
182C22C2×#13711yescharacteristic
183C22#57511yescharacteristic
184C4—22no—
185C4—22no—
186C4—22no—
187C4—41no—
188C4—44no—
189C4—44no—
190C4—82no—
191C4—82no—
192C4#48314yes—
193C4#52912yes—
194C4#53312yes—
195C2—24no—
196C2#285511yescharacteristic
197C2C2×#47611yescharacteristic
198C2#307811yescharacteristic
199C2C2×#48311yescharacteristic
200C2#309411yescharacteristic
201C2#310111yescharacteristic
202C2#311211yescharacteristic
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. #1987C2semidirect
4. #1948C2semidirect
5. #1920C2semidirect
50. C2×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#28572
C2×C4×C4⋊C42
#19872
#19482
#19202
#18532
#18232
#17972