Finite Group Databasebeta

← #29640

Subgroups — #29640

Order 256.

538 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22×#125C211yesmaximal, characteristic, complemented
2C2×#481C211yesmaximal, characteristic
3C2×#480C211yesmaximal, characteristic
4#2854C211yesmaximal, characteristic
5#2059C214yesmaximal, complemented
6#2025C214yesmaximal, complemented
7#1824C211yesmaximal, characteristic, complemented
8#1823C211yesmaximal, characteristic, complemented
9#1817C211yesmaximal, characteristic, complemented
10C2×#125—24no—
11C2×#125—24no—
12C2×#125—22no—
13C2×#125C2214yes—
14C2×#123—21no—
15C2×#121C2211yescharacteristic
16C22×C4⋊C4—21no—
17C22×C4⋊C4—21no—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C42C2211yescharacteristic
21#481—24no—
22#480—24no—
23C2×#107C2212yes—
24C2×#106C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#390C2212yes—
30#389C2212yes—
31#388—22no—
32#386—21no—
33#380C2211yescharacteristic
34#340C2214yes—
35#340C2214yes—
36#336C2214yes—
37C22×D4—21no—
38C22×D4C2311yescharacteristic
39C23×C4—21no—
40C23×C4—21no—
41C23×C4D411yescharacteristic
42C23×C4D411yescharacteristic
43C23×C4C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2312yes—
46#125—216no—
47#125—28no—
48#125—28no—
49#123—44no—
50#121—22no—
51#121—22no—
52C2×C4⋊C4—28no—
53C2×C4⋊C4—28no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4D414yes—
62C2×C4⋊C4D414yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C42—22no—
72C2×C42—22no—
73C2×C42D412yes—
74C2×C42C2312yes—
75#107—24no—
76#106—24no—
77#102—24no—
78#102—24no—
79C22⋊C8—22no—
80C22⋊C8—22no—
81#95—21no—
82#95—22no—
83C24—21no—
84C24C2×D411yescharacteristic
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—24no—
88C2×D4—42no—
89C2×D4—42no—
90C2×D4—42no—
91C22×C4—24no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—24no—
95C22×C4—24no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—42no—
103C22×C4—44no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4C2×D412yes—
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D412yes—
110C22×C4C2×D412yes—
111C22×C4C2411yescharacteristic
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—24no—
115C2×C8—24no—
116C4⋊C4—216no—
117C4⋊C4—216no—
118C4⋊C4—24no—
119C4⋊C4—24no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—24no—
125C4⋊C4—44no—
126C4⋊C4C2×D414yes—
127C4⋊C4C2×D414yes—
128C22⋊C4—44no—
129C22⋊C4—44no—
130C22⋊C4—42no—
131C22⋊C4—42no—
132C42—24no—
133C42—22no—
134C42—22no—
135C23—22no—
136C23—22no—
137C23—41no—
138C23—42no—
139C23—41no—
140C23—41no—
141C23—42no—
142C23#12012yes—
143C23#13512yes—
144C23AGL(1,ℤ/8)12yes—
145C23C22×D411yescharacteristic
146D4—42no—
147D4—42no—
148D4—84no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—44no—
161C2×C4—48no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—44no—
166C2×C4—48no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4#12012yes—
174C2×C4C22×D411yescharacteristic
175C2×C4C22×D411yescharacteristic
176C2×C42−1+414yes—
177C8—42no—
178C8—42no—
179C22—22no—
180C22—24no—
181C22—24no—
182C22—22no—
183C22—81no—
184C22—82no—
185C22—82no—
186C22—81no—
187C22—81no—
188C22#44914yes—
189C22C2×#12011yescharacteristic
190C22#53612yes—
191C22C2×#13511yescharacteristic
192C22C2×AGL(1,ℤ/8)11yescharacteristic
193C22#57512yes—
194C4—22no—
195C4—42no—
196C4—44no—
197C4—42no—
198C4—44no—
199C4—82no—
200C4—81no—
201C4#53612yes—
202C4#53612yes—
203C2—24no—
204C2—161no—
205C2#261411yescharacteristic
206C2C2×#44911yescharacteristic
207C2#300511yescharacteristic
208C2#323212yes—
209C2#323412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
1. C22×#125C2semidirect
5. #2059C2semidirect
6. #2025C2semidirect
7. #1824C2semidirect
8. #1823C2semidirect
9. #1817C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1252
C2×#4812
C2×#4802
#28542
#20592
#20252
#18242
#18232
#18172