Finite Group Databasebeta

← #12125

Subgroups — #12125

Order 256.

220 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 37 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#2841C211yesmaximal, characteristic, complemented
2#2022C211yesmaximal, characteristic
3#1613C211yesmaximal, characteristic
4#1601C211yesmaximal, characteristic
5#1352C211yesmaximal, characteristic, complemented
6#1351C211yesmaximal, characteristic, complemented
7#1317C211yesmaximal, characteristic, complemented
8#534C2211yescharacteristic
9C2×#123—21no—
10#481—21no—
11#448—21no—
12C2×#99—21no—
13#400—21no—
14#399C2211yescharacteristic
15#398—21no—
16#398—21no—
17#396—21no—
18#395—21no—
19#393C2211yescharacteristic
20#381C2211yescharacteristic
21#380—21no—
22#342C2211yescharacteristic
23#327—21no—
24#327—21no—
25#323C2211yescharacteristic
26#323C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×D4—21no—
29C23×C4D411yescharacteristic
30C2×SD16—41no—
31#123—42no—
32#123—41no—
33#123—42no—
34#121—21no—
35#121—21no—
36#120—41no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—41no—
44C2×C4⋊C4D411yescharacteristic
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—41no—
49C2×C22⋊C4D411yescharacteristic
50C2×C22⋊C4C2311yescharacteristic
51#106—41no—
52#103—41no—
53#102—41no—
54#99—42no—
55C22⋊C8—21no—
56C22⋊C8—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63#95D411yescharacteristic
64#95D411yescharacteristic
65C24—21no—
66C24C2×D411yescharacteristic
67C2×Q8—41no—
68C2×Q8—81no—
69C2×D4—21no—
70C2×D4—21no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4—41no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4SD1612yescomplemented
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91SD16—82no—
92C2×C8—41no—
93C2×C8—41no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—41no—
97C4⋊C4—44no—
98C4⋊C4—81no—
99C4⋊C4—81no—
100C4⋊C4—81no—
101C4⋊C4—81no—
102C4⋊C4—81no—
103C4⋊C4—81no—
104C22⋊C4—22no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—44no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—81no—
112C22⋊C4—81no—
113C23—21no—
114C23—21no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—81no—
121C23—81no—
122C23#12011yescharacteristic
123C23C2×SD1611yescharacteristic
124C23#13711yescharacteristic
125Q8—42no—
126Q8—81no—
127D4—81no—
128D4—81no—
129D4—81no—
130D4—81no—
131D4—81no—
132D4—81no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—81no—
161C2×C4—81no—
162C2×C4—81no—
163C8—81no—
164C8—81no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—81no—
174C22—82no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—161no—
179C22#44811yescharacteristic
180C22#45311yescharacteristic
181C22B(4,𝔽2)11yescharacteristic
182C4—42no—
183C4—41no—
184C4—41no—
185C4—81no—
186C4—81no—
187C4—81no—
188C4—82no—
189C4—81no—
190C4—81no—
191C4—81no—
192C2—21no—
193C2—21no—
194C2—41no—
195C2—41no—
196C2—161no—
197C2#159611yescharacteristic
198C2#220011yescharacteristic
199C2#220111yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radical#534C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #2841C2semidirect
5. #1352C2semidirect
6. #1351C2semidirect
7. #1317C2semidirect
88. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
#28412
#20222
#16132
#16012
#13522
#13512
#13172