Finite Group Databasebeta

← #12199

Subgroups — #12199

Order 256.

217 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 35 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#2832C211yesmaximal, characteristic, complemented
2#2007C211yesmaximal, characteristic, complemented
3#1604C211yesmaximal, characteristic, complemented
4#1598C211yesmaximal, characteristic
5#1354C211yesmaximal, characteristic, complemented
6#1345C211yesmaximal, characteristic, complemented
7#1319C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#537C2211yescharacteristic
10#448—21no—
11#447—21no—
12C2×#104—21no—
13#400—21no—
14#397—21no—
15#395C2211yescharacteristic
16#394—21no—
17#394C2211yescharacteristic
18#393—21no—
19#393—21no—
20#391C2211yescharacteristic
21#388—21no—
22#343C2211yescharacteristic
23#327—21no—
24#327—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C2×#41—21no—
28C22×Q8D411yescharacteristic
29C22×Q8D411yescharacteristic
30C22×D4D411yescharacteristic
31#137—42no—
32#137—42no—
33C2×Q16—41no—
34C2×SD16—41no—
35C2×SD16—41no—
36C2×D8—41no—
37C2×M16—21no—
38C22⋊Q8—41no—
39#121—21no—
40#121—21no—
41#121—41no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4C2311yescharacteristic
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C42D411yescharacteristic
52#104—42no—
53#103—41no—
54#102—41no—
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—
66#41—41no—
67#41—41no—
68#41—41no—
69#41—41no—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8—41no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—81no—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4—21no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—81no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—41no—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92Q16—44no—
93SD16—44no—
94SD16—82no—
95D8—82no—
96M16—22no—
97M16—41no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—41no—
104C4⋊C4—81no—
105C4⋊C4—81no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C22⋊C4—81no—
111C22⋊C4—81no—
112C22⋊C4—81no—
113C22⋊C4—81no—
114C22⋊C4—81no—
115C42—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23#12011yescharacteristic
125Q8—42no—
126Q8—41no—
127Q8—41no—
128Q8—42no—
129Q8—81no—
130Q8—81no—
131D4—41no—
132D4—41no—
133D4—42no—
134D4—81no—
135D4—81no—
136D4—81no—
137D4—81no—
138D4—81no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
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—41no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4#13511yescharacteristic
164C2×C4AGL(1,ℤ/8)11yescharacteristic
165C8—42no—
166C8—81no—
167C8—81no—
168C22—21no—
169C22—21no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22#44911yescharacteristic
181C22#45311yescharacteristic
182C22B(4,𝔽2)11yescharacteristic
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—81no—
188C4—81no—
189C4—81no—
190C4—81no—
191C4—81no—
192C4—81no—
193C2—21no—
194C2—21no—
195C2—161no—
196C2—161no—
197C2#160011yescharacteristic
198C2#219911yescharacteristic
199C2#220311yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radical#1598C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2832C2semidirect
2. #2007C2semidirect
3. #1604C2semidirect
5. #1354C2semidirect
6. #1345C2semidirect
7. #1319C2semidirect

Maximal subgroups

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

subgroupindex
#28322
#20072
#16042
#15982
#13542
#13452
#13192