Finite Group Databasebeta

← #12144

Subgroups — #12144

Order 256.

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

normal subgroups only

This group has 199 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#2795C211yesmaximal, characteristic, complemented
2#2039C211yesmaximal, characteristic, complemented
3#1616C211yesmaximal, characteristic, complemented
4#1598C211yesmaximal, characteristic
5#1353C211yesmaximal, characteristic, complemented
6#1345C211yesmaximal, characteristic, complemented
7#1325C211yesmaximal, characteristic, complemented
8#537C2211yescharacteristic
9#481—21no—
10#447—21no—
11C2×#104—21no—
12#417—21no—
13#400C2211yescharacteristic
14#396—21no—
15#394—21no—
16#394C2211yescharacteristic
17#388—21no—
18#388—21no—
19#388—21no—
20#387—21no—
21#384C2211yescharacteristic
22#343C2211yescharacteristic
23#327—21no—
24#327—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C2×#41—21no—
28C22×D4D411yescharacteristic
29C2×D8—41no—
30C2×M16—21no—
31C22⋊Q8—41no—
32#121—21no—
33#121—21no—
34#121—41no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4D411yescharacteristic
39C2×C4⋊C4D411yescharacteristic
40C2×C4⋊C4C2311yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C42—21no—
48C2×C42D411yescharacteristic
49#106—41no—
50#104—42no—
51#103—41no—
52#102—41no—
53#102—41no—
54#102—41no—
55C22⋊C8—21no—
56C22⋊C8—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95D411yescharacteristic
62#95D411yescharacteristic
63C24—21no—
64#41—41no—
65#41—41no—
66#41—41no—
67#41—41no—
68C2×Q8—21no—
69C2×D4—21no—
70C2×D4—21no—
71C2×D4—21no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—81no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—41no—
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84D8—82no—
85M16—41no—
86M16—41no—
87C2×C8—41no—
88C2×C8—41no—
89C2×C8—41no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—42no—
96C4⋊C4—81no—
97C4⋊C4—81no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—81no—
105C22⋊C4—81no—
106C22⋊C4—81no—
107C42—21no—
108C42—21no—
109C42—81no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23#12011yescharacteristic
119Q8—41no—
120Q8—41no—
121D4—41no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—81no—
126D4—81no—
127D4—81no—
128D4—81no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
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—81no—
152C2×C4—81no—
153C2×C4#13511yescharacteristic
154C2×C4AGL(1,ℤ/8)11yescharacteristic
155C8—81no—
156C8—81no—
157C8—81no—
158C22—21no—
159C22—21no—
160C22—81no—
161C22—81no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22#44911yescharacteristic
171C22#45411yescharacteristic
172C22B(4,𝔽2)11yescharacteristic
173C4—21no—
174C4—21no—
175C4—42no—
176C4—41no—
177C4—41no—
178C4—81no—
179C4—81no—
180C4—81no—
181C4—81no—
182C4—81no—
183C2—21no—
184C2—21no—
185C2—161no—
186C2—161no—
187C2#159711yescharacteristic
188C2#220011yescharacteristic
189C2#220411yescharacteristic
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. #2795C2semidirect
2. #2039C2semidirect
3. #1616C2semidirect
5. #1353C2semidirect
6. #1345C2semidirect
7. #1325C2semidirect

Maximal subgroups

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

subgroupindex
#27952
#20392
#16162
#15982
#13532
#13452
#13252