Finite Group Databasebeta

← #12201

Subgroups — #12201

Order 256.

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

normal subgroups only

This group has 190 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#2833C211yesmaximal, characteristic, complemented
2#2007C211yesmaximal, characteristic
3#1616C211yesmaximal, characteristic
4#1610C211yesmaximal, characteristic
5#1353C211yesmaximal, characteristic, complemented
6#1346C211yesmaximal, characteristic, complemented
7#1319C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#537C2211yescharacteristic
10#481—21no—
11#480—21no—
12C2×#104—21no—
13#401—21no—
14#400—21no—
15#400—21no—
16#400C2211yescharacteristic
17#398—21no—
18#398C2211yescharacteristic
19#395—21no—
20#391C2211yescharacteristic
21#387—21no—
22#343C2211yescharacteristic
23#327—21no—
24#327—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C2×#41—21no—
28C22×Q8D411yescharacteristic
29#137—42no—
30#137—42no—
31C2×Q16—41no—
32C2×SD16—41no—
33C2×M16—21no—
34C22⋊Q8—41no—
35#121—21no—
36#121—21no—
37#121—41no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4C2311yescharacteristic
45C2×C42D411yescharacteristic
46#107—41no—
47#106—41no—
48#104—42no—
49#102—41no—
50#102—41no—
51C22⋊C8—21no—
52C22⋊C8—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63#95D411yescharacteristic
64#95D411yescharacteristic
65#41—41no—
66#41—41no—
67#41—41no—
68#41—41no—
69C2×Q8—21no—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—41no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—41no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88Q16—44no—
89SD16—44no—
90M16—22no—
91M16—41no—
92C2×C8—41no—
93C2×C8—41no—
94C2×C8—41no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—81no—
103C4⋊C4—81no—
104C4⋊C4—81no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C42—41no—
108C23—41no—
109C23#12011yescharacteristic
110Q8—42no—
111Q8—41no—
112Q8—41no—
113Q8—81no—
114D4—41no—
115D4—41no—
116D4—81no—
117D4—81no—
118D4—81no—
119D4—81no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—41no—
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—81no—
145C2×C4—81no—
146C2×C4#13511yescharacteristic
147C2×C4AGL(1,ℤ/8)11yescharacteristic
148C8—42no—
149C8—81no—
150C8—81no—
151C22—21no—
152C22—21no—
153C22—81no—
154C22—81no—
155C22—81no—
156C22#44911yescharacteristic
157C22#45311yescharacteristic
158C22B(4,𝔽2)11yescharacteristic
159C4—21no—
160C4—21no—
161C4—41no—
162C4—41no—
163C4—81no—
164C4—81no—
165C4—81no—
166C4—81no—
167C4—81no—
168C4—81no—
169C4—81no—
170C2—21no—
171C2—21no—
172C2—161no—
173C2#160011yescharacteristic
174C2#220111yescharacteristic
175C2#220311yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radical#537C22
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. #2833C2semidirect
5. #1353C2semidirect
6. #1346C2semidirect
7. #1319C2semidirect

Maximal subgroups

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

subgroupindex
#28332
#20072
#16162
#16102
#13532
#13462
#13192