Finite Group Databasebeta

← #7804

Subgroups — #7804

Order 256.

211 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 49 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#2804C211yesmaximal, characteristic
2#2096C211yesmaximal, characteristic, complemented
3#2072C211yesmaximal, characteristic, complemented
4#2071C211yesmaximal, characteristic
5#1272C211yesmaximal, characteristic, complemented
6#1272C211yesmaximal, characteristic, complemented
7#1270C211yesmaximal, characteristic
8C2×#106—21no—
9C2×#103—21no—
10C2×#103—21no—
11C2×#102—21no—
12#400—21no—
13#393—21no—
14#393—21no—
15#391C2211yescharacteristic
16#391C2211yescharacteristic
17#390C2211yescharacteristic
18#386—21no—
19#385—21no—
20#385—21no—
21#384C2211yescharacteristic
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×Q8D411yescharacteristic
29C22×D4D411yescharacteristic
30C23×C4—21no—
31C22×C8—21no—
32C22×C8—21no—
33C22×C8—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4D411yescharacteristic
39C2×C4⋊C4D411yescharacteristic
40C2×C4⋊C4D411yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C42C2311yescharacteristic
47#106—42no—
48#103—42no—
49#103—42no—
50#102—42no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58C24—21no—
59C2×Q8—21no—
60C2×Q8—21no—
61C2×Q8—21no—
62C2×Q8—21no—
63C2×Q8—41no—
64C2×Q8—41no—
65C2×Q8—41no—
66C2×Q8—41no—
67C2×D4—21no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4SD1612yes—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—41no—
77C22×C4—81no—
78C22×C4—81no—
79C22×C4—81no—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C2×C8—22no—
84C2×C8—41no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8—41no—
88C2×C8—41no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8—41no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—81no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—81no—
107C22⋊C4—81no—
108C22⋊C4—81no—
109C42—41no—
110C23—22no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23#12011yescharacteristic
118Q8—42no—
119Q8—42no—
120Q8—81no—
121Q8—81no—
122D4—42no—
123D4—81no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
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—41no—
145C2×C4—84no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4—81no—
149C2×C4C2×SD1611yescharacteristic
150C2×C4#13511yescharacteristic
151C2×C4#13511yescharacteristic
152C2×C4AGL(1,ℤ/8)11yescharacteristic
153C2×C4#13711yescharacteristic
154C2×C4#13711yescharacteristic
155C8—42no—
156C8—81no—
157C8—81no—
158C22—42no—
159C22—42no—
160C22—42no—
161C22—81no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22#44811yescharacteristic
166C22#44911yescharacteristic
167C22#45211yescharacteristic
168C22#45311yescharacteristic
169C22#45511yescharacteristic
170C22#45511yescharacteristic
171C22#45811yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—81no—
179C4—81no—
180C4—81no—
181C4—81no—
182C4—81no—
183C4—161no—
184C2—82no—
185C2#160811yescharacteristic
186C2#161211yescharacteristic
187C2#161811yescharacteristic
188C2#162211yescharacteristic
189C2#164011yescharacteristic
190C2#164311yescharacteristic
191C2#165911yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2096C2semidirect
3. #2072C2semidirect
5. #1272C2semidirect
6. #1272C2semidirect

Maximal subgroups

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

subgroupindex
#28042
#20962
#20722
#20712
#12722
#12702