Finite Group Databasebeta

← #7820

Subgroups — #7820

Order 256.

198 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 47 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#2808C211yesmaximal, characteristic
2#2093C211yesmaximal, characteristic
3#2084C211yesmaximal, characteristic, complemented
4#2072C211yesmaximal, characteristic, complemented
5#1272C211yesmaximal, characteristic, complemented
6#1271C211yesmaximal, characteristic, complemented
7#1270C211yesmaximal, characteristic
8C2×#107—21no—
9C2×#106—21no—
10C2×#103—21no—
11C2×#102—21no—
12#397—21no—
13#394—21no—
14#393—21no—
15#391C2211yescharacteristic
16#390C2211yescharacteristic
17#389C2211yescharacteristic
18#388—21no—
19#388—21no—
20#387—21no—
21#384C2211yescharacteristic
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C22×D4D411yescharacteristic
29C22×C8—21no—
30C22×C8—21no—
31C22×C8—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4D411yescharacteristic
37C2×C4⋊C4D411yescharacteristic
38C2×C4⋊C4D411yescharacteristic
39C2×C4⋊C4D411yescharacteristic
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C42—21no—
47C2×C42C2311yescharacteristic
48#107—42no—
49#106—42no—
50#103—42no—
51#102—42no—
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—41no—
62C2×Q8—41no—
63C2×D4—21no—
64C2×D4—21no—
65C2×D4—41no—
66C2×D4—41no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—41no—
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C22×C4C2×D411yescharacteristic
76C2×C8—22no—
77C2×C8—22no—
78C2×C8—41no—
79C2×C8—41no—
80C2×C8—41no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—41no—
84C2×C8—41no—
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—81no—
103C22⋊C4—81no—
104C22⋊C4—81no—
105C22⋊C4—81no—
106C42—41no—
107C42—81no—
108C23—41no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23#12011yescharacteristic
116Q8—42no—
117Q8—81no—
118D4—42no—
119D4—81no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
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—81no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4#13511yescharacteristic
145C2×C4#13511yescharacteristic
146C2×C4#13511yescharacteristic
147C2×C4AGL(1,ℤ/8)11yescharacteristic
148C2×C4AGL(1,ℤ/8)11yescharacteristic
149C2×C4#13711yescharacteristic
150C8—42no—
151C8—42no—
152C8—81no—
153C22—81no—
154C22—81no—
155C22—81no—
156C22—81no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22#44911yescharacteristic
161C22#44911yescharacteristic
162C22#45211yescharacteristic
163C22#45311yescharacteristic
164C22#45511yescharacteristic
165C22#45611yescharacteristic
166C22#45811yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—81no—
174C4—81no—
175C4—81no—
176C4—81no—
177C4—81no—
178C4—161no—
179C2—161no—
180C2#160911yescharacteristic
181C2#161211yescharacteristic
182C2#161811yescharacteristic
183C2#162011yescharacteristic
184C2#164611yescharacteristic
185C2#165211yescharacteristic
186C2#166111yescharacteristic
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
3. #2084C2semidirect
4. #2072C2semidirect
5. #1272C2semidirect
6. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#28082
#20932
#20842
#20722
#12722
#12712
#12702