Finite Group Databasebeta

← #12150

Subgroups — #12150

Order 256.

216 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 37 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#2722C211yesmaximal, characteristic, complemented
2#2044C211yesmaximal, characteristic
3#1607C211yesmaximal, characteristic
4#1607C211yesmaximal, characteristic
5#1344C211yesmaximal, characteristic, complemented
6#1344C211yesmaximal, characteristic, complemented
7#1326C211yesmaximal, characteristic, complemented
8#534C2211yescharacteristic
9C2×#126—21no—
10#480—21no—
11#480—21no—
12C2×#99—21no—
13#398—21no—
14#398—21no—
15#397C2211yescharacteristic
16#397C2211yescharacteristic
17#389—21no—
18#387—21no—
19#386—21no—
20#385C2211yescharacteristic
21#380—21no—
22#342C2211yescharacteristic
23#327—21no—
24#327—21no—
25#323C2211yescharacteristic
26#323C2211yescharacteristic
27C22×D4—21no—
28C23×C4D411yescharacteristic
29#126—42no—
30#126—42no—
31#123—41no—
32#121—21no—
33#121—21no—
34#120—41no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4D411yescharacteristic
42C2×C4⋊C4D411yescharacteristic
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—41no—
47C2×C22⋊C4D411yescharacteristic
48C2×C22⋊C4C2311yescharacteristic
49C2×C42—21no—
50#107—41no—
51#107—41no—
52#102—41no—
53#102—41no—
54#99—42no—
55C22⋊C8—21no—
56C22⋊C8—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—41no—
63#95D411yescharacteristic
64#95D411yescharacteristic
65C24—21no—
66C24C2×D411yescharacteristic
67C2×D4—21no—
68C2×D4—21no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—41no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—42no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C2×C8—41no—
88C2×C8—41no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—42no—
96C4⋊C4—42no—
97C4⋊C4—42no—
98C4⋊C4—81no—
99C22⋊C4—21no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—82no—
110C22⋊C4D812yes—
111C42—42no—
112C23—21no—
113C23—21no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—81no—
120C23—81no—
121C23#12011yescharacteristic
122C23C2×D811yescharacteristic
123C23AGL(1,ℤ/8)11yescharacteristic
124D4—81no—
125D4—81no—
126D4—81no—
127D4—81no—
128D4—81no—
129D4—81no—
130C2×C4—22no—
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—42no—
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—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4—81no—
159C2×C4—81no—
160C8—81no—
161C8—81no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—81no—
171C22—82no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—161no—
176C22#44711yescharacteristic
177C22#45411yescharacteristic
178C22B(4,𝔽2)11yescharacteristic
179C4—42no—
180C4—41no—
181C4—42no—
182C4—41no—
183C4—81no—
184C4—81no—
185C4—81no—
186C4—81no—
187C4—81no—
188C4—161no—
189C2—21no—
190C2—21no—
191C2—41no—
192C2—41no—
193C2—161no—
194C2#159811yescharacteristic
195C2#219711yescharacteristic
196C2#220111yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radical#534C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2722C2semidirect
5. #1344C2semidirect
6. #1344C2semidirect
7. #1326C2semidirect

Maximal subgroups

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

subgroupindex
#27222
#20442
#16072
#13442
#13262