Finite Group Databasebeta

← #12092

Subgroups — #12092

Order 256.

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

normal subgroups only

This group has 272 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#2600C211yesmaximal, characteristic, complemented
2#2022C211yesmaximal, characteristic, complemented
3#1607C211yesmaximal, characteristic, complemented
4#1595C211yesmaximal, characteristic
5#1344C211yesmaximal, characteristic, complemented
6#1343C211yesmaximal, characteristic, complemented
7#1317C211yesmaximal, characteristic, complemented
8#534C2211yescharacteristic
9C2×#123—21no—
10C2×C4×D4—21no—
11#480—21no—
12#447—21no—
13C2×#99—21no—
14#397C2211yescharacteristic
15#394—21no—
16#392C2211yescharacteristic
17#388—21no—
18#387—21no—
19#386—21no—
20#385—21no—
21#381C2211yescharacteristic
22#342C2211yescharacteristic
23#327—21no—
24#327—21no—
25#323C2211yescharacteristic
26#323C2211yescharacteristic
27C22×D4—21no—
28C22×D4—21no—
29C22×D4D411yescharacteristic
30C23×C4—21no—
31C23×C4D411yescharacteristic
32C2×D8—41no—
33#123—42no—
34#123—41no—
35#123—42no—
36#121—21no—
37#121—21no—
38#120—41no—
39C4×D4—42no—
40C4×D4—42no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—41no—
44C2×C4⋊C4D411yescharacteristic
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C22⋊C4—41no—
52C2×C22⋊C4D411yescharacteristic
53C2×C22⋊C4C2311yescharacteristic
54C2×C42—21no—
55#107—41no—
56#102—41no—
57#102—41no—
58#99—42no—
59C22⋊C8—21no—
60C22⋊C8—21no—
61#95—21no—
62#95—21no—
63#95—21no—
64#95—21no—
65#95D411yescharacteristic
66#95D411yescharacteristic
67C24—21no—
68C24—21no—
69C24C2×D411yescharacteristic
70C2×D4—21no—
71C2×D4—21no—
72C2×D4—42no—
73C2×D4—41no—
74C2×D4—42no—
75C2×D4—41no—
76C2×D4—41no—
77C2×D4—42no—
78C2×D4—41no—
79C2×D4—41no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4—41no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—42no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—81no—
96C22×C4—81no—
97C22×C4D812yescomplemented
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100D8—82no—
101C2×C8—41no—
102C2×C8—41no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—41no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—44no—
109C22⋊C4—22no—
110C22⋊C4—41no—
111C22⋊C4—42no—
112C22⋊C4—42no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—42no—
116C22⋊C4—44no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—81no—
122C42—42no—
123C23—21no—
124C23—21no—
125C23—22no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—81no—
138C23—81no—
139C23#12011yescharacteristic
140C23C2×D811yescharacteristic
141C23AGL(1,ℤ/8)11yescharacteristic
142D4—42no—
143D4—44no—
144D4—44no—
145D4—81no—
146D4—81no—
147D4—81no—
148D4—81no—
149D4—81no—
150D4—81no—
151D4—81no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—81no—
177C2×C4—84no—
178C2×C4—81no—
179C2×C4—81no—
180C8—81no—
181C8—81no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—41no—
187C22—41no—
188C22—42no—
189C22—42no—
190C22—41no—
191C22—41no—
192C22—42no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22—82no—
197C22—81no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—161no—
203C22#44711yescharacteristic
204C22#45311yescharacteristic
205C22B(4,𝔽2)11yescharacteristic
206C4—42no—
207C4—42no—
208C4—41no—
209C4—41no—
210C4—81no—
211C4—81no—
212C4—82no—
213C4—81no—
214C4—81no—
215C2—21no—
216C2—21no—
217C2—41no—
218C2—41no—
219C2—82no—
220C2—161no—
221C2#159511yescharacteristic
222C2#219711yescharacteristic
223C2#219811yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radical#1595C2
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. #2600C2semidirect
2. #2022C2semidirect
3. #1607C2semidirect
5. #1344C2semidirect
6. #1343C2semidirect
7. #1317C2semidirect
97. C22×C4D8semidirect

Maximal subgroups

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

subgroupindex
#26002
#20222
#16072
#15952
#13442
#13432
#13172