Finite Group Databasebeta

← #11566

Subgroups — #11566

Order 256.

247 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 51 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#2544C211yesmaximal, characteristic
2#1465C211yesmaximal, characteristic, complemented
3#1459C211yesmaximal, characteristic, complemented
4#1319C212yesmaximal, complemented
5#1318C212yesmaximal, complemented
6C2×#127C411yescharacteristic
7C2×#124C411yescharacteristic
8C2×#124C411yescharacteristic
9C2×C4×D4C2211yescharacteristic
10#407—21no—
11C2×C22⋊C8—21no—
12#394—21no—
13#394—21no—
14#391C2211yescharacteristic
15#390C2211yescharacteristic
16#386—21no—
17#386—21no—
18C4×C4⋊C4C411yescharacteristic
19#324C2212yes—
20#324C2212yes—
21#323—21no—
22#323—21no—
23C22×Q8C2×C411yescharacteristic
24C22×D4—21no—
25C22×D4—21no—
26C22×D4C2×C411yescharacteristic
27C22×D4C2×C411yescharacteristic
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C2×M16—41no—
31C22×C8—41no—
32#127—42no—
33#124—41no—
34#124—41no—
35#124—42no—
36C4×D4—41no—
37C4×D4—41no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4D411yescharacteristic
47C2×C22⋊C4D411yescharacteristic
48C2×C42C2×C411yescharacteristic
49C2×C42C2×C411yescharacteristic
50C2×C42C2×C411yescharacteristic
51C22⋊C8—22no—
52C22⋊C8—22no—
53C22⋊C8—41no—
54C22⋊C8—41no—
55#95—21no—
56#95—21no—
57#95—21no—
58C24—21no—
59C24C22⋊C411yescharacteristic
60C24C22⋊C411yescharacteristic
61C2×Q8—22no—
62C2×Q8—41no—
63C2×Q8—41no—
64C2×D4—22no—
65C2×D4—22no—
66C2×D4—22no—
67C2×D4—41no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4—42no—
73C2×D4—42no—
74C2×D4—81no—
75C2×D4—81no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—81no—
84C22×C4—81no—
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22×C411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90M16—82no—
91C2×C8—42no—
92C2×C8—42no—
93C2×C8—82no—
94C4⋊C4—41no—
95C4⋊C4—42no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—81no—
104C22⋊C4—81no—
105C42—21no—
106C42—21no—
107C42—41no—
108C42—41no—
109C23—21no—
110C23—21no—
111C23—21no—
112C23—21no—
113C23—21no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23#10012yes—
126C23C2×C22⋊C411yescharacteristic
127Q8—82no—
128D4—44no—
129D4—82no—
130D4—84no—
131D4—84no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—42no—
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—81no—
153C2×C4—81no—
154C2×C4#9911yescharacteristic
155C2×C4#9911yescharacteristic
156C2×C4#13112yes—
157C8—82no—
158C8—82no—
159C22—21no—
160C22—21no—
161C22—22no—
162C22—42no—
163C22—42no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22#40811yescharacteristic
179C22C2×#9911yescharacteristic
180C22C2×#10011yescharacteristic
181C4—21no—
182C4—21no—
183C4—22no—
184C4—42no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—81no—
189C4—81no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—81no—
194C2—161no—
195C2#147111yescharacteristic
196C2#212711yescharacteristic
197C2#213211yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC2×#127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1465C2semidirect
3. #1459C2semidirect
4. #1319C2semidirect
5. #1318C2semidirect

Maximal subgroups

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

subgroupindex
#25442
#14652
#14592
#13192
#13182