Finite Group Databasebeta

← #11758

Subgroups — #11758

Order 256.

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

normal subgroups only

This group has 243 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#2610C211yesmaximal, characteristic
2#1740C211yesmaximal, characteristic, complemented
3#1464C211yesmaximal, characteristic, complemented
4#1351C212yesmaximal, complemented
5#1343C212yesmaximal
6C2×#124C411yescharacteristic, complemented
7C2×#117—21no—
8C2×#117C2211yescharacteristic
9#432—21no—
10#395C411yescharacteristic, complemented
11#394C411yescharacteristic, complemented
12#393C2211yescharacteristic
13#392C2211yescharacteristic
14#390—21no—
15#388—21no—
16#388—21no—
17#386—21no—
18#376C411yescharacteristic, complemented
19#342—21no—
20#342C2212yes—
21#324—21no—
22#323C2212yes—
23C22×Q8C2×C411yescharacteristic
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C23×C4D411yescharacteristic
27#124—41no—
28#124—41no—
29#117—24no—
30#117—41no—
31#117—41no—
32#117—42no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—41no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4D411yescharacteristic
37C2×C22⋊C4—22no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4D411yescharacteristic
45C2×C22⋊C4C2311yescharacteristic
46C2×C42—21no—
47C2×C42—41no—
48C2×C42C2×C411yescharacteristic
49C2×C42D411yescharacteristic
50C4⋊C8—41no—
51C22⋊C8—22no—
52#97—41no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95C2×C411yescharacteristic
57#95C2×C411yescharacteristic
58C24—21no—
59C24C2×D411yescharacteristic
60C2×Q8—41no—
61C2×Q8—81no—
62C2×D4—41no—
63C2×D4—42no—
64C2×D4—41no—
65C2×D4—41no—
66C2×D4—81no—
67C2×D4—81no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—22no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—42no—
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22×C411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C2×C8—42no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—44no—
87C4⋊C4—81no—
88C22⋊C4—24no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—42no—
92C22⋊C4—42no—
93C22⋊C4—42no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—42no—
97C22⋊C4—81no—
98C22⋊C4—81no—
99C42—21no—
100C42—21no—
101C42—21no—
102C42—21no—
103C42—42no—
104C42—44no—
105C23—21no—
106C23—21no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23C2×C22⋊C411yescharacteristic
117C23AGL(1,ℤ/8)11yescharacteristic
118C23#13711yescharacteristic
119Q8—82no—
120D4—82no—
121D4—82no—
122D4—82no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—44no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4#9911yescharacteristic
152C2×C4#9911yescharacteristic
153C2×C4#10412yes—
154C8—82no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22—41no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22C2×#9911yescharacteristic
172C22#41711yescharacteristic
173C22C2×#10411yescharacteristic
174C4—21no—
175C4—21no—
176C4—22no—
177C4—42no—
178C4—41no—
179C4—41no—
180C4—41no—
181C4—41no—
182C4—81no—
183C4—84no—
184C4—81no—
185C2—21no—
186C2—21no—
187C2—41no—
188C2—41no—
189C2—161no—
190C2#150411yescharacteristic
191C2#212211yescharacteristic
192C2#212611yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#392C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1740C2semidirect
3. #1464C2semidirect
4. #1351C2semidirect
6. C2×#124C4semidirect
10. #395C4semidirect
11. #394C4semidirect
18. #376C4semidirect

Maximal subgroups

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

subgroupindex
#26102
#17402
#14642
#13512
#13432