Finite Group Databasebeta

← #11775

Subgroups — #11775

Order 256.

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

normal subgroups only

This group has 266 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#2543C211yesmaximal, characteristic
2C4×#99C211yesmaximal, characteristic, complemented
3#1466C211yesmaximal, characteristic, complemented
4#1344C212yesmaximal, complemented
5#1343C212yesmaximal, complemented
6C2×#121—21no—
7C2×#121C411yescharacteristic, complemented
8C2×C4×D4C2211yescharacteristic
9#432—21no—
10C2×#99—21no—
11#398C411yescharacteristic, complemented
12#397C2211yescharacteristic
13#394C411yescharacteristic, complemented
14#392C2211yescharacteristic
15#388—21no—
16#386—21no—
17#386—21no—
18C4×C22⋊C4—21no—
19C2×#95C411yescharacteristic, complemented
20#342C2212yes—
21#323—21no—
22#323C2212yes—
23C22×D4—21no—
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C22×D4D411yescharacteristic
27C23×C4—21no—
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30#121—44no—
31#121—41no—
32#121—41no—
33C4×D4—41no—
34C4×D4—41no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C22⋊C4—22no—
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—41no—
47C2×C42D411yescharacteristic
48C4⋊C8—41no—
49#99—24no—
50C22⋊C8—22no—
51#97—41no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—41no—
56#95C2×C411yescharacteristic
57#95C2×C411yescharacteristic
58C24—21no—
59C24C22⋊C411yescharacteristic
60C24C2×D411yescharacteristic
61C2×D4—22no—
62C2×D4—22no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4—42no—
66C2×D4—42no—
67C2×D4—42no—
68C2×D4—42no—
69C2×D4—42no—
70C2×D4—81no—
71C2×D4—81no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—81no—
83C22×C4—81no—
84C22×C4—81no—
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22⋊C411yescharacteristic
88C22×C4C22×C411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C2×C8—42no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—42no—
94C22⋊C4—24no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—42no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—81no—
104C42—21no—
105C42—21no—
106C42—82no—
107C23—21no—
108C23—21no—
109C23—21no—
110C23—21no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23#10412yes—
123C23C2×C22⋊C411yescharacteristic
124C23AGL(1,ℤ/8)12yes—
125D4—44no—
126D4—84no—
127D4—82no—
128D4—84no—
129C2×C4—22no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—44no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4—84no—
154C2×C4—82no—
155C2×C4—82no—
156C2×C4#9911yescharacteristic
157C2×C4#9911yescharacteristic
158C8—82no—
159C22—21no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22C2×#9911yescharacteristic
179C22#41811yescharacteristic
180C22C2×#10411yescharacteristic
181C4—21no—
182C4—21no—
183C4—42no—
184C4—41no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—81no—
189C4—84no—
190C4—81no—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—42no—
195C2—161no—
196C2#150711yescharacteristic
197C2#212111yescharacteristic
198C2#212611yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2543C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#99C2semidirect
3. #1466C2semidirect
4. #1344C2semidirect
5. #1343C2semidirect
7. C2×#121C4semidirect
11. #398C4semidirect
13. #394C4semidirect
19. C2×#95C4semidirect

Maximal subgroups

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

subgroupindex
#25432
C4×#992
#14662
#13442
#13432