Finite Group Databasebeta

← #7642

Subgroups — #7642

Order 256.

239 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 71 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#2293C211yesmaximal, characteristic
2#1839C211yesmaximal, characteristic, complemented
3#1831C211yesmaximal, characteristic, complemented
4#1271C212yesmaximal, complemented
5#1270C212yesmaximal
6C2×C4×D4C411yescharacteristic
7C2×#117—21no—
8C2×C4⋊C8—21no—
9#390C411yescharacteristic
10#390C2211yescharacteristic
11#389C411yescharacteristic
12#389C2211yescharacteristic
13#388—21no—
14#388—21no—
15C4×C4⋊C4C411yescharacteristic
16C4×C22⋊C4—21no—
17C43C2211yescharacteristic
18#336—21no—
19#336—21no—
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C22×D4C2×C411yescharacteristic
24C23×C4—21no—
25C22×C8—22no—
26C22×C8—22no—
27C4×D4—22no—
28C4×D4—22no—
29#117—42no—
30#117—42no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C42—21no—
42C2×C42C2×C411yescharacteristic
43C2×C42C2×C411yescharacteristic
44C2×C42D411yescharacteristic
45C2×C42D411yescharacteristic
46C2×C42D411yescharacteristic
47C2×C42D411yescharacteristic
48C2×C42C2311yescharacteristic
49C4⋊C8—42no—
50#95—21no—
51#95—21no—
52#95—21no—
53C24—21no—
54C2×D4—21no—
55C2×D4—41no—
56C2×D4—41no—
57C2×D4M1612yes—
58C22×C4—24no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—42no—
63C22×C4—42no—
64C22×C4C22⋊C411yescharacteristic
65C22×C4C22⋊C411yescharacteristic
66C22×C4C22⋊C411yescharacteristic
67C22×C4C22⋊C411yescharacteristic
68C22×C4C22×C411yescharacteristic
69C22×C4C2×D411yescharacteristic
70C22×C4C2×D411yescharacteristic
71C2×C8—42no—
72C2×C8—42no—
73C2×C8—42no—
74C2×C8—42no—
75C2×C8—42no—
76C2×C8—42no—
77C4⋊C4—21no—
78C4⋊C4—41no—
79C4⋊C4—41no—
80C4⋊C4—42no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4M1612yes—
84C22⋊C4—24no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C22⋊C4—41no—
88C22⋊C4—41no—
89C22⋊C4—42no—
90C42—21no—
91C42—21no—
92C42—22no—
93C42—41no—
94C42—41no—
95C42—41no—
96C42—41no—
97C42—42no—
98C42D812yes—
99C42SD1612yes—
100C23—22no—
101C23—41no—
102C23—41no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23—41no—
107C23C2×C22⋊C411yescharacteristic
108D4—42no—
109D4—42no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—22no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—48no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4#10212yes—
137C2×C4#10212yes—
138C2×C4C2×M1611yescharacteristic
139C2×C4C2×M1611yescharacteristic
140C2×C4C2×D811yescharacteristic
141C2×C4C2×SD1611yescharacteristic
142C2×C4#13512yes—
143C8—82no—
144C8—82no—
145C22—42no—
146C22—42no—
147C22—42no—
148C22—81no—
149C22—81no—
150C22—81no—
151C22—81no—
152C22#40711yescharacteristic
153C22#41011yescharacteristic
154C22#41311yescharacteristic
155C22C2×#10211yescharacteristic
156C22#41611yescharacteristic
157C22#42111yescharacteristic
158C22#42111yescharacteristic
159C4—22no—
160C4—21no—
161C4—21no—
162C4—22no—
163C4—21no—
164C4—21no—
165C4—22no—
166C4—42no—
167C4—41no—
168C4—41no—
169C4—81no—
170C4—81no—
171C2—82no—
172C2#146411yescharacteristic
173C2#148611yescharacteristic
174C2#148811yescharacteristic
175C2#151711yescharacteristic
176C2#152111yescharacteristic
177C2#153511yescharacteristic
178C2#154011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1839C2semidirect
3. #1831C2semidirect
4. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#22932
#18392
#18312
#12712
#12702