Finite Group Databasebeta

← #11554

Subgroups — #11554

Order 256.

204 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 49 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#2500C211yesmaximal, characteristic
2#1464C211yesmaximal, characteristic, complemented
3#1458C211yesmaximal, characteristic, complemented
4#1325C212yesmaximal, complemented
5#1318C212yesmaximal, complemented
6C2×#121C411yescharacteristic
7C2×#117C2211yescharacteristic
8#432—21no—
9#431—21no—
10#394—21no—
11#394—21no—
12#390C411yescharacteristic
13#390C2211yescharacteristic
14#388—21no—
15#388—21no—
16#384C2211yescharacteristic
17#382C411yescharacteristic
18C4×C22⋊C4C411yescharacteristic
19#324C2212yes—
20#324C2212yes—
21#323—21no—
22#323—21no—
23C22×D4—21no—
24C22×D4C2×C411yescharacteristic
25C23×C4C2×C411yescharacteristic
26#121—41no—
27#121—41no—
28#117—41no—
29#117—41no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4C2×C411yescharacteristic
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2311yescharacteristic
35C2×C22⋊C4—21no—
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4D411yescharacteristic
42C2×C22⋊C4D411yescharacteristic
43C2×C42C2×C411yescharacteristic
44C2×C42C2×C411yescharacteristic
45C2×C42D411yescharacteristic
46C2×C42D411yescharacteristic
47C4⋊C8—41no—
48C4⋊C8—41no—
49C22⋊C8—22no—
50C22⋊C8—22no—
51#97—41no—
52C4×C8—41no—
53#95—21no—
54#95—21no—
55#95—21no—
56C24—21no—
57C24C22⋊C411yescharacteristic
58C2×D4—22no—
59C2×D4—41no—
60C2×D4—41no—
61C2×D4—42no—
62C2×D4—81no—
63C2×D4—81no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—22no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—41no—
71C22×C4C22⋊C411yescharacteristic
72C22×C4C22⋊C411yescharacteristic
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22×C411yescharacteristic
75C22×C4C2×D411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C2×C8—42no—
78C2×C8—42no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—81no—
83C4⋊C4—81no—
84C22⋊C4—41no—
85C22⋊C4—41no—
86C22⋊C4—42no—
87C22⋊C4—42no—
88C22⋊C4—42no—
89C22⋊C4—41no—
90C22⋊C4—41no—
91C22⋊C4—81no—
92C22⋊C4—81no—
93C42—21no—
94C42—21no—
95C42—21no—
96C42—21no—
97C42—22no—
98C42—41no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—41no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23—41no—
111C23C2×C22⋊C411yescharacteristic
112D4—82no—
113D4—84no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
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—41no—
137C2×C4#9911yescharacteristic
138C2×C4#9911yescharacteristic
139C2×C4#13112yes—
140C8—82no—
141C8—82no—
142C22—21no—
143C22—21no—
144C22—41no—
145C22—41no—
146C22—41no—
147C22—81no—
148C22—81no—
149C22—81no—
150C22—81no—
151C22—81no—
152C22—81no—
153C22—81no—
154C22—81no—
155C22—81no—
156C22#40811yescharacteristic
157C22C2×#9911yescharacteristic
158C22#41311yescharacteristic
159C4—21no—
160C4—21no—
161C4—22no—
162C4—42no—
163C4—41no—
164C4—41no—
165C4—41no—
166C4—41no—
167C4—81no—
168C4—81no—
169C2—21no—
170C2—21no—
171C2—81no—
172C2—161no—
173C2#146811yescharacteristic
174C2#212811yescharacteristic
175C2#213211yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC2×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1464C2semidirect
3. #1458C2semidirect
4. #1325C2semidirect
5. #1318C2semidirect

Maximal subgroups

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

subgroupindex
#25002
#14642
#14582
#13252
#13182