Finite Group Databasebeta

← #11549

Subgroups — #11549

Order 256.

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

normal subgroups only

This group has 248 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#2501C211yesmaximal, characteristic
2#1463C211yesmaximal, characteristic, complemented
3C2×#324C211yesmaximal, characteristic, complemented
4#1325C212yesmaximal, complemented
5#1318C212yesmaximal, complemented
6C2×#123—21no—
7C2×#123—21no—
8C2×#121C411yescharacteristic
9C22×C4⋊C4C2211yescharacteristic
10#407—21no—
11C2×C22⋊C8—21no—
12#399—21no—
13#399—21no—
14#390C411yescharacteristic
15#390C2211yescharacteristic
16#384C411yescharacteristic
17#384C2211yescharacteristic
18C4×C22⋊C4C411yescharacteristic
19#324—21no—
20#324—21no—
21#324C2212yes—
22#324C2212yes—
23C22×D4—21no—
24C22×D4C2×C411yescharacteristic
25C23×C4C2×C411yescharacteristic
26C23×C4D411yescharacteristic
27C23×C4D411yescharacteristic
28C2×M16—41no—
29C22×C8—41no—
30#123—44no—
31#123—44no—
32#121—41no—
33#121—41no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—41no—
39C2×C4⋊C4—41no—
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4C2311yescharacteristic
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C42C2×C411yescharacteristic
52C2×C42C2×C411yescharacteristic
53C22⋊C8—22no—
54C22⋊C8—22no—
55C22⋊C8—41no—
56C22⋊C8—41no—
57#95—21no—
58C24—21no—
59C24C22⋊C411yescharacteristic
60C2×D4—22no—
61C2×D4—41no—
62C2×D4—41no—
63C2×D4—42no—
64C2×D4—42no—
65C2×D4—41no—
66C2×D4—41no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—22no—
76C22×C4—21no—
77C22×C4—41no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22⋊C411yescharacteristic
83C22×C4C22×C411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D411yescharacteristic
86M16—82no—
87C2×C8—42no—
88C2×C8—42no—
89C2×C8—82no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—42no—
96C4⋊C4—42no—
97C4⋊C4—42no—
98C4⋊C4—42no—
99C22⋊C4—24no—
100C22⋊C4—42no—
101C22⋊C4—42no—
102C22⋊C4—42no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C42—22no—
106C42—41no—
107C23—22no—
108C23—21no—
109C23—21no—
110C23—41no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23#9911yescharacteristic
119C23#9911yescharacteristic
120C23C2×C22⋊C411yescharacteristic
121D4—82no—
122D4—82no—
123D4—82no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—41no—
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—81no—
151C2×C4—81no—
152C2×C4#13112yes—
153C8—82no—
154C8—82no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—42no—
163C22—42no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22#35312yes—
172C22#40811yescharacteristic
173C22C2×#9911yescharacteristic
174C22#41311yescharacteristic
175C4—22no—
176C4—42no—
177C4—42no—
178C4—41no—
179C4—41no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—81no—
184C2—21no—
185C2—21no—
186C2—22no—
187C2—41no—
188C2—161no—
189C2#146811yescharacteristic
190C2C2×#35311yescharacteristic
191C2#213311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC2×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1463C2semidirect
3. C2×#324C2semidirect
4. #1325C2semidirect
5. #1318C2semidirect

Maximal subgroups

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

subgroupindex
#25012
#14632
C2×#3242
#13252
#13182