Finite Group Databasebeta

← #7213

Subgroups — #7213

Order 256.

280 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 41 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#1819C211yesmaximal, characteristic
2#1462C211yesmaximal, characteristic
3#1437C211yesmaximal, characteristic
4C22×C22⋊C4—21no—
5C22×C22⋊C4C2211yescharacteristic
6C22×C42C411yescharacteristic, complemented
7#407—21no—
8#379—21no—
9C2×#95—21no—
10C2×#95C411yescharacteristic, complemented
11#342—21no—
12#342C412yescomplemented
13#323—21no—
14#323C412yescomplemented
15C25D411yescharacteristic
16C23×C4—21no—
17C23×C4—21no—
18C23×C4C2×C411yescharacteristic
19C23×C4D411yescharacteristic
20C2×M16—41no—
21C2×C22⋊C4—24no—
22C2×C22⋊C4—22no—
23C2×C22⋊C4—41no—
24C2×C22⋊C4—41no—
25C2×C22⋊C4—42no—
26C2×C22⋊C4—42no—
27C2×C22⋊C4—41no—
28C2×C22⋊C4—41no—
29C2×C22⋊C4C2×C411yescharacteristic
30C2×C22⋊C4C2×C411yescharacteristic
31C2×C22⋊C4D411yescharacteristic
32C2×C22⋊C4Q811yescharacteristic
33C2×C42—41no—
34C2×C42—42no—
35C22⋊C8—22no—
36C22⋊C8—41no—
37#95—41no—
38#95—42no—
39C24—22no—
40C24—21no—
41C24—21no—
42C24—41no—
43C24—41no—
44C24—41no—
45C24—41no—
46C24—41no—
47C24C22⋊C411yescharacteristic
48C24C22⋊C411yescharacteristic
49C24C4⋊C411yescharacteristic
50C22×C4—22no—
51C22×C4—22no—
52C22×C4—22no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—22no—
58C22×C4—41no—
59C22×C4—41no—
60C22×C4—41no—
61C22×C4—41no—
62C22×C4—41no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—42no—
73C22×C4C4211yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C4⋊C411yescharacteristic
76C22×C4C4⋊C411yescharacteristic
77M16—81no—
78M16—81no—
79C2×C8—42no—
80C22⋊C4—41no—
81C22⋊C4—41no—
82C22⋊C4—41no—
83C22⋊C4—41no—
84C22⋊C4—48no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C22⋊C4—81no—
88C22⋊C4—81no—
89C22⋊C4—81no—
90C22⋊C4—81no—
91C42—42no—
92C42—42no—
93C23—21no—
94C23—21no—
95C23—21no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23—21no—
100C23—21no—
101C23—21no—
102C23—21no—
103C23—21no—
104C23—42no—
105C23—42no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23—42no—
111C23—42no—
112C23—42no—
113C23—42no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23—81no—
119C23—81no—
120C23—81no—
121C23—81no—
122C23—81no—
123C23—81no—
124C23#9511yescharacteristic
125C23#9911yescharacteristic
126C23#9911yescharacteristic
127C23#10412yes—
128C2×C4—24no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—41no—
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—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4—81no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4—82no—
157C2×C4—82no—
158C2×C4—82no—
159C8—82no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—22no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—82no—
184C22—82no—
185C22—82no—
186C22—82no—
187C22—81no—
188C22H(3,ℤ/4)11yescharacteristic
189C22#33911yescharacteristic
190C22#34211yescharacteristic
191C22AΣL(1,𝔽16)11yescharacteristic
192C22#35211yescharacteristic
193C22#35311yescharacteristic
194C22#35411yescharacteristic
195C4—44no—
196C4—41no—
197C4—41no—
198C4—41no—
199C4—41no—
200C4—82no—
201C4—162no—
202C2—21no—
203C2—21no—
204C2—21no—
205C2—21no—
206C2—21no—
207C2—21no—
208C2—42no—
209C2—81no—
210C2#128411yescharacteristic
211C2#139011yescharacteristic
212C2#139311yescharacteristic
subgroupG/H
center, socleC22#342
commutatorC22×C4C42
Frattini, cosocleC22×C22⋊C4C22
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C22×C42C4semidirect
10. C2×#95C4semidirect
12. #342C4semidirect
14. #323C4semidirect

Maximal subgroups

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

subgroupindex
#18192
#14622
#14372