Finite Group Databasebeta

← #11796

Subgroups — #11796

Order 256.

534 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C2×#392C211yesmaximal, characteristic
2#1819C211yesmaximal, characteristic, complemented
3#1462C211yesmaximal, characteristic, complemented
4#1343C214yesmaximal, complemented
5C23×D4—21no—
6C23×D4C411yescharacteristic, complemented
7C22×C22⋊C4—21no—
8C22×C22⋊C4C2211yescharacteristic
9#407—21no—
10#392—21no—
11#392—21no—
12#392C412yescomplemented
13#392C2212yes—
14#379—21no—
15C2×#95C411yescharacteristic, complemented
16#342—21no—
17#342C2212yescomplemented
18#323—21no—
19#323C2212yescomplemented
20C25—21no—
21C25D411yescharacteristic
22C22×D4—22no—
23C22×D4—22no—
24C22×D4—21no—
25C22×D4—41no—
26C22×D4—41no—
27C22×D4—44no—
28C22×D4—44no—
29C22×D4—44no—
30C22×D4—44no—
31C22×D4C2×C412yes—
32C23×C4—21no—
33C23×C4C2×C411yescharacteristic
34C23×C4D411yescharacteristic
35C2×M16—41no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—41no—
40C2×C22⋊C4—41no—
41C2×C22⋊C4—44no—
42C2×C22⋊C4—41no—
43C2×C22⋊C4—41no—
44C2×C22⋊C4C2×C411yescharacteristic
45C2×C22⋊C4D411yescharacteristic
46C2×C22⋊C4D411yescharacteristic
47C2×C22⋊C4C2311yescharacteristic
48C22⋊C8—22no—
49C22⋊C8—41no—
50#95—21no—
51#95C2×C412yes—
52C24—24no—
53C24—22no—
54C24—22no—
55C24—21no—
56C24—21no—
57C24—41no—
58C24—42no—
59C24—42no—
60C24—42no—
61C24—42no—
62C24—42no—
63C24—42no—
64C24—42no—
65C24—41no—
66C24—41no—
67C24C22⋊C411yescharacteristic
68C24C22⋊C411yescharacteristic
69C24C2×D411yescharacteristic
70C2×D4—42no—
71C2×D4—44no—
72C2×D4—44no—
73C2×D4—42no—
74C2×D4—44no—
75C2×D4—44no—
76C2×D4—44no—
77C2×D4—44no—
78C2×D4—44no—
79C2×D4—48no—
80C2×D4—432no—
81C2×D4—44no—
82C2×D4—44no—
83C2×D4—44no—
84C22×C4—24no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22⋊C411yescharacteristic
97C22×C4C22×C411yescharacteristic
98C22×C4C2×D411yescharacteristic
99M16—82no—
100C2×C8—42no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—41no—
105C22⋊C4—44no—
106C22⋊C4—44no—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—82no—
110C22⋊C4—82no—
111C23—22no—
112C23—22no—
113C23—21no—
114C23—21no—
115C23—21no—
116C23—21no—
117C23—24no—
118C23—21no—
119C23—21no—
120C23—21no—
121C23—42no—
122C23—42no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—42no—
128C23—48no—
129C23—48no—
130C23—42no—
131C23—44no—
132C23—42no—
133C23—48no—
134C23—42no—
135C23—44no—
136C23—42no—
137C23—42no—
138C23—42no—
139C23—42no—
140C23—81no—
141C23—81no—
142C23—84no—
143C23—84no—
144C23—84no—
145C23—82no—
146C23—82no—
147C23—82no—
148C23—82no—
149C23#9911yescharacteristic
150C23#9911yescharacteristic
151C23C2×C22⋊C411yescharacteristic
152C23AGL(1,ℤ/8)12yes—
153D4—44no—
154D4—416no—
155D4—416no—
156D4—44no—
157D4—48no—
158C2×C4—24no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—48no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—81no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C8—82no—
178C22—24no—
179C22—24no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—22no—
187C22—42no—
188C22—42no—
189C22—44no—
190C22—44no—
191C22—41no—
192C22—41no—
193C22—44no—
194C22—41no—
195C22—41no—
196C22—41no—
197C22—81no—
198C22—84no—
199C22—84no—
200C22—81no—
201C22—82no—
202C22—82no—
203C22—88no—
204C22—88no—
205C22—82no—
206C22—82no—
207C22AΣL(1,𝔽16)12yes—
208C22#35312yes—
209C22C2×#9911yescharacteristic
210C22#41811yescharacteristic
211C22#42111yescharacteristic
212C4—44no—
213C4—41no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—162no—
218C2—21no—
219C2—21no—
220C2—22no—
221C2—22no—
222C2—42no—
223C2—81no—
224C2—84no—
225C2#151011yescharacteristic
226C2C2×AΣL(1,𝔽16)11yescharacteristic
227C2C2×#35311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC2×#392C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1819C2semidirect
3. #1462C2semidirect
4. #1343C2semidirect
6. C23×D4C4semidirect
12. #392C4semidirect
15. C2×#95C4semidirect
17. #342C22semidirect
19. #323C22semidirect

Maximal subgroups

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

subgroupindex
C2×#3922
#18192
#14622
#13432