Finite Group Databasebeta

← #11541

Subgroups — #11541

Order 256.

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

normal subgroups only

This group has 350 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#2405C211yesmaximal, characteristic
2#1462C211yesmaximal, characteristic, complemented
3C2×#323C211yesmaximal, characteristic, complemented
4#1326C212yesmaximal, complemented
5#1316C212yesmaximal, complemented
6C23×D4C411yescharacteristic
7C2×#123—21no—
8C22×C22⋊C4C2211yescharacteristic
9#407—21no—
10C2×C22⋊C8—21no—
11#394—21no—
12#394—21no—
13#386—21no—
14#385C411yescharacteristic
15#385C2211yescharacteristic
16#379C411yescharacteristic
17#379C411yescharacteristic
18#379C2211yescharacteristic
19#323—21no—
20#323—21no—
21#323C2212yes—
22#323C2212yes—
23C25C2×C411yescharacteristic
24C25C2×C411yescharacteristic
25C22×D4—21no—
26C22×D4—21no—
27C22×D4—44no—
28C22×D4—41no—
29C22×D4—41no—
30C23×C4C2×C411yescharacteristic
31C23×C4D411yescharacteristic
32C23×C4D411yescharacteristic
33C2×M16—41no—
34C22×C8—41no—
35#123—44no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—41no—
45C2×C22⋊C4—41no—
46C2×C22⋊C4C2×C411yescharacteristic
47C2×C22⋊C4C2×C411yescharacteristic
48C2×C22⋊C4C2×C411yescharacteristic
49C2×C22⋊C4D411yescharacteristic
50C2×C22⋊C4D411yescharacteristic
51C2×C22⋊C4C2311yescharacteristic
52C22⋊C8—22no—
53C22⋊C8—22no—
54C22⋊C8—41no—
55C22⋊C8—41no—
56#95—21no—
57#95—21no—
58C24—21no—
59C24—21no—
60C24—21no—
61C24—22no—
62C24—22no—
63C24—41no—
64C24—42no—
65C24—41no—
66C24—42no—
67C24—41no—
68C24—81no—
69C24—81no—
70C24C22⋊C411yescharacteristic
71C24C22⋊C411yescharacteristic
72C24C22×C411yescharacteristic
73C2×D4—24no—
74C2×D4—42no—
75C2×D4—42no—
76C2×D4—42no—
77C2×D4—42no—
78C2×D4—44no—
79C2×D4—44no—
80C2×D4—44no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4—44no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—42no—
92C22×C4—41no—
93C22×C4—42no—
94C22×C4—81no—
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22⋊C411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99M16—82no—
100C2×C8—42no—
101C2×C8—42no—
102C2×C8—82no—
103C4⋊C4—42no—
104C4⋊C4—42no—
105C22⋊C4—22no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C22⋊C4—42no—
113C22⋊C4—41no—
114C22⋊C4—81no—
115C22⋊C4—81no—
116C22⋊C4—81no—
117C23—22no—
118C23—21no—
119C23—21no—
120C23—21no—
121C23—21no—
122C23—22no—
123C23—21no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—42no—
128C23—42no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—42no—
135C23—41no—
136C23—41no—
137C23—42no—
138C23—42no—
139C23—42no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—81no—
144C23—81no—
145C23—82no—
146C23—84no—
147C23—84no—
148C23—84no—
149C23—84no—
150C23—82no—
151C23#9911yescharacteristic
152C23#9911yescharacteristic
153C23C2×C22⋊C411yescharacteristic
154C23#13112yes—
155D4—44no—
156D4—48no—
157D4—44no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—81no—
180C2×C4—81no—
181C8—82no—
182C8—82no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—22no—
190C22—21no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—41no—
198C22—41no—
199C22—42no—
200C22—41no—
201C22—41no—
202C22—41no—
203C22—84no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—84no—
209C22—84no—
210C22—84no—
211C22—81no—
212C22—81no—
213C22AΣL(1,𝔽16)12yes—
214C22#40811yescharacteristic
215C22C2×#9911yescharacteristic
216C22#41311yescharacteristic
217C4—41no—
218C4—42no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—81no—
223C4—81no—
224C2—21no—
225C2—21no—
226C2—22no—
227C2—41no—
228C2—42no—
229C2—82no—
230C2—81no—
231C2#146811yescharacteristic
232C2C2×AΣL(1,𝔽16)11yescharacteristic
233C2#212111yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC23×D4C4
LCSG ▷ C24 ▷ C22 ▷ 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. #1462C2semidirect
3. C2×#323C2semidirect
4. #1326C2semidirect
5. #1316C2semidirect

Maximal subgroups

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

subgroupindex
#24052
#14622
C2×#3232
#13262
#13162