Finite Group Databasebeta

← #11839

Subgroups — #11839

Order 256.

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

normal subgroups only

This group has 349 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#2404C211yesmaximal, characteristic
2#1819C211yesmaximal, characteristic, complemented
3C2×#323C211yesmaximal, characteristic, complemented
4#1351C212yesmaximal, complemented
5#1343C212yesmaximal, complemented
6C22×#41C411yescharacteristic, complemented
7C2×#120—21no—
8C22×C22⋊C4C2211yescharacteristic
9C2×C22⋊C8—21no—
10#393C411yescharacteristic, complemented
11#393C2211yescharacteristic
12#392C411yescharacteristic, complemented
13#392C2211yescharacteristic
14#386—21no—
15#386—21no—
16#386—21no—
17#381C411yescharacteristic, complemented
18#379—21no—
19#342—21no—
20#342C2212yes—
21#323—21no—
22#323C2212yes—
23C25D411yescharacteristic
24C2×#41—41no—
25C2×#41—41no—
26C2×#41—44no—
27C22×Q8C2×C411yescharacteristic
28C22×D4—21no—
29C22×D4—21no—
30C22×D4—21no—
31C22×D4C2×C411yescharacteristic
32C23×C4—21no—
33C23×C4C2×C411yescharacteristic
34C23×C4D411yescharacteristic
35C22×C8—41no—
36#120—44no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—41no—
44C2×C22⋊C4—41no—
45C2×C22⋊C4—41no—
46C2×C22⋊C4—41no—
47C2×C22⋊C4C2×C411yescharacteristic
48C2×C22⋊C4D411yescharacteristic
49C2×C22⋊C4D411yescharacteristic
50C2×C22⋊C4C2311yescharacteristic
51C22⋊C8—22no—
52C22⋊C8—41no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95C2×C411yescharacteristic
57#95C2×C411yescharacteristic
58C24—21no—
59C24—21no—
60C24—21no—
61C24—22no—
62C24—41no—
63C24—42no—
64C24—81no—
65C24C22⋊C411yescharacteristic
66C24C22⋊C411yescharacteristic
67C24C2×D411yescharacteristic
68#41—44no—
69#41—44no—
70#41—44no—
71#41—44no—
72C2×Q8—42no—
73C2×Q8—41no—
74C2×D4—24no—
75C2×D4—41no—
76C2×D4—42no—
77C2×D4—41no—
78C2×D4—41no—
79C2×D4—42no—
80C2×D4—42no—
81C2×D4—42no—
82C2×D4—42no—
83C2×D4—42no—
84C2×D4—81no—
85C2×D4—81no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—81no—
99C22×C4C22⋊C411yescharacteristic
100C22×C4C22⋊C411yescharacteristic
101C22×C4C22×C411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C2×C8—42no—
104C2×C8—82no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C22⋊C4—42no—
113C22⋊C4—42no—
114C22⋊C4—81no—
115C22⋊C4—81no—
116C22⋊C4—82no—
117C22⋊C4—82no—
118C23—22no—
119C23—21no—
120C23—21no—
121C23—21no—
122C23—21no—
123C23—22no—
124C23—21no—
125C23—21no—
126C23—21no—
127C23—41no—
128C23—42no—
129C23—42no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—42no—
137C23—41no—
138C23—42no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—81no—
143C23—81no—
144C23—84no—
145C23—84no—
146C23#9911yescharacteristic
147C23#9911yescharacteristic
148C23C2×C22⋊C411yescharacteristic
149C23#13512yes—
150Q8—42no—
151Q8—42no—
152D4—44no—
153D4—42no—
154D4—42no—
155D4—44no—
156D4—84no—
157D4—84no—
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—48no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—81no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C8—82no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—22no—
191C22—21no—
192C22—42no—
193C22—42no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—81no—
205C22—84no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22—84no—
210C22—81no—
211C22—81no—
212C22AΣL(1,𝔽16)12yes—
213C22C2×#9911yescharacteristic
214C22#41611yescharacteristic
215C22#42111yescharacteristic
216C4—41no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—41no—
221C4—81no—
222C4—81no—
223C4—162no—
224C2—21no—
225C2—21no—
226C2—22no—
227C2—42no—
228C2—41no—
229C2—81no—
230C2—82no—
231C2#151611yescharacteristic
232C2C2×AΣL(1,𝔽16)11yescharacteristic
233C2#212711yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2404C2
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. C2×#323C2semidirect
4. #1351C2semidirect
5. #1343C2semidirect
6. C22×#41C4semidirect
10. #393C4semidirect
12. #392C4semidirect
17. #381C4semidirect

Maximal subgroups

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

subgroupindex
#24042
#18192
C2×#3232
#13512
#13432