Finite Group Databasebeta

← #11753

Subgroups — #11753

Order 256.

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

normal subgroups only

This group has 389 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
1#2404C211yesmaximal, characteristic
2C2×#342C211yesmaximal, characteristic, complemented
3#1462C211yesmaximal, characteristic, complemented
4#1351C212yesmaximal, complemented
5#1343C212yesmaximal, complemented
6C22×#41C411yescharacteristic, complemented
7C2×#120—21no—
8C22×C22⋊C4—21no—
9C22×C22⋊C4C2211yescharacteristic
10#407—21no—
11#393C411yescharacteristic, complemented
12#393C2211yescharacteristic
13#392C411yescharacteristic, complemented
14#392C2211yescharacteristic
15#386—21no—
16#386—21no—
17#386—21no—
18#381C411yescharacteristic, complemented
19#342—21no—
20#342C2212yes—
21#323—21no—
22#323C2212yes—
23C25D411yescharacteristic
24C2×#41—41no—
25C2×#41—41no—
26C2×#41—44no—
27C22×Q8C2×C411yescharacteristic, complemented
28C22×D4—21no—
29C22×D4—21no—
30C22×D4—21no—
31C22×D4C2×C411yescharacteristic, complemented
32C23×C4—21no—
33C23×C4—41no—
34C23×C4C2×C411yescharacteristic
35C23×C4D411yescharacteristic
36C2×M16—41no—
37#120—44no—
38C2×C22⋊C4—24no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—41no—
46C2×C22⋊C4—41no—
47C2×C22⋊C4—41no—
48C2×C22⋊C4—42no—
49C2×C22⋊C4C2×C411yescharacteristic
50C2×C22⋊C4D411yescharacteristic
51C2×C22⋊C4D411yescharacteristic
52C2×C22⋊C4C2311yescharacteristic
53C22⋊C8—22no—
54C22⋊C8—41no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95C2×C411yescharacteristic, complemented
59#95C2×C411yescharacteristic, complemented
60C24—21no—
61C24—21no—
62C24—22no—
63C24—21no—
64C24—21no—
65C24—21no—
66C24—42no—
67C24—81no—
68C24C22⋊C411yescharacteristic
69C24C22⋊C411yescharacteristic
70C24C2×D411yescharacteristic
71#41—44no—
72#41—44no—
73#41—44no—
74#41—44no—
75C2×Q8—42no—
76C2×Q8—41no—
77C2×D4—24no—
78C2×D4—41no—
79C2×D4—42no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—42no—
83C2×D4—42no—
84C2×D4—42no—
85C2×D4—42no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—81no—
89C2×D4—81no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—44no—
102C22×C4—42no—
103C22×C4—81no—
104C22×C4—82no—
105C22×C4—82no—
106C22×C4C22⋊C411yescharacteristic
107C22×C4C22⋊C411yescharacteristic
108C22×C4C22×C411yescharacteristic
109C22×C4C2×D411yescharacteristic
110M16—82no—
111C2×C8—42no—
112C22⋊C4—24no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—42no—
118C22⋊C4—42no—
119C22⋊C4—44no—
120C22⋊C4—42no—
121C22⋊C4—42no—
122C22⋊C4—81no—
123C22⋊C4—81no—
124C23—22no—
125C23—22no—
126C23—22no—
127C23—22no—
128C23—22no—
129C23—21no—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23—22no—
136C23—41no—
137C23—42no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—42no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—81no—
150C23—81no—
151C23—84no—
152C23—82no—
153C23—82no—
154C23#9911yescharacteristic
155C23#9911yescharacteristic
156C23#10412yes—
157C23C2×C22⋊C411yescharacteristic
158C23AGL(1,ℤ/8)11yescharacteristic
159C23#13711yescharacteristic
160Q8—42no—
161Q8—42no—
162D4—44no—
163D4—42no—
164D4—42no—
165D4—44no—
166D4—84no—
167D4—82no—
168D4—82no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—48no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—44no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—81no—
192C2×C4—88no—
193C2×C4—82no—
194C2×C4—82no—
195C8—82no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—22no—
205C22—21no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—41no—
210C22—41no—
211C22—44no—
212C22—44no—
213C22—42no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—42no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22—81no—
224C22—82no—
225C22—82no—
226C22—81no—
227C22—81no—
228C22AΣL(1,𝔽16)12yes—
229C22C2×#9911yescharacteristic
230C22#41711yescharacteristic
231C22C2×#10411yescharacteristic
232C4—41no—
233C4—42no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—81no—
238C4—81no—
239C4—84no—
240C2—21no—
241C2—21no—
242C2—22no—
243C2—41no—
244C2—41no—
245C2—42no—
246C2—41no—
247C2—82no—
248C2#150411yescharacteristic
249C2C2×AΣL(1,𝔽16)11yescharacteristic
250C2#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. C2×#342C2semidirect
3. #1462C2semidirect
4. #1351C2semidirect
5. #1343C2semidirect
6. C22×#41C4semidirect
11. #393C4semidirect
13. #392C4semidirect
18. #381C4semidirect
27. C22×Q8C2×C4semidirect
31. C22×D4C2×C4semidirect
58. #95C2×C4semidirect
59. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#24042
C2×#3422
#14622
#13512
#13432