Finite Group Databasebeta

← #18123

Subgroups — #18123

Order 256.

604 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 127 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#3449C211yesmaximal, characteristic
2#3103C212yesmaximal, complemented
3#2396C211yesmaximal, characteristic, complemented
4#1616C212yesmaximal, complemented
5#1613C212yesmaximal, complemented
6#1604C212yesmaximal, complemented
7#1601C212yesmaximal, complemented
8#1502C212yesmaximal, complemented
9#1458C211yesmaximal, characteristic, complemented
10C22×#41—21no—
11#537—21no—
12#537C2212yes—
13#536—22no—
14#535—22no—
15#534—21no—
16#534C2212yes—
17#532—21no—
18#532C2212yes—
19#525—24no—
20C2×C4×Q8C2211yescharacteristic, complemented
21C2×#117C2211yescharacteristic
22#481C2214yescomplemented
23#475—22no—
24#460—22no—
25#448C2214yescomplemented
26C4×SD16—24no—
27#431C2212yescomplemented
28#400C2211yescharacteristic, complemented
29#399C2211yescharacteristic
30#398—21no—
31#396—21no—
32#395C2211yescharacteristic, complemented
33#393C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2211yescharacteristic, complemented
36C4×C22⋊C4—21no—
37#327C2214yescomplemented
38#327C2214yescomplemented
39#324C2212yescomplemented
40#323C2212yes—
41C2×#41—22no—
42C2×#41—22no—
43C2×#41—22no—
44C2×#41—22no—
45C2×#41—44no—
46C2×#41—44no—
47C22×Q8—21no—
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4—22no—
51C23×C4—22no—
52C23×C4D411yescharacteristic
53C2×SD16—24no—
54C4⋊Q8—21no—
55C4⋊Q8D412yes—
56#127—21no—
57#127D412yes—
58#124—22no—
59#124—22no—
60#123—24no—
61#123—24no—
62C22⋊Q8—24no—
63C22⋊Q8—24no—
64#121—24no—
65#121—24no—
66#121—22no—
67#121—22no—
68#121—22no—
69#121C2314yes—
70#120—24no—
71#120—24no—
72C4×Q8—22no—
73C4×Q8—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4D414yes—
78#117—22no—
79#117—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4D411yescharacteristic
90C2×C22⋊C4C2311yescharacteristic
91C2×C42—21no—
92C2×C42—21no—
93C2×C42D411yescharacteristic
94C2×C42C2311yescharacteristic
95#106—24no—
96C4⋊C8—22no—
97#103—24no—
98#102—24no—
99C22⋊C8C2314yes—
100C4×C8—22no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95C2311yescharacteristic
106#95C2311yescharacteristic
107C24—22no—
108C24C2×D411yescharacteristic
109#41—44no—
110#41—48no—
111#41—48no—
112#41—44no—
113#41—44no—
114#41—44no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—42no—
119C2×Q8—44no—
120C2×D4—24no—
121C2×D4—22no—
122C2×D4—24no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4C2×D414yes—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—24no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—48no—
142C22×C4SD1614yescomplemented
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2411yescharacteristic
149SD16—48no—
150C2×C8—24no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4C2×D414yes—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—22no—
165C22⋊C4—24no—
166C22⋊C4—22no—
167C22⋊C4—24no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C42—21no—
171C42—21no—
172C42—41no—
173C42—41no—
174C42—42no—
175C42C2×D412yes—
176C42C2×D412yes—
177C23—21no—
178C23—21no—
179C23—24no—
180C23—41no—
181C23—41no—
182C23—44no—
183C23—82no—
184C23—82no—
185C23C2×SD1612yes—
186C23C22×D411yescharacteristic
187Q8—24no—
188Q8—42no—
189Q8—44no—
190Q8—44no—
191D4—44no—
192D4—44no—
193D4—48no—
194D4—44no—
195D4—44no—
196D4—44no—
197D4—82no—
198D4—82no—
199D4—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—48no—
212C2×C4—41no—
213C2×C4—44no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—48no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4#12012yes—
227C2×C4#12011yescharacteristic
228C2×C4#12011yescharacteristic
229C2×C4C2×SD1612yes—
230C2×C4C2×SD1612yes—
231C2×C4AGL(1,ℤ/8)12yes—
232C2×C4C22×D412yes—
233C8—44no—
234C22—21no—
235C22—21no—
236C22—22no—
237C22—42no—
238C22—41no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—44no—
243C22—81no—
244C22—88no—
245C22C2×#12011yescharacteristic
246C22C22×SD1611yescharacteristic
247C22C2×AGL(1,ℤ/8)11yescharacteristic
248C4—22no—
249C4—22no—
250C4—22no—
251C4—42no—
252C4—44no—
253C4—41no—
254C4—42no—
255C4—41no—
256C4—81no—
257C4—81no—
258C4#45012yes—
259C4#45012yes—
260C2—21no—
261C2—21no—
262C2—42no—
263C2—84no—
264C2C2×#45011yescharacteristic
265C2#301811yescharacteristic
266C2#302511yescharacteristic
subgroupG/H
centerC2×C4#120
commutator, Frattini, cosocleC22×C4C24
socleC22C2×#120
prime radical#3449C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3103C2semidirect
3. #2396C2semidirect
4. #1616C2semidirect
5. #1613C2semidirect
6. #1604C2semidirect
7. #1601C2semidirect
8. #1502C2semidirect
9. #1458C2semidirect
20. C2×C4×Q8C22semidirect
22. #481C22semidirect
25. #448C22semidirect
27. #431C22semidirect
28. #400C22semidirect
32. #395C22semidirect
35. C4×C4⋊C4C22semidirect
37. #327C22semidirect
38. #327C22semidirect
39. #324C22semidirect
142. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
#34492
#31032
#23962
#16162
#16132
#16042
#16012
#15022
#14582