Finite Group Databasebeta

← #18152

Subgroups — #18152

Order 256.

620 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 115 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, complemented
2#3109C212yesmaximal, complemented
3#2598C211yesmaximal, characteristic, complemented
4#1616C212yesmaximal, complemented
5#1613C212yesmaximal, complemented
6#1598C212yesmaximal, complemented
7#1595C212yesmaximal, complemented
8#1514C212yesmaximal, complemented
9#1464C211yesmaximal, 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×D4C2211yescharacteristic, complemented
21C2×#117C2211yescharacteristic
22#481C2214yescomplemented
23#476—22no—
24#459—22no—
25#447C2214yescomplemented
26#442—24no—
27#432C2212yescomplemented
28#400C2211yescharacteristic, complemented
29#399C2211yescharacteristic
30#394C2211yescharacteristic, complemented
31#392C2211yescharacteristic
32#388—21no—
33#386—21no—
34#386—21no—
35#385—21no—
36#383C2211yescharacteristic, complemented
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×D4—21no—
49C22×D4—22no—
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C23×C4—22no—
53C23×C4—21no—
54C23×C4D411yescharacteristic
55C2×D8—24no—
56C4⋊Q8—21no—
57C4⋊Q8D412yescomplemented
58#127—21no—
59#127D412yescomplemented
60#124—22no—
61#124—22no—
62#123—24no—
63#123—24no—
64C22⋊Q8—24no—
65C22⋊Q8—24no—
66#121—24no—
67#121—24no—
68#121—22no—
69#121—22no—
70#121—22no—
71#121C2314yes—
72#120—24no—
73#120—24no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4D414yescomplemented
80#117—22no—
81#117—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4D411yescharacteristic
92C2×C22⋊C4C2311yescharacteristic
93C2×C42D411yescharacteristic
94C2×C42C2311yescharacteristic
95#106—24no—
96C4⋊C8—22no—
97#102—24no—
98#102—24no—
99C22⋊C8C2314yes—
100#97—22no—
101#95—21no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95C2311yescharacteristic
106#95C2311yescharacteristic
107C24—22no—
108C24—21no—
109C24C2×D411yescharacteristic
110#41—48no—
111#41—48no—
112#41—44no—
113#41—44no—
114#41—44no—
115#41—44no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—44no—
119C2×D4—24no—
120C2×D4—22no—
121C2×D4—24no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—44no—
128C2×D4—42no—
129C2×D4—44no—
130C2×D4—44no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—81no—
134C2×D4C2×D414yes—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—48no—
146C22×C4—42no—
147C22×C4—81no—
148C22×C4—81no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2411yescharacteristic
155D8—48no—
156C2×C8—24no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—81no—
164C4⋊C4C2×D414yes—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—22no—
168C22⋊C4—24no—
169C22⋊C4—22no—
170C22⋊C4—24no—
171C22⋊C4—42no—
172C22⋊C4—41no—
173C22⋊C4—41no—
174C22⋊C4—42no—
175C22⋊C4—81no—
176C22⋊C4—81no—
177C42—21no—
178C42—21no—
179C42C2×D412yes—
180C42C2×D412yes—
181C23—21no—
182C23—21no—
183C23—22no—
184C23—24no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—44no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—82no—
195C23—82no—
196C23AGL(1,ℤ/8)12yes—
197C23C22×D411yescharacteristic
198Q8—44no—
199Q8—44no—
200D4—24no—
201D4—44no—
202D4—42no—
203D4—44no—
204D4—44no—
205D4—48no—
206D4—44no—
207D4—44no—
208D4—82no—
209D4—82no—
210D4—82no—
211D4—84no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—24no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—48no—
224C2×C4—48no—
225C2×C4—44no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—84no—
238C2×C4#12011yescharacteristic
239C2×C4#12011yescharacteristic
240C2×C4#12012yes—
241C2×C4AGL(1,ℤ/8)12yes—
242C2×C4C22×D412yes—
243C8—44no—
244C22—21no—
245C22—21no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—41no—
251C22—41no—
252C22—44no—
253C22—44no—
254C22—42no—
255C22—44no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—88no—
260C22—81no—
261C22—81no—
262C22C2×#12011yescharacteristic
263C22C2×AGL(1,ℤ/8)11yescharacteristic
264C22C2×AGL(1,ℤ/8)11yescharacteristic
265C4—21no—
266C4—22no—
267C4—21no—
268C4—22no—
269C4—22no—
270C4—44no—
271C4—41no—
272C4—41no—
273C4—81no—
274C4—81no—
275C4—81no—
276C2—21no—
277C2—21no—
278C2—42no—
279C2—82no—
280C2—84no—
281C2#300311yescharacteristic
282C2#302111yescharacteristic
283C2#302511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3449C2semidirect
2. #3109C2semidirect
3. #2598C2semidirect
4. #1616C2semidirect
5. #1613C2semidirect
6. #1598C2semidirect
7. #1595C2semidirect
8. #1514C2semidirect
9. #1464C2semidirect
20. C2×C4×D4C22semidirect
22. #481C22semidirect
25. #447C22semidirect
27. #432C22semidirect
28. #400C22semidirect
30. #394C22semidirect
36. #383C22semidirect
37. #327C22semidirect
38. #327C22semidirect
39. #324C22semidirect
57. C4⋊Q8D4semidirect
59. #127D4semidirect
79. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
#34492
#31092
#25982
#16162
#16132
#15982
#15952
#15142
#14642