Finite Group Databasebeta

← #18405

Subgroups — #18405

Order 256.

595 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 595 conjugacy classes of subgroups — too many to draw. Only its 121 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#3506C211yesmaximal, characteristic, complemented
2#3208C211yesmaximal, characteristic, complemented
3#3205C211yesmaximal, characteristic, complemented
4#2541C211yesmaximal, characteristic, complemented
5#1600C218yesmaximal, complemented
6#1503C211yesmaximal, characteristic, complemented
7#1502C211yesmaximal, characteristic, complemented
8#1460C211yesmaximal, characteristic, complemented
9#559—22no—
10#556—21no—
11#555—21no—
12#553—22no—
13#550—21no—
14#547—21no—
15#538C2214yes—
16#535—21no—
17#533C2211yescharacteristic
18#532C2211yescharacteristic
19C2×#127—21no—
20C2×#127C2211yescharacteristic
21#525—22no—
22C2×#121—22no—
23#520—21no—
24C2×C4×D4C2211yescharacteristic
25#493—21no—
26#492—21no—
27#486—21no—
28#486—21no—
29#460—22no—
30#459—22no—
31#450C2214yes—
32#447C2214yes—
33#431C2211yescharacteristic
34#431C2211yescharacteristic
35#392C2214yes—
36#390—21no—
37C4×C4⋊C4C2211yescharacteristic
38#328C2214yes—
39#327C2214yes—
40#323C2214yes—
41C2×#41—22no—
42C2×#41—21no—
43C22×D4—24no—
44C22×D4—21no—
45C22×D4—21no—
46C22×D4D411yescharacteristic
47C22×D4C2312yes—
48C23×C4D412yes—
49C2×SD16—24no—
50C2×D8—24no—
51C4⋊Q8D411yescharacteristic, complemented
52C4⋊Q8D411yescharacteristic, complemented
53C4⋊Q8D411yescharacteristic, complemented
54#127—22no—
55#127—21no—
56#127—21no—
57#127—21no—
58#127—21no—
59#127—44no—
60#127D411yescharacteristic, complemented
61#126—24no—
62#125—21no—
63#124—22no—
64#124—24no—
65#124—21no—
66#123—24no—
67#123—24no—
68C22⋊Q8—22no—
69C22⋊Q8—21no—
70C22⋊Q8C2312yes—
71#121—22no—
72#121—22no—
73#121—22no—
74#121—24no—
75#121—21no—
76#121—48no—
77#121C2312yes—
78#120—24no—
79C4×Q8—22no—
80C4×Q8—21no—
81C4×Q8D412yescomplemented
82C4×D4—22no—
83C4×D4—24no—
84C4×D4—24no—
85C4×D4—21no—
86C4×D4—21no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4D412yescomplemented
91#117—22no—
92#117—22no—
93#117—21no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4D411yescharacteristic
97C2×C22⋊C4—24no—
98C2×C22⋊C4C2312yes—
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101C4⋊C8—21no—
102C4⋊C8—21no—
103#102—24no—
104#102—24no—
105C22⋊C8C2312yes—
106C22⋊C8C2312yes—
107C4×C8—21no—
108C4×C8—21no—
109#95C2312yes—
110C24—22no—
111C24C2×D412yes—
112#41—42no—
113#41—42no—
114#41—84no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8C2×D412yes—
118C2×D4—22no—
119C2×D4—24no—
120C2×D4—24no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—44no—
130C2×D4—48no—
131C2×D4—48no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—82no—
139C2×D4—84no—
140C2×D4C2×D412yes—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—44no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D412yes—
149C22×C4C2×D411yescharacteristic
150C22×C4C2411yescharacteristic
151SD16—84no—
152D8—84no—
153C2×C8—22no—
154C2×C8—22no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4C2×D412yes—
166C4⋊C4C2×D412yes—
167C22⋊C4—24no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—48no—
176C42—24no—
177C42—22no—
178C42—42no—
179C42C2×D411yescharacteristic
180C42C2×D411yescharacteristic
181C42C2×D411yescharacteristic
182C42C2×D411yescharacteristic
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—42no—
188C23—42no—
189C23—44no—
190C23—44no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23C22×D411yescharacteristic
195Q8—42no—
196Q8—42no—
197Q8—42no—
198D4—42no—
199D4—44no—
200D4—44no—
201D4—42no—
202D4—42no—
203D4—42no—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—82no—
208D4—84no—
209D4—84no—
210D4—816no—
211D4—82no—
212D4—82no—
213D4—82no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—82no—
227C2×C4—84no—
228C2×C4#12011yescharacteristic
229C2×C4#12011yescharacteristic
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C2×C4#13512yes—
233C2×C4AGL(1,ℤ/8)12yes—
234C2×C4C22×D411yescharacteristic
235C2×C4C22×D411yescharacteristic
236C8—42no—
237C8—42no—
238C22—21no—
239C22—21no—
240C22—44no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—82no—
248C22—84no—
249C22—84no—
250C22—82no—
251C22—82no—
252C22—82no—
253C22C2×#12011yescharacteristic
254C22C2×#13511yescharacteristic
255C22C2×AGL(1,ℤ/8)11yescharacteristic
256C4—21no—
257C4—21no—
258C4—21no—
259C4—44no—
260C4—42no—
261C4—42no—
262C4—42no—
263C4—42no—
264C4#44912yes—
265C4#44912yes—
266C4#45312yes—
267C2—21no—
268C2—21no—
269C2—82no—
270C2—82no—
271C2—162no—
272C2C2×#44911yescharacteristic
273C2C2×#45311yescharacteristic
274C2#302711yescharacteristic
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. #3506C2semidirect
2. #3208C2semidirect
3. #3205C2semidirect
4. #2541C2semidirect
5. #1600C2semidirect
6. #1503C2semidirect
7. #1502C2semidirect
8. #1460C2semidirect
51. C4⋊Q8D4semidirect
52. C4⋊Q8D4semidirect
53. C4⋊Q8D4semidirect
60. #127D4semidirect
81. C4×Q8D4semidirect
90. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
#35062
#32082
#32052
#25412
#16002
#15032
#15022
#14602