Finite Group Databasebeta

← C2×#1642 #18512

Subgroups — C2×#1642 #18512

Order 256.

645 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 645 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#550C211yesmaximal, characteristic
2C2×#486C211yesmaximal, characteristic, complemented
3C2×#475C211yesmaximal, characteristic, complemented
4C2×#461C211yesmaximal, characteristic, complemented
5#1642C218yesmaximal, direct product factor
6C2×#331C211yesmaximal, characteristic, complemented
7C2×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×D4⋊C8C211yesmaximal, characteristic, complemented
9C22×#41—21no—
10C22×SD16—21no—
11#550—22no—
12#550—22no—
13#550—22no—
14#550C2214yes—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#127—21no—
17C2×#127C2211yescharacteristic
18C2×#121—22no—
19C2×#121—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23#486—22no—
24#486C2214yescomplemented
25#475—22no—
26#475C2214yescomplemented
27#461—22no—
28#461C2214yescomplemented
29C2×#106—21no—
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103—21no—
33C2×#103—21no—
34C2×#102—22no—
35C2×C4×C8C2211yescharacteristic
36#331C2214yescomplemented
37Q8⋊C8C2214yescomplemented
38D4⋊C8C2214yescomplemented
39C2×#41—24no—
40C2×#41—24no—
41C2×#41—44no—
42C2×#41—44no—
43C22×Q8—21no—
44C22×Q8D411yescharacteristic
45C22×D4—22no—
46C22×D4—22no—
47C22×D4—21no—
48C22×D4—21no—
49C22×D4D411yescharacteristic
50C22×D4D412yes—
51C23×C4—22no—
52C23×C4—21no—
53C2×SD16—24no—
54C2×SD16—44no—
55C22×C8—21no—
56C22×C8—21no—
57C22×C8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8C2312yes—
62#127—24no—
63#127—24no—
64#127—21no—
65#127—21no—
66#127D412yes—
67#127C2312yes—
68#121—24no—
69#121—28no—
70#121—28no—
71#121—24no—
72C4×Q8—21no—
73C4×Q8—21no—
74C4×Q8D412yes—
75C4×Q8C2312yes—
76C4×D4—24no—
77C4×D4—24no—
78C4×D4—21no—
79C4×D4—21no—
80C4×D4D412yes—
81C4×D4C2312yes—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C42—21no—
90C2×C42C2311yescharacteristic
91#106—24no—
92C4⋊C8—21no—
93C4⋊C8—21no—
94C4⋊C8C2312yes—
95C4⋊C8C2312yes—
96#103—24no—
97#103—24no—
98#102—28no—
99C4×C8—21no—
100C4×C8C2312yes—
101C24—21no—
102C24—22no—
103#41—48no—
104#41—416no—
105#41—48no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—41no—
110C2×Q8—42no—
111C2×Q8—42no—
112C2×Q8SD1614yescomplemented
113C2×Q8C2×D412yes—
114C2×D4—28no—
115C2×D4—28no—
116C2×D4—21no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—48no—
125C2×D4—44no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—84no—
131C2×D4C2×D412yes—
132C2×D4C2×D414yes—
133C22×C4—21no—
134C22×C4—28no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—48no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146SD16—48no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4C2×D412yes—
162C4⋊C4C2×D412yes—
163C22⋊C4—28no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C42—22no—
167C42—22no—
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C42C2×D411yescharacteristic
171C42C2411yescharacteristic
172C23—22no—
173C23—24no—
174C23—42no—
175C23—44no—
176C23—44no—
177C23—42no—
178C23—44no—
179C23—42no—
180C23#12011yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—42no—
184Q8—42no—
185Q8—82no—
186Q8C2×SD1614yescomplemented
187D4—22no—
188D4—22no—
189D4—44no—
190D4—416no—
191D4—42no—
192D4—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—44no—
197D4—44no—
198D4—82no—
199D4—88no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—416no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4#12011yescharacteristic
214C2×C4#12011yescharacteristic
215C2×C4#12011yescharacteristic
216C2×C4C2×SD1612yes—
217C2×C4#13512yes—
218C2×C4AGL(1,ℤ/8)12yes—
219C2×C4AGL(1,ℤ/8)12yes—
220C2×C4C22×D411yescharacteristic
221C2×C4C22×D411yescharacteristic
222C2×C4C22×D411yescharacteristic
223C8—42no—
224C8—42no—
225C8—42no—
226C22—44no—
227C22—42no—
228C22—42no—
229C22—44no—
230C22—44no—
231C22—42no—
232C22—84no—
233C22—88no—
234C22#44912yes—
235C22#45012yes—
236C22#45512yes—
237C22C2×#12011yescharacteristic
238C4—24no—
239C4—21no—
240C4—21no—
241C4—42no—
242C4—42no—
243C4—82no—
244C4#44912yes—
245C4#45012yes—
246C4C22×SD1611yescharacteristic
247C4C2×#13511yescharacteristic
248C4C2×AGL(1,ℤ/8)11yescharacteristic
249C4C2×AGL(1,ℤ/8)11yescharacteristic
250C2—82no—
251C2—84no—
252C2#164214yesdirect product factor
253C2C2×#45011yescharacteristic
254C2C2×#44911yescharacteristic
255C2C2×#45511yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#550C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#486C2semidirect
3. C2×#475C2semidirect
4. C2×#461C2semidirect
5. #1642C2direct (N × H)
6. C2×#331C2semidirect
7. C2×Q8⋊C8C2semidirect
8. C2×D4⋊C8C2semidirect
24. #486C22semidirect
26. #475C22semidirect
28. #461C22semidirect
36. #331C22semidirect
37. Q8⋊C8C22semidirect
38. D4⋊C8C22semidirect
112. C2×Q8SD16semidirect
186. Q8C2×SD16semidirect
252. C2#1642direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5502
C2×#4862
C2×#4752
C2×#4612
#16422
C2×#3312
C2×Q8⋊C82
C2×D4⋊C82