Finite Group Databasebeta

← C2×#1628 #18492

Subgroups — C2×#1628 #18492

Order 256.

531 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 135 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×#556C211yesmaximal, characteristic, complemented
2C2×#486C211yesmaximal, characteristic, complemented
3C2×#460C211yesmaximal, characteristic, complemented
4C2×#459C211yesmaximal, characteristic
5#1628C218yesmaximal, direct product factor
6C2×#332C211yesmaximal, characteristic, complemented
7C2×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×D4⋊C8C211yesmaximal, characteristic, complemented
9C22×SD16—21no—
10C22×D8—21no—
11#556—22no—
12#556—22no—
13#556—22no—
14#556C2214yescomplemented
15C2×C4⋊Q8C2211yescharacteristic
16C2×#127C2211yescharacteristic
17C2×#126—22no—
18C2×#124—21no—
19C2×C22⋊Q8—21no—
20C2×C4×Q8—21no—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117—21no—
24#486—22no—
25#486C2214yescomplemented
26#460—22no—
27#460C2214yescomplemented
28#459—22no—
29#459C2214yes—
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#102—22no—
33C2×#102—21no—
34C2×#102—21no—
35C2×C4×C8C2211yescharacteristic
36#332C2214yescomplemented
37Q8⋊C8C2214yescomplemented
38D4⋊C8C2214yescomplemented
39C22×Q8—21no—
40C22×Q8D411yescharacteristic
41C22×D4—21no—
42C22×D4—21no—
43C22×D4—21no—
44C22×D4D411yescharacteristic
45C23×C4—21no—
46C2×SD16—24no—
47C2×SD16—44no—
48C2×D8—24no—
49C2×D8—44no—
50C22×C8—21no—
51C22×C8—21no—
52C22×C8—21no—
53C4⋊Q8—21no—
54C4⋊Q8—21no—
55C4⋊Q8D412yes—
56C4⋊Q8C2312yes—
57#127—21no—
58#127—21no—
59#127—21no—
60#127C2312yes—
61#126—28no—
62#126—28no—
63#124—24no—
64#124—24no—
65C22⋊Q8—24no—
66C22⋊Q8—24no—
67C4×Q8—24no—
68C4×Q8—24no—
69C4×Q8—21no—
70C4×Q8—21no—
71C4×Q8D412yes—
72C4×Q8C2312yes—
73C4×D4—21no—
74C4×D4—21no—
75C4×D4D412yes—
76C4×D4C2312yes—
77#117—24no—
78#117—24no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D412yes—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C42—22no—
89C2×C42—21no—
90C2×C42C2311yescharacteristic
91C4⋊C8—21no—
92C4⋊C8—21no—
93C4⋊C8C2312yes—
94C4⋊C8C2312yes—
95#102—24no—
96#102—24no—
97#102—28no—
98C4×C8—21no—
99C4×C8C2312yes—
100C24—21no—
101C24—41no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—42no—
108C2×Q8—42no—
109C2×Q8C2×D412yes—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—44no—
117C2×D4—41no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4C2×D412yes—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136SD16—48no—
137D8—48no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—42no—
144C4⋊C4—28no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4C2×D412yes—
153C4⋊C4C2×D412yes—
154C4⋊C4C2×D414yes—
155C22⋊C4—28no—
156C22⋊C4—28no—
157C22⋊C4—22no—
158C22⋊C4—22no—
159C42—28no—
160C42—22no—
161C42—22no—
162C42C2×D411yescharacteristic
163C42C2×D411yescharacteristic
164C42C2×D411yescharacteristic
165C42C2411yescharacteristic
166C23—22no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—82no—
172C23—82no—
173C23—82no—
174C23#12011yescharacteristic
175Q8—22no—
176Q8—22no—
177Q8—42no—
178Q8—42no—
179Q8—42no—
180Q8—44no—
181D4—22no—
182D4—22no—
183D4—44no—
184D4—42no—
185D4—42no—
186D4—42no—
187D4—42no—
188D4—42no—
189D4—82no—
190D4—82no—
191D4—82no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4#12011yescharacteristic
205C2×C4#12011yescharacteristic
206C2×C4#12011yescharacteristic
207C2×C4#13512yes—
208C2×C4#13512yes—
209C2×C4AGL(1,ℤ/8)12yes—
210C2×C4AGL(1,ℤ/8)12yes—
211C2×C4C22×D411yescharacteristic
212C2×C4C22×D411yescharacteristic
213C2×C4C22×D411yescharacteristic
214C8—42no—
215C8—42no—
216C8—42no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—84no—
221C22—82no—
222C22—82no—
223C22—82no—
224C22—164no—
225C22#44912yes—
226C22#44912yes—
227C22#45412yes—
228C22C2×#12011yescharacteristic
229C4—21no—
230C4—21no—
231C4—42no—
232C4—42no—
233C4—44no—
234C4—42no—
235C4#44912yes—
236C4#44912yes—
237C4C2×#13511yescharacteristic
238C4C2×#13511yescharacteristic
239C4C2×AGL(1,ℤ/8)11yescharacteristic
240C4C2×AGL(1,ℤ/8)11yescharacteristic
241C2—82no—
242C2—162no—
243C2#162814yesdirect product factor
244C2C2×#44911yescharacteristic
245C2C2×#44911yescharacteristic
246C2C2×#45411yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#459C2
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
1. C2×#556C2semidirect
2. C2×#486C2semidirect
3. C2×#460C2semidirect
5. #1628C2direct (N × H)
6. C2×#332C2semidirect
7. C2×Q8⋊C8C2semidirect
8. C2×D4⋊C8C2semidirect
14. #556C22semidirect
25. #486C22semidirect
27. #460C22semidirect
36. #332C22semidirect
37. Q8⋊C8C22semidirect
38. D4⋊C8C22semidirect
243. C2#1628direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4862
C2×#4602
C2×#4592
#16282
C2×#3322
C2×Q8⋊C82
C2×D4⋊C82