Finite Group Databasebeta

← C2×#1609 #18105

Subgroups — C2×#1609 #18105

Order 256.

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

normal subgroups only

This group has 530 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
1C2×#539C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic, complemented
3C2×#480C211yesmaximal, characteristic
4C2×#398C211yesmaximal, characteristic, complemented
5#1609C218yesmaximal, direct product factor
6C2×#328C211yesmaximal, characteristic, complemented
7C2×#327C211yesmaximal, characteristic
8C2×#324C211yesmaximal, characteristic, complemented
9#539—22no—
10#539—22no—
11#539—22no—
12#539C2214yes—
13C2×#126—21no—
14C2×#125—21no—
15C2×#123—21no—
16C2×#123—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#483—22no—
25#483C2214yescomplemented
26#480—22no—
27#480C2214yes—
28C2×#107—21no—
29C2×#106—21no—
30C2×#103—21no—
31C2×#102—21no—
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34#398—22no—
35#398C2214yescomplemented
36C2×#95—21no—
37C2×#95C2211yescharacteristic
38#328C2214yescomplemented
39#327C2214yes—
40#324C2214yescomplemented
41C22×Q8D411yescharacteristic
42C22×D4—21no—
43C22×D4D411yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4—21no—
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C22×C8—21no—
51C22×C8—21no—
52#126—24no—
53#126—24no—
54#125—24no—
55#125—24no—
56#123—24no—
57#123—24no—
58#123—24no—
59#123—24no—
60C22⋊Q8—24no—
61C22⋊Q8—24no—
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64C22⋊Q8D412yes—
65C22⋊Q8C2312yes—
66#121—21no—
67#121—21no—
68#121D412yes—
69#121C2312yes—
70C4×D4—24no—
71C4×D4—24no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4D412yes—
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2312yes—
91C2×C4⋊C4C2312yes—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C42—21no—
98#107—24no—
99#106—24no—
100#103—24no—
101#102—24no—
102C22⋊C8—21no—
103C22⋊C8—21no—
104C22⋊C8C2312yes—
105C22⋊C8C2312yes—
106#95—24no—
107#95—21no—
108#95C2312yes—
109C24—21no—
110C24C2×D411yescharacteristic
111C2×Q8—21no—
112C2×Q8—42no—
113C2×Q8C2×D412yes—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—21no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4C2×D412yes—
121C22×C4—24no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—82no—
143C22×C4C2×D412yes—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D412yes—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2411yescharacteristic
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—42no—
154C2×C8—42no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—48no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4C2×D412yes—
172C4⋊C4C2×D412yes—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C42—24no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—42no—
185C23—42no—
186C23—42no—
187C23#12011yescharacteristic
188C23#12011yescharacteristic
189C23#13512yes—
190C23AGL(1,ℤ/8)12yes—
191C23C22×D411yescharacteristic
192Q8—42no—
193Q8—81no—
194D4—42no—
195D4—42no—
196D4—42no—
197D4—44no—
198D4—81no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—48no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—81no—
227C2×C4—84no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C2×C4C22×D411yescharacteristic
233C2×C4C22×D411yescharacteristic
234C8—42no—
235C8—42no—
236C22—24no—
237C22—24no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—84no—
246C22#44912yes—
247C22#44912yes—
248C22#45612yes—
249C22#45812yes—
250C22C2×#12011yescharacteristic
251C22C2×#13511yescharacteristic
252C22C2×AGL(1,ℤ/8)11yescharacteristic
253C4—21no—
254C4—21no—
255C4—21no—
256C4—21no—
257C4—44no—
258C4—42no—
259C4—42no—
260C4—42no—
261C4—42no—
262C4—42no—
263C4—82no—
264C2—21no—
265C2—21no—
266C2—22no—
267C2—82no—
268C2#160914yesdirect product factor
269C2C2×#44911yescharacteristic
270C2C2×#45611yescharacteristic
271C2C2×#45811yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#483C2semidirect
4. C2×#398C2semidirect
5. #1609C2direct (N × H)
6. C2×#328C2semidirect
8. C2×#324C2semidirect
25. #483C22semidirect
35. #398C22semidirect
38. #328C22semidirect
40. #324C22semidirect
268. C2#1609direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5392
C2×#4832
C2×#4802
C2×#3982
#16092
C2×#3282
C2×#3272
C2×#3242