Finite Group Databasebeta

← C2×#1520 #17565

Subgroups — C2×#1520 #17565

Order 256.

761 conjugacy classes of subgroups; only the 239 normal subgroups are shown.

normal subgroups only

This group has 761 conjugacy classes of subgroups — too many to draw. Only its 239 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
1C22×C22⋊Q8C211yesmaximal, characteristic
2C22×C22⋊C8C211yesmaximal, characteristic
3C2×#380C211yesmaximal, characteristic
4#1520C218yesmaximal, direct product factor
5C2×#328C214yesmaximal, complemented
6C23×Q8C411yescharacteristic, complemented
7C24×C4C2211yescharacteristic
8C23×C8—21no—
9C2×C22⋊Q8—21no—
10C2×C22⋊Q8—22no—
11C2×C22⋊Q8—21no—
12C2×C22⋊Q8—22no—
13C2×C22⋊Q8C412yescomplemented
14C2×C22⋊Q8C414yescomplemented
15C2×C22⋊Q8C2212yes—
16C2×C22⋊Q8C2214yes—
17C22×C4⋊C4—21no—
18C22×C4⋊C4—21no—
19C22×C4⋊C4C411yescharacteristic, complemented
20C22×C22⋊C4—21no—
21C22×C22⋊C4—21no—
22C2×C22⋊C8—21no—
23C2×C22⋊C8—22no—
24C2×C22⋊C8C2212yes—
25C2×C22⋊C8C2214yes—
26#380—22no—
27#380C2214yes—
28C2×#95C2212yes—
29#328C22116yescomplemented
30C25D411yescharacteristic
31C22×Q8—22no—
32C22×Q8—21no—
33C22×Q8—44no—
34C22×Q8C2×C412yes—
35C22×Q8C2×C414yescomplemented
36C23×C4—21no—
37C23×C4—21no—
38C23×C4—22no—
39C23×C4—44no—
40C23×C4C2×C411yescharacteristic
41C23×C4C2×C412yes—
42C23×C4C2×C411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4D412yes—
46C23×C4D411yescharacteristic
47C23×C4D412yes—
48C23×C4C2311yescharacteristic
49C23×C4C2312yes—
50C22×C8—24no—
51C22×C8—22no—
52C22×C8—44no—
53C22⋊Q8—24no—
54C22⋊Q8—24no—
55C22⋊Q8—24no—
56C22⋊Q8—22no—
57C22⋊Q8—24no—
58C22⋊Q8—22no—
59C22⋊Q8—24no—
60C22⋊Q8C2×C414yes—
61C22⋊Q8C2×C418yescomplemented
62C22⋊Q8C2314yes—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—44no—
71C2×C4⋊C4C2×C412yes—
72C2×C4⋊C4C2×C414yescomplemented
73C2×C22⋊C4—24no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—44no—
79C22⋊C8—24no—
80C22⋊C8—22no—
81C22⋊C8C2314yes—
82#95—22no—
83#95C2314yes—
84C24—42no—
85C24—42no—
86C24C22⋊C411yescharacteristic
87C24C22⋊C411yescharacteristic
88C24C22⋊C412yes—
89C24SD1614yes—
90C24Q1614yes—
91C24C2×D411yescharacteristic
92C24C2×D412yes—
93C2×Q8—22no—
94C2×Q8—24no—
95C2×Q8—48no—
96C2×Q8—48no—
97C2×Q8—44no—
98C2×Q8—44no—
99C2×Q8C22×C414yes—
100C22×C4—24no—
101C22×C4—24no—
102C22×C4—24no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—44no—
109C22×C4—44no—
110C22×C4—44no—
111C22×C4—44no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—44no—
118C22×C4—44no—
119C22×C4—44no—
120C22×C4C22⋊C412yes—
121C22×C4C22⋊C412yes—
122C22×C4C22⋊C412yes—
123C22×C4C22⋊C411yescharacteristic
124C22×C4C22⋊C412yes—
125C22×C4C22⋊C412yes—
126C22×C4C22⋊C411yescharacteristic
127C22×C4C22×C412yes—
128C22×C4C22×C411yescharacteristic
129C22×C4C22×C411yescharacteristic
130C22×C4C22×C412yes—
131C22×C4C2×D412yes—
132C22×C4C2×D412yes—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D412yes—
136C22×C4C2×D411yescharacteristic
137C22×C4C2411yescharacteristic
138C2×C8—24no—
139C2×C8—44no—
140C2×C8—48no—
141C4⋊C4—24no—
142C4⋊C4—28no—
143C4⋊C4—22no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—48no—
147C4⋊C4C22×C414yes—
148C22⋊C4—24no—
149C22⋊C4—28no—
150C22⋊C4—24no—
151C22⋊C4—48no—
152C23—22no—
153C23—22no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—48no—
162C23—48no—
163C23—42no—
164C23#10314yes—
165C23#10314yes—
166C23#10318yes—
167C23C2×C22⋊C411yescharacteristic
168C23C2×C22⋊C412yes—
169C23C2×C22⋊C412yes—
170C23C2×C22⋊C411yescharacteristic
171C23C2×SD1612yes—
172C23C2×SD1614yes—
173C23C2×Q1612yes—
174C23C2×Q1614yes—
175C23C22×D411yescharacteristic
176Q8—44no—
177Q8—44no—
178Q8—48no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—48no—
199C2×C4—48no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—88no—
206C2×C4#9914yes—
207C2×C4C2×C22⋊C411yescharacteristic
208C2×C4C2×C22⋊C412yes—
209C2×C4C2×C22⋊C411yescharacteristic
210C2×C4C2×C22⋊C412yes—
211C2×C4C23×C411yescharacteristic
212C2×C4C22×D411yescharacteristic
213C8—44no—
214C22—22no—
215C22—21no—
216C22—22no—
217C22—21no—
218C22—22no—
219C22—21no—
220C22—21no—
221C22—22no—
222C22—24no—
223C22—28no—
224C22—28no—
225C22—24no—
226C22—48no—
227C22—44no—
228C22—44no—
229C22C2×#9912yes—
230C22C2×#10314yes—
231C22C2×#10314yes—
232C22C2×#10312yes—
233C22C2×#10312yes—
234C22#42112yes—
235C22C22×C22⋊C411yescharacteristic
236C22C22×SD1611yescharacteristic
237C22C22×Q1611yescharacteristic
238C4—21no—
239C4—21no—
240C4—22no—
241C4—21no—
242C4—21no—
243C4—44no—
244C4—44no—
245C4—84no—
246C4C2×#9914yes—
247C2—21no—
248C2—21no—
249C2—24no—
250C2—22no—
251C2—24no—
252C2#152014yesdirect product factor
253C2C22×#9911yescharacteristic
254C2C22×#10311yescharacteristic
255C2C2×#42111yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1520C2direct (N × H)
5. C2×#328C2semidirect
6. C23×Q8C4semidirect
13. C2×C22⋊Q8C4semidirect
14. C2×C22⋊Q8C4semidirect
19. C22×C4⋊C4C4semidirect
29. #328C22semidirect
35. C22×Q8C2×C4semidirect
61. C22⋊Q8C2×C4semidirect
72. C2×C4⋊C4C2×C4semidirect
252. C2#1520direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C22×C22⋊C82
C2×#3802
#15202
C2×#3282