Finite Group Databasebeta

← C22×#2089 #6432237

Subgroups — C22×#2089 #6432237

Order 512.

2426 conjugacy classes of subgroups.

The subgroup lattice has 253 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C212yesmaximal
2C23×#107C211yesmaximal, characteristic
3C22×#424C211yesmaximal, characteristic
4C2×#2089C2124yesmaximal, direct product factor
5C22×#343C212yesmaximal
6C22×#341C211yesmaximal, characteristic
7C23×M16C2211yescharacteristic
8C23×M16C2212yes—
9C24×C8C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C23×C4⋊C4C2212yes—
12C2×#428—212no—
13C2×#428C22124yes—
14C22×#107—22no—
15C22×#107—26no—
16C22×#107—22no—
17C22×#107C22112yes—
18C22×#106—22no—
19C2×#424—26no—
20C2×#424C22112yes—
21C22×C4⋊C8—22no—
22#2089C22164yesdirect product factor
23C2×#343C22124yes—
24C2×#341C22112yes—
25C24×C4—22no—
26C24×C4C2311yescharacteristic
27C22×M16—26no—
28C22×M16—28no—
29C22×M16—23no—
30C22×M16—28no—
31C22×M16Q8116yes—
32C22×M16C2316yes—
33C22×M16C23112yes—
34C23×C8—22no—
35C23×C8—28no—
36C23×C8—21no—
37C23×C8D412yes—
38C23×C8D416yes—
39C23×C8C2316yes—
40C2×#117—23no—
41C2×#117—23no—
42C2×#117D416yes—
43C2×#117C2316yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—26no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—26no—
48C22×C4⋊C4—26no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C23112yes—
52C22×C22⋊C4—21no—
53C22×C42—21no—
54#428—248no—
55#428C23132yes—
56C2×#107—224no—
57C2×#107—224no—
58C2×#107—224no—
59C2×#107C23116yes—
60C2×#106—224no—
61#424—224no—
62#424C23116yes—
63C2×C4⋊C8—224no—
64#343C23132yes—
65#341C23116yes—
66C25#4111yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—212no—
70C23×C4—46no—
71C23×C4—42no—
72C23×C4—88no—
73C23×C4C2×D413yes—
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D413yes—
76C23×C4C2×D411yescharacteristic
77C23×C4C2×Q818yes—
78C23×C4#4113yes—
79C23×C4#4118yes—
80C23×C4C2413yes—
81C2×M16—248no—
82C2×M16—212no—
83C2×M16—224no—
84C2×M16—26no—
85C2×M16—224no—
86C2×M16—412no—
87C2×M16C2×Q8148yes—
88C2×M16C2414yes—
89C2×M16C2418yes—
90C22×C8—216no—
91C22×C8—28no—
92C22×C8—224no—
93C22×C8—212no—
94C22×C8—224no—
95C22×C8—26no—
96C22×C8—44no—
97C22×C8C2×D4112yes—
98C22×C8C2×D4112yes—
99C22×C8C2414yes—
100#117—212no—
101#117—26no—
102#117—26no—
103#117C2×D4112yes—
104#117C2414yes—
105C2×C4⋊C4—23no—
106C2×C4⋊C4—23no—
107C2×C4⋊C4—212no—
108C2×C4⋊C4—212no—
109C2×C4⋊C4—212no—
110C2×C4⋊C4—212no—
111C2×C4⋊C4—212no—
112C2×C4⋊C4—46no—
113C2×C4⋊C4—46no—
114C2×C4⋊C4—412no—
115C2×C4⋊C4C2×D416yes—
116C2×C4⋊C4C2×D416yes—
117C2×C4⋊C4C2418yes—
118C2×C22⋊C4—26no—
119C2×C22⋊C4—26no—
120C2×C42—26no—
121C2×C42—26no—
122#107—232no—
123#107—232no—
124#106—232no—
125C4⋊C8—232no—
126C24—24no—
127C24—24no—
128C24#12113yes—
129C24#12111yescharacteristic
130C24#12518yes—
131C24C2×#4113yes—
132C22×C4—26no—
133C22×C4—26no—
134C22×C4—28no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—412no—
138C22×C4—412no—
139C22×C4—824no—
140C22×C4—88no—
141C22×C4—824no—
142C22×C4#12113yes—
143C22×C4#12113yes—
144C22×C4#12116yes—
145C22×C4C22⋊Q818yes—
146C22×C4C22⋊Q8124yes—
147C22×C4#12314yes—
148C22×C4#123112yes—
149C22×C4#125124yes—
150C22×C4C4⋊Q8112yes—
151C22×C4C4⋊Q814yes—
152C22×C4C22×D413yes—
153C22×C4C22×D413yes—
154C22×C4C22×D413yes—
155C22×C4C22×D413yes—
156C22×C4C22×Q8112yes—
157C22×C4C2×#4113yes—
158C22×C4C2×#41112yes—
159C22×C4C2511yescharacteristic
160M16—248no—
161M16—28no—
162M16—28no—
163M16—412no—
164M16C22×Q8116yes—
165C2×C8—248no—
166C2×C8—224no—
167C2×C8—224no—
168C2×C8—28no—
169C2×C8—28no—
170C2×C8—24no—
171C2×C8—412no—
172C2×C8C22×D418yes—
173C4⋊C4—26no—
174C4⋊C4—26no—
175C4⋊C4—28no—
176C4⋊C4—28no—
177C4⋊C4—412no—
178C4⋊C4—412no—
179C4⋊C4C22×D414yes—
180C4⋊C4C22×D414yes—
181C22⋊C4—212no—
182C22⋊C4—24no—
183C42—212no—
184C42—24no—
185C23—232no—
186C23—26no—
187C23—26no—
188C23—26no—
189C23—26no—
190C23#398112yes—
191C23#39814yes—
192C23#47214yes—
193C23#47314yes—
194C23C2×#12113yes—
195C23C2×#12113yes—
196C23C2×#125112yes—
197C23C22×#4111yescharacteristic
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—412no—
201C2×C4—412no—
202C2×C4—48no—
203C2×C4—824no—
204C2×C4—824no—
205C2×C4—88no—
206C2×C4#398124yes—
207C2×C4#398112yes—
208C2×C4#398112yes—
209C2×C4C2×#12113yes—
210C2×C4C2×#12113yes—
211C2×C4C2×C22⋊Q8112yes—
212C2×C4C2×C22⋊Q8112yes—
213C2×C4C2×#12316yes—
214C2×C4C2×#12316yes—
215C2×C4C2×#125112yes—
216C2×C4C2×C4⋊Q816yes—
217C2×C4C2×C4⋊Q816yes—
218C2×C4C23×D411yescharacteristic
219C2×C4C23×D411yescharacteristic
220C2×C4C23×Q812yes—
221C2×C4C22×#4112yes—
222C8—216no—
223C8—28no—
224C8—44no—
225C22—23no—
226C22—21no—
227C22—23no—
228C22—21no—
229C22—224no—
230C22—224no—
231C22#2089116yesdirect product factor
232C22C2×#39816yes—
233C22C2×#39816yes—
234C22C2×#47216yes—
235C22C2×#47316yes—
236C22C22×#12111yescharacteristic
237C22C22×#12512yes—
238C4—44no—
239C4—44no—
240C4—88no—
241C4C2×#39816yes—
242C4C2×#39816yes—
243C4C22×C22⋊Q812yes—
244C4C22×#12311yescharacteristic
245C4C22×C4⋊Q811yescharacteristic
246C2—26no—
247C2—22no—
248C2C2×#2089112yesdirect product factor
249C2C22×#39811yescharacteristic
250C2C22×#47211yescharacteristic
251C2C22×#47311yescharacteristic
subgroupG/H
center, socleC24#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#2089C2direct (N × H)
22. #2089C22direct (N × H)
231. C22#2089direct (N × H)
248. C2C2×#2089direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4282
C23×#1072
C22×#4242
C2×#20892
C22×#3432
C22×#3412