Finite Group Databasebeta

← C22×#1346 #481453

Subgroups — C22×#1346 #481453

Order 512.

1558 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1558 conjugacy classes of subgroups — too many to draw. Only its 322 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×#398C211yesmaximal, characteristic
2C22×#343C211yesmaximal, characteristic
3C22×#324C211yesmaximal, characteristic
4C2×#1346C2112yesmaximal, direct product factor
5C23×M16—21no—
6C23×C4⋊C4—21no—
7C23×C4⋊C4C411yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C22×C22⋊C8—21no—
10C2×#398—23no—
11C2×#398—23no—
12C2×#398C416yes—
13C2×#398C2216yes—
14C22×#95—21no—
15C22×#95C411yescharacteristic
16C2×#343—23no—
17C2×#343C2216yes—
18C2×#324—23no—
19C2×#324C2216yes—
20#1346C22116yesdirect product factor
21C24×C4—21no—
22C24×C4—21no—
23C24×C4C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C24×C4D411yescharacteristic
26C22×M16—28no—
27C22×M16—26no—
28C22×M16—43no—
29C22×M16—44no—
30C23×C8—41no—
31C23×C8—41no—
32C22×C4⋊C4—23no—
33C22×C4⋊C4—26no—
34C22×C4⋊C4—26no—
35C22×C4⋊C4—43no—
36C22×C4⋊C4—41no—
37C22×C4⋊C4—43no—
38C22×C4⋊C4—41no—
39C22×C4⋊C4—44no—
40C22×C4⋊C4—44no—
41C22×C4⋊C4C2×C416yes—
42C22×C4⋊C4C2×C413yes—
43C22×C4⋊C4D413yes—
44C22×C4⋊C4D413yes—
45C22×C4⋊C4C2313yes—
46C2×C22⋊C8—26no—
47C2×C22⋊C8—43no—
48#398—212no—
49#398—26no—
50#398—26no—
51#398C2×C4112yes—
52#398C2314yes—
53C2×#95—23no—
54C2×#95—26no—
55C2×#95—26no—
56C2×#95C2×C416yes—
57#343—26no—
58#343C2314yes—
59#324—26no—
60#324C2314yes—
61C25C22⋊C411yescharacteristic
62C23×C4—23no—
63C23×C4—21no—
64C23×C4—24no—
65C23×C4—28no—
66C23×C4—23no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—26no—
70C23×C4—23no—
71C23×C4—26no—
72C23×C4—44no—
73C23×C4—44no—
74C23×C4—44no—
75C23×C4—44no—
76C23×C4—43no—
77C23×C4—41no—
78C23×C4—44no—
79C23×C4—44no—
80C23×C4C22⋊C413yes—
81C23×C4C22⋊C413yes—
82C23×C4C22⋊C413yes—
83C23×C4D814yes—
84C23×C4SD1614yes—
85C23×C4C22×C413yes—
86C23×C4C2×D413yes—
87C23×C4C2×D413yes—
88C2×M16—224no—
89C2×M16—24no—
90C2×M16—424no—
91C2×M16—46no—
92C2×M16—412no—
93C22×C8—48no—
94C22×C8—46no—
95C22×C8—46no—
96C22×C8—84no—
97C2×C4⋊C4—212no—
98C2×C4⋊C4—26no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—43no—
101C2×C4⋊C4—43no—
102C2×C4⋊C4—43no—
103C2×C4⋊C4—43no—
104C2×C4⋊C4—43no—
105C2×C4⋊C4—43no—
106C2×C4⋊C4—43no—
107C2×C4⋊C4—43no—
108C2×C4⋊C4—46no—
109C2×C4⋊C4—46no—
110C2×C4⋊C4—46no—
111C2×C4⋊C4—46no—
112C2×C4⋊C4—424no—
113C2×C4⋊C4—412no—
114C2×C4⋊C4—412no—
115C2×C4⋊C4C22⋊C416yes—
116C2×C4⋊C4C22×C414yes—
117C2×C4⋊C4C22×C413yes—
118C2×C4⋊C4C2×D413yes—
119C2×C4⋊C4C2×D413yes—
120C2×C4⋊C4C2411yescharacteristic
121C22⋊C8—24no—
122C22⋊C8—46no—
123#95—212no—
124#95—26no—
125#95—24no—
126#95C22×C414yes—
127C24—24no—
128C24—24no—
129C24#9913yes—
130C24#9911yescharacteristic
131C24#10214yes—
132C24#10414yes—
133C24C2×C22⋊C413yes—
134C22×C4—24no—
135C22×C4—212no—
136C22×C4—212no—
137C22×C4—24no—
138C22×C4—23no—
139C22×C4—23no—
140C22×C4—23no—
141C22×C4—23no—
142C22×C4—26no—
143C22×C4—26no—
144C22×C4—224no—
145C22×C4—24no—
146C22×C4—23no—
147C22×C4—23no—
148C22×C4—23no—
149C22×C4—23no—
150C22×C4—23no—
151C22×C4—23no—
152C22×C4—23no—
153C22×C4—23no—
154C22×C4—24no—
155C22×C4—432no—
156C22×C4—412no—
157C22×C4—412no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—412no—
161C22×C4—46no—
162C22×C4—46no—
163C22×C4—46no—
164C22×C4—46no—
165C22×C4—46no—
166C22×C4—46no—
167C22×C4—412no—
168C22×C4—412no—
169C22×C4—86no—
170C22×C4—812no—
171C22×C4—84no—
172C22×C4#102112yes—
173C22×C4C2×C22⋊C413yes—
174C22×C4C2×C22⋊C413yes—
175C22×C4C2×C22⋊C413yes—
176C22×C4C2×D816yes—
177C22×C4C2×SD1616yes—
178C22×C4C23×C411yescharacteristic
179C22×C4C22×D411yescharacteristic
180C22×C4C22×D411yescharacteristic
181M16—28no—
182M16—424no—
183M16—44no—
184C2×C8—44no—
185C2×C8—424no—
186C2×C8—44no—
187C2×C8—812no—
188C4⋊C4—46no—
189C4⋊C4—43no—
190C4⋊C4—43no—
191C4⋊C4—43no—
192C4⋊C4—41no—
193C4⋊C4—44no—
194C4⋊C4—412no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—424no—
198C23—216no—
199C23—216no—
200C23—26no—
201C23—26no—
202C23—26no—
203C23—26no—
204C23#327112yes—
205C23#32714yes—
206C23#35214yes—
207C23#35614yes—
208C23C2×#9913yes—
209C23C2×#9913yes—
210C23C2×#10216yes—
211C23C2×#10416yes—
212C23C22×C22⋊C411yescharacteristic
213C2×C4—21no—
214C2×C4—23no—
215C2×C4—21no—
216C2×C4—23no—
217C2×C4—26no—
218C2×C4—26no—
219C2×C4—212no—
220C2×C4—212no—
221C2×C4—26no—
222C2×C4—26no—
223C2×C4—28no—
224C2×C4—23no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—23no—
228C2×C4—44no—
229C2×C4—448no—
230C2×C4—46no—
231C2×C4—46no—
232C2×C4—46no—
233C2×C4—46no—
234C2×C4—46no—
235C2×C4—46no—
236C2×C4—44no—
237C2×C4—43no—
238C2×C4—43no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—83no—
245C2×C4—83no—
246C2×C4—812no—
247C2×C4—812no—
248C2×C4C2×#10216yes—
249C2×C4C22×D811yescharacteristic
250C2×C4C22×SD1611yescharacteristic
251C8—48no—
252C8—84no—
253C22—21no—
254C22—23no—
255C22—21no—
256C22—23no—
257C22—212no—
258C22—224no—
259C22—212no—
260C22#1346116yesdirect product factor
261C22C2×#32716yes—
262C22C2×#32716yes—
263C22C2×#35216yes—
264C22C2×#35616yes—
265C22C22×#9911yescharacteristic
266C22C22×#10211yescharacteristic
267C22C22×#10411yescharacteristic
268C4—21no—
269C4—21no—
270C4—26no—
271C4—48no—
272C4—41no—
273C4—43no—
274C4—41no—
275C4—43no—
276C4—84no—
277C2—21no—
278C2—21no—
279C2—26no—
280C2C2×#1346112yesdirect product factor
281C2C22×#32711yescharacteristic
282C2C22×#35211yescharacteristic
283C2C22×#35611yescharacteristic
subgroupG/H
center, socleC24#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1346C2direct (N × H)
20. #1346C22direct (N × H)
260. C22#1346direct (N × H)
280. C2C2×#1346direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3982
C22×#3432
C22×#3242
C2×#13462