Finite Group Databasebeta

← C4×#2913 #6369937

Subgroups — C4×#2913 #6369937

Order 512.

3228 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2913C212yesmaximal
2C2×C4×#131C211yesmaximal, characteristic
3C4×#520C212yesmaximal
4#20455C216yesmaximal, complemented
5#20453C216yesmaximal, complemented
6C4×#428C216yesmaximal, complemented
7C4×#427C213yesmaximal, complemented
8C2×C4×#107C213yesmaximal, complemented
9#20348C212yesmaximal
10C22×#131C2211yescharacteristic
11C22×#131C2211yescharacteristic
12C2×#520C2214yes—
13C2×C4×#41C2211yescharacteristic
14#2913C4132yesdirect product factor
15C2×#428C22112yes—
16C2×#427C2216yes—
17C22×#107C2216yes—
18C4×#131C4116yescomplemented
19C2×#405C2213yes—
20C2×#405C2213yes—
21C2×C4×M16C2213yes—
22C22×C4×C8C2213yes—
23#2276—22no—
24#2275—26no—
25#2274C22112yes—
26C42×Q8—22no—
27C42×D4—26no—
28C4×#117C2216yes—
29C2×C4×C4⋊C4C2216yes—
30#1776C22124yes—
31C4×#107C2213yes—
32C4×#107C2216yes—
33C4×#107C2216yes—
34C4×#107C2216yes—
35C4×#107C2211yescharacteristic
36C4×#107C2216yes—
37C4×#106C2216yes—
38C4×#106C2216yes—
39#1736C22112yes—
40#1735C2216yes—
41C2×#340C2216yes—
42C22×#41C2311yescharacteristic
43C2×#131—28no—
44C2×#131C2×C418yes—
45C2×#131C2×C418yes—
46C2×#131C2×C4116yes—
47C22×M16C2313yes—
48C22×M16C2313yes—
49C23×C8C2313yes—
50C23×C8C2313yes—
51#520—216no—
52#520C2×C4132yes—
53C4×#41C2×C418yes—
54C4×#41Q818yes—
55C2×C4×Q8—24no—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×D4—212no—
59C2×C4×D4D413yes—
60C2×C4×D4D413yes—
61C2×#117C23112yes—
62C2×#117C2313yes—
63C22×C4⋊C4C23112yes—
64C22×C42—26no—
65C22×C42C2313yes—
66#428C2×C4196yescomplemented
67#427C2×C4148yescomplemented
68C2×#107C2×C4148yescomplemented
69C2×#107C2316yes—
70C2×#107C23112yes—
71C2×#107C2312yes—
72C2×#106C23112yes—
73#405C2×C4124yescomplemented
74#405C2×C4124yescomplemented
75C4×M16C2×C4124yescomplemented
76C2×#97C2316yes—
77C2×C4×C8C2×C4124yescomplemented
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2313yes—
80C2×C4×C8C2313yes—
81C2×C4×C8C2313yes—
82C4×C4⋊C4—212no—
83C4×C4⋊C4—212no—
84C4×C4⋊C4—26no—
85C4×C4⋊C4—26no—
86C4×C4⋊C4C2316yes—
87C4×C4⋊C4C23112yes—
88C4×C4⋊C4C2312yes—
89C4×C4⋊C4C23112yes—
90C4×C22⋊C4—26no—
91C4×C22⋊C4—26no—
92C4×C22⋊C4—224no—
93#376—26no—
94C2×#95—26no—
95C43—26no—
96#340C2316yes—
97#340C23112yes—
98#340C2312yes—
99#340C23112yes—
100C2×#41C22×C414yes—
101C2×#41C22×C418yes—
102C2×#41C2×Q814yes—
103C2×#41#4118yes—
104C22×Q8C2×D411yescharacteristic
105C22×D4C2×D413yes—
106C23×C4—212no—
107C23×C4C2×D413yes—
108C23×C4C2×D413yes—
109C23×C4C2413yes—
110#131—232no—
111#131C42164yescomplemented
112C2×M16—212no—
113C2×M16C22×C4112yes—
114C2×M16C22×C4112yes—
115C2×M16C22×C4124yes—
116C22×C8—212no—
117C22×C8C22×C4112yes—
118C22×C8C22×C4112yes—
119C22×C8C22×C4124yes—
120C22×C8C2411yescharacteristic
121C22×C8C2413yes—
122C22×C8C2411yescharacteristic
123C22×C8C2413yes—
124C4×Q8—232no—
125C4×Q8C4⋊C418yes—
126C4×Q8C4⋊C418yes—
127C4×D4—296no—
128C4×D4C4⋊C4124yes—
129C4×D4C4⋊C4124yes—
130#117—224no—
131#117C22×C4148yes—
132#117C22×C4112yes—
133#117C2×Q8112yes—
134C2×C4⋊C4—212no—
135C2×C4⋊C4—224no—
136C2×C4⋊C4C22×C4148yes—
137C2×C4⋊C4C2×D413yes—
138C2×C4⋊C4C2×D413yes—
139C2×C4⋊C4C24112yes—
140C2×C4⋊C4C2414yes—
141C2×C22⋊C4—212no—
142C2×C22⋊C4C2×D416yes—
143C2×C42—212no—
144C2×C42—22no—
145C2×C42—26no—
146C2×C42—212no—
147C2×C42—212no—
148C2×C42—812no—
149C2×C42C22×C4112yes—
150C2×C42C2×D413yes—
151C2×C42C2×D413yes—
152C2×C42C2×Q8112yes—
153C2×C42C2413yes—
154C2×C42C2411yescharacteristic
155#107C22×C4124yes—
156#107C22×C4148yes—
157#107C22×C418yes—
158#106C22×C4148yes—
159#97C22×C4124yes—
160C4×C8C22×C414yes—
161C4×C8C22×C4112yes—
162C4×C8C22×C4112yes—
163C4×C8C22×C4112yes—
164#95—26no—
165#95—212no—
166#95—22no—
167#95—212no—
168C24C22×D413yes—
169#41C2×C42116yes—
170#41#117116yes—
171#41C4×Q8132yes—
172C2×Q8C2×C4⋊C414yes—
173C2×Q8C4×D418yes—
174C2×D4C2×C4⋊C4112yes—
175C2×D4C4×D4124yes—
176C22×C4—26no—
177C22×C4—26no—
178C22×C4—24no—
179C22×C4—212no—
180C22×C4—248no—
181C22×C4—46no—
182C22×C4—812no—
183C22×C4—812no—
184C22×C4C2×C4⋊C4112yes—
185C22×C4C2×C4⋊C4112yes—
186C22×C4C23×C416yes—
187C22×C4C23×C4112yes—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D413yes—
190C22×C4C22×D413yes—
191C22×C4C22×D413yes—
192C22×C4C22×D411yescharacteristic
193C22×C4C22×Q816yes—
194C22×C4C2×#41112yes—
195C22×C4C2511yescharacteristic
196M16—224no—
197M16C2×C42148yescomplemented
198C2×C8—224no—
199C2×C8—22no—
200C2×C8—26no—
201C2×C8C2×C42148yescomplemented
202C2×C8C23×C412yes—
203C2×C8C23×C416yes—
204C2×C8C23×C412yes—
205C2×C8C23×C416yes—
206C2×C8C23×C4112yes—
207C2×C8C23×C414yes—
208C4⋊C4—248no—
209C4⋊C4—212no—
210C4⋊C4—24no—
211C4⋊C4C2×C4⋊C4112yes—
212C4⋊C4C2×C4⋊C4112yes—
213C4⋊C4C23×C418yes—
214C4⋊C4C23×C4124yes—
215C22⋊C4—248no—
216C22⋊C4C2×C4⋊C4124yes—
217C42—248no—
218C42—412no—
219C42—44no—
220C42—812no—
221C42—812no—
222C42C2×C4⋊C4112yes—
223C42C2×C4⋊C4112yes—
224C42C23×C412yes—
225C42C23×C416yes—
226C42C22×Q812yes—
227C42C22×Q816yes—
228C23—26no—
229C23—23no—
230C23—23no—
231C23C22×C4⋊C416yes—
232C23C2×C4×D4112yes—
233C23C23×D411yescharacteristic
234Q8C4×C4⋊C4116yes—
235D4C4×C4⋊C4148yes—
236C2×C4—26no—
237C2×C4—22no—
238C2×C4—23no—
239C2×C4—23no—
240C2×C4—224no—
241C2×C4—28no—
242C2×C4—224no—
243C2×C4—412no—
244C2×C4—44no—
245C2×C4—412no—
246C2×C4—412no—
247C2×C4—44no—
248C2×C4—848no—
249C2×C4C22×C42112yes—
250C2×C4C22×C4⋊C412yes—
251C2×C4C22×C4⋊C416yes—
252C2×C4C22×C4⋊C416yes—
253C2×C4C22×C4⋊C412yes—
254C2×C4C22×C4⋊C416yes—
255C2×C4C2×#117112yes—
256C2×C4C2×C4×D4112yes—
257C2×C4C2×C4×Q8124yes—
258C2×C4#57612yes—
259C2×C4#57812yes—
260C2×C4C24×C411yescharacteristic
261C2×C4C24×C412yes—
262C2×C4C23×Q811yescharacteristic
263C2×C4C22×#4112yes—
264C8—22no—
265C8—26no—
266C8C22×C4214yes—
267C8C22×C42112yes—
268C22—212no—
269C22—212no—
270C22—26no—
271C22—26no—
272C22C2×C4×C4⋊C4112yes—
273C22C23×C4⋊C411yescharacteristic
274C22C22×C4×D412yes—
275C22C2×#57611yescharacteristic
276C22C2×#57811yescharacteristic
277C22#358412yes—
278C4—26no—
279C4—22no—
280C4—26no—
281C4—424no—
282C4—48no—
283C4C2×C4×C4⋊C4112yes—
284C4#291314yesdirect product factor
285C4C23×C4211yescharacteristic
286C4C22×#11711yescharacteristic
287C4C22×C4×Q812yes—
288C2—212no—
289C2C22×C4×C4⋊C411yescharacteristic
290C2C2×#291311yescharacteristic
291C2#6010811yescharacteristic
292C2C4×#57612yes—
293C2C4×#57812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC4C23×C42
Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC22×#41C23
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20455C2semidirect
5. #20453C2semidirect
6. C4×#428C2semidirect
7. C4×#427C2semidirect
8. C2×C4×#107C2semidirect
14. #2913C4direct (N × H)
18. C4×#131C4semidirect
66. #428C2×C4semidirect
67. #427C2×C4semidirect
68. C2×#107C2×C4semidirect
73. #405C2×C4semidirect
74. #405C2×C4semidirect
75. C4×M16C2×C4semidirect
77. C2×C4×C8C2×C4semidirect
111. #131C42semidirect
197. M16C2×C42semidirect
201. C2×C8C2×C42semidirect
284. C4#2913direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29132
C2×C4×#1312
C4×#5202
#204552
#204532
C4×#4282
C4×#4272
C2×C4×#1072
#203482