Finite Group Databasebeta

← C22×#1808 #6370805

Subgroups — C22×#1808 #6370805

Order 512.

6354 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C25×C8C211yesmaximal, characteristic
2C23×#107C212yesmaximal, complemented
3C22×#380C212yesmaximal
4C2×#1808C2124yesmaximal, direct product factor
5C22×#340C212yesmaximal, complemented
6C25×C4C2211yescharacteristic
7C24×C8—216no—
8C24×C8C4116yescomplemented
9C24×C8C2211yescharacteristic
10C24×C8C22112yes—
11C24×C8C2211yescharacteristic
12C23×C4⋊C4C2214yes—
13C23×C22⋊C4—22no—
14C22×#107—216no—
15C22×#107C22124yescomplemented
16C2×#380—212no—
17C2×#380C22124yes—
18C22×#95—22no—
19#1808C22164yesdirect product factor
20C2×#340C22124yescomplemented
21C26D411yescharacteristic
22C24×C4—28no—
23C24×C4—28no—
24C24×C4—24no—
25C24×C4C2×C418yes—
26C24×C4D416yes—
27C24×C4D411yescharacteristic
28C24×C4Q818yes—
29C24×C4C2316yes—
30C24×C4C2311yescharacteristic
31C23×C8—28no—
32C23×C8—2128no—
33C23×C8—296no—
34C23×C8C2×C418yes—
35C23×C8C2×C418yes—
36C23×C8C2×C4196yescomplemented
37C23×C8D4116yes—
38C23×C8C2316yes—
39C23×C8C23116yes—
40C23×C8C2316yes—
41C22×C4⋊C4—216no—
42C22×C4⋊C4—216no—
43C22×C4⋊C4—212no—
44C22×C4⋊C4C23124yes—
45C22×C22⋊C4—224no—
46C22×C22⋊C4—416no—
47C2×#107—296no—
48C2×#107C23132yescomplemented
49#380—248no—
50#380C23132yes—
51C2×#95—224no—
52#340C23132yescomplemented
53C25—28no—
54C25—28no—
55C25C4⋊C418yes—
56C25D818yes—
57C25Q1618yes—
58C25C2×D416yes—
59C25C2×D411yescharacteristic
60C23×C4—28no—
61C23×C4—232no—
62C23×C4—232no—
63C23×C4—248no—
64C23×C4—264no—
65C23×C4—224no—
66C23×C4—224no—
67C23×C4—224no—
68C23×C4—416no—
69C23×C4—416no—
70C23×C4—416no—
71C23×C4C4⋊C418yes—
72C23×C4C4⋊C4148yes—
73C23×C4C22×C4124yes—
74C23×C4C22×C414yes—
75C23×C4C2×D416yes—
76C23×C4C2×D4112yes—
77C23×C4C2×D418yes—
78C23×C4C2×D413yes—
79C23×C4C2×Q8124yes—
80C23×C4C2×Q814yes—
81C23×C4#4118yes—
82C23×C4C2414yes—
83C23×C4C2413yes—
84C22×C8—232no—
85C22×C8—224no—
86C22×C8—2384no—
87C22×C8—264no—
88C22×C8C22⋊C4164yescomplemented
89C22×C8C22×C4124yes—
90C22×C8C22×C4124yes—
91C22×C8C22×C4164yescomplemented
92C22×C8C2×D4148yes—
93C22×C8C2414yes—
94C22×C8C2414yes—
95C2×C4⋊C4—248no—
96C2×C4⋊C4—248no—
97C2×C4⋊C4—224no—
98C2×C4⋊C4—448no—
99C2×C4⋊C4C24116yes—
100C2×C22⋊C4—232no—
101C2×C22⋊C4—496no—
102#107—264no—
103#95—232no—
104C24—224no—
105C24—224no—
106C24—232no—
107C24—232no—
108C24—264no—
109C24—264no—
110C24#107164yescomplemented
111C24C2×C4⋊C4124yes—
112C24C2×C4⋊C414yes—
113C24#12118yes—
114C24C2×D8124yes—
115C24C2×D814yes—
116C24C2×Q16124yes—
117C24C2×Q1614yes—
118C24#13518yes—
119C24C22×D414yes—
120C24C22×D413yes—
121C22×C4—224no—
122C22×C4—232no—
123C22×C4—248no—
124C22×C4—248no—
125C22×C4—248no—
126C22×C4—296no—
127C22×C4—28no—
128C22×C4—28no—
129C22×C4—2192no—
130C22×C4—216no—
131C22×C4—448no—
132C22×C4—448no—
133C22×C4—448no—
134C22×C4—864no—
135C22×C4C2×C22⋊C4116yes—
136C22×C4C2×C4⋊C4124yes—
137C22×C4C2×C4⋊C4148yes—
138C22×C4C2×C4⋊C4112yes—
139C22×C4#117116yes—
140C22×C4#121124yes—
141C22×C4C22⋊Q8132yes—
142C22×C4C23×C418yes—
143C22×C4C23×C416yes—
144C22×C4C22×D414yes—
145C22×C4C22×D4112yes—
146C22×C4C22×D413yes—
147C22×C4C22×Q818yes—
148C22×C4C22×Q816yes—
149C22×C4C2×#41112yes—
150C22×C4C2511yescharacteristic
151C2×C8—248no—
152C2×C8—28no—
153C2×C8—2128no—
154C2×C8C2×C22⋊C4196yescomplemented
155C2×C8C23×C418yes—
156C2×C8C23×C418yes—
157C2×C8C22×D4116yes—
158C4⋊C4—216no—
159C4⋊C4—216no—
160C4⋊C4—448no—
161C22⋊C4—464no—
162C23—2256no—
163C23—248no—
164C23—28no—
165C23—248no—
166C23—296no—
167C23—296no—
168C23—28no—
169C23#380116yes—
170C23C2×#107116yes—
171C23C2×#107196yescomplemented
172C23#427116yes—
173C23#466132yes—
174C23#467132yes—
175C23C22×C4⋊C418yes—
176C23C22×C4⋊C416yes—
177C23C2×#121112yes—
178C23C22×D818yes—
179C23C22×D816yes—
180C23C22×Q1618yes—
181C23C22×Q1616yes—
182C23C2×#135112yes—
183C23C23×D411yescharacteristic
184C2×C4—28no—
185C2×C4—248no—
186C2×C4—28no—
187C2×C4—28no—
188C2×C4—216no—
189C2×C4—296no—
190C2×C4—416no—
191C2×C4—416no—
192C2×C4—416no—
193C2×C4—896no—
194C2×C4#380148yes—
195C2×C4C22×C22⋊C4112yes—
196C2×C4C22×C4⋊C418yes—
197C2×C4C22×C4⋊C416yes—
198C2×C4C2×#117112yes—
199C2×C4C2×#121112yes—
200C2×C4C2×C22⋊Q8124yes—
201C2×C4C24×C411yescharacteristic
202C2×C4C23×D412yes—
203C2×C4C23×Q811yescharacteristic
204C2×C4C22×#4112yes—
205C8—28no—
206C8C22×C22⋊C4116yescomplemented
207C22—216no—
208C22—216no—
209C22—2192no—
210C22—28no—
211C22—28no—
212C22#1808164yesdirect product factor
213C22C2×#380112yes—
214C22C22×#107112yes—
215C22C22×#107116yescomplemented
216C22C2×#427112yes—
217C22C2×#466124yes—
218C22C2×#467124yes—
219C22C23×C4⋊C411yescharacteristic
220C22C22×#12112yes—
221C22C23×D811yescharacteristic
222C22C23×Q1611yescharacteristic
223C22C22×#13512yes—
224C4—28no—
225C4—816no—
226C4C2×#380112yes—
227C4C23×C22⋊C411yescharacteristic
228C4C22×#11711yescharacteristic
229C4C22×C22⋊Q812yes—
230C2—216no—
231C2C2×#1808124yesdirect product factor
232C2C22×#38011yescharacteristic
233C2C23×#10711yescharacteristic
234C2C22×#42711yescharacteristic
235C2C22×#46612yes—
236C2C22×#46712yes—
subgroupG/H
center, socleC25C2×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC26D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C25 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C23×#107C2semidirect
4. C2×#1808C2direct (N × H)
5. C22×#340C2semidirect
8. C24×C8C4semidirect
15. C22×#107C22semidirect
19. #1808C22direct (N × H)
20. C2×#340C22semidirect
36. C23×C8C2×C4semidirect
48. C2×#107C23semidirect
52. #340C23semidirect
88. C22×C8C22⋊C4semidirect
91. C22×C8C22×C4semidirect
110. C24#107semidirect
154. C2×C8C2×C22⋊C4semidirect
171. C23C2×#107semidirect
206. C8C22×C22⋊C4semidirect
212. C22#1808direct (N × H)
215. C22C22×#107semidirect
231. C2C2×#1808direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C82
C23×#1072
C22×#3802
C2×#18082
C22×#3402