Finite Group Databasebeta

← C22×#4874 #11664784

Subgroups — C22×#4874 #11664784

Order 768.

2398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C212yesmaximal
2C22×C3×#107C211yesmaximal, characteristic
3C22×#818C211yesmaximal, characteristic
4C22×C8×Dic3C211yesmaximal, characteristic
5C22×#807C212yesmaximal
6C2×#4874C2124yesmaximal, direct product factor
7C22×#500—31nomaximal, Sylow
8C23×Dic6—22no—
9C23×C3×C8C2211yescharacteristic
10C22×C3×C4⋊C4C2212yes—
11C22×C4⋊Dic3C2212yes—
12C22×Dic3⋊C4—22no—
13C22×C4×Dic3C2211yescharacteristic
14C23×C3⋊C8C2211yescharacteristic
15C2×#888—212no—
16C2×#888C22124yes—
17C2×C3×#107C22112yes—
18C2×#818C22112yes—
19C2×C8×Dic3C22112yes—
20C2×#807C22124yes—
21#4874C22164yesdirect product factor
22C22×C4⋊Q8—32no—
23C2×#500—324no—
24C22×#107—31no—
25C22×#107—32no—
26C22×#107D311yescharacteristic
27C22×C4×C8—31no—
28C23×C3×C4—22no—
29C23×C3×C4C2311yescharacteristic
30C23×Dic3—22no—
31C23×Dic3D411yescharacteristic
32C23×Dic3D411yescharacteristic
33C22×Dic6—224no—
34C22×Dic6—416no—
35C22×C3×C8Q818yescomplemented
36C22×C3×C8C2316yes—
37C2×C3×C4⋊C4—26no—
38C2×C3×C4⋊C4C23112yes—
39C2×C4⋊Dic3—26no—
40C2×C4⋊Dic3C23112yes—
41C2×Dic3⋊C4—224no—
42C2×C4×Dic3D416yes—
43C2×C4×Dic3C2316yes—
44C22×C3⋊C8Q818yescomplemented
45C22×C3⋊C8C2316yes—
46#888—248no—
47#888C23132yes—
48C3×#107C23116yes—
49#818C23116yes—
50C8×Dic3C23116yes—
51#807C23132yes—
52C23×Q8—62no—
53C23×C8—31no—
54C23×C8C2×D311yescharacteristic
55C2×C4⋊Q8—324no—
56C2×C4⋊Q8—612no—
57C22×C4⋊C4—32no—
58C22×C4⋊C4—62no—
59C22×C4⋊C4C2×D312yes—
60C22×C42—31no—
61#500—364no—
62C2×#107—312no—
63C2×#107—324no—
64C2×#107C2×D3112yes—
65C2×C4×C8—312no—
66C24×C3C2×D411yescharacteristic
67C22×C3×C4—212no—
68C22×C3×C4—48no—
69C22×C3×C4C2×D413yes—
70C22×C3×C4C2×Q814yes—
71C22×C3×C4C2×Q814yes—
72C22×C3×C4C2413yes—
73C22×Dic3—212no—
74C22×Dic3—48no—
75C22×Dic3D818yes—
76C22×Dic3Q1618yes—
77C22×Dic3C2×D416yes—
78C22×Dic3C2×D416yes—
79C2×Dic6—232no—
80C2×Dic6—496no—
81C2×C3×C8C2×Q8124yescomplemented
82C2×C3×C8C2414yes—
83C3×C4⋊C4—212no—
84C3×C4⋊C4C2418yes—
85C4⋊Dic3—212no—
86C4⋊Dic3C2418yes—
87Dic3⋊C4—232no—
88C4×Dic3C2×D4112yes—
89C4×Dic3C2414yes—
90C2×C3⋊C8C2×Q8124yescomplemented
91C2×C3⋊C8C2414yes—
92C22×Q8—624no—
93C22×Q8—1216no—
94C23×C4—22no—
95C23×C4—31no—
96C23×C4—31no—
97C23×C4—62no—
98C23×C4C22×D311yescharacteristic
99C22×C8—38no—
100C22×C8—36no—
101C22×C8Dic618yescomplemented
102C22×C8C22×D316yes—
103C4⋊Q8—332no—
104C4⋊Q8—648no—
105C2×C4⋊C4—26no—
106C2×C4⋊C4—312no—
107C2×C4⋊C4—624no—
108C2×C4⋊C4—66no—
109C2×C4⋊C4C22×D3112yes—
110C2×C42—36no—
111C2×C42—36no—
112#107—316no—
113#107—332no—
114#107C22×D3116yes—
115C4×C8—316no—
116C23×C3C4⋊Q814yes—
117C23×C3C2×D814yes—
118C23×C3C2×Q1614yes—
119C23×C3C22×D413yes—
120C2×C3×C4—28no—
121C2×C3×C4—424no—
122C2×C3×C4C4⋊Q8112yes—
123C2×C3×C4C22×D413yes—
124C2×C3×C4C22×Q816yes—
125C2×C3×C4C22×Q816yes—
126C2×C3×C4C2511yescharacteristic
127C2×Dic3—28no—
128C2×Dic3—424no—
129C2×Dic3C2×D8124yes—
130C2×Dic3C2×Q16124yes—
131C2×Dic3C22×D414yes—
132C2×Dic3C22×D414yes—
133Dic6—464no—
134C3×C8C22×Q818yescomplemented
135C3⋊C8C22×Q818yescomplemented
136C24D3×D411yescharacteristic
137C2×Q8—632no—
138C2×Q8—1296no—
139C22×C4—212no—
140C22×C4—38no—
141C22×C4—38no—
142C22×C4—36no—
143C22×C4—36no—
144C22×C4—48no—
145C22×C4—612no—
146C22×C4—128no—
147C22×C4C2×Dic614yes—
148C22×C4D3×D413yes—
149C22×C4D3×Q814yes—
150C22×C4C23×D313yes—
151C2×C8—324no—
152C2×C8—34no—
153C2×C8C2×Dic6124yescomplemented
154C2×C8C23×D314yes—
155C4⋊C4—212no—
156C4⋊C4—38no—
157C4⋊C4—632no—
158C4⋊C4—612no—
159C4⋊C4C23×D318yes—
160C42—312no—
161C42—34no—
162C22×C3#500116yescomplemented
163C22×C3C2×C4⋊Q816yes—
164C22×C3C22×D816yes—
165C22×C3C22×Q1616yes—
166C22×C3C23×D411yescharacteristic
167C3×C4—48no—
168C3×C4C2×C4⋊Q816yes—
169C3×C4C23×Q811yescharacteristic
170C3×C4C23×Q811yescharacteristic
171Dic3—48no—
172Dic3C22×D818yes—
173Dic3C22×Q1618yes—
174C23#88814yes—
175C23D3×D814yes—
176C23D3×Q1614yes—
177C23C2×D3×D413yes—
178Q8—1264no—
179C2×C4—28no—
180C2×C4—34no—
181C2×C4—34no—
182C2×C4—324no—
183C2×C4—324no—
184C2×C4—424no—
185C2×C4—68no—
186C2×C4—1224no—
187C2×C4#888112yes—
188C2×C4C22×Dic616yes—
189C2×C4C2×D3×D413yes—
190C2×C4C2×D3×Q816yes—
191C2×C4C24×D311yescharacteristic
192C8—38no—
193C8C22×Dic618yescomplemented
194C2×C3C2×#500112yescomplemented
195C2×C3C22×C4⋊Q811yescharacteristic
196C2×C3C23×D811yescharacteristic
197C2×C3C23×Q1611yescharacteristic
198C22#4874116yesdirect product factor
199C22C2×#88816yes—
200C22C2×D3×D816yes—
201C22C2×D3×Q1616yes—
202C22C22×D3×D411yescharacteristic
203C4—38no—
204C4—38no—
205C4—48no—
206C4—128no—
207C4C2×#88816yes—
208C4C23×Dic611yescharacteristic
209C4C22×D3×Q811yescharacteristic
210C3C22×#50011yescharacteristic, complemented, Sylow
211C2C2×#4874112yesdirect product factor
212C2C22×#88811yescharacteristic
213C2C22×D3×D811yescharacteristic
214C2C22×D3×Q1611yescharacteristic
subgroupG/H
centerC24D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC22×C3×#107C2
socle, prime radicalC24×C3C2×D4
hypercenterC22×#107D3
hypocenterC3C22×#500
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
6. C2×#4874C2direct (N × H)
21. #4874C22direct (N × H)
35. C22×C3×C8Q8semidirect
44. C22×C3⋊C8Q8semidirect
81. C2×C3×C8C2×Q8semidirect
90. C2×C3⋊C8C2×Q8semidirect
101. C22×C8Dic6semidirect
134. C3×C8C22×Q8semidirect
135. C3⋊C8C22×Q8semidirect
153. C2×C8C2×Dic6semidirect
162. C22×C3#500semidirect
193. C8C22×Dic6semidirect
194. C2×C3C2×#500semidirect
198. C22#4874direct (N × H)
210. C3C22×#500semidirect
211. C2C2×#4874direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8882
C22×C3×#1072
C22×#8182
C22×C8×Dic32
C22×#8072
C2×#48742
C22×#5003