Finite Group Databasebeta

← #72245799

Subgroups — #72245799

Order 2016.

326 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 62 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
1#11759630C211yesmaximal, characteristic, complemented
2C3⋊D3×D28C211yesmaximal, characteristic
3#11759200C212yesmaximal, complemented
4C7×#3710C211yesmaximal, characteristic, complemented
5C3×#66565C212yesmaximal, complemented
6D3×#1132—32nomaximal
7C2×D7×C3⋊D3—21no—
8C4⋊(C3⋊D21)C2211yescharacteristic
9C4×C7×C3⋊D3C2211yescharacteristic
10C3×Dic42C2212yescomplemented
11C32×D28C2211yescharacteristic
12#92696—21no—
13#92693—21no—
14#92677—21no—
15C3×C7×C3⋊C8C2212yescomplemented
16D3×D28—32no—
17D3×D28—61no—
18D3×Dic14—32no—
19#66623—32no—
20D3×C7×C8—32no—
21C3×#1132—32no—
22#66566—32no—
23#66565D312yescomplemented
24#63930—71nomaximal, Hall
25C2×C3⋊D21—21no—
26C2×C7×C3⋊D3D411yescharacteristic
27C2×C32×D7—21no—
28D7×C3⋊D3—42no—
29C32×C4×C7C2311yescharacteristic
30C7×C3⋊Dic3D411yescharacteristic
31C3×Dic21—22no—
32C2×#1132—91noHall
33C2×D3×D7—62no—
34C2×D3×D7—61no—
35C2×D3×D7—61no—
36D84—32no—
37D84—61no—
38Dic42C2×D312yescomplemented
39C4×D3×C7—32no—
40C4×D3×C7—61no—
41C3×D28—21no—
42C3×D28C2×D312yes—
43C3×Dic14—32no—
44#4248—62no—
45#4246—62no—
46#4246—61no—
47#4246—61no—
48D3×Dic7—62no—
49C3×C7×C8—32no—
50C7×C3⋊C8C2×D312yescomplemented
51D4×C3⋊D3—71no—
52#3797—71no—
53C3×#214—72no—
54#3717—72no—
55#3710D711yescharacteristic, complemented
56C2×C32×C7C2×D411yescharacteristic
57C3⋊D21—41no—
58C7×C3⋊D3SD1612yescomplemented
59C32×D7—41no—
60C2×D28—91no—
61C2×Dic14—91no—
62C2×C7×C8—91no—
63#1132—32no—
64#1132—92no—
65D3×SD16—212no—
66C2×D21—61no—
67C2×D21—61no—
68C2×D21—62no—
69C2×D3×C7—32no—
70C2×D3×C7—61no—
71C2×C3×D7—21no—
72C2×C3×D7—21no—
73C2×C3×D7—22no—
74D3×D7—124no—
75D3×D7—122no—
76D3×D7—122no—
77C3×C4×C7—21no—
78C3×C4×C7C22×D312yes—
79Dic21—22no—
80C3×Dic7—62no—
81C7×Dic3—32no—
82C7×Dic3—61no—
83C22×C3⋊D3—141no—
84C32×D4—71no—
85#638—141no—
86C4⋊(C3⋊D3)—71no—
87C4×C3⋊D3C2×D711yescharacteristic
88C3×Dic6—72no—
89#627—141no—
90#624—141no—
91C3×C3⋊C8C2×D712yescomplemented
92C32×C7C2×SD1611yescharacteristic, complemented, Hall
93C22×D7—181no—
94C2×C4×C7—91no—
95C2×Dic7—181no—
96D28—91no—
97D28—181no—
98D28D3211yescharacteristic
99Dic14—32no—
100Dic14—181no—
101C7×C8—32no—
102D3×Q8—212no—
103D3×D4—212no—
104D3×D4—421no—
105C3×SD16—212no—
106#215—212no—
107#214—72no—
108#204—212no—
109D3×C8—32no—
110C2×C3×C7—21no—
111C2×C3×C7D3×D412yes—
112D21—121no—
113D21—121no—
114D21—122no—
115C3×D7—41no—
116C3×D7—41no—
117C3×D7—42no—
118D3×C7—34no—
119D3×C7—62no—
120C22×C32—141no—
121C2×C3⋊D3—141no—
122C2×C3⋊D3—282no—
123C2×C3⋊D3D2811yescharacteristic
124C32×C4C22×D711yescharacteristic
125C3⋊Dic3D2811yescharacteristic
126C3×Dic3—142no—
127C2×SD16—631noSylow
128C22×C7—91no—
129C2×D7—21no—
130C2×D7—181no—
131C2×D7—362no—
132C4×C7—91no—
133C4×C7C2×D3211yescharacteristic
134Dic7—62no—
135C22×D3—422no—
136C22×D3—421no—
137C22×D3—421no—
138C3×Q8—212no—
139C3×D4—72no—
140C3×D4—141no—
141#67—422no—
142#67—421no—
143#67—421no—
144D12—212no—
145D12—421no—
146C4×D3—32no—
147C4×D3—61no—
148C4×D3—422no—
149Dic6—72no—
150Dic6—422no—
151C3×C8—32no—
152C3⋊C8D3×D712yescomplemented
153C3×C7—21no—
154C3×C7D3×SD1612yescomplemented
155C2×C32—281no—
156C2×C32C2×D2811yescharacteristic
157C3⋊D3—281no—
158C3⋊D3#113212yescomplemented
159C2×Q8—631no—
160C2×D4—631no—
161SD16—212no—
162SD16—632no—
163C2×C8—91no—
164C2×C7—92no—
165C2×C7#381111yescharacteristic
166D7—41no—
167D7—361no—
168C22×C3—141no—
169C22×C3—141no—
170C22×C3—142no—
171C2×D3—32no—
172C2×D3—61no—
173C2×D3—421no—
174C2×D3—421no—
175C2×D3—422no—
176C2×D3—844no—
177C2×D3—842no—
178C2×D3—842no—
179C3×C4—21no—
180C3×C4—422no—
181C3×C4C2×D3×D712yes—
182Dic3—32no—
183Dic3—61no—
184Dic3—142no—
185C32C2×#113211yescharacteristic, complemented, Sylow
186C23—1261no—
187Q8—212no—
188Q8—1261no—
189D4—71no—
190D4—631no—
191D4—1261no—
192C2×C4—91no—
193C2×C4—1261no—
194C8—32no—
195C7#6393011yescharacteristic, complemented, Sylow
196C2×C3—21no—
197C2×C3—281no—
198C2×C3—281no—
199C2×C3—282no—
200C2×C3D3×D2812yes—
201D3—34no—
202D3—62no—
203D3—841no—
204D3—841no—
205D3—842no—
206C22—91no—
207C22—141no—
208C22—1261no—
209C22—2522no—
210C4—91no—
211C4—422no—
212C4D32×D711yescharacteristic
213C3—21no—
214C3D3×#113212yescomplemented
215C2—92no—
216C2—281no—
217C2—2521no—
218C2#1175942811yescharacteristic
subgroupG/H
centerC2#11759428
commutator, Fitting, cosocleC32×C4×C7C23
Frattini, hypercenterC4D32×D7
socleC2×C32×C7C2×D4
prime radicalC3⋊D3×D28C2
hypocenterC32×C7C2×SD16
LCSG ▷ C32×C4×C7 ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #11759630C2semidirect
3. #11759200C2semidirect
4. C7×#3710C2semidirect
5. C3×#66565C2semidirect
10. C3×Dic42C22semidirect
15. C3×C7×C3⋊C8C22semidirect
23. #66565D3semidirect
38. Dic42C2×D3semidirect
50. C7×C3⋊C8C2×D3semidirect
55. #3710D7semidirect
58. C7×C3⋊D3SD16semidirect
91. C3×C3⋊C8C2×D7semidirect
92. C32×C7C2×SD16semidirect
152. C3⋊C8D3×D7semidirect
154. C3×C7D3×SD16semidirect
158. C3⋊D3#1132semidirect
185. C32C2×#1132semidirect
195. C7#63930semidirect
214. C3D3×#1132semidirect

Maximal subgroups

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

subgroupindex
#117596302
C3⋊D3×D282
#117592002
C7×#37102
C3×#665652
D3×#11323
#639307