Finite Group Databasebeta

← #13163816

Subgroups — #13163816

Order 1728.

2268 conjugacy classes of subgroups; only the 407 normal subgroups are shown.

normal subgroups only

This group has 2268 conjugacy classes of subgroups — too many to draw. Only its 407 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
1C2×#89080C211yesmaximal, characteristic, complemented
2C2×C33×SD16C211yesmaximal, characteristic, complemented
3#11723979C211yesmaximal, characteristic, complemented
4C2×#88860C211yesmaximal, characteristic, complemented
5#11722205C211yesmaximal, characteristic, complemented
6#11722191C211yesmaximal, characteristic, complemented
7#11722190C211yesmaximal, characteristic, complemented
8#10592871—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10C2×C33×D4C2211yescharacteristic
11C2×#6312—21no—
12#89080—22no—
13#89080—22no—
14C2×C4⋊(C3⋊(C3⋊D3))—21no—
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)—21no—
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
18C33×SD16—22no—
19C2×C33×C8C2211yescharacteristic
20#88864—21no—
21#88860—22no—
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
23C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
24C2×#3832—313no—
25C2×C32×SD16D3113yescomplemented
26#64147—313no—
27C2×#3753—313no—
28#63662—313no—
29#63648—313no—
30#63647—313no—
31C23×C33—21no—
32C22×C3⋊(C3⋊D3)—21no—
33C22×C3⋊(C3⋊D3)D411yescharacteristic
34C33×Q8—21no—
35C33×Q8D412yescomplemented
36C33×D4—21no—
37C33×D4—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40#6312—42no—
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—24no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))D412yescomplemented
46C4×C3⋊(C3⋊D3)—24no—
47C4×C3⋊(C3⋊D3)—21no—
48C4×C3⋊(C3⋊D3)—21no—
49C33×C8—21no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51#5172—913no—
52C2×C32×Q8C2×D3113yes—
53C2×C32×D4C2×D3113yes—
54C2×#638—613no—
55#3832—626no—
56#3832—626no—
57C2×C4⋊(C3⋊D3)—313no—
58C2×C4⋊(C3⋊D3)—613no—
59C2×C4×C3⋊D3—313no—
60C2×C4×C3⋊D3—613no—
61C32×SD16—226no—
62C2×C32×C8C2×D3113yes—
63#3757—613no—
64#3753—626no—
65C4⋊(C3⋊Dic3)—313no—
66C2×C3⋊(C3⋊C8)—313no—
67C22×C33—41no—
68C22×C33—41no—
69C22×C33—41no—
70C22×C33C2×D411yescharacteristic
71C2×C3⋊(C3⋊D3)—21no—
72C2×C3⋊(C3⋊D3)—22no—
73C2×C3⋊(C3⋊D3)—21no—
74C2×C3⋊(C3⋊D3)—21no—
75C2×C3⋊(C3⋊D3)—41no—
76C2×C3⋊(C3⋊D3)—41no—
77C33×C4—22no—
78C33×C4C2×D411yescharacteristic
79C33×C4C2×D411yescharacteristic
80C3⋊(C3⋊Dic3)—21no—
81C3⋊(C3⋊Dic3)—41no—
82C2×#239—913no—
83C2×C3×SD16C3⋊D3113yescomplemented
84#938—913no—
85C2×#213—913no—
86#835—913no—
87#821—913no—
88(C2×D3)⋊C8—913no—
89C23×C32—213no—
90C22×C3⋊D3—313no—
91C22×C3⋊D3—613no—
92C32×Q8—213no—
93C32×Q8#67126yescomplemented
94C32×D4—213no—
95C32×D4—213no—
96C2×C32×C4—213no—
97C2×C32×C4C22×D3113yes—
98#638—1226no—
99C2×C3⋊Dic3—313no—
100C2×C3⋊Dic3—613no—
101C4⋊(C3⋊D3)—326no—
102C4⋊(C3⋊D3)—652no—
103C4⋊(C3⋊D3)—613no—
104C4×C3⋊D3—652no—
105C4×C3⋊D3—613no—
106C4×C3⋊D3—613no—
107C32×C8—213no—
108C3⋊(C3⋊C8)—613no—
109#449—271noSylow
110C2×C33—81no—
111C2×C33#12011yescharacteristic
112C2×C33#13511yescharacteristic
113C2×C33AGL(1,ℤ/8)11yescharacteristic
114C3⋊(C3⋊D3)—41no—
115C3⋊(C3⋊D3)—42no—
116C2×C3×Q8C2×C3⋊D3113yes—
117C2×C3×D4C2×C3⋊D3113yes—
118C2×#67—1813no—
119#239—1826no—
120#239—1826no—
121C2×D12—913no—
122C2×D12—1813no—
123C2×C4×D3—913no—
124C2×C4×D3—1813no—
125C3×SD16—226no—
126C2×C3×C8C2×C3⋊D3113yes—
127#217—1813no—
128#213—1826no—
129C4⋊Dic3—913no—
130C2×C3⋊C8—913no—
131C22×C32—413no—
132C22×C32—413no—
133C22×C32—413no—
134C22×C32D3×D4113yes—
135C2×C3⋊D3—613no—
136C2×C3⋊D3—613no—
137C2×C3⋊D3—626no—
138C2×C3⋊D3—613no—
139C2×C3⋊D3—1213no—
140C2×C3⋊D3—1213no—
141C32×C4—226no—
142C32×C4D3×D4113yes—
143C32×C4C2×#67113yes—
144C3⋊Dic3—613no—
145C3⋊Dic3—1213no—
146C2×#41—271no—
147C2×SD16C3⋊(C3⋊D3)11yescharacteristic, complemented
148C2×D8—271no—
149#121—271no—
150#103—271no—
151#102—271no—
152C22⋊C8—271no—
153C33#44911yescharacteristic, complemented, Sylow
154C23×C3—213no—
155C22×D3—913no—
156C22×D3—1813no—
157C3×Q8—213no—
158C3×Q8#638126yescomplemented
159C3×D4—213no—
160C3×D4—213no—
161C2×C3×C4—213no—
162C2×C3×C4C22×C3⋊D3113yes—
163#67—3626no—
164C2×Dic3—913no—
165C2×Dic3—1813no—
166D12—926no—
167D12—1852no—
168D12—1813no—
169C4×D3—1852no—
170C4×D3—1813no—
171C4×D3—1813no—
172C3×C8—213no—
173C3⋊C8—1813no—
174C2×C32—813no—
175C2×C32#914113yes—
176C2×C32#916113yes—
177C2×C32#937113yes—
178C3⋊D3—1213no—
179C3⋊D3—1226no—
180#41—542no—
181#41—542no—
182C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
183C2×D4—271no—
184C2×D4—541no—
185C2×D4—541no—
186C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
187C22×C4—271no—
188C22×C4—541no—
189SD16—22no—
190D8—542no—
191C2×C8—271no—
192C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
193C4⋊C4—271no—
194C22⋊C4—541no—
195C22×C3—413no—
196C22×C3—413no—
197C22×C3—413no—
198C22×C3D4×C3⋊D3113yes—
199C2×D3—1813no—
200C2×D3—1813no—
201C2×D3—1826no—
202C2×D3—1813no—
203C2×D3—3613no—
204C2×D3—3613no—
205C3×C4—226no—
206C3×C4D4×C3⋊D3113yes—
207C3×C4C2×#638113yes—
208Dic3—1813no—
209Dic3—3613no—
210C32#5172113yescomplemented
211C23—21no—
212C23—271no—
213C23—541no—
214Q8—21no—
215Q8#631212yescomplemented
216D4—21no—
217D4—21no—
218D4—272no—
219D4—544no—
220D4—541no—
221D4—1082no—
222C2×C4—21no—
223C2×C4—271no—
224C2×C4—544no—
225C2×C4—541no—
226C2×C4—541no—
227C2×C4—541no—
228C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
229C8—21no—
230C8—541no—
231C2×C3—813no—
232C2×C3#64123113yes—
233C2×C3#64125113yes—
234C2×C3#64146113yes—
235D3—3613no—
236D3—3626no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—541no—
241C22—541no—
242C22—542no—
243C22—541no—
244C22—1081no—
245C22—1081no—
246C22D4×C3⋊(C3⋊D3)11yescharacteristic
247C4—22no—
248C4—541no—
249C4—1081no—
250C4D4×C3⋊(C3⋊D3)11yescharacteristic
251C4C2×#631211yescharacteristic
252C3#10592871113yescomplemented
253C2—81no—
254C2—1081no—
255C2—1082no—
256C2#1172395511yescharacteristic
257C2#1172395711yescharacteristic
258C2#1172397811yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC2×C33×SD16C2
socleC22×C33C2×D4
hypercenterC2×SD16C3⋊(C3⋊D3)
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#89080C2semidirect
2. C2×C33×SD16C2semidirect
3. #11723979C2semidirect
4. C2×#88860C2semidirect
5. #11722205C2semidirect
6. #11722191C2semidirect
7. #11722190C2semidirect
25. C2×C32×SD16D3semidirect
35. C33×Q8D4semidirect
45. C4⋊(C3⋊(C3⋊D3))D4semidirect
83. C2×C3×SD16C3⋊D3semidirect
93. C32×Q8#67semidirect
147. C2×SD16C3⋊(C3⋊D3)semidirect
153. C33#449semidirect
158. C3×Q8#638semidirect
210. C32#5172semidirect
215. Q8#6312semidirect
252. C3#10592871semidirect

Maximal subgroups

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

subgroupindex
C2×#890802
C2×C33×SD162
#117239792
C2×#888602
#117222052
#117221912
#117221902
#105928713