Finite Group Databasebeta

← #13163456

Subgroups — #13163456

Order 1728.

2128 conjugacy classes of subgroups; only the 407 normal subgroups are shown.

normal subgroups only

This group has 2128 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
2C2×#88863C211yesmaximal, characteristic, complemented
3C2×#88851C211yesmaximal, characteristic, complemented
4#11723938C211yesmaximal, characteristic, complemented
5C33×#103C211yesmaximal, characteristic, complemented
6#11722190C211yesmaximal, characteristic, complemented
7#11722179C211yesmaximal, characteristic, complemented
8#10592507—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10#89080—22no—
11#89080—22no—
12C2×C4⋊(C3⋊(C3⋊D3))—21no—
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
14C2×C4×C3⋊(C3⋊D3)—21no—
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×#6308C2211yescharacteristic
17C2×C33×C8C2211yescharacteristic
18C33×C4⋊C4C2211yescharacteristic
19#88863—22no—
20#88859—21no—
21C4⋊(C3⋊(C3⋊Dic3))—21no—
22C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23#88851—22no—
24C2×#3832—313no—
25C2×#3756—313no—
26C2×#3744—313no—
27#64106—313no—
28C32×#103D3113yescomplemented
29#63647—313no—
30#63636—313no—
31C22×C3⋊(C3⋊D3)—21no—
32C22×C3⋊(C3⋊D3)D411yescharacteristic
33C33×Q8—21no—
34C33×Q8D412yes—
35C2×C33×C4—21no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C4⋊(C3⋊(C3⋊D3))—24no—
41C4⋊(C3⋊(C3⋊D3))—21no—
42C4⋊(C3⋊(C3⋊D3))D412yes—
43C4×C3⋊(C3⋊D3)—24no—
44C4×C3⋊(C3⋊D3)—21no—
45C4×C3⋊(C3⋊D3)—21no—
46#6308—21no—
47#6308—21no—
48C33×C8—21no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50#4808—913no—
51C2×C32×Q8C2×D3113yes—
52#3832—626no—
53#3832—626no—
54C2×C4⋊(C3⋊D3)—313no—
55C2×C4⋊(C3⋊D3)—613no—
56C2×C4×C3⋊D3—313no—
57C2×C4×C3⋊D3—613no—
58C2×#634—313no—
59C2×C32×C8C2×D3113yes—
60C32×C4⋊C4C2×D3113yes—
61#3756—626no—
62#3752—613no—
63C4⋊(C3⋊Dic3)—613no—
64C2×C3⋊(C3⋊C8)—313no—
65#3744—626no—
66C22×C33C2×D411yescharacteristic
67C2×C3⋊(C3⋊D3)—21no—
68C2×C3⋊(C3⋊D3)—22no—
69C2×C3⋊(C3⋊D3)—21no—
70C2×C3⋊(C3⋊D3)—21no—
71C2×C3⋊(C3⋊D3)—41no—
72C2×C3⋊(C3⋊D3)—41no—
73C33×C4—22no—
74C33×C4—41no—
75C33×C4C2×D411yescharacteristic
76C33×C4C2×D411yescharacteristic
77C3⋊(C3⋊Dic3)—21no—
78C3⋊(C3⋊Dic3)—41no—
79C2×#239—913no—
80C2×#216—913no—
81C2×#204—913no—
82#897—913no—
83C3×#103C3⋊D3113yescomplemented
84(C2×D3)⋊C8—913no—
85#809—913no—
86C22×C3⋊D3—313no—
87C22×C3⋊D3—613no—
88C32×Q8—213no—
89C32×Q8D12126yes—
90C2×C32×C4—213no—
91C2×C32×C4—213no—
92C2×C32×C4C22×D3113yes—
93C2×C3⋊Dic3—313no—
94C2×C3⋊Dic3—613no—
95C4⋊(C3⋊D3)—326no—
96C4⋊(C3⋊D3)—652no—
97C4⋊(C3⋊D3)—613no—
98C4×C3⋊D3—652no—
99C4×C3⋊D3—613no—
100C4×C3⋊D3—613no—
101#634—613no—
102#634—613no—
103C32×C8—213no—
104C3⋊(C3⋊C8)—613no—
105#452—271noSylow
106C2×C33#12011yescharacteristic
107C2×C33#13511yescharacteristic
108C2×C33#13711yescharacteristic
109C3⋊(C3⋊D3)—41no—
110C3⋊(C3⋊D3)—42no—
111C2×C3×Q8C2×C3⋊D3113yes—
112#239—1826no—
113#239—1826no—
114C2×D12—913no—
115C2×D12—1813no—
116C2×C4×D3—913no—
117C2×C4×D3—1813no—
118C2×Dic6—913no—
119C2×C3×C8C2×C3⋊D3113yes—
120C3×C4⋊C4C2×C3⋊D3113yes—
121#216—1826no—
122(C2×D3)⋊C4—1813no—
123C4⋊Dic3—1813no—
124C2×C3⋊C8—913no—
125#204—1826no—
126C22×C32D3×D4113yes—
127C2×C3⋊D3—626no—
128C2×C3⋊D3—613no—
129C2×C3⋊D3—613no—
130C2×C3⋊D3—613no—
131C2×C3⋊D3—1213no—
132C2×C3⋊D3—1213no—
133C32×C4—226no—
134C32×C4—413no—
135C32×C4C2×D12113yes—
136C32×C4D3×D4113yes—
137C3⋊Dic3—613no—
138C3⋊Dic3—1213no—
139C2×#41—271no—
140C2×Q16—271no—
141C2×SD16—271no—
142C22⋊Q8—271no—
143#103C3⋊(C3⋊D3)11yescharacteristic, complemented
144#102—271no—
145C22⋊C8—271no—
146C33#45211yescharacteristic, complemented, Sylow
147C22×D3—913no—
148C22×D3—1813no—
149C3×Q8—213no—
150C3×Q8C4⋊(C3⋊D3)126yes—
151C2×C3×C4—213no—
152C2×C3×C4—213no—
153C2×C3×C4C22×C3⋊D3113yes—
154C2×Dic3—913no—
155C2×Dic3—1813no—
156D12—926no—
157D12—1852no—
158D12—1813no—
159C4×D3—1852no—
160C4×D3—1813no—
161C4×D3—1813no—
162Dic6—1813no—
163Dic6—1813no—
164C3×C8—213no—
165C3⋊C8—1813no—
166C2×C32#882113yes—
167C2×C32#916113yes—
168C2×C32#918113yes—
169C3⋊D3—1213no—
170C3⋊D3—1226no—
171#41—542no—
172#41—542no—
173C2×Q8—271no—
174C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
175C2×D4—271no—
176C2×D4—541no—
177C22×C4—271no—
178C22×C4—541no—
179Q16—542no—
180SD16—542no—
181C2×C8—271no—
182C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
183C4⋊C4—541no—
184C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
185C22⋊C4—541no—
186C22×C3D4×C3⋊D3113yes—
187C2×D3—1826no—
188C2×D3—1813no—
189C2×D3—1813no—
190C2×D3—1813no—
191C2×D3—3613no—
192C2×D3—3613no—
193C3×C4—226no—
194C3×C4—413no—
195C3×C4C2×C4⋊(C3⋊D3)113yes—
196C3×C4D4×C3⋊D3113yes—
197Dic3—1813no—
198Dic3—3613no—
199C32#4808113yescomplemented
200C23—271no—
201C23—541no—
202Q8—21no—
203Q8—541no—
204Q8—541no—
205Q8C4⋊(C3⋊(C3⋊D3))12yes—
206D4—272no—
207D4—544no—
208D4—541no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—271no—
212C2×C4—544no—
213C2×C4—541no—
214C2×C4—541no—
215C2×C4—541no—
216C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
217C8—21no—
218C8—541no—
219C2×C3#64091113yes—
220C2×C3#64125113yes—
221C2×C3#64127113yes—
222D3—3613no—
223D3—3626no—
224C22—542no—
225C22—541no—
226C22—541no—
227C22—541no—
228C22—1081no—
229C22—1081no—
230C22D4×C3⋊(C3⋊D3)11yescharacteristic
231C4—22no—
232C4—41no—
233C4—541no—
234C4—1081no—
235C4C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
236C4D4×C3⋊(C3⋊D3)11yescharacteristic
237C3#10592507113yescomplemented
238C2—1081no—
239C2—1082no—
240C2#1172392311yescharacteristic
241C2#1172395711yescharacteristic
242C2#1172395911yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC2×#89080C2
hypercenter#103C3⋊(C3⋊D3)
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×#88863C2semidirect
3. C2×#88851C2semidirect
4. #11723938C2semidirect
5. C33×#103C2semidirect
6. #11722190C2semidirect
7. #11722179C2semidirect
28. C32×#103D3semidirect
83. C3×#103C3⋊D3semidirect
143. #103C3⋊(C3⋊D3)semidirect
146. C33#452semidirect
199. C32#4808semidirect
237. C3#10592507semidirect

Maximal subgroups

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

subgroupindex
C2×#890802
C2×#888632
C2×#888512
#117239382
C33×#1032
#117221902
#117221792
#105925073