Finite Group Databasebeta

← #13163420

Subgroups — #13163420

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×#89078C211yesmaximal, characteristic, complemented
2C2×#88860C211yesmaximal, characteristic, complemented
3C2×#88851C211yesmaximal, characteristic, complemented
4#11723936C211yesmaximal, characteristic, complemented
5C33×#102C211yesmaximal, characteristic, complemented
6#11722190C211yesmaximal, characteristic, complemented
7#11722180C211yesmaximal, characteristic, complemented
8#10592480—313nomaximal
9C2×C33×D4C2211yescharacteristic
10C2×#6312—21no—
11C22×C3⋊(C3⋊Dic3)—21no—
12#89078—22no—
13#89078—22no—
14C2×C4⋊(C3⋊(C3⋊D3))—21no—
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×#6308C2211yescharacteristic
18C2×C33×C8C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20#88860—22no—
21#88859—21no—
22C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23#88851—22no—
24C2×#3830—313no—
25C2×#3753—313no—
26C2×#3744—313no—
27#64104—313no—
28C32×#102D3113yescomplemented
29#63647—313no—
30#63637—313no—
31C23×C33—21no—
32C22×C3⋊(C3⋊D3)—21no—
33C22×C3⋊(C3⋊D3)D411yescharacteristic
34C33×D4—21no—
35C33×D4D412yescomplemented
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38#6312—24no—
39C2×C3⋊(C3⋊Dic3)—24no—
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—42no—
45C4×C3⋊(C3⋊D3)—21no—
46C4×C3⋊(C3⋊D3)—21no—
47#6308—21no—
48#6308D412yescomplemented
49C33×C8—21no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51#4781—913no—
52C2×C32×D4C2×D3113yes—
53C2×#638—613no—
54C22×C3⋊Dic3—613no—
55#3830—626no—
56#3830—626no—
57C2×C4⋊(C3⋊D3)—313no—
58C2×C4⋊(C3⋊D3)—613no—
59C2×C4×C3⋊D3—313no—
60C2×#634—313no—
61C2×C32×C8C2×D3113yes—
62C32×C4⋊C4C2×D3113yes—
63#3753—626no—
64#3752—613no—
65C2×C3⋊(C3⋊C8)—313no—
66#3744—626no—
67C22×C33—22no—
68C22×C33—41no—
69C22×C33—41no—
70C22×C33C2×D411yescharacteristic
71C2×C3⋊(C3⋊D3)—21no—
72C2×C3⋊(C3⋊D3)—21no—
73C2×C3⋊(C3⋊D3)—21no—
74C2×C3⋊(C3⋊D3)—41no—
75C2×C3⋊(C3⋊D3)—41no—
76C2×C3⋊(C3⋊D3)—41no—
77C33×C4—41no—
78C33×C4C2×D411yescharacteristic
79C33×C4C2×D411yescharacteristic
80C3⋊(C3⋊Dic3)—22no—
81C3⋊(C3⋊Dic3)—21no—
82C2×#237—913no—
83C2×#213—913no—
84C2×#204—913no—
85#895—913no—
86C3×#102C3⋊D3113yescomplemented
87(C2×D3)⋊C8—913no—
88#810—913no—
89C23×C32—213no—
90C22×C3⋊D3—313no—
91C22×C3⋊D3—613no—
92C32×D4—213no—
93C32×D4D12126yescomplemented
94C2×C32×C4—213no—
95C2×C32×C4C22×D3113yes—
96#638—652no—
97C2×C3⋊Dic3—313no—
98C2×C3⋊Dic3—652no—
99C2×C3⋊Dic3—613no—
100C4⋊(C3⋊D3)—613no—
101C4⋊(C3⋊D3)—613no—
102C4⋊(C3⋊D3)—1226no—
103C4×C3⋊D3—613no—
104C4×C3⋊D3—613no—
105#634—326no—
106#634—613no—
107C32×C8—213no—
108C3⋊(C3⋊C8)—613no—
109#449—271noSylow
110C2×C33—42no—
111C2×C33#12011yescharacteristic
112C2×C33#13511yescharacteristic
113C2×C33AGL(1,ℤ/8)11yescharacteristic
114C3⋊(C3⋊D3)—41no—
115C3⋊(C3⋊D3)—81no—
116C2×C3×D4C2×C3⋊D3113yes—
117C2×#67—1813no—
118C22×Dic3—1813no—
119#237—1826no—
120#237—1826no—
121C2×D12—913no—
122C2×D12—1813no—
123C2×C4×D3—913no—
124C2×Dic6—913no—
125C2×C3×C8C2×C3⋊D3113yes—
126C3×C4⋊C4C2×C3⋊D3113yes—
127#213—1826no—
128(C2×D3)⋊C4—1813no—
129C2×C3⋊C8—913no—
130#204—1826no—
131C22×C32—226no—
132C22×C32—413no—
133C22×C32—413no—
134C22×C32D3×D4113yes—
135C2×C3⋊D3—613no—
136C2×C3⋊D3—613no—
137C2×C3⋊D3—613no—
138C2×C3⋊D3—1213no—
139C2×C3⋊D3—1213no—
140C2×C3⋊D3—1213no—
141C32×C4—413no—
142C32×C4C2×D12113yes—
143C32×C4D3×D4113yes—
144C3⋊Dic3—626no—
145C3⋊Dic3—613no—
146C2×#41—271no—
147C2×SD16—271no—
148C2×D8—271no—
149#121—271no—
150#103—271no—
151#102C3⋊(C3⋊D3)11yescharacteristic, complemented
152C22⋊C8—271no—
153C33#44911yescharacteristic, complemented, Sylow
154C23×C3—213no—
155C22×D3—913no—
156C22×D3—1813no—
157C3×D4—213no—
158C3×D4C4⋊(C3⋊D3)126yescomplemented
159C2×C3×C4—213no—
160C2×C3×C4C22×C3⋊D3113yes—
161#67—1852no—
162C2×Dic3—913no—
163C2×Dic3—1852no—
164C2×Dic3—1813no—
165D12—1813no—
166D12—1813no—
167D12—3626no—
168C4×D3—1813no—
169C4×D3—1813no—
170Dic6—926no—
171Dic6—1813no—
172C3×C8—213no—
173C3⋊C8—1813no—
174C2×C32—426no—
175C2×C32#882113yes—
176C2×C32#911113yes—
177C2×C32#916113yes—
178C3⋊D3—1213no—
179C3⋊D3—2413no—
180#41—542no—
181#41—542no—
182C2×Q8—271no—
183C2×D4—271no—
184C2×D4—541no—
185C2×D4—541no—
186C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
187C22×C4—271no—
188C22×C4—541no—
189SD16—542no—
190D8—542no—
191C2×C8—271no—
192C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
193C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
194C22⋊C4—541no—
195C22×C3—226no—
196C22×C3—413no—
197C22×C3—413no—
198C22×C3D4×C3⋊D3113yes—
199C2×D3—1813no—
200C2×D3—1813no—
201C2×D3—1813no—
202C2×D3—3613no—
203C2×D3—3613no—
204C2×D3—3613no—
205C3×C4—413no—
206C3×C4C2×C4⋊(C3⋊D3)113yes—
207C3×C4D4×C3⋊D3113yes—
208Dic3—1826no—
209Dic3—1813no—
210C32#4781113yescomplemented
211C23—21no—
212C23—271no—
213C23—541no—
214Q8—272no—
215Q8—541no—
216D4—21no—
217D4—544no—
218D4—541no—
219D4—541no—
220D4—1082no—
221D4C4⋊(C3⋊(C3⋊D3))12yescomplemented
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—426no—
232C2×C3#64091113yes—
233C2×C3#64120113yes—
234C2×C3#64125113yes—
235D3—3613no—
236D3—7213no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—541no—
241C22—541no—
242C22—541no—
243C22—1081no—
244C22—1081no—
245C22—1081no—
246C22D4×C3⋊(C3⋊D3)11yescharacteristic
247C4—41no—
248C4—541no—
249C4—542no—
250C4C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
251C4D4×C3⋊(C3⋊D3)11yescharacteristic
252C3#10592480113yescomplemented
253C2—42no—
254C2—1081no—
255C2—2161no—
256C2#1172392311yescharacteristic
257C2#1172395211yescharacteristic
258C2#1172395711yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#102C2
socleC22×C33C2×D4
hypercenter#102C3⋊(C3⋊D3)
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
1. C2×#89078C2semidirect
2. C2×#88860C2semidirect
3. C2×#88851C2semidirect
4. #11723936C2semidirect
5. C33×#102C2semidirect
6. #11722190C2semidirect
7. #11722180C2semidirect
28. C32×#102D3semidirect
35. C33×D4D4semidirect
48. #6308D4semidirect
86. C3×#102C3⋊D3semidirect
93. C32×D4D12semidirect
151. #102C3⋊(C3⋊D3)semidirect
153. C33#449semidirect
158. C3×D4C4⋊(C3⋊D3)semidirect
210. C32#4781semidirect
221. D4C4⋊(C3⋊(C3⋊D3))semidirect
252. C3#10592480semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
C2×#888602
C2×#888512
#117239362
C33×#1022
#117221902
#117221802
#105924803