Finite Group Databasebeta

← #13163467

Subgroups — #13163467

Order 1728.

2044 conjugacy classes of subgroups; only the 351 normal subgroups are shown.

normal subgroups only

This group has 2044 conjugacy classes of subgroups — too many to draw. Only its 351 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#11724685C211yesmaximal, characteristic, complemented
2#11723990C211yesmaximal, characteristic, complemented
3#11723949C211yesmaximal, characteristic, complemented
4#11723916C211yesmaximal, characteristic, complemented
5C33×#104C211yesmaximal, characteristic, complemented
6#11722194C211yesmaximal, characteristic, complemented
7#11722175C211yesmaximal, characteristic, complemented
8#10592526—313nomaximal
9C33×#41C2211yescharacteristic
10Q8×C3⋊(C3⋊D3)—21no—
11#89078—21no—
12#89078—22no—
13#89076—21no—
14#89076C2211yescharacteristic
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×#6308—21no—
17C2×#6308C2211yescharacteristic
18C33×M16C2211yescharacteristic
19C33×C42C2211yescharacteristic
20#88862—21no—
21#88860—21no—
22#88859—22no—
23#88855C2211yescharacteristic
24#88852—21no—
25#88851—21no—
26#64367—313no—
27#64158—313no—
28#64117—313no—
29#64084—313no—
30C32×#104D3113yescomplemented
31#63651—313no—
32#63632—313no—
33C22×C3⋊(C3⋊D3)—21no—
34C33×Q8D411yescharacteristic
35C33×D4—21no—
36C33×D4D411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40#6312—22no—
41#6312—21no—
42C2×C3⋊(C3⋊Dic3)—22no—
43C2×C3⋊(C3⋊Dic3)D412yes—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
47C4×C3⋊(C3⋊D3)—22no—
48C4×C3⋊(C3⋊D3)—21no—
49#6308—22no—
50#6308—21no—
51#6308—21no—
52#6308D411yescharacteristic
53C33×C8—21no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55#4827—913no—
56C32×#41C2×D3113yes—
57Q8×C3⋊D3—613no—
58#3830—626no—
59#3830—613no—
60#3828—313no—
61#3828—613no—
62C2×C4⋊(C3⋊D3)—313no—
63C2×#634—313no—
64C2×#634—613no—
65C32×M16C2×D3113yes—
66C32×C42C2×D3113yes—
67#3755—613no—
68#3753—613no—
69#3752—626no—
70#3748—313no—
71#3745—613no—
72#3744—613no—
73C22×C33—21no—
74C22×C33C2×D411yescharacteristic
75C2×C3⋊(C3⋊D3)—21no—
76C2×C3⋊(C3⋊D3)—42no—
77C2×C3⋊(C3⋊D3)—41no—
78C33×C4—21no—
79C33×C4—41no—
80C33×C4C2×D411yescharacteristic
81C33×C4C2×D411yescharacteristic
82C3⋊(C3⋊Dic3)—22no—
83C3⋊(C3⋊Dic3)—21no—
84#1010—913no—
85#949—913no—
86#908—913no—
87#875—913no—
88C3×#104C3⋊D3113yescomplemented
89#824—913no—
90#805—913no—
91C22×C3⋊D3—613no—
92C32×Q8D12113yes—
93C32×D4—213no—
94C32×D4D12113yes—
95C2×C32×C4—213no—
96C2×C32×C4—213no—
97C2×C32×C4C22×D3113yes—
98#638—626no—
99#638—613no—
100C2×C3⋊Dic3—326no—
101C2×C3⋊Dic3—626no—
102C4⋊(C3⋊D3)—313no—
103C4⋊(C3⋊D3)—613no—
104C4⋊(C3⋊D3)—613no—
105C4×C3⋊D3—626no—
106C4×C3⋊D3—613no—
107#634—313no—
108#634—626no—
109#634—613no—
110#634—613no—
111C32×C8—213no—
112C3⋊(C3⋊C8)—613no—
113#454—271noSylow
114C2×C33—21no—
115C2×C33—41no—
116C2×C33#12011yescharacteristic
117C3⋊(C3⋊D3)—41no—
118C3⋊(C3⋊D3)—81no—
119C3×#41C2×C3⋊D3113yes—
120D3×Q8—1813no—
121#237—1826no—
122#237—1813no—
123#235—913no—
124#235—1813no—
125C2×D12—913no—
126C2×Dic6—913no—
127C2×Dic6—1813no—
128C3×M16C2×C3⋊D3113yes—
129C3×C42C2×C3⋊D3113yes—
130#215—1813no—
131#213—1813no—
132(C2×D3)⋊C4—1826no—
133#208—913no—
134D24—1813no—
135#204—1813no—
136C22×C32—213no—
137C22×C32D3×D4113yes—
138C2×C3⋊D3—613no—
139C2×C3⋊D3—1226no—
140C2×C3⋊D3—1213no—
141C32×C4—213no—
142C32×C4—413no—
143C32×C4C2×D12113yes—
144C32×C4D3×D4113yes—
145C3⋊Dic3—613no—
146C3⋊Dic3—626no—
1472−1+4—271no—
148AGL(1,ℤ/8)—271no—
149AGL(1,ℤ/8)—271no—
150#124—271no—
151#104—271no—
152#104C3⋊(C3⋊D3)11yescharacteristic, complemented
153#101—271no—
154C33#45411yescharacteristic, complemented, Sylow
155C22×D3—1813no—
156C3×Q8C4⋊(C3⋊D3)113yes—
157C3×D4—213no—
158C3×D4C4⋊(C3⋊D3)113yes—
159C2×C3×C4—213no—
160C2×C3×C4—213no—
161C2×C3×C4C22×C3⋊D3113yes—
162#67—1826no—
163#67—1813no—
164C2×Dic3—926no—
165C2×Dic3—1826no—
166D12—913no—
167D12—1813no—
168D12—1813no—
169C4×D3—1826no—
170C4×D3—1813no—
171Dic6—913no—
172Dic6—1826no—
173Dic6—1813no—
174Dic6—1813no—
175C3×C8—213no—
176C3⋊C8—1813no—
177C2×C32—213no—
178C2×C32—413no—
179C2×C32#882113yes—
180C3⋊D3—1213no—
181C3⋊D3—2413no—
182#41—271no—
183#41—541no—
184#41—542no—
185#41—541no—
186#41C2×C3⋊(C3⋊D3)11yescharacteristic
187C2×Q8—271no—
188C2×Q8—541no—
189C2×Q8—541no—
190C2×D4—271no—
191SD16—541no—
192SD16—541no—
193D8—541no—
194D8—541no—
195M16—271no—
196M16C2×C3⋊(C3⋊D3)11yescharacteristic
197C22⋊C4—542no—
198C42C2×C3⋊(C3⋊D3)11yescharacteristic
199C22×C3—213no—
200C22×C3D4×C3⋊D3113yes—
201C2×D3—1813no—
202C2×D3—3626no—
203C2×D3—3613no—
204C3×C4—213no—
205C3×C4—413no—
206C3×C4C2×C4⋊(C3⋊D3)113yes—
207C3×C4D4×C3⋊D3113yes—
208Dic3—1813no—
209Dic3—1826no—
210C32#4827113yescomplemented
211C23—541no—
212Q8—271no—
213Q8—542no—
214Q8—541no—
215Q8—541no—
216Q8C4⋊(C3⋊(C3⋊D3))11yescharacteristic
217D4—21no—
218D4—271no—
219D4—542no—
220D4—541no—
221D4—541no—
222D4—541no—
223D4C4⋊(C3⋊(C3⋊D3))11yescharacteristic
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—272no—
227C2×C4—542no—
228C2×C4—542no—
229C2×C4—541no—
230C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
231C8—21no—
232C8—541no—
233C2×C3—213no—
234C2×C3—413no—
235C2×C3#64091113yes—
236D3—3613no—
237D3—7213no—
238C22—21no—
239C22—541no—
240C22—1082no—
241C22—1081no—
242C22D4×C3⋊(C3⋊D3)11yescharacteristic
243C4—21no—
244C4—41no—
245C4—542no—
246C4—541no—
247C4C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
248C4D4×C3⋊(C3⋊D3)11yescharacteristic
249C3#10592526113yescomplemented
250C2—21no—
251C2—41no—
252C2—1081no—
253C2—2161no—
254C2#1172392311yescharacteristic
subgroupG/H
centerC2#11723923
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#104C2
socleC2×C33#120
hypercenter#104C3⋊(C3⋊D3)
hypocenterC33#454
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
1. #11724685C2semidirect
2. #11723990C2semidirect
3. #11723949C2semidirect
4. #11723916C2semidirect
5. C33×#104C2semidirect
6. #11722194C2semidirect
7. #11722175C2semidirect
30. C32×#104D3semidirect
88. C3×#104C3⋊D3semidirect
152. #104C3⋊(C3⋊D3)semidirect
154. C33#454semidirect
210. C32#4827semidirect
249. C3#10592526semidirect

Maximal subgroups

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

subgroupindex
#117246852
#117239902
#117239492
#117239162
C33×#1042
#117221942
#117221752
#105925263