Finite Group Databasebeta

← #13163466

Subgroups — #13163466

Order 1728.

1988 conjugacy classes of subgroups; only the 351 normal subgroups are shown.

normal subgroups only

This group has 1988 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
2#11723992C211yesmaximal, characteristic, complemented
3#11723950C211yesmaximal, characteristic, complemented
4#11723910C211yesmaximal, characteristic, complemented
5C33×#104C211yesmaximal, characteristic, complemented
6#11722194C211yesmaximal, characteristic, complemented
7#11722175C211yesmaximal, characteristic, complemented
8#10592525—313nomaximal
9C33×#41C2211yescharacteristic
10Q8×C3⋊(C3⋊D3)—21no—
11#89078—21no—
12#89078—22no—
13#89076—21no—
14#89076C2211yescharacteristic
15C2×#6308—21no—
16C2×#6308C2211yescharacteristic
17C2×#6308C2211yescharacteristic
18C33×M16C2211yescharacteristic
19C33×C42C2211yescharacteristic
20#88863—21no—
21#88861—21no—
22C4⋊(C3⋊(C3⋊Dic3))—22no—
23#88855C2211yescharacteristic
24#88853—21no—
25#88851—21no—
26#64367—313no—
27#64160—313no—
28#64118—313no—
29#64078—313no—
30C32×#104D3113yescomplemented
31#63651—313no—
32#63632—313no—
33C33×Q8D411yescharacteristic
34C33×D4—21no—
35C33×D4D411yescharacteristic
36C2×C33×C4—21no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39#6312—22no—
40#6312—21no—
41C2×C3⋊(C3⋊Dic3)—22no—
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)D412yes—
44C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
45C4×C3⋊(C3⋊D3)—22no—
46C4×C3⋊(C3⋊D3)—21no—
47#6308—22no—
48#6308—21no—
49#6308—21no—
50#6308—21no—
51#6308—21no—
52#6308D411yescharacteristic
53C33×C8—21no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55#4826—913no—
56C32×#41C2×D3113yes—
57Q8×C3⋊D3—613no—
58#3830—626no—
59#3830—613no—
60#3828—313no—
61#3828—613no—
62C2×#634—313no—
63C2×#634—313no—
64C2×#634—613no—
65C32×M16C2×D3113yes—
66C32×C42C2×D3113yes—
67#3756—613no—
68#3754—613no—
69C4⋊(C3⋊Dic3)—626no—
70#3748—313no—
71#3746—613no—
72#3744—613no—
73C22×C33—21no—
74C22×C33C2×D411yescharacteristic
75C2×C3⋊(C3⋊D3)—21no—
76C33×C4—22no—
77C33×C4—21no—
78C33×C4C2×D411yescharacteristic
79C33×C4C2×D411yescharacteristic
80C3⋊(C3⋊Dic3)—22no—
81C3⋊(C3⋊Dic3)—21no—
82C3⋊(C3⋊Dic3)—41no—
83#1010—913no—
84#951—913no—
85#909—913no—
86#869—913no—
87C3×#104C3⋊D3113yescomplemented
88#824—913no—
89#805—913no—
90C32×Q8D12113yes—
91C32×D4—213no—
92C32×D4D12113yes—
93C2×C32×C4—213no—
94C2×C32×C4—213no—
95C2×C32×C4C22×D3113yes—
96#638—626no—
97#638—613no—
98C2×C3⋊Dic3—326no—
99C2×C3⋊Dic3—626no—
100C2×C3⋊Dic3—613no—
101C4⋊(C3⋊D3)—313no—
102C4×C3⋊D3—626no—
103C4×C3⋊D3—613no—
104#634—313no—
105#634—626no—
106#634—613no—
107#634—613no—
108#634—613no—
109#634—613no—
110C32×C8—213no—
111C3⋊(C3⋊C8)—613no—
112#456—271noSylow
113C2×C33—21no—
114C2×C33—41no—
115C2×C33#12011yescharacteristic
116C3⋊(C3⋊D3)—41no—
117C3×#41C2×C3⋊D3113yes—
118D3×Q8—1813no—
119#237—1826no—
120#237—1813no—
121#235—913no—
122#235—1813no—
123C2×Dic6—913no—
124C2×Dic6—913no—
125C2×Dic6—1813no—
126C3×M16C2×C3⋊D3113yes—
127C3×C42C2×C3⋊D3113yes—
128#216—1813no—
129#214—1813no—
130C4⋊Dic3—1826no—
131#208—913no—
132Dic12—1813no—
133#204—1813no—
134C22×C32—213no—
135C22×C32D3×D4113yes—
136C2×C3⋊D3—613no—
137C32×C4—226no—
138C32×C4—213no—
139C32×C4C2×D12113yes—
140C32×C4D3×D4113yes—
141C3⋊Dic3—613no—
142C3⋊Dic3—626no—
143C3⋊Dic3—1213no—
1442−1+4—271no—
145#137—271no—
146#137—271no—
147C4⋊Q8—271no—
148#104—271no—
149#104C3⋊(C3⋊D3)11yescharacteristic, complemented
150#101—271no—
151C33#45611yescharacteristic, complemented, Sylow
152C3×Q8C4⋊(C3⋊D3)113yes—
153C3×D4—213no—
154C3×D4C4⋊(C3⋊D3)113yes—
155C2×C3×C4—213no—
156C2×C3×C4—213no—
157C2×C3×C4C22×C3⋊D3113yes—
158#67—1826no—
159#67—1813no—
160C2×Dic3—926no—
161C2×Dic3—1826no—
162C2×Dic3—1813no—
163D12—913no—
164C4×D3—1826no—
165C4×D3—1813no—
166Dic6—913no—
167Dic6—1826no—
168Dic6—1813no—
169Dic6—1813no—
170Dic6—1813no—
171Dic6—1813no—
172C3×C8—213no—
173C3⋊C8—1813no—
174C2×C32—213no—
175C2×C32—413no—
176C2×C32#882113yes—
177C3⋊D3—1213no—
178#41—271no—
179#41—541no—
180#41—542no—
181#41—541no—
182#41C2×C3⋊(C3⋊D3)11yescharacteristic
183C2×Q8—271no—
184C2×Q8—271no—
185C2×Q8—541no—
186C2×Q8—541no—
187Q16—541no—
188Q16—541no—
189SD16—541no—
190SD16—541no—
191M16—271no—
192M16C2×C3⋊(C3⋊D3)11yescharacteristic
193C4⋊C4—542no—
194C42C2×C3⋊(C3⋊D3)11yescharacteristic
195C22×C3—213no—
196C22×C3D4×C3⋊D3113yes—
197C2×D3—1813no—
198C3×C4—226no—
199C3×C4—213no—
200C3×C4C2×C4⋊(C3⋊D3)113yes—
201C3×C4D4×C3⋊D3113yes—
202Dic3—1813no—
203Dic3—1826no—
204Dic3—3613no—
205C32#4826113yescomplemented
206Q8—271no—
207Q8—542no—
208Q8—541no—
209Q8—541no—
210Q8—541no—
211Q8—541no—
212Q8C4⋊(C3⋊(C3⋊D3))11yescharacteristic
213D4—21no—
214D4—271no—
215D4—542no—
216D4—541no—
217D4C4⋊(C3⋊(C3⋊D3))11yescharacteristic
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—272no—
221C2×C4—542no—
222C2×C4—542no—
223C2×C4—541no—
224C2×C4—541no—
225C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
226C8—21no—
227C8—541no—
228C2×C3—213no—
229C2×C3—413no—
230C2×C3#64091113yes—
231D3—3613no—
232C22—21no—
233C22—541no—
234C22D4×C3⋊(C3⋊D3)11yescharacteristic
235C4—22no—
236C4—21no—
237C4—542no—
238C4—541no—
239C4—1081no—
240C4C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
241C4D4×C3⋊(C3⋊D3)11yescharacteristic
242C3#10592525113yescomplemented
243C2—21no—
244C2—41no—
245C2—1081no—
246C2#1172392311yescharacteristic
subgroupG/H
centerC2#11723923
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#104C2
socleC2×C33#120
prime radical#11724685C2
hypercenter#104C3⋊(C3⋊D3)
hypocenterC33#456
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
2. #11723992C2semidirect
3. #11723950C2semidirect
4. #11723910C2semidirect
5. C33×#104C2semidirect
6. #11722194C2semidirect
7. #11722175C2semidirect
30. C32×#104D3semidirect
87. C3×#104C3⋊D3semidirect
149. #104C3⋊(C3⋊D3)semidirect
151. C33#456semidirect
205. C32#4826semidirect
242. C3#10592525semidirect

Maximal subgroups

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

subgroupindex
#117246852
#117239922
#117239502
#117239102
C33×#1042
#117221942
#117221752
#105925253