Finite Group Databasebeta

← #13163845

Subgroups — #13163845

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
2C33×#137C211yesmaximal, characteristic, complemented
3#11723985C211yesmaximal, characteristic, complemented
4#11723983C211yesmaximal, characteristic, complemented
5#11722207C211yesmaximal, characteristic, complemented
6#11722195C211yesmaximal, characteristic, complemented
7#11722194C211yesmaximal, characteristic, complemented
8#10592904—313nomaximal
9C33×#41C2211yescharacteristic
10C2×C33×Q8C2211yescharacteristic
11Q8×C3⋊(C3⋊D3)—21no—
12#89078—21no—
13#89078—22no—
14#89076—21no—
15#89076C2211yescharacteristic
16C2×#6308—21no—
17C2×#6308C2211yescharacteristic
18C33×Q16—21no—
19C33×SD16—21no—
20C33×M16C2211yescharacteristic
21#88863—21no—
22#88862—21no—
23#88857—22no—
24C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
25#88855C2211yescharacteristic
26#64367—313no—
27C32×#137D3113yescomplemented
28#64153—313no—
29#64151—313no—
30#63664—313no—
31#63652—313no—
32#63651—313no—
33C33×Q8—21no—
34C33×Q8—21no—
35C33×Q8D411yescharacteristic
36C33×D4—21no—
37C33×D4D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41#6312—22no—
42#6312—21no—
43C2×C3⋊(C3⋊Dic3)—22no—
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊Dic3)D412yes—
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#5205—913no—
56C32×#41C2×D3113yes—
57C2×C32×Q8C2×D3113yes—
58Q8×C3⋊D3—613no—
59#3830—626no—
60#3830—613no—
61#3828—313no—
62#3828—613no—
63C2×#634—313no—
64C2×#634—613no—
65C32×Q16—213no—
66C32×SD16—213no—
67C32×M16C2×D3113yes—
68#3756—613no—
69#3755—613no—
70#3750—626no—
71C4×C3⋊Dic3—313no—
72#3748—313no—
73C22×C33—21no—
74C22×C33C2×D411yescharacteristic
75C2×C3⋊(C3⋊D3)—21no—
76C33×C4—21no—
77C33×C4—41no—
78C33×C4C2×D411yescharacteristic
79C33×C4C2×D411yescharacteristic
80C3⋊(C3⋊Dic3)—22no—
81C3⋊(C3⋊Dic3)—22no—
82C3⋊(C3⋊Dic3)—21no—
83#1010—913no—
84C3×#137C3⋊D3113yescomplemented
85#944—913no—
86#942—913no—
87#837—913no—
88#825—913no—
89#824—913no—
90C32×Q8—213no—
91C32×Q8—213no—
92C32×Q8#67113yes—
93C32×D4—213no—
94C32×D4#67113yes—
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—
102C2×C3⋊Dic3—613no—
103C4⋊(C3⋊D3)—313no—
104C4×C3⋊D3—626no—
105C4×C3⋊D3—613no—
106#634—313no—
107#634—626no—
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—
118C2×C3×Q8C2×C3⋊D3113yes—
119D3×Q8—1813no—
120#237—1826no—
121#237—1813no—
122#235—913no—
123#235—1813no—
124C2×Dic6—913no—
125C2×Dic6—1813no—
126C3×Q16—213no—
127C3×SD16—213no—
128C3×M16C2×C3⋊D3113yes—
129#216—1813no—
130#215—1813no—
131Dic3⋊C4—1826no—
132C4×Dic3—913no—
133#208—913no—
134C22×C32—213no—
135C22×C32D3×D4113yes—
136C2×C3⋊D3—613no—
137C32×C4—213no—
138C32×C4—413no—
139C32×C4D3×D4113yes—
140C32×C4C2×#67113yes—
141C3⋊Dic3—626no—
142C3⋊Dic3—613no—
143C3⋊Dic3—626no—
1442−1+4—271no—
145#137—271no—
146#137C3⋊(C3⋊D3)11yescharacteristic, complemented
147C4⋊Q8—271no—
148#104—271no—
149#104—271no—
150#101—271no—
151C33#45611yescharacteristic, complemented, Sylow
152C3×Q8—213no—
153C3×Q8—213no—
154C3×Q8#638113yes—
155C3×D4—213no—
156C3×D4#638113yes—
157C2×C3×C4—213no—
158C2×C3×C4—213no—
159C2×C3×C4C22×C3⋊D3113yes—
160#67—1826no—
161#67—1813no—
162C2×Dic3—926no—
163C2×Dic3—1826no—
164C2×Dic3—1813no—
165D12—913no—
166C4×D3—1826no—
167C4×D3—1813no—
168Dic6—913no—
169Dic6—1826no—
170Dic6—1813no—
171Dic6—1813no—
172C3×C8—213no—
173C3⋊C8—1813no—
174C2×C32—213no—
175C2×C32—413no—
176C2×C32#937113yes—
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—541no—
185C2×Q8—541no—
186C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
187Q16—21no—
188Q16—541no—
189SD16—21no—
190SD16—541no—
191M16—271no—
192M16C2×C3⋊(C3⋊D3)11yescharacteristic
193C4⋊C4—542no—
194C42—271no—
195C22×C3—213no—
196C22×C3D4×C3⋊D3113yes—
197C2×D3—1813no—
198C3×C4—213no—
199C3×C4—413no—
200C3×C4D4×C3⋊D3113yes—
201C3×C4C2×#638113yes—
202Dic3—1826no—
203Dic3—1813no—
204Dic3—1826no—
205C32#5205113yescomplemented
206Q8—21no—
207Q8—21no—
208Q8—271no—
209Q8—542no—
210Q8—541no—
211Q8—541no—
212Q8#631211yescharacteristic
213D4—21no—
214D4—271no—
215D4—542no—
216D4—541no—
217D4#631211yescharacteristic
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#64146113yes—
231D3—3613no—
232C22—21no—
233C22—541no—
234C22D4×C3⋊(C3⋊D3)11yescharacteristic
235C4—21no—
236C4—41no—
237C4—542no—
238C4—542no—
239C4—541no—
240C4D4×C3⋊(C3⋊D3)11yescharacteristic
241C4C2×#631211yescharacteristic
242C3#10592904113yescomplemented
243C2—21no—
244C2—41no—
245C2—1081no—
246C2#1172397811yescharacteristic
subgroupG/H
centerC2#11723978
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#137C2
socleC2×C33#120
prime radical#11724685C2
hypercenter#137C3⋊(C3⋊D3)
hypocenterC33#456
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. C33×#137C2semidirect
3. #11723985C2semidirect
4. #11723983C2semidirect
5. #11722207C2semidirect
6. #11722195C2semidirect
7. #11722194C2semidirect
27. C32×#137D3semidirect
84. C3×#137C3⋊D3semidirect
146. #137C3⋊(C3⋊D3)semidirect
151. C33#456semidirect
205. C32#5205semidirect
242. C3#10592904semidirect

Maximal subgroups

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

subgroupindex
#117246852
C33×#1372
#117239852
#117239832
#117222072
#117221952
#117221942
#105929043