Finite Group Databasebeta

← #13163453

Subgroups — #13163453

Order 1728.

1540 conjugacy classes of subgroups; only the 351 normal subgroups are shown.

normal subgroups only

This group has 1540 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#11723987C211yesmaximal, characteristic
2C2×#88863C211yesmaximal, characteristic, complemented
3C2×#88850C211yesmaximal, characteristic
4#11723938C211yesmaximal, characteristic
5C33×#103C211yesmaximal, characteristic, complemented
6#11722186C211yesmaximal, characteristic, complemented
7#11722178C211yesmaximal, characteristic, complemented
8#10592515—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14#88863—22no—
15#88859—21no—
16#88859—21no—
17C4⋊(C3⋊(C3⋊Dic3))—21no—
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19#88857—21no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21#88850—22no—
22#64155—313no—
23C2×#3756—313no—
24C2×#3743—313no—
25#64106—313no—
26C32×#103D3113yescomplemented
27#63643—313no—
28#63635—313no—
29C22×C3⋊(C3⋊D3)D411yescharacteristic
30C33×Q8—21no—
31C33×Q8—21no—
32C2×C33×C4—21no—
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3⋊(C3⋊Dic3)—21no—
36C2×C3⋊(C3⋊Dic3)—21no—
37C2×C3⋊(C3⋊Dic3)D411yescharacteristic
38C4×C3⋊(C3⋊D3)—21no—
39C4×C3⋊(C3⋊D3)—21no—
40#6308—21no—
41#6308—21no—
42C33×C8—21no—
43C3⋊(C3⋊(C3⋊C8))D412yes—
44#4816—913no—
45C2×C32×Q8C2×D3113yes—
46C2×C4×C3⋊D3—313no—
47C2×#634—313no—
48C2×C32×C8C2×D3113yes—
49C32×C4⋊C4C2×D3113yes—
50#3756—626no—
51#3752—613no—
52#3752—613no—
53C4⋊(C3⋊Dic3)—313no—
54C4⋊(C3⋊Dic3)—613no—
55#3750—613no—
56C2×C3⋊(C3⋊C8)—313no—
57#3743—626no—
58C22×C33C2×D411yescharacteristic
59C2×C3⋊(C3⋊D3)—21no—
60C2×C3⋊(C3⋊D3)—21no—
61C2×C3⋊(C3⋊D3)—21no—
62C33×C4—41no—
63C33×C4—41no—
64C33×C4C2×D411yescharacteristic
65C33×C4#4111yescharacteristic
66C3⋊(C3⋊Dic3)—21no—
67C3⋊(C3⋊Dic3)—41no—
68C3⋊(C3⋊Dic3)—41no—
69#946—913no—
70C2×#216—913no—
71C2×ΓL(1,𝔽25)—913no—
72#897—913no—
73C3×#103C3⋊D3113yescomplemented
74#816—913no—
75#808—913no—
76C22×C3⋊D3—313no—
77C32×Q8—213no—
78C32×Q8—213no—
79C2×C32×C4—213no—
80C2×C32×C4—213no—
81C2×C32×C4C22×D3113yes—
82C2×C3⋊Dic3—313no—
83C2×C3⋊Dic3—613no—
84C2×C3⋊Dic3—613no—
85C4×C3⋊D3—613no—
86C4×C3⋊D3—613no—
87#634—613no—
88#634—613no—
89C32×C8—213no—
90C3⋊(C3⋊C8)—326no—
91#470—271noSylow
92C2×C33#12111yescharacteristic
93C2×C33#13711yescharacteristic
94C2×C33#13711yescharacteristic
95C3⋊(C3⋊D3)—41no—
96C2×C3×Q8C2×C3⋊D3113yes—
97C2×C4×D3—913no—
98C2×Dic6—913no—
99C2×C3×C8C2×C3⋊D3113yes—
100C3×C4⋊C4C2×C3⋊D3113yes—
101#216—1826no—
102(C2×D3)⋊C4—1813no—
103(C2×D3)⋊C4—1813no—
104C4⋊Dic3—913no—
105C4⋊Dic3—1813no—
106Dic3⋊C4—1813no—
107C2×C3⋊C8—913no—
108ΓL(1,𝔽25)—1826no—
109C22×C32D3×D4113yes—
110C2×C3⋊D3—613no—
111C2×C3⋊D3—613no—
112C2×C3⋊D3—613no—
113C32×C4—413no—
114C32×C4—413no—
115C32×C4#235113yes—
116C32×C4D3×D4113yes—
117C3⋊Dic3—613no—
118C3⋊Dic3—1213no—
119C3⋊Dic3—1213no—
120C2×Q16—271no—
121C2×M16—271no—
122C22⋊Q8—271no—
123C22⋊Q8—271no—
124#106—271no—
125#103—271no—
126#103C3⋊(C3⋊D3)11yescharacteristic, complemented
127C33#47011yescharacteristic, complemented, Sylow
128C22×D3—913no—
129C3×Q8—213no—
130C3×Q8—213no—
131C2×C3×C4—213no—
132C2×C3×C4—213no—
133C2×C3×C4C22×C3⋊D3113yes—
134C2×Dic3—913no—
135C2×Dic3—1813no—
136C2×Dic3—1813no—
137C4×D3—1813no—
138C4×D3—1813no—
139Dic6—1813no—
140Dic6—1813no—
141C3×C8—213no—
142C3⋊C8—926no—
143C2×C32#884113yes—
144C2×C32#915113yes—
145C2×C32#918113yes—
146C3⋊D3—1213no—
147C2×Q8—271no—
148C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
149C22×C4—271no—
150Q16—542no—
151M16—542no—
152C2×C8—271no—
153C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
154C4⋊C4—271no—
155C4⋊C4—541no—
156C4⋊C4—541no—
157C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
158C22⋊C4—541no—
159C22⋊C4—541no—
160C22×C3D4×C3⋊D3113yes—
161C2×D3—1813no—
162C2×D3—1813no—
163C2×D3—1813no—
164C3×C4—413no—
165C3×C4—413no—
166C3×C4#3828113yes—
167C3×C4D4×C3⋊D3113yes—
168Dic3—1813no—
169Dic3—3613no—
170Dic3—3613no—
171C32#4816113yescomplemented
172C23—271no—
173Q8—21no—
174Q8—21no—
175Q8—541no—
176Q8—541no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—271no—
180C2×C4—541no—
181C2×C4—541no—
182C2×C4—541no—
183C2×C4—541no—
184C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
185C8—21no—
186C8—272no—
187C2×C3#64093113yes—
188C2×C3#64124113yes—
189C2×C3#64127113yes—
190D3—3613no—
191C22—541no—
192C22—541no—
193C22—541no—
194C22D4×C3⋊(C3⋊D3)11yescharacteristic
195C4—41no—
196C4—41no—
197C4—541no—
198C4—1081no—
199C4—1081no—
200C4#8907611yescharacteristic
201C4D4×C3⋊(C3⋊D3)11yescharacteristic
202C3#10592515113yescomplemented
203C2—1081no—
204C2#1172392511yescharacteristic
205C2#1172395611yescharacteristic
206C2#1172395911yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenter#103C3⋊(C3⋊D3)
hypocenterC33#470
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
5. C33×#103C2semidirect
6. #11722186C2semidirect
7. #11722178C2semidirect
26. C32×#103D3semidirect
73. C3×#103C3⋊D3semidirect
126. #103C3⋊(C3⋊D3)semidirect
127. C33#470semidirect
171. C32#4816semidirect
202. C3#10592515semidirect

Maximal subgroups

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

subgroupindex
#117239872
C2×#888632
C2×#888502
#117239382
C33×#1032
#117221862
#117221782
#105925153