Finite Group Databasebeta

← #13163463

Subgroups — #13163463

Order 1728.

1456 conjugacy classes of subgroups; only the 357 normal subgroups are shown.

normal subgroups only

This group has 1456 conjugacy classes of subgroups — too many to draw. Only its 357 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#11723991C211yesmaximal, characteristic
2#11723989C211yesmaximal, characteristic, complemented
3#11723948C211yesmaximal, characteristic, complemented
4C33×#104C211yesmaximal, characteristic, complemented
5#11722192C211yesmaximal, characteristic, complemented
6#11722175C211yesmaximal, characteristic, complemented
7#11722173C211yesmaximal, characteristic, complemented
8#10592522—313nomaximal
9C33×#41C2211yescharacteristic
10#89076C2211yescharacteristic
11C33×M16C2211yescharacteristic
12C33×C42C2211yescharacteristic
13#88863C411yescharacteristic, complemented
14#88862C411yescharacteristic, complemented
15#88861C411yescharacteristic, complemented
16#88860C411yescharacteristic, complemented
17#88855—21no—
18#88855C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))—21no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22#88850—21no—
23C8×C3⋊(C3⋊D3)—21no—
24#64159—313no—
25#64157—313no—
26#64116—313no—
27C32×#104D3113yescomplemented
28#63649—313no—
29#63632—313no—
30#63630—313no—
31C33×Q8C2×C411yescharacteristic
32C33×D4—21no—
33C33×D4C2×C411yescharacteristic
34C2×C33×C4—21no—
35C2×C33×C4—21no—
36C2×C33×C4C2311yescharacteristic
37#6312—21no—
38C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
39C4×C3⋊(C3⋊D3)—21no—
40#6308C2×C411yescharacteristic
41C33×C8—21no—
42C3⋊(C3⋊(C3⋊C8))—21no—
43C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
44C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
45C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
46C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
47#4823—913no—
48C32×#41C2×D3113yes—
49#3828—313no—
50C32×M16C2×D3113yes—
51C32×C42C2×D3113yes—
52#3756—313no—
53#3755—313no—
54#3754—313no—
55#3753—313no—
56#3748—313no—
57#3748—613no—
58C2×C3⋊(C3⋊C8)—313no—
59C2×C3⋊(C3⋊C8)—313no—
60C2×C3⋊(C3⋊C8)—613no—
61#3743—613no—
62C8×C3⋊D3—613no—
63C22×C33—21no—
64C22×C33#4111yescharacteristic
65C2×C3⋊(C3⋊D3)—21no—
66C33×C4—21no—
67C33×C4—41no—
68C33×C4C22×C411yescharacteristic
69C33×C4C2×D411yescharacteristic
70C3⋊(C3⋊Dic3)—21no—
71#950—913no—
72#948—913no—
73#907—913no—
74C3×#104C3⋊D3113yescomplemented
75#822—913no—
76#805—913no—
77#803—913no—
78C32×Q8C4×D3113yes—
79C32×D4—213no—
80C32×D4C4×D3113yes—
81C2×C32×C4—213no—
82C2×C32×C4—213no—
83C2×C32×C4C22×D3113yes—
84#638—613no—
85C4⋊(C3⋊D3)—313no—
86C4×C3⋊D3—613no—
87#634—313no—
88C32×C8—213no—
89C3⋊(C3⋊C8)—313no—
90C3⋊(C3⋊C8)—313no—
91C3⋊(C3⋊C8)—313no—
92C3⋊(C3⋊C8)—313no—
93C3⋊(C3⋊C8)—613no—
94#444—271noSylow
95C2×C33—21no—
96C2×C33—41no—
97C2×C33C4×D411yescharacteristic
98C3⋊(C3⋊D3)—41no—
99C3×#41C2×C3⋊D3113yes—
100#235—913no—
101C3×M16C2×C3⋊D3113yes—
102C3×C42C2×C3⋊D3113yes—
103#216—913no—
104#215—913no—
105#214—913no—
106#213—913no—
107#208—913no—
108#208—1813no—
109C2×C3⋊C8—913no—
110C2×C3⋊C8—913no—
111C2×C3⋊C8—1813no—
112ΓL(1,𝔽25)—1813no—
113D3×C8—1813no—
114C22×C32—213no—
115C22×C32#237113yes—
116C2×C3⋊D3—613no—
117C32×C4—213no—
118C32×C4—413no—
119C32×C4C2×C4×D3113yes—
120C32×C4D3×D4113yes—
121C3⋊Dic3—613no—
122#135—271no—
123#131—271no—
124#131—271no—
125#108—271no—
126#104—271no—
127#104C3⋊(C3⋊D3)11yescharacteristic, complemented
128#97—271no—
129C33#44411yescharacteristic, complemented, Sylow
130C3×Q8C4×C3⋊D3113yes—
131C3×D4—213no—
132C3×D4C4×C3⋊D3113yes—
133C2×C3×C4—213no—
134C2×C3×C4—213no—
135C2×C3×C4C22×C3⋊D3113yes—
136#67—1813no—
137D12—913no—
138C4×D3—1813no—
139Dic6—913no—
140C3×C8—213no—
141C3⋊C8—913no—
142C3⋊C8—913no—
143C3⋊C8—913no—
144C3⋊C8—913no—
145C3⋊C8—1813no—
146C2×C32—213no—
147C2×C32—413no—
148C2×C32#67⋊C4113yes—
149C3⋊D3—1213no—
150#41—271no—
151#41C2×C3⋊(C3⋊D3)11yescharacteristic
152Q16—271no—
153SD16—271no—
154SD16—271no—
155D8—271no—
156M16—271no—
157M16—541no—
158M16—541no—
159M16C2×C3⋊(C3⋊D3)11yescharacteristic
160C2×C8—271no—
161C2×C8—271no—
162C2×C8—541no—
163C2×C8—541no—
164C42C2×C3⋊(C3⋊D3)11yescharacteristic
165C22×C3—213no—
166C22×C3#3830113yes—
167C2×D3—1813no—
168C3×C4—213no—
169C3×C4—413no—
170C3×C4C2×C4×C3⋊D3113yes—
171C3×C4D4×C3⋊D3113yes—
172Dic3—1813no—
173C32#4823113yescomplemented
174Q8—271no—
175Q8C4×C3⋊(C3⋊D3)11yescharacteristic
176D4—21no—
177D4—271no—
178D4—541no—
179D4C4×C3⋊(C3⋊D3)11yescharacteristic
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—541no—
183C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
184C8—21no—
185C8—271no—
186C8—271no—
187C8—271no—
188C8—271no—
189C8—541no—
190C2×C3—213no—
191C2×C3—413no—
192C2×C3#64090113yes—
193D3—3613no—
194C22—21no—
195C22—541no—
196C22#8907811yescharacteristic
197C4—21no—
198C4—41no—
199C4—541no—
200C4C2×C4×C3⋊(C3⋊D3)11yescharacteristic
201C4D4×C3⋊(C3⋊D3)11yescharacteristic
202C3#10592522113yescomplemented
203C2—21no—
204C2—41no—
205C2—1081no—
206C2#1172392211yescharacteristic
subgroupG/H
centerC4D4×C3⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#104C2
socleC2×C33C4×D4
prime radical#11723991C2
cosocleC2×C33×C4C23
hypercenter#104C3⋊(C3⋊D3)
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #11723989C2semidirect
3. #11723948C2semidirect
4. C33×#104C2semidirect
5. #11722192C2semidirect
6. #11722175C2semidirect
7. #11722173C2semidirect
13. #88863C4semidirect
14. #88862C4semidirect
15. #88861C4semidirect
16. #88860C4semidirect
27. C32×#104D3semidirect
74. C3×#104C3⋊D3semidirect
127. #104C3⋊(C3⋊D3)semidirect
129. C33#444semidirect
173. C32#4823semidirect
202. C3#10592522semidirect

Maximal subgroups

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

subgroupindex
#117239912
#117239892
#117239482
C33×#1042
#117221922
#117221752
#117221732
#105925223