Finite Group Databasebeta

← #13163886

Subgroups — #13163886

Order 1728.

1988 conjugacy classes of subgroups; only the 459 normal subgroups are shown.

normal subgroups only

This group has 1988 conjugacy classes of subgroups — too many to draw. Only its 459 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
1C33×2−1+4C211yesmaximal, characteristic
2#11723992C212yesmaximal, complemented
3#11723985C211yesmaximal, characteristic, complemented
4#11722207C212yesmaximal, complemented
5#11722206C211yesmaximal, characteristic, complemented
6#10592945—313nomaximal
7C33×#41—22no—
8C33×#41—22no—
9C33×#41C2212yes—
10C2×C33×Q8—22no—
11C2×C33×Q8C2211yescharacteristic
12C2×#6308C2211yescharacteristic
13#88863—22no—
14#88861—22no—
15#88857—22no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17#88855C2212yes—
18C32×2−1+4D3113yes—
19#64160—326no—
20#64153—313no—
21#63664—326no—
22#63663—313no—
23C33×Q8—22no—
24C33×Q8—22no—
25C33×Q8D412yes—
26C33×D4—22no—
27C33×D4—22no—
28C33×D4D412yes—
29C2×C33×C4—24no—
30C2×C33×C4—22no—
31C2×C33×C4D412yes—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3⋊(C3⋊Dic3)—21no—
35#6308—22no—
36C3⋊(C3⋊(C3⋊C8))—22no—
37#5246—913no—
38C32×#41—226no—
39C32×#41—226no—
40C32×#41C2×D3126yes—
41C2×C32×Q8—226no—
42C2×C32×Q8C2×D3113yes—
43C2×#634—313no—
44#3756—626no—
45#3754—626no—
46#3750—626no—
47C4×C3⋊Dic3—313no—
48#3748—326no—
49C22×C33—22no—
50C22×C33C2×D411yescharacteristic
51C33×C4—22no—
52C33×C4—22no—
53C33×C4C2×D412yes—
54C3⋊(C3⋊Dic3)—22no—
55C3⋊(C3⋊Dic3)—41no—
56C3×2−1+4C3⋊D3113yes—
57#951—926no—
58#944—913no—
59#837—926no—
60#836—913no—
61C32×Q8—226no—
62C32×Q8—226no—
63C32×Q8#67126yes—
64C32×D4—226no—
65C32×D4—226no—
66C32×D4#67126yes—
67C2×C32×C4—252no—
68C2×C32×C4—226no—
69C2×C32×C4#67126yes—
70C2×C32×C4C22×D3113yes—
71C2×C3⋊Dic3—613no—
72C2×C3⋊Dic3—613no—
73#634—626no—
74C3⋊(C3⋊C8)—626no—
75#456—271noSylow
76C2×C33—21no—
77C2×C33—42no—
78C2×C33#12011yescharacteristic
79C3×#41—226no—
80C3×#41—226no—
81C3×#41C2×C3⋊D3126yes—
82C2×C3×Q8—226no—
83C2×C3×Q8C2×C3⋊D3113yes—
84C2×Dic6—913no—
85#216—1826no—
86#214—1826no—
87Dic3⋊C4—1826no—
88C4×Dic3—913no—
89#208—926no—
90C22×C32—226no—
91C22×C32C2×#67113yes—
92C32×C4—226no—
93C32×C4—226no—
94C32×C4C2×#67126yes—
95C3⋊Dic3—626no—
96C3⋊Dic3—1213no—
972−1+4C3⋊(C3⋊D3)11yescharacteristic
98#137—272no—
99C4⋊Q8—271no—
100#104—272no—
101#101—271no—
102C33#45611yescharacteristic, complemented, Sylow
103C3×Q8—226no—
104C3×Q8—226no—
105C3×Q8#638126yes—
106C3×D4—226no—
107C3×D4—226no—
108C3×D4#638126yes—
109C2×C3×C4—252no—
110C2×C3×C4—226no—
111C2×C3×C4#638126yes—
112C2×C3×C4C22×C3⋊D3113yes—
113C2×Dic3—1813no—
114C2×Dic3—1813no—
115Dic6—1826no—
116C3⋊C8—1826no—
117C2×C32—213no—
118C2×C32—426no—
119C2×C32#953113yes—
120#41—22no—
121#41—22no—
122#41C2×C3⋊(C3⋊D3)12yes—
123C2×Q8—22no—
124C2×Q8—271no—
125C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
126Q16—542no—
127SD16—542no—
128M16—272no—
129C4⋊C4—542no—
130C42—271no—
131C22×C3—226no—
132C22×C3C2×#638113yes—
133C3×C4—226no—
134C3×C4—226no—
135C3×C4C2×#638126yes—
136Dic3—1826no—
137Dic3—3613no—
138C32#5246113yescomplemented
139Q8—22no—
140Q8—22no—
141Q8—542no—
142Q8#631212yes—
143D4—22no—
144D4—22no—
145D4#631212yes—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—541no—
149C2×C4—541no—
150C2×C4#631212yes—
151C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
152C8—542no—
153C2×C3—213no—
154C2×C3—426no—
155C2×C3#64162113yes—
156C22—22no—
157C22C2×#631211yescharacteristic
158C4—22no—
159C4—22no—
160C4—542no—
161C4—1081no—
162C4C2×#631212yes—
163C3#10592945113yescomplemented
164C2—21no—
165C2—42no—
166C2#1172399411yescharacteristic
subgroupG/H
centerC2#11723994
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
Fitting, prime radicalC33×2−1+4C2
socleC2×C33#120
hypercenter2−1+4C3⋊(C3⋊D3)
hypocenterC33#456
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
2. #11723992C2semidirect
3. #11723985C2semidirect
4. #11722207C2semidirect
5. #11722206C2semidirect
102. C33#456semidirect
138. C32#5246semidirect
163. C3#10592945semidirect

Maximal subgroups

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

subgroupindex
C33×2−1+42
#117239922
#117239852
#117222072
#117222062
#105929453