Finite Group Databasebeta

← #13163883

Subgroups — #13163883

Order 1728.

2128 conjugacy classes of subgroups; only the 459 normal subgroups are shown.

normal subgroups only

This group has 2128 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×H(5,𝔽2)C211yesmaximal, characteristic
2#11723992C212yesmaximal, complemented
3#11723977C211yesmaximal, characteristic, complemented
4#11722207C212yesmaximal, complemented
5#11722203C211yesmaximal, characteristic, complemented
6#10592941—313nomaximal
7C33×#41—22no—
8C33×#41C2212yes—
9C2×C33×D4—22no—
10C2×C33×D4—22no—
11C2×C33×D4C2211yescharacteristic
12C2×#6308C2211yescharacteristic, complemented
13#88864—22no—
14#88863—22no—
15#88861—22no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
17#88855C2212yescomplemented
18C32×H(5,𝔽2)D3113yes—
19#64160—326no—
20#64145—313no—
21#63664—326no—
22#63660—313no—
23C23×C33—22no—
24C23×C33D412yes—
25C33×Q8D412yes—
26C33×D4—24no—
27C33×D4—22no—
28C33×D4—22no—
29C33×D4D412yes—
30C2×C33×C4—22no—
31C2×C33×C4—22no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3⋊(C3⋊Dic3)—21no—
35#6308—22no—
36C3⋊(C3⋊(C3⋊C8))—22no—
37#5242—913no—
38C32×#41—226no—
39C32×#41C2×D3126yes—
40C2×C32×D4—226no—
41C2×C32×D4—226no—
42C2×C32×D4C2×D3113yes—
43C2×#634—313no—
44#3757—626no—
45#3756—626no—
46#3754—626no—
47C4×C3⋊Dic3—313no—
48#3748—326no—
49C22×C33—22no—
50C22×C33—22no—
51C22×C33—42no—
52C22×C33—82no—
53C22×C33C2×D411yescharacteristic
54C33×C4—22no—
55C33×C4C2×D412yes—
56C3⋊(C3⋊Dic3)—41no—
57C3⋊(C3⋊Dic3)—41no—
58C3×H(5,𝔽2)C3⋊D3113yes—
59#951—926no—
60#936—913no—
61#837—926no—
62#833—913no—
63C23×C32—226no—
64C23×C32#67126yes—
65C32×Q8#67126yes—
66C32×D4—252no—
67C32×D4—226no—
68C32×D4—226no—
69C32×D4#67126yes—
70C2×C32×C4—226no—
71C2×C32×C4—226no—
72C2×C32×C4C22×D3113yes—
73C2×C3⋊Dic3—613no—
74C2×C3⋊Dic3—613no—
75#634—626no—
76C3⋊(C3⋊C8)—626no—
77#455—271noSylow
78C2×C33—21no—
79C2×C33—42no—
80C2×C33—42no—
81C2×C33#12011yescharacteristic
82C3×#41—226no—
83C3×#41C2×C3⋊D3126yes—
84C2×C3×D4—226no—
85C2×C3×D4—226no—
86C2×C3×D4C2×C3⋊D3113yes—
87C2×Dic6—913no—
88#217—1826no—
89#216—1826no—
90#214—1826no—
91C4×Dic3—913no—
92#208—926no—
93C22×C32—226no—
94C22×C32—226no—
95C22×C32—426no—
96C22×C32—826no—
97C22×C32C2×#67113yes—
98C32×C4—226no—
99C32×C4C2×#67126yes—
100C3⋊Dic3—1213no—
101C3⋊Dic3—1213no—
102H(5,𝔽2)C3⋊(C3⋊D3)11yescharacteristic
103#137—272no—
104#124—271no—
105#104—272no—
106#100—271no—
107C33#45511yescharacteristic, complemented, Sylow
108C23×C3—226no—
109C23×C3#638126yes—
110C3×Q8#638126yes—
111C3×D4—252no—
112C3×D4—226no—
113C3×D4—226no—
114C3×D4#638126yes—
115C2×C3×C4—226no—
116C2×C3×C4—226no—
117C2×C3×C4C22×C3⋊D3113yes—
118C2×Dic3—1813no—
119C2×Dic3—1813no—
120Dic6—1826no—
121C3⋊C8—1826no—
122C2×C32—213no—
123C2×C32—426no—
124C2×C32—426no—
125C2×C32#953113yes—
126#41—22no—
127#41C2×C3⋊(C3⋊D3)12yes—
128C2×Q8—271no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
132Q16—542no—
133SD16—542no—
134M16—272no—
135C22⋊C4—542no—
136C42—271no—
137C22×C3—226no—
138C22×C3—226no—
139C22×C3—426no—
140C22×C3—826no—
141C22×C3C2×#638113yes—
142C3×C4—226no—
143C3×C4C2×#638126yes—
144Dic3—3613no—
145Dic3—3613no—
146C32#5242113yescomplemented
147C23—22no—
148C23#631212yes—
149Q8—542no—
150Q8#631212yes—
151D4—24no—
152D4—22no—
153D4—22no—
154D4#631212yes—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—541no—
158C2×C4—541no—
159C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
160C8—542no—
161C2×C3—213no—
162C2×C3—426no—
163C2×C3—426no—
164C2×C3#64162113yes—
165C22—22no—
166C22—22no—
167C22—42no—
168C22—82no—
169C22C2×#631211yescharacteristic
170C4—22no—
171C4—1081no—
172C4—1081no—
173C4C2×#631212yes—
174C3#10592941113yescomplemented
175C2—21no—
176C2—42no—
177C2—42no—
178C2#1172399411yescharacteristic
subgroupG/H
centerC2#11723994
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
Fitting, prime radicalC33×H(5,𝔽2)C2
socleC2×C33#120
hypercenterH(5,𝔽2)C3⋊(C3⋊D3)
hypocenterC33#455
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. #11723992C2semidirect
3. #11723977C2semidirect
4. #11722207C2semidirect
5. #11722203C2semidirect
12. C2×#6308C22semidirect
16. C4×C3⋊(C3⋊Dic3)C22semidirect
17. #88855C22semidirect
107. C33#455semidirect
146. C32#5242semidirect
174. C3#10592941semidirect

Maximal subgroups

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

subgroupindex
C33×H(5,𝔽2)2
#117239922
#117239772
#117222072
#117222032
#105929413