Finite Group Databasebeta

← #13163686

Subgroups — #13163686

Order 1728.

1596 conjugacy classes of subgroups; only the 461 normal subgroups are shown.

normal subgroups only

This group has 1596 conjugacy classes of subgroups — too many to draw. Only its 461 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×C22⋊Q8C211yesmaximal, characteristic
2C2×#88863C211yesmaximal, characteristic, complemented
3#11723965C211yesmaximal, characteristic
4C22×C3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
5#11722205C211yesmaximal, characteristic, complemented
6#11722180C211yesmaximal, characteristic, complemented
7#11722177C211yesmaximal, characteristic, complemented
8#10592750—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C33×C4⋊C4—21no—
13C33×C4⋊C4C2211yescharacteristic
14C33×C22⋊C4—21no—
15#88864—21no—
16#88863—22no—
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18#88857—21no—
19C2×C3⋊(C3⋊(C3⋊C8))—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C32×C22⋊Q8D3113yes—
23C2×#3756—313no—
24#64133—313no—
25C22×C3⋊(C3⋊C8)—313no—
26#63662—313no—
27#63637—313no—
28#63634—313no—
29C23×C33D411yescharacteristic
30C33×Q8—21no—
31C33×Q8—21no—
32C2×C33×C4—21no—
33C2×C33×C4—21no—
34C2×C33×C4—21no—
35C2×C33×C4—21no—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40#6308—21no—
41#6308—21no—
42C3⋊(C3⋊(C3⋊C8))—21no—
43C3⋊(C3⋊(C3⋊C8))D412yes—
44#5051—913no—
45C2×C32×Q8C2×D3113yes—
46C22×C32×C4C2×D3113yes—
47C2×#634—313no—
48C32×C4⋊C4—213no—
49C32×C4⋊C4C2×D3113yes—
50C32×C22⋊C4—213no—
51#3757—613no—
52#3756—626no—
53C4⋊(C3⋊Dic3)—313no—
54#3750—613no—
55C2×C3⋊(C3⋊C8)—313no—
56C2×C3⋊(C3⋊C8)—313no—
57C2×C3⋊(C3⋊C8)—626no—
58C22×C33—21no—
59C22×C33—21no—
60C22×C33Q1612yescomplemented
61C22×C33C2×D411yescharacteristic
62C33×C4—21no—
63C33×C4—41no—
64C33×C4—41no—
65C33×C4C2×D411yescharacteristic
66C33×C4#4111yescharacteristic
67C3⋊(C3⋊Dic3)—41no—
68C3⋊(C3⋊Dic3)—41no—
69C3×C22⋊Q8C3⋊D3113yes—
70C2×#216—913no—
71#924—913no—
72C22×C3⋊C8—913no—
73#835—913no—
74#810—913no—
75#807—913no—
76C23×C32#67113yes—
77C32×Q8—213no—
78C32×Q8—213no—
79C2×C32×C4—213no—
80C2×C32×C4—213no—
81C2×C32×C4—213no—
82C2×C32×C4—213no—
83C2×C32×C4#67113yes—
84C2×C32×C4C22×D3113yes—
85C2×C3⋊Dic3—613no—
86C2×C3⋊Dic3—613no—
87#634—613no—
88#634—613no—
89C3⋊(C3⋊C8)—326no—
90C3⋊(C3⋊C8)—613no—
91#467—271noSylow
92C2×C33—22no—
93C2×C33#12111yescharacteristic
94C2×C33C2×Q1611yescharacteristic
95C2×C33#13511yescharacteristic
96C2×C3×Q8C2×C3⋊D3113yes—
97C22×C3×C4C2×C3⋊D3113yes—
98C2×Dic6—913no—
99C3×C4⋊C4—213no—
100C3×C4⋊C4C2×C3⋊D3113yes—
101C3×C22⋊C4—213no—
102#217—1813no—
103#216—1826no—
104C4⋊Dic3—913no—
105Dic3⋊C4—1813no—
106C2×C3⋊C8—913no—
107C2×C3⋊C8—913no—
108C2×C3⋊C8—1826no—
109C22×C32—213no—
110C22×C32—213no—
111C22×C32#216126yescomplemented
112C22×C32C2×#67113yes—
113C32×C4—213no—
114C32×C4—413no—
115C32×C4—413no—
116C32×C4D3×D4113yes—
117C32×C4#237113yes—
118C3⋊Dic3—1213no—
119C3⋊Dic3—1213no—
120C2×Q16—271no—
121C22×C8—271no—
122C22⋊Q8—271no—
123C22⋊Q8C3⋊(C3⋊D3)11yescharacteristic
124#107—271no—
125#103—271no—
126#103—271no—
127C33#46711yescharacteristic, complemented, Sylow
128C23×C3#638113yes—
129C3×Q8—213no—
130C3×Q8—213no—
131C2×C3×C4—213no—
132C2×C3×C4—213no—
133C2×C3×C4—213no—
134C2×C3×C4—213no—
135C2×C3×C4#638113yes—
136C2×C3×C4C22×C3⋊D3113yes—
137C2×Dic3—1813no—
138C2×Dic3—1813no—
139Dic6—1813no—
140Dic6—1813no—
141C3⋊C8—926no—
142C3⋊C8—1813no—
143C2×C32—226no—
144C2×C32#939113yes—
145C2×C32C2×#216113yes—
146C2×C32#950113yes—
147C2×Q8—271no—
148C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
149C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
150Q16—542no—
151C2×C8—271no—
152C2×C8—271no—
153C2×C8—542no—
154C4⋊C4—21no—
155C4⋊C4—271no—
156C4⋊C4—541no—
157C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—541no—
160C22×C3—213no—
161C22×C3—213no—
162C22×C3#3756126yescomplemented
163C22×C3C2×#638113yes—
164C3×C4—213no—
165C3×C4—413no—
166C3×C4—413no—
167C3×C4D4×C3⋊D3113yes—
168C3×C4#3830113yes—
169Dic3—3613no—
170Dic3—3613no—
171C32#5051113yescomplemented
172C23#631211yescharacteristic
173Q8—21no—
174Q8—21no—
175Q8—541no—
176Q8—541no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—541no—
182C2×C4—541no—
183C2×C4#631211yescharacteristic
184C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
185C8—272no—
186C8—541no—
187C2×C3—226no—
188C2×C3#64148113yes—
189C2×C3C2×#3756113yes—
190C2×C3#64159113yes—
191C22—21no—
192C22—21no—
193C22#8886312yescomplemented
194C22C2×#631211yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—1081no—
199C4—1081no—
200C4D4×C3⋊(C3⋊D3)11yescharacteristic
201C4#8907811yescharacteristic
202C3#10592750113yescomplemented
203C2—22no—
204C2#1172398011yescharacteristic
205C2C2×#8886311yescharacteristic
206C2#1172399111yescharacteristic
subgroupG/H
centerC22C2×#6312
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×C22⋊Q8C2
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC22⋊Q8C3⋊(C3⋊D3)
hypocenterC33#467
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#88863C2semidirect
5. #11722205C2semidirect
6. #11722180C2semidirect
7. #11722177C2semidirect
60. C22×C33Q16semidirect
111. C22×C32#216semidirect
127. C33#467semidirect
162. C22×C3#3756semidirect
171. C32#5051semidirect
193. C22#88863semidirect
202. C3#10592750semidirect

Maximal subgroups

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

subgroupindex
C33×C22⋊Q82
C2×#888632
#117239652
C22×C3⋊(C3⋊(C3⋊C8))2
#117222052
#117221802
#117221772
#105927503