Finite Group Databasebeta

← #13163679

Subgroups — #13163679

Order 1728.

1680 conjugacy classes of subgroups; only the 405 normal subgroups are shown.

normal subgroups only

This group has 1680 conjugacy classes of subgroups — too many to draw. Only its 405 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×#121C211yesmaximal, characteristic
2C2×#88861C211yesmaximal, characteristic, complemented
3#11723965C211yesmaximal, characteristic, complemented
4C2×#88855C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722180C211yesmaximal, characteristic, complemented
7#11722177C211yesmaximal, characteristic, complemented
8#10592741—313nomaximal
9C2×C33×D4—21no—
10C2×C33×D4C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C2×#6308C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C33×C22⋊C4—21no—
15#88864—21no—
16#88861—22no—
17C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
18#88857—21no—
19#88855—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C32×#121D3113yes—
23C2×#3754—313no—
24#64133—313no—
25C2×#3748—313no—
26#63659—313no—
27#63637—313no—
28#63634—313no—
29C23×C33—21no—
30C23×C33D411yescharacteristic
31C33×D4—21no—
32C33×D4—21no—
33C33×D4—42no—
34C2×C33×C4—21no—
35C2×C33×C4—21no—
36C2×C33×C4—21no—
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)—21no—
41#6308—21no—
42#6308—21no—
43C3⋊(C3⋊(C3⋊C8))—21no—
44C3⋊(C3⋊(C3⋊C8))D412yescomplemented
45#5042—913no—
46C2×C32×D4—213no—
47C2×C32×D4C2×D3113yes—
48C22×C32×C4C2×D3113yes—
49C2×#634—313no—
50C32×C4⋊C4C2×D3113yes—
51C32×C22⋊C4—213no—
52#3757—613no—
53#3754—626no—
54C4⋊(C3⋊Dic3)—313no—
55#3750—613no—
56#3748—626no—
57C2×C3⋊(C3⋊C8)—313no—
58C2×C3⋊(C3⋊C8)—313no—
59C22×C33—21no—
60C22×C33—21no—
61C22×C33—21no—
62C22×C33—41no—
63C22×C33—41no—
64C22×C33—41no—
65C22×C33C2×D411yescharacteristic
66C33×C4—21no—
67C33×C4—41no—
68C33×C4C2×D411yescharacteristic
69C33×C4#4111yescharacteristic
70C3⋊(C3⋊Dic3)—41no—
71C3⋊(C3⋊Dic3)—41no—
72C3×#121C3⋊D3113yes—
73C2×#214—913no—
74#924—913no—
75C2×#208—913no—
76#832—913no—
77#810—913no—
78#807—913no—
79C23×C32—213no—
80C23×C32#67113yes—
81C32×D4—213no—
82C32×D4—213no—
83C32×D4—426no—
84C2×C32×C4—213no—
85C2×C32×C4—213no—
86C2×C32×C4—213no—
87C2×C32×C4#67113yes—
88C2×C32×C4C22×D3113yes—
89C2×C3⋊Dic3—613no—
90C2×C3⋊Dic3—613no—
91#634—613no—
92#634—613no—
93C3⋊(C3⋊C8)—326no—
94C3⋊(C3⋊C8)—613no—
95#468—271noSylow
96C2×C33—41no—
97C2×C33—81no—
98C2×C33#12111yescharacteristic
99C2×C33AGL(1,ℤ/8)11yescharacteristic
100C2×C33#13711yescharacteristic
101C2×C3×D4—213no—
102C2×C3×D4C2×C3⋊D3113yes—
103C22×C3×C4C2×C3⋊D3113yes—
104C2×Dic6—913no—
105C3×C4⋊C4C2×C3⋊D3113yes—
106C3×C22⋊C4—213no—
107#217—1813no—
108#214—1826no—
109C4⋊Dic3—913no—
110Dic3⋊C4—1813no—
111#208—1826no—
112C2×C3⋊C8—913no—
113C2×C3⋊C8—913no—
114C22×C32—213no—
115C22×C32—213no—
116C22×C32—213no—
117C22×C32—413no—
118C22×C32—413no—
119C22×C32—413no—
120C22×C32C2×#67113yes—
121C32×C4—213no—
122C32×C4—413no—
123C32×C4D3×D4113yes—
124C32×C4#237113yes—
125C3⋊Dic3—1213no—
126C3⋊Dic3—1213no—
127C2×SD16—271no—
128C2×M16—271no—
129C22⋊Q8—271no—
130#121C3⋊(C3⋊D3)11yescharacteristic
131#107—271no—
132#103—271no—
133#102—271no—
134C33#46811yescharacteristic, complemented, Sylow
135C23×C3—213no—
136C23×C3#638113yes—
137C3×D4—213no—
138C3×D4—213no—
139C3×D4—426no—
140C2×C3×C4—213no—
141C2×C3×C4—213no—
142C2×C3×C4—213no—
143C2×C3×C4#638113yes—
144C2×C3×C4C22×C3⋊D3113yes—
145C2×Dic3—1813no—
146C2×Dic3—1813no—
147Dic6—1813no—
148Dic6—1813no—
149C3⋊C8—926no—
150C3⋊C8—1813no—
151C2×C32—413no—
152C2×C32—813no—
153C2×C32#932113yes—
154C2×C32#939113yes—
155C2×C32#951113yes—
156C2×Q8—271no—
157C2×D4—21no—
158C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
159C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
160SD16—542no—
161M16—542no—
162C2×C8—271no—
163C2×C8—271no—
164C4⋊C4—271no—
165C4⋊C4—541no—
166C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—541no—
169C22×C3—213no—
170C22×C3—213no—
171C22×C3—213no—
172C22×C3—413no—
173C22×C3—413no—
174C22×C3—413no—
175C22×C3C2×#638113yes—
176C3×C4—213no—
177C3×C4—413no—
178C3×C4D4×C3⋊D3113yes—
179C3×C4#3830113yes—
180Dic3—3613no—
181Dic3—3613no—
182C32#5042113yescomplemented
183C23—21no—
184C23#631211yescharacteristic
185Q8—541no—
186Q8—541no—
187D4—21no—
188D4—21no—
189D4—42no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—541no—
194C2×C4—541no—
195C2×C4#631211yescharacteristic
196C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
197C8—272no—
198C8—541no—
199C2×C3—413no—
200C2×C3—813no—
201C2×C3#64141113yes—
202C2×C3#64148113yes—
203C2×C3#64160113yes—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22C2×#631211yescharacteristic
211C4—21no—
212C4—41no—
213C4—1081no—
214C4—1081no—
215C4D4×C3⋊(C3⋊D3)11yescharacteristic
216C4#8907811yescharacteristic
217C3#10592741113yescomplemented
218C2—41no—
219C2—81no—
220C2#1172397311yescharacteristic
221C2#1172398011yescharacteristic
222C2#1172399211yescharacteristic
subgroupG/H
centerC22C2×#6312
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
Fitting, prime radicalC33×#121C2
socleC22×C33C2×D4
hypercenter#121C3⋊(C3⋊D3)
hypocenterC33#468
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. C2×#88861C2semidirect
3. #11723965C2semidirect
4. C2×#88855C2semidirect
5. #11722202C2semidirect
6. #11722180C2semidirect
7. #11722177C2semidirect
44. C3⋊(C3⋊(C3⋊C8))D4semidirect
134. C33#468semidirect
182. C32#5042semidirect
217. C3#10592741semidirect

Maximal subgroups

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

subgroupindex
C33×#1212
C2×#888612
#117239652
C2×#888552
#117222022
#117221802
#117221772
#105927413