Finite Group Databasebeta

← #13163689

Subgroups — #13163689

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×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#88862C211yesmaximal, characteristic, complemented
3#11723971C211yesmaximal, characteristic
4C2×#88855C211yesmaximal, characteristic, complemented
5#11722205C211yesmaximal, characteristic, complemented
6#11722179C211yesmaximal, characteristic, complemented
7#11722177C211yesmaximal, characteristic, complemented
8#10592748—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C2×#6312—21no—
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C33×C4⋊C4—21no—
14C33×C4⋊C4C2211yescharacteristic
15C33×C22⋊C4—21no—
16#88862—22no—
17#88859—21no—
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19#88855—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C32×C22⋊Q8D3113yescomplemented
23C2×#3755—313no—
24#64139—313no—
25C2×#3748—313no—
26#63662—313no—
27#63636—313no—
28#63634—313no—
29C23×C33D411yescharacteristic
30C22×C3⋊(C3⋊D3)—21no—
31C33×Q8—21no—
32C33×Q8—21no—
33C2×C33×C4—21no—
34C2×C33×C4—21no—
35C2×C33×C4—21no—
36C2×C33×C4—21no—
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39#6312—42no—
40C2×C3⋊(C3⋊Dic3)—21no—
41C4⋊(C3⋊(C3⋊D3))—21no—
42C4⋊(C3⋊(C3⋊D3))—21no—
43C3⋊(C3⋊(C3⋊C8))—21no—
44C3⋊(C3⋊(C3⋊C8))D412yescomplemented
45#5049—913no—
46C2×C32×Q8C2×D3113yes—
47C22×C32×C4C2×D3113yes—
48C2×#638—613no—
49C2×C4⋊(C3⋊D3)—313no—
50C32×C4⋊C4—213no—
51C32×C4⋊C4C2×D3113yes—
52C32×C22⋊C4—213no—
53#3755—626no—
54#3752—613no—
55C4⋊(C3⋊Dic3)—313no—
56#3748—626no—
57C2×C3⋊(C3⋊C8)—313no—
58C2×C3⋊(C3⋊C8)—313no—
59C22×C33—21no—
60C22×C33—21no—
61C22×C33—21no—
62C22×C33C2×D411yescharacteristic
63C2×C3⋊(C3⋊D3)—41no—
64C2×C3⋊(C3⋊D3)—41no—
65C2×C3⋊(C3⋊D3)—41no—
66C33×C4—21no—
67C33×C4—41no—
68C33×C4—41no—
69C33×C4C2×D411yescharacteristic
70C33×C4#4111yescharacteristic
71C3⋊(C3⋊Dic3)—41no—
72C3×C22⋊Q8C3⋊D3113yescomplemented
73C2×#215—913no—
74#930—913no—
75C2×#208—913no—
76#835—913no—
77#809—913no—
78#807—913no—
79C23×C32#67113yes—
80C22×C3⋊D3—613no—
81C32×Q8—213no—
82C32×Q8—213no—
83C2×C32×C4—213no—
84C2×C32×C4—213no—
85C2×C32×C4—213no—
86C2×C32×C4—213no—
87C2×C32×C4#67113yes—
88C2×C32×C4C22×D3113yes—
89#638—1226no—
90C2×C3⋊Dic3—613no—
91C4⋊(C3⋊D3)—613no—
92C4⋊(C3⋊D3)—613no—
93C3⋊(C3⋊C8)—326no—
94C3⋊(C3⋊C8)—613no—
95#468—271noSylow
96C2×C33—41no—
97C2×C33#12111yescharacteristic
98C2×C33AGL(1,ℤ/8)11yescharacteristic
99C2×C33#13711yescharacteristic
100C3⋊(C3⋊D3)—81no—
101C2×C3×Q8C2×C3⋊D3113yes—
102C22×C3×C4C2×C3⋊D3113yes—
103C2×#67—1813no—
104C2×D12—913no—
105C3×C4⋊C4—213no—
106C3×C4⋊C4C2×C3⋊D3113yes—
107C3×C22⋊C4—213no—
108#215—1826no—
109(C2×D3)⋊C4—1813no—
110C4⋊Dic3—913no—
111#208—1826no—
112C2×C3⋊C8—913no—
113C2×C3⋊C8—913no—
114C22×C32—213no—
115C22×C32—213no—
116C22×C32—213no—
117C22×C32C2×#67113yes—
118C2×C3⋊D3—1213no—
119C2×C3⋊D3—1213no—
120C2×C3⋊D3—1213no—
121C32×C4—213no—
122C32×C4—413no—
123C32×C4—413no—
124C32×C4D3×D4113yes—
125C32×C4#237113yes—
126C3⋊Dic3—1213no—
127C2×SD16—271no—
128C2×M16—271no—
129C22⋊Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
130#121—271no—
131#107—271no—
132#103—271no—
133#102—271no—
134C33#46811yescharacteristic, complemented, Sylow
135C23×C3#638113yes—
136C22×D3—1813no—
137C3×Q8—213no—
138C3×Q8—213no—
139C2×C3×C4—213no—
140C2×C3×C4—213no—
141C2×C3×C4—213no—
142C2×C3×C4—213no—
143C2×C3×C4#638113yes—
144C2×C3×C4C22×C3⋊D3113yes—
145#67—3626no—
146C2×Dic3—1813no—
147D12—1813no—
148D12—1813no—
149C3⋊C8—926no—
150C3⋊C8—1813no—
151C2×C32—413no—
152C2×C32#939113yes—
153C2×C32#942113yes—
154C2×C32#949113yes—
155C3⋊D3—2413no—
156C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
157C2×D4—271no—
158C2×D4—541no—
159C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
160SD16—542no—
161M16—542no—
162C2×C8—271no—
163C2×C8—271no—
164C4⋊C4—21no—
165C4⋊C4—271no—
166C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—541no—
169C22×C3—213no—
170C22×C3—213no—
171C22×C3—213no—
172C22×C3C2×#638113yes—
173C2×D3—3613no—
174C2×D3—3613no—
175C2×D3—3613no—
176C3×C4—213no—
177C3×C4—413no—
178C3×C4—413no—
179C3×C4D4×C3⋊D3113yes—
180C3×C4#3830113yes—
181Dic3—3613no—
182C32#5049113yescomplemented
183C23—541no—
184C23#631211yescharacteristic
185Q8—21no—
186Q8—21no—
187D4—541no—
188D4—541no—
189D4—1082no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—541no—
195C2×C4#631211yescharacteristic
196C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
197C8—272no—
198C8—541no—
199C2×C3—413no—
200C2×C3#64148113yes—
201C2×C3#64151113yes—
202C2×C3#64158113yes—
203D3—7213no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—1081no—
208C22—1081no—
209C22—1081no—
210C22C2×#631211yescharacteristic
211C4—21no—
212C4—41no—
213C4—41no—
214C4—1081no—
215C4D4×C3⋊(C3⋊D3)11yescharacteristic
216C4#8907811yescharacteristic
217C3#10592748113yescomplemented
218C2—41no—
219C2—2161no—
220C2#1172398011yescharacteristic
221C2#1172398311yescharacteristic
222C2#1172399011yescharacteristic
subgroupG/H
centerC22C2×#6312
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×C22⋊Q8C2
socleC22×C33C2×D4
prime radical#11723971C2
hypercenterC22⋊Q8C3⋊(C3⋊D3)
hypocenterC33#468
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
1. C33×C22⋊Q8C2semidirect
2. C2×#88862C2semidirect
4. C2×#88855C2semidirect
5. #11722205C2semidirect
6. #11722179C2semidirect
7. #11722177C2semidirect
22. C32×C22⋊Q8D3semidirect
44. C3⋊(C3⋊(C3⋊C8))D4semidirect
72. C3×C22⋊Q8C3⋊D3semidirect
129. C22⋊Q8C3⋊(C3⋊D3)semidirect
134. C33#468semidirect
182. C32#5049semidirect
217. C3#10592748semidirect

Maximal subgroups

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

subgroupindex
C33×C22⋊Q82
C2×#888622
#117239712
C2×#888552
#117222052
#117221792
#117221772
#105927483