Finite Group Databasebeta

← C2×C32×2−1+4⋊C3 #13181703

Subgroups — C2×C32×2−1+4⋊C3 #13181703

Order 1728.

1912 conjugacy classes of subgroups; only the 562 normal subgroups are shown.

normal subgroups only

This group has 1912 conjugacy classes of subgroups — too many to draw. Only its 562 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
1C32×2−1+4⋊C3C214yesmaximal, direct product factor
2C2×C32×#41⋊C3C211yesmaximal, characteristic, complemented
3C22×C32×SL(2,𝔽3)C212yesmaximal, complemented
4C2×C32×2−1+4C311yesmaximal, characteristic, complemented
5C2×C3×2−1+4⋊C3C3112yesmaximal, direct product factor
6C2×C33×D4—41nomaximal
7C32×#41⋊C3C2212yescomplemented
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C2×C32×SL(2,𝔽3)C2214yescomplemented
10C32×2−1+4—34no—
11C32×2−1+4C2×C314yescomplemented
12C2×C32×#41—31no—
13C2×C32×#41—32no—
14C2×C32×#41C2×C311yescharacteristic, complemented
15C22×C32×Q8—31no—
16C22×C32×Q8C2×C312yescomplemented
17C3×2−1+4⋊C3C2×C3148yesdirect product factor
18C2×C3×#41⋊C3C2×C3112yescomplemented
19C22×C3×SL(2,𝔽3)C2×C3124yescomplemented
20C23×C33—42no—
21C33×D4—44no—
22C2×C33×C4—41no—
23C32×SL(2,𝔽3)C2311yescharacteristic
24C2×C3×2−1+4C3214yescomplemented
25C2×2−1+4⋊C3C3219yesdirect product factor
26C32×#41—38no—
27C32×#41—34no—
28C32×#41—34no—
29C32×#41—38no—
30C32×#41—32no—
31C32×#41C22×C312yescomplemented
32C2×C32×Q8—34no—
33C2×C32×Q8—34no—
34C2×C32×Q8—31no—
35C2×C32×Q8—32no—
36C2×C32×Q8—34no—
37C2×C32×Q8C22×C311yescharacteristic
38C2×C32×Q8C22×C314yescomplemented
39C2×C32×D4—32no—
40C2×C32×D4—31no—
41C2×C32×D4—412no—
42C2×C32×D4A411yescharacteristic
43C22×C32×C4—32no—
44C22×C32×C4—32no—
45C22×C32×C4—31no—
46C3×#41⋊C3C22×C3124yescomplemented
47C2×C3×SL(2,𝔽3)C22×C3112yes—
48C2×C3×SL(2,𝔽3)C22×C3148yescomplemented
49C22×C33—41no—
50C22×C33—44no—
51C22×C33—84no—
52C33×C4—42no—
53C3×2−1+4—316no—
54C3×2−1+4C2×C32116yescomplemented
55C2×C3×#41—34no—
56C2×C3×#41—38no—
57C2×C3×#41C2×C3214yescomplemented
58C22×C3×Q8—34no—
59C22×C3×Q8C2×C3218yescomplemented
602−1+4⋊C3C2×C32136yesdirect product factor
61C2×#41⋊C3C2×C3219yescomplemented
62C22×SL(2,𝔽3)C2×C32118yescomplemented
63C23×C32—31no—
64C23×C32—424no—
65C23×C32C2×A412yes—
66C32×Q8—31no—
67C32×Q8—34no—
68C32×Q8—34no—
69C32×Q8—34no—
70C32×Q8C23×C311yescharacteristic
71C32×D4—32no—
72C32×D4—32no—
73C32×D4—38no—
74C32×D4—448no—
75C32×D4C2×A414yes—
76C2×C32×C4—38no—
77C2×C32×C4—34no—
78C2×C32×C4—34no—
79C2×C32×C4—32no—
80C2×C32×C4—31no—
81C2×C32×C4—32no—
82C2×C32×C4—32no—
83C2×C32×C4—412no—
84C2×C32×C4C2×A411yescharacteristic
85C3×SL(2,𝔽3)C23×C3112yes—
86C2×2−1+4C3311yescharacteristic, complemented, Sylow
87C2×C33—42no—
88C2×C33—41no—
89C2×C33—84no—
90C3×#41—332no—
91C3×#41—316no—
92C3×#41—316no—
93C3×#41—332no—
94C3×#41—38no—
95C3×#41C22×C3218yescomplemented
96C2×C3×Q8—316no—
97C2×C3×Q8—316no—
98C2×C3×Q8—34no—
99C2×C3×Q8—38no—
100C2×C3×Q8—316no—
101C2×C3×Q8C22×C3214yes—
102C2×C3×Q8C22×C32116yescomplemented
103C2×C3×D4—34no—
104C2×C3×D4—38no—
105C2×C3×D4—49no—
106C2×C3×D4C3×A414yes—
107C22×C3×C4—38no—
108C22×C3×C4—38no—
109C22×C3×C4—34no—
110#41⋊C3C22×C32118yescomplemented
111C2×SL(2,𝔽3)C22×C3219yes—
112C2×SL(2,𝔽3)C22×C32136yescomplemented
113C22×C32—24no—
114C22×C32—32no—
115C22×C32—412no—
116C22×C32—448no—
117C22×C32—62no—
118C22×C32—848no—
119C22×C32C22×A414yes—
120C22×C32C22×A411yescharacteristic
121C32×C4—34no—
122C32×C4—31no—
123C32×C4—31no—
124C32×C4—424no—
125C32×C4C22×A412yes—
1262−1+4—34no—
1272−1+4C2×C3314yescomplemented
128C2×#41—31no—
129C2×#41—32no—
130C2×#41C2×C3311yescharacteristic, complemented
131C22×Q8—31no—
132C22×Q8C2×C3312yescomplemented
133C33—41noSylow
134C23×C3—34no—
135C23×C3—418no—
136C23×C3C2×C3×A418yes—
137C3×Q8—34no—
138C3×Q8—316no—
139C3×Q8—316no—
140C3×Q8—316no—
141C3×Q8C23×C3214yes—
142C3×D4—38no—
143C3×D4—38no—
144C3×D4—332no—
145C3×D4—436no—
146C3×D4C2×C3×A4116yes—
147C2×C3×C4—332no—
148C2×C3×C4—38no—
149C2×C3×C4—34no—
150C2×C3×C4—316no—
151C2×C3×C4—316no—
152C2×C3×C4—38no—
153C2×C3×C4—38no—
154C2×C3×C4—49no—
155C2×C3×C4C2×C3×A414yes—
156SL(2,𝔽3)C23×C3219yes—
157C2×C32—24no—
158C2×C32—424no—
159C2×C32—412no—
160C2×C32—62no—
161C2×C32—848no—
162C2×C322−1+4⋊C312yesdirect product factor
163C2×C32C23×A411yescharacteristic
164#41—38no—
165#41—34no—
166#41—34no—
167#41—38no—
168#41—32no—
169#41C22×C3312yescomplemented
170C2×Q8—34no—
171C2×Q8—34no—
172C2×Q8—31no—
173C2×Q8—32no—
174C2×Q8—34no—
175C2×Q8C22×C3311yescharacteristic
176C2×Q8C22×C3314yescomplemented
177C2×D4—31no—
178C2×D4—32no—
179C2×D4C32×A411yescharacteristic
180C22×C4—32no—
181C22×C4—32no—
182C22×C4—31no—
183C22×C3—216no—
184C22×C3—38no—
185C22×C3—49no—
186C22×C3—436no—
187C22×C3—68no—
188C22×C3—836no—
189C22×C3C22×C3×A4116yes—
190C22×C3C22×C3×A414yes—
191C3×C4—34no—
192C3×C4—316no—
193C3×C4—34no—
194C3×C4—418no—
195C3×C4C22×C3×A418yes—
196C32—412no—
197C32C2×2−1+4⋊C311yescharacteristic, direct product factor
198C23—31no—
199C23C2×C32×A412yes—
200Q8—31no—
201Q8—34no—
202Q8—34no—
203Q8—34no—
204Q8C23×C3311yescharacteristic
205D4—32no—
206D4—32no—
207D4—38no—
208D4C2×C32×A414yes—
209C2×C4—38no—
210C2×C4—31no—
211C2×C4—32no—
212C2×C4—34no—
213C2×C4—34no—
214C2×C4—32no—
215C2×C4—32no—
216C2×C4C2×C32×A411yescharacteristic
217C2×C3—216no—
218C2×C3—418no—
219C2×C3—49no—
220C2×C3—68no—
221C2×C3—836no—
222C2×C3C3×2−1+4⋊C318yesdirect product factor
223C2×C3C23×C3×A414yes—
224C22—24no—
225C22—32no—
226C22—62no—
227C22C22×C32×A414yes—
228C22C22×C32×A411yescharacteristic
229C4—31no—
230C4—34no—
231C4—31no—
232C4C22×C32×A412yes—
233C3—49no—
234C3C2×C3×2−1+4⋊C314yesdirect product factor
235C2—24no—
236C2—62no—
237C2C32×2−1+4⋊C312yesdirect product factor
238C2C23×C32×A411yescharacteristic
subgroupG/H
center, socleC22×C32C22×A4
commutator, cosocle, hypocenterQ8C23×C33
FrattiniC2C23×C32×A4
FittingC2×C32×2−1+4C3
hypercenterC2×C32×D4A4
LCSG ▷ Q8
UCSC1 ◁ C22×C32 ◁ C2×C32×D4

Complements

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

NH ≅ G/Ntype
1. C32×2−1+4⋊C3C2direct (N × H)
2. C2×C32×#41⋊C3C2semidirect
3. C22×C32×SL(2,𝔽3)C2semidirect
4. C2×C32×2−1+4C3semidirect
5. C2×C3×2−1+4⋊C3C3direct (N × H)
7. C32×#41⋊C3C22semidirect
9. C2×C32×SL(2,𝔽3)C22semidirect
11. C32×2−1+4C2×C3semidirect
14. C2×C32×#41C2×C3semidirect
16. C22×C32×Q8C2×C3semidirect
17. C3×2−1+4⋊C3C2×C3direct (N × H)
18. C2×C3×#41⋊C3C2×C3semidirect
19. C22×C3×SL(2,𝔽3)C2×C3semidirect
24. C2×C3×2−1+4C32semidirect
25. C2×2−1+4⋊C3C32direct (N × H)
31. C32×#41C22×C3semidirect
38. C2×C32×Q8C22×C3semidirect
46. C3×#41⋊C3C22×C3semidirect
48. C2×C3×SL(2,𝔽3)C22×C3semidirect
54. C3×2−1+4C2×C32semidirect
57. C2×C3×#41C2×C32semidirect
59. C22×C3×Q8C2×C32semidirect
60. 2−1+4⋊C3C2×C32direct (N × H)
61. C2×#41⋊C3C2×C32semidirect
62. C22×SL(2,𝔽3)C2×C32semidirect
86. C2×2−1+4C33semidirect
95. C3×#41C22×C32semidirect
102. C2×C3×Q8C22×C32semidirect
110. #41⋊C3C22×C32semidirect
112. C2×SL(2,𝔽3)C22×C32semidirect
127. 2−1+4C2×C33semidirect
130. C2×#41C2×C33semidirect
132. C22×Q8C2×C33semidirect
162. C2×C322−1+4⋊C3direct (N × H)
169. #41C22×C33semidirect
176. C2×Q8C22×C33semidirect
197. C32C2×2−1+4⋊C3direct (N × H)
222. C2×C3C3×2−1+4⋊C3direct (N × H)
234. C3C2×C3×2−1+4⋊C3direct (N × H)
237. C2C32×2−1+4⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C32×2−1+4⋊C32
C2×C32×#41⋊C32
C22×C32×SL(2,𝔽3)2
C2×C32×2−1+43
C2×C3×2−1+4⋊C33
C2×C33×D44