Finite Group Databasebeta

← #13179612

Subgroups — #13179612

Order 1728.

4732 conjugacy classes of subgroups; only the 931 normal subgroups are shown.

normal subgroups only

This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 931 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
1C22×#6312C211yesmaximal, characteristic
2C2×C33×C22⋊C4C211yesmaximal, characteristic, complemented
3C2×#88864C211yesmaximal, characteristic, complemented
4C4×#6312C214yesmaximal, complemented
5#11723922C214yesmaximal, complemented
6C22⋊C4×C3⋊(C3⋊D3)C212yesmaximal, complemented
7#11723918C212yesmaximal, complemented
8#10595957—313nomaximal
9C24×C33C2211yescharacteristic
10C23×C3⋊(C3⋊D3)C2211yescharacteristic
11C22×C33×C4C2212yescomplemented
12C2×#6312C418yescomplemented
13C2×#6312C2212yes—
14C2×#6312C2212yes—
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3⋊(C3⋊Dic3)C2212yescomplemented
17C2×C4×C3⋊(C3⋊D3)C2214yescomplemented
18C33×C22⋊C4C2212yes—
19C33×C22⋊C4C2212yescomplemented
20#88864C2212yes—
21#88864C2212yescomplemented
22#88859C2214yescomplemented
23#88857C2214yescomplemented
24C4×C3⋊(C3⋊Dic3)C2214yescomplemented
25C22×#638—313no—
26C2×C32×C22⋊C4D3113yescomplemented
27C2×#3757—313no—
28C4×#638—352no—
29#64090—352no—
30C22⋊C4×C3⋊D3—326no—
31#64086—326no—
32C23×C33—42no—
33C23×C33C2×C414yescomplemented
34C23×C33C2311yescharacteristic
35C23×C33C2311yescharacteristic
36C23×C33C2311yescharacteristic
37C22×C3⋊(C3⋊D3)—42no—
38C22×C3⋊(C3⋊D3)C2×C414yescomplemented
39C22×C3⋊(C3⋊D3)C2312yes—
40C2×C33×C4—42no—
41C2×C33×C4C2314yes—
42#6312C2×C418yescomplemented
43#6312C2×C418yescomplemented
44C2×C3⋊(C3⋊Dic3)—42no—
45C2×C3⋊(C3⋊Dic3)C2×C414yescomplemented
46C2×C3⋊(C3⋊Dic3)C2314yes—
47C2×C3⋊(C3⋊Dic3)C2312yes—
48C4×C3⋊(C3⋊D3)—44no—
49#5486—913no—
50C24×C32C2×D3113yes—
51C23×C3⋊D3—313no—
52C22×C32×C4C2×D3126yescomplemented
53C2×#638—3104no—
54C2×#638—326no—
55C2×#638—326no—
56C22×C3⋊Dic3—313no—
57C22×C3⋊Dic3—326no—
58C2×C4×C3⋊D3—352no—
59C32×C22⋊C4C2×D3126yes—
60C32×C22⋊C4C2×D3126yescomplemented
61#3757—326no—
62#3757—326no—
63#3752—352no—
64#3750—352no—
65C4×C3⋊Dic3—352no—
66C22×C33—22no—
67C22×C33—22no—
68C22×C33—22no—
69C22×C33—44no—
70C22×C33C22×C412yes—
71C22×C33C22×C412yes—
72C22×C33C22×C412yes—
73C22×C33C2411yescharacteristic
74C2×C3⋊(C3⋊D3)—24no—
75C2×C3⋊(C3⋊D3)—44no—
76C2×C3⋊(C3⋊D3)C22×C414yes—
77C33×C4—24no—
78C3⋊(C3⋊Dic3)—24no—
79C3⋊(C3⋊Dic3)C22×C414yes—
80C22×#67—913no—
81C2×C3×C22⋊C4C3⋊D3113yescomplemented
82C2×#217—913no—
83C4×#67—952no—
84#67⋊C4—952no—
85D3×C22⋊C4—926no—
86#877—926no—
87C23×C32—426no—
88C23×C32C4×D3152yescomplemented
89C23×C32C22×D3113yes—
90C23×C32C22×D3113yes—
91C23×C32C22×D3113yes—
92C22×C3⋊D3—326no—
93C22×C3⋊D3—352no—
94C22×C3⋊D3—1226no—
95C2×C32×C4—426no—
96C2×C32×C4C22×D3152yes—
97#638—3104no—
98#638—3104no—
99C2×C3⋊Dic3—352no—
100C2×C3⋊Dic3—326no—
101C2×C3⋊Dic3—352no—
102C2×C3⋊Dic3—1226no—
103C4×C3⋊D3—1252no—
104#518—271noSylow
105C2×C33—22no—
106C2×C33—22no—
107C2×C33—22no—
108C2×C33C23×C411yescharacteristic
109C2×C33H(5,𝔽2)12yes—
110C3⋊(C3⋊D3)—24no—
111C24×C3C2×C3⋊D3113yes—
112C23×D3—913no—
113C22×C3×C4C2×C3⋊D3126yescomplemented
114C2×#67—9104no—
115C2×#67—926no—
116C2×#67—926no—
117C22×Dic3—913no—
118C22×Dic3—926no—
119C2×C4×D3—952no—
120C3×C22⋊C4C2×C3⋊D3126yes—
121C3×C22⋊C4C2×C3⋊D3126yescomplemented
122#217—926no—
123#217—926no—
124(C2×D3)⋊C4—952no—
125Dic3⋊C4—952no—
126C4×Dic3—952no—
127C22×C32—226no—
128C22×C32—226no—
129C22×C32—226no—
130C22×C32—452no—
131C22×C32C2×C4×D3126yes—
132C22×C32C2×C4×D3126yes—
133C22×C32C2×C4×D3126yes—
134C22×C32C23×D3113yes—
135C2×C3⋊D3—352no—
136C2×C3⋊D3—652no—
137C2×C3⋊D3—1252no—
138C32×C4—252no—
139C3⋊Dic3—352no—
140C3⋊Dic3—652no—
141C22×D4—271no—
142C4×D4—274no—
143C4×D4—274no—
144#117—272no—
145C2×C22⋊C4—271no—
146C2×C22⋊C4—272no—
147C2×C22⋊C4C3⋊(C3⋊D3)11yescharacteristic, complemented
148C33#51811yescharacteristic, complemented, Sylow
149C23×C3—426no—
150C23×C3C4×C3⋊D3152yescomplemented
151C23×C3C22×C3⋊D3113yes—
152C23×C3C22×C3⋊D3113yes—
153C23×C3C22×C3⋊D3113yes—
154C22×D3—926no—
155C22×D3—952no—
156C22×D3—3626no—
157C2×C3×C4—426no—
158C2×C3×C4C22×C3⋊D3152yes—
159#67—9104no—
160#67—9104no—
161C2×Dic3—926no—
162C2×Dic3—952no—
163C2×Dic3—952no—
164C2×Dic3—3626no—
165C4×D3—3652no—
166C2×C32—226no—
167C2×C32—226no—
168C2×C32—226no—
169C2×C32C22×C4×D3113yes—
170C2×C32#1004126yes—
171C3⋊D3—652no—
172C24—271no—
173C24C2×C3⋊(C3⋊D3)11yescharacteristic
174C2×D4—278no—
175C2×D4—272no—
176C2×D4—272no—
177C22×C4—271no—
178C22×C4—272no—
179C22×C4—274no—
180C22×C4C2×C3⋊(C3⋊D3)12yescomplemented
181C4⋊C4—274no—
182C22⋊C4—272no—
183C22⋊C4—272no—
184C22⋊C4—274no—
185C22⋊C4C2×C3⋊(C3⋊D3)12yes—
186C22⋊C4C2×C3⋊(C3⋊D3)12yescomplemented
187C42—274no—
188C22×C3—226no—
189C22×C3—226no—
190C22×C3—226no—
191C22×C3—452no—
192C22×C3C2×C4×C3⋊D3126yes—
193C22×C3C2×C4×C3⋊D3126yes—
194C22×C3C2×C4×C3⋊D3126yes—
195C22×C3C23×C3⋊D3113yes—
196C2×D3—952no—
197C2×D3—1852no—
198C2×D3—3652no—
199C3×C4—252no—
200Dic3—952no—
201Dic3—1852no—
202C32#5486113yescomplemented
203C23—42no—
204C23—272no—
205C23—274no—
206C23—1082no—
207C23C4×C3⋊(C3⋊D3)14yescomplemented
208C23C22×C3⋊(C3⋊D3)11yescharacteristic
209C23C22×C3⋊(C3⋊D3)11yescharacteristic
210C23C22×C3⋊(C3⋊D3)11yescharacteristic
211D4—278no—
212D4—278no—
213C2×C4—42no—
214C2×C4—272no—
215C2×C4—274no—
216C2×C4—274no—
217C2×C4—1084no—
218C2×C4—1082no—
219C2×C4C22×C3⋊(C3⋊D3)14yes—
220C2×C3—226no—
221C2×C3—226no—
222C2×C3—226no—
223C2×C3C22×C4×C3⋊D3113yes—
224C2×C3#64361126yes—
225D3—1852no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—44no—
230C22—274no—
231C22—544no—
232C22—1084no—
233C22C2×C4×C3⋊(C3⋊D3)12yes—
234C22C2×C4×C3⋊(C3⋊D3)12yes—
235C22C2×C4×C3⋊(C3⋊D3)12yes—
236C22C23×C3⋊(C3⋊D3)11yescharacteristic
237C4—24no—
238C4—274no—
239C4—544no—
240C3#10595957113yescomplemented
241C2—22no—
242C2—22no—
243C2—22no—
244C2—544no—
245C2C22×C4×C3⋊(C3⋊D3)11yescharacteristic
246C2#1172467912yes—
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutatorC2×C33C23×C4
FittingC2×C33×C22⋊C4C2
socle, cosocleC22×C33C24
prime radicalC22×#6312C2
hypercenterC2×C22⋊C4C3⋊(C3⋊D3)
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C33×C22⋊C4C2semidirect
3. C2×#88864C2semidirect
4. C4×#6312C2semidirect
5. #11723922C2semidirect
6. C22⋊C4×C3⋊(C3⋊D3)C2semidirect
7. #11723918C2semidirect
11. C22×C33×C4C22semidirect
12. C2×#6312C4semidirect
16. C22×C3⋊(C3⋊Dic3)C22semidirect
17. C2×C4×C3⋊(C3⋊D3)C22semidirect
19. C33×C22⋊C4C22semidirect
21. #88864C22semidirect
22. #88859C22semidirect
23. #88857C22semidirect
24. C4×C3⋊(C3⋊Dic3)C22semidirect
26. C2×C32×C22⋊C4D3semidirect
33. C23×C33C2×C4semidirect
38. C22×C3⋊(C3⋊D3)C2×C4semidirect
42. #6312C2×C4semidirect
43. #6312C2×C4semidirect
45. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
52. C22×C32×C4C2×D3semidirect
60. C32×C22⋊C4C2×D3semidirect
81. C2×C3×C22⋊C4C3⋊D3semidirect
88. C23×C32C4×D3semidirect
113. C22×C3×C4C2×C3⋊D3semidirect
121. C3×C22⋊C4C2×C3⋊D3semidirect
147. C2×C22⋊C4C3⋊(C3⋊D3)semidirect
148. C33#518semidirect
150. C23×C3C4×C3⋊D3semidirect
180. C22×C4C2×C3⋊(C3⋊D3)semidirect
186. C22⋊C4C2×C3⋊(C3⋊D3)semidirect
202. C32#5486semidirect
207. C23C4×C3⋊(C3⋊D3)semidirect
240. C3#10595957semidirect

Maximal subgroups

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

subgroupindex
C22×#63122
C2×C33×C22⋊C42
C2×#888642
C4×#63122
#117239222
C22⋊C4×C3⋊(C3⋊D3)2
#117239182
#105959573