Finite Group Databasebeta

← C2×#11724336 #13181133

Subgroups — C2×#11724336 #13181133

Order 1728.

1256 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×C4⋊#1113C212yesmaximal, complemented
2C2×C3×#3787C211yesmaximal, characteristic, complemented
3C2×#88941C214yesmaximal, complemented
4#11724336C218yesmaximal, direct product factor
5C2×#64225—31nomaximal
6C22×#1113—22no—
7C2×C3×C4⋊(C3⋊D3)—22no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×#6295C2214yescomplemented
10C4⋊#1113—24no—
11C4⋊#1113C2218yescomplemented
12C2×C3×#622C2212yescomplemented
13C3×#3787C2214yescomplemented
14#88941C22116yescomplemented
15C2×#3802—32no—
16C2×D3×D12—62no—
17C2×#3787D311yescharacteristic, complemented
18C2×#3774—34no—
19#64225—38no—
20C22×C3×C3⋊D3—24no—
21C22×C3×C3⋊D3D411yescharacteristic
22C2×#1113—28no—
23C2×#1113—48no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C3×C4⋊(C3⋊D3)—28no—
27C3×C4×C3⋊D3D412yes—
28C3×C4×C3⋊D3C2312yes—
29#6295—24no—
30#6295C2318yescomplemented
31C3×#622C2314yescomplemented
32C22×D24—91nomaximal
33C22×D32—64no—
34C22×D32—62no—
35C2×C4⋊(C3⋊D3)—22no—
36C2×C4×C3⋊D3C2×D311yescharacteristic
37C2×C3×D12—62no—
38C2×C3×D12—62no—
39C2×C3×D12—122no—
40C2×C3×C4×D3—62no—
41C2×#626—64no—
42C2×#625—34no—
43#3802—38no—
44#3802—64no—
45D3×D12—68no—
46D3×D12—68no—
47C2×#622C2×D312yescomplemented
48#3787C2×D314yescomplemented
49#3774—316no—
50C22×C33C2×D411yescharacteristic
51C2×C3×C3⋊D3—28no—
52C2×C3×C3⋊D3—48no—
53C2×C3×C3⋊D3D814yescomplemented
54C2×C3×C3⋊D3C2×D412yes—
55#1113—416no—
56C33×C4C2×D412yes—
57C3×C3⋊Dic3C2×D411yescharacteristic
58C3×C3⋊Dic3C2411yescharacteristic
59C2×D3×D4—182no—
60C22×D12—92no—
61C22×C3×C8—91no—
62C2×D24—98no—
63C2×D24—94no—
64C22×C3⋊D3—24no—
65C22×C3⋊D3D1211yescharacteristic
66C22×C3×D3—62no—
67C22×C3×D3—64no—
68C22×C3×D3—64no—
69C22×C3×D3—124no—
70C2×D32—616no—
71C2×D32—68no—
72C2×D32—1216no—
73C2×D32—128no—
74C2×C32×C4—22no—
75C2×C32×C4—42no—
76C2×C32×C4D1211yescharacteristic
77C2×C3⋊Dic3C22×D311yescharacteristic
78C4⋊(C3⋊D3)—28no—
79C4×C3⋊D3D1212yes—
80C4×C3⋊D3C22×D312yes—
81C2×C3×Dic3—62no—
82C3×D12—68no—
83C3×D12—68no—
84C3×D12—128no—
85C3×C4×D3—64no—
86C3×C4×D3—64no—
87#626—68no—
88#626—68no—
89#625—38no—
90#625—64no—
91#622C22×D314yescomplemented
92C22×D8—271noSylow
93C2×C33C2×D812yescomplemented
94C2×C33C22×D411yescharacteristic
95C3×C3⋊D3—48no—
96C3×C3⋊D3C2×D814yescomplemented
97C23×D3—182no—
98C23×D3—184no—
99C2×C3×D4—182no—
100C22×C3×C4—91no—
101C2×#67—184no—
102D3×D4—188no—
103D3×D4—188no—
104C2×D12—62no—
105C2×D12—62no—
106C2×D12—98no—
107C2×D12—94no—
108C2×D12—122no—
109C2×D12—184no—
110C2×C4×D3—62no—
111C2×C3×C8—94no—
112C2×C3×C8—92no—
113D24—916no—
114C22×C32—22no—
115C22×C32—42no—
116C22×C32C2×D1211yescharacteristic
117C2×C3⋊D3—28no—
118C2×C3⋊D3—48no—
119C2×C3⋊D3D2414yescomplemented
120C2×C3⋊D3C2×D1212yes—
121C2×C3×D3—68no—
122C2×C3×D3—64no—
123C2×C3×D3—68no—
124C2×C3×D3—68no—
125C2×C3×D3—128no—
126C2×C3×D3—128no—
127C2×C3×D3—128no—
128C2×C3×D3—248no—
129D32—1232no—
130D32—1216no—
131C32×C4—24no—
132C32×C4—44no—
133C32×C4C2×D1212yes—
134C3⋊Dic3C2×D1211yescharacteristic
135C3⋊Dic3C23×D311yescharacteristic
136C3×Dic3—62no—
137C3×Dic3—62no—
138C22×D4—272no—
139C2×D8—278no—
140C2×D8—274no—
141C22×C8—91no—
142C33C22×D811yescharacteristic, complemented, Sylow
143C23×C3—91no—
144C23×C3—184no—
145C22×D3—62no—
146C22×D3—64no—
147C22×D3—64no—
148C22×D3—124no—
149C22×D3—188no—
150C22×D3—1816no—
151C22×D3—3616no—
152C22×D3—368no—
153C3×D4—188no—
154C2×C3×C4—22no—
155C2×C3×C4—42no—
156C2×C3×C4—91no—
157C2×C3×C4—92no—
158C2×C3×C4—92no—
159C2×C3×C4GO+(4,𝔽2)11yescharacteristic
160#67—188no—
161#67—188no—
162C2×Dic3—62no—
163D12—68no—
164D12—68no—
165D12—98no—
166D12—128no—
167D12—184no—
168C4×D3—64no—
169C4×D3—64no—
170C3×C8—94no—
171C2×C32—22no—
172C2×C32—24no—
173C2×C32—44no—
174C2×C32—42no—
175C2×C32C2×D2412yescomplemented
176C2×C32C22×D1211yescharacteristic
177C3⋊D3—48no—
178C3⋊D3C2×D2414yescomplemented
179C3×D3—68no—
180C3×D3—128no—
181C3×D3—128no—
182C3×D3—248no—
183C24—542no—
184C2×D4—182no—
185C2×D4—278no—
186C2×D4—274no—
187C2×D4—544no—
188C22×C4—91no—
189D8—2716no—
190C2×C8—94no—
191C2×C8—92no—
192C22×C3—22no—
193C22×C3—42no—
194C22×C3—92no—
195C22×C3—94no—
196C22×C3—188no—
197C22×C3—368no—
198C22×C3C2×GO+(4,𝔽2)11yescharacteristic
199C2×D3—64no—
200C2×D3—68no—
201C2×D3—68no—
202C2×D3—68no—
203C2×D3—128no—
204C2×D3—128no—
205C2×D3—128no—
206C2×D3—248no—
207C2×D3—3632no—
208C2×D3—3616no—
209C3×C4—24no—
210C3×C4—44no—
211C3×C4—91no—
212C3×C4—91no—
213C3×C4C2×GO+(4,𝔽2)12yes—
214Dic3—62no—
215Dic3—62no—
216C32—22no—
217C32—42no—
218C32C22×D2411yescharacteristic, complemented
219C23—91no—
220C23—184no—
221C23—548no—
222C23—1088no—
223D4—188no—
224D4—278no—
225D4—544no—
226C2×C4—91no—
227C2×C4—92no—
228C2×C4—92no—
229C2×C4#157⋊D311yescharacteristic
230C8—94no—
231C2×C3—24no—
232C2×C3—22no—
233C2×C3—44no—
234C2×C3—42no—
235C2×C3—94no—
236C2×C3—368no—
237C2×C3#6422512yescomplemented
238C2×C3C22×GO+(4,𝔽2)11yescharacteristic
239D3—68no—
240D3—128no—
241D3—128no—
242D3—248no—
243C22—94no—
244C22—92no—
245C22—188no—
246C22—368no—
247C22—10816no—
248C22C2×#157⋊D311yescharacteristic
249C4—91no—
250C4—91no—
251C4C2×#157⋊D312yes—
252C3—22no—
253C3—42no—
254C3C2×#6422511yescharacteristic, complemented
255C2—94no—
256C2—368no—
257C2#1172433612yesdirect product factor
258C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC22C2×#157⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33C22×D8
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#1113C2semidirect
2. C2×C3×#3787C2semidirect
3. C2×#88941C2semidirect
4. #11724336C2direct (N × H)
9. C2×#6295C22semidirect
11. C4⋊#1113C22semidirect
12. C2×C3×#622C22semidirect
13. C3×#3787C22semidirect
14. #88941C22semidirect
17. C2×#3787D3semidirect
30. #6295C23semidirect
31. C3×#622C23semidirect
47. C2×#622C2×D3semidirect
48. #3787C2×D3semidirect
53. C2×C3×C3⋊D3D8semidirect
91. #622C22×D3semidirect
93. C2×C33C2×D8semidirect
96. C3×C3⋊D3C2×D8semidirect
119. C2×C3⋊D3D24semidirect
142. C33C22×D8semidirect
175. C2×C32C2×D24semidirect
178. C3⋊D3C2×D24semidirect
218. C32C22×D24semidirect
237. C2×C3#64225semidirect
254. C3C2×#64225semidirect
257. C2#11724336direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C2×C3×#37872
C2×#889412
#117243362
C2×#642253
C22×D249