Finite Group Databasebeta

← C2×C3×#64237 #13181050

Subgroups — C2×C3×#64237 #13181050

Order 1728.

968 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 178 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×C3×#3805C212yesmaximal, complemented
2C22×C3×#622C211yesmaximal, characteristic, complemented
3C2×C3×#3776C211yesmaximal, characteristic, complemented
4C2×C3×#3775C212yesmaximal, complemented
5C3×#64237C218yesmaximal, direct product factor
6C2×C3×#3774C211yesmaximal, characteristic, complemented
7C2×#64237C311yesmaximal, characteristic, direct product factor
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×C32×#67—22no—
10C2×C3×#627C2212yes—
11C2×C3×#625C2212yes—
12C3×#3805—24no—
13C3×#3805C2218yescomplemented
14C2×C3×D3×Dic3—22no—
15C2×C3×#622C2211yescharacteristic
16C2×C3×#622C2211yescharacteristic
17C2×C3×#622C2214yescomplemented
18C3×#3776C2214yescomplemented
19C3×#3775C2218yescomplemented
20C3×#3774C2214yescomplemented
21C2×C3×#237—62no—
22C2×#3805C2×C312yescomplemented
23C22×#622C2×C311yescharacteristic, complemented
24C2×#3776C2×C311yescharacteristic, complemented
25C2×#3775C2×C312yescomplemented
26#64237C2×C318yesdirect product factor
27C2×#3774C2×C311yescharacteristic, complemented
28C23×C33D411yescharacteristic
29C22×C32×D3—22no—
30C2×C3×C3⋊Dic3D411yescharacteristic
31C2×C3×C3⋊Dic3D412yes—
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C2×C3×C3⋊Dic3C2312yes—
34C32×#67—28no—
35C2×C32×Dic3—22no—
36C3×#627—22no—
37C3×#627C2314yes—
38C3×#625—22no—
39C3×#625C2314yes—
40C3×D3×Dic3—28no—
41C3×#622C2312yes—
42C3×#622C2312yes—
43C2×C3×#135—91nomaximal
44C2×C32×D4—62no—
45C22×C3⋊Dic3C22×C311yescharacteristic
46C2×C3×#67—22no—
47C2×C3×#67—22no—
48C2×C3×#67—42no—
49C2×C3×#67—62no—
50C22×C3×Dic3—62no—
51C22×C3×Dic3—62no—
52C3×#237—68no—
53C3×#237—68no—
54C2×C3×C4×D3—62no—
55C2×C3×Dic6—62no—
56C2×#627C22×C312yes—
57C2×#625C22×C312yes—
58#3805—24no—
59#3805C22×C318yescomplemented
60C2×D3×Dic3—22no—
61C2×#622C22×C311yescharacteristic
62C2×#622C22×C311yescharacteristic
63C2×#622C22×C314yescomplemented
64#3776C22×C314yescomplemented
65#3775C22×C318yescomplemented
66#3774C22×C314yescomplemented
67C22×C33—22no—
68C22×C33C2×D412yes—
69C22×C33C2×D411yescharacteristic
70C2×C32×D3—24no—
71C2×C32×D3—44no—
72C3×C3⋊Dic3C2×D412yes—
73C3×C3⋊Dic3C2×D411yescharacteristic
74C3×C3⋊Dic3C2411yescharacteristic
75C32×Dic3—24no—
76C2×C3×#41—92no—
77C2×#237—62no—
78C3×#135—98no—
79C2×C3×Q16—91no—
80C2×C3×SD16—92no—
81C2×C3×D8—91no—
82C22×C3×C8—91no—
83C23×C32—22no—
84C23×C32—42no—
85C23×C32—62no—
86C23×C32C3×D411yescharacteristic
87C22×C3×D3—22no—
88C22×C3×D3—22no—
89C22×C3×D3—42no—
90C32×D4—68no—
91C2×C32×C4—62no—
92C2×C3⋊Dic3C3×D411yescharacteristic
93C2×C3⋊Dic3C3×D412yes—
94C2×C3⋊Dic3C23×C311yescharacteristic
95C2×C3⋊Dic3C23×C312yes—
96C3×#67—28no—
97C3×#67—28no—
98C3×#67—48no—
99C3×#67—64no—
100C3×#67—64no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—42no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—64no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—64no—
108C2×C3×Dic3—68no—
109C3×C4×D3—68no—
110C3×Dic6—64no—
111C3×Dic6—64no—
112#627—22no—
113#627C23×C314yes—
114#625—22no—
115#625C23×C314yes—
116D3×Dic3—28no—
117#622C23×C312yes—
118#622C23×C312yes—
119C2×#135—91noSylow
120C2×C33—22no—
121C2×C33#13512yescomplemented
122C2×C33C22×D411yescharacteristic
123C32×D3—44no—
124C3×#41—98no—
125C3×#41—184no—
126C2×C3×Q8—92no—
127C2×C3×D4—62no—
128C2×C3×D4—62no—
129C2×C3×D4—92no—
130C2×C3×D4—122no—
131C22×C3×C4—91no—
132C22×C3×C4—182no—
133C2×#67—22no—
134C2×#67—62no—
135C22×Dic3—62no—
136C22×Dic3—62no—
137#237—68no—
138#237—68no—
139C2×C4×D3—62no—
140C2×Dic6—62no—
141C3×Q16—94no—
142C3×SD16—98no—
143C3×D8—94no—
144C2×C3×C8—91no—
145C2×C3×C8—91no—
146C2×C3×C8—94no—
147C22×C32—24no—
148C22×C32—22no—
149C22×C32—22no—
150C22×C32—42no—
151C22×C32—44no—
152C22×C32—44no—
153C22×C32—64no—
154C22×C32—84no—
155C22×C32—124no—
156C22×C32C2×C3×D412yes—
157C22×C32C2×C3×D411yescharacteristic
158C2×C3×D3—24no—
159C2×C3×D3—24no—
160C2×C3×D3—44no—
161C2×C3×D3—44no—
162C2×C3×D3—44no—
163C2×C3×D3—84no—
164C32×C4—64no—
165C3⋊Dic3C2×C3×D412yes—
166C3⋊Dic3C2×C3×D411yescharacteristic
167C3⋊Dic3C24×C311yescharacteristic
168C3×Dic3—24no—
169C3×Dic3—24no—
170C3×Dic3—44no—
171C3×Dic3—62no—
172C3×Dic3—64no—
173C3×Dic3—62no—
174C2×#41—92no—
175#135—98no—
176C2×Q16—91no—
177C2×SD16—92no—
178C2×D8—91no—
179C22×C8—91no—
180C33C2×#13511yescharacteristic, complemented, Sylow
181C23×C3—22no—
182C23×C3—42no—
183C23×C3—62no—
184C23×C3—62no—
185C23×C3—122no—
186C23×C3GO+(4,𝔽2)11yescharacteristic
187C22×D3—22no—
188C3×Q8—94no—
189C3×Q8—182no—
190C3×D4—68no—
191C3×D4—68no—
192C3×D4—94no—
193C3×D4—128no—
194C3×D4—182no—
195C2×C3×C4—62no—
196C2×C3×C4—62no—
197C2×C3×C4—92no—
198C2×C3×C4—92no—
199C2×C3×C4—91no—
200C2×C3×C4—91no—
201C2×C3×C4—122no—
202C2×C3×C4—188no—
203#67—28no—
204#67—64no—
205#67—64no—
206C2×Dic3—22no—
207C2×Dic3—62no—
208C2×Dic3—64no—
209C2×Dic3—62no—
210C2×Dic3—64no—
211C2×Dic3—68no—
212C4×D3—68no—
213Dic6—64no—
214Dic6—64no—
215C3×C8—92no—
216C3×C8—92no—
217C2×C32—22no—
218C2×C32—24no—
219C2×C32—22no—
220C2×C32—44no—
221C2×C32—42no—
222C2×C32—44no—
223C2×C32—84no—
224C2×C32—124no—
225C2×C32C3×#13512yescomplemented
226C2×C32C22×C3×D411yescharacteristic
227C3×D3—44no—
228C3×D3—44no—
229C3×D3—84no—
230#41—98no—
231#41—184no—
232C2×Q8—92no—
233C2×D4—62no—
234C2×D4—92no—
235C22×C4—91no—
236C22×C4—182no—
237Q16—94no—
238SD16—98no—
239D8—94no—
240C2×C8—91no—
241C2×C8—91no—
242C2×C8—94no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—24no—
246C22×C3—44no—
247C22×C3—42no—
248C22×C3—44no—
249C22×C3—64no—
250C22×C3—64no—
251C22×C3—84no—
252C22×C3—124no—
253C22×C3—124no—
254C22×C3—124no—
255C22×C3—244no—
256C22×C3C2×GO+(4,𝔽2)12yes—
257C22×C3C2×GO+(4,𝔽2)11yescharacteristic
258C2×D3—24no—
259C2×D3—44no—
260C3×C4—64no—
261C3×C4—64no—
262C3×C4—91no—
263C3×C4—92no—
264C3×C4—91no—
265C3×C4—124no—
266Dic3—24no—
267Dic3—62no—
268Dic3—62no—
269Dic3—64no—
270C32—22no—
271C32—42no—
272C32C2×C3×#13511yescharacteristic, complemented
273C23—62no—
274C23C3×GO+(4,𝔽2)11yescharacteristic
275Q8—94no—
276Q8—182no—
277D4—68no—
278D4—94no—
279D4—182no—
280C2×C4—62no—
281C2×C4—91no—
282C2×C4—92no—
283C2×C4—91no—
284C2×C4—92no—
285C2×C4—188no—
286C8—92no—
287C8—92no—
288C2×C3—22no—
289C2×C3—24no—
290C2×C3—22no—
291C2×C3—44no—
292C2×C3—44no—
293C2×C3—42no—
294C2×C3—84no—
295C2×C3—124no—
296C2×C3—124no—
297C2×C3—244no—
298C2×C3#6423712yesdirect product factor
299C2×C3C22×GO+(4,𝔽2)11yescharacteristic
300D3—44no—
301C22—22no—
302C22—64no—
303C22—124no—
304C22C2×C3×GO+(4,𝔽2)12yes—
305C22C2×C3×GO+(4,𝔽2)11yescharacteristic
306C4—64no—
307C4—91no—
308C4—92no—
309C4—91no—
310C3—22no—
311C3—42no—
312C3C2×#6423711yescharacteristic, direct product factor
313C2—22no—
314C2—124no—
315C2C3×#6423712yesdirect product factor
316C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C2×C3×#135
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3805C2semidirect
2. C22×C3×#622C2semidirect
3. C2×C3×#3776C2semidirect
4. C2×C3×#3775C2semidirect
5. C3×#64237C2direct (N × H)
6. C2×C3×#3774C2semidirect
7. C2×#64237C3direct (N × H)
13. C3×#3805C22semidirect
17. C2×C3×#622C22semidirect
18. C3×#3776C22semidirect
19. C3×#3775C22semidirect
20. C3×#3774C22semidirect
22. C2×#3805C2×C3semidirect
23. C22×#622C2×C3semidirect
24. C2×#3776C2×C3semidirect
25. C2×#3775C2×C3semidirect
26. #64237C2×C3direct (N × H)
27. C2×#3774C2×C3semidirect
59. #3805C22×C3semidirect
63. C2×#622C22×C3semidirect
64. #3776C22×C3semidirect
65. #3775C22×C3semidirect
66. #3774C22×C3semidirect
121. C2×C33#135semidirect
180. C33C2×#135semidirect
225. C2×C32C3×#135semidirect
272. C32C2×C3×#135semidirect
298. C2×C3#64237direct (N × H)
312. C3C2×#64237direct (N × H)
315. C2C3×#64237direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38052
C22×C3×#6222
C2×C3×#37762
C2×C3×#37752
C3×#642372
C2×C3×#37742
C2×#642373
C2×C3×#1359