Finite Group Databasebeta

← C2×C3×#64223 #13181015

Subgroups — C2×C3×#64223 #13181015

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×#3798C212yesmaximal, complemented
2C2×C3×#3787C211yesmaximal, characteristic, complemented
3C2×C3×#3776C211yesmaximal, characteristic, complemented
4C2×C3×#3775C212yesmaximal, complemented
5C2×C3×#3774C211yesmaximal, characteristic, complemented
6C3×#64223C218yesmaximal, direct product factor
7C2×#64223C311yesmaximal, characteristic, direct product factor
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×C4×D3—22no—
10C2×C3×#627C2212yes—
11C2×C3×#626—22no—
12C2×C3×#625C2212yes—
13C3×#3798—24no—
14C3×#3798C2218yescomplemented
15C2×C3×#622C2211yescharacteristic
16C2×C3×#622C2211yescharacteristic
17C3×#3787C2214yescomplemented
18C3×#3776C2214yescomplemented
19C3×#3775C2218yescomplemented
20C3×#3774C2214yescomplemented
21C2×C3×#235—62no—
22C2×#3798C2×C312yescomplemented
23C2×#3787C2×C311yescharacteristic, complemented
24C2×#3776C2×C311yescharacteristic, complemented
25C2×#3775C2×C312yescomplemented
26C2×#3774C2×C311yescharacteristic, complemented
27#64223C2×C318yesdirect product factor
28C22×C3×C3⋊D3D411yescharacteristic
29C22×C32×D3—22no—
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C3×C4×C3⋊D3D412yes—
33C3×C4×C3⋊D3C2312yes—
34C2×C32×Dic3—22no—
35C32×C4×D3—28no—
36C3×#627—22no—
37C3×#627C2314yes—
38C3×#626—28no—
39C3×#625—22no—
40C3×#625C2314yes—
41C3×#622C2312yes—
42C3×#622C2312yes—
43C2×C3×#135—91nomaximal
44C22×C32×C4—62no—
45C2×C4×C3⋊D3C22×C311yescharacteristic
46C2×C3×#67—62no—
47C2×C3×#67—62no—
48C3×#235—68no—
49C3×#235—68no—
50C2×C3×D12—62no—
51C2×C3×C4×D3—22no—
52C2×C3×C4×D3—22no—
53C2×C3×C4×D3—42no—
54C2×C3×C4×D3—62no—
55C2×C3×Dic6—62no—
56C2×#627C22×C312yes—
57C2×#626—22no—
58C2×#625C22×C312yes—
59#3798—24no—
60#3798C22×C318yescomplemented
61C2×#622C22×C311yescharacteristic
62C2×#622C22×C311yescharacteristic
63#3787C22×C314yescomplemented
64#3776C22×C314yescomplemented
65#3775C22×C318yescomplemented
66#3774C22×C314yescomplemented
67C22×C33C2×D411yescharacteristic
68C2×C3×C3⋊D3—22no—
69C2×C3×C3⋊D3C2×D412yes—
70C2×C32×D3—24no—
71C2×C32×D3—44no—
72C33×C4C2×D412yes—
73C3×C3⋊Dic3C2×D411yescharacteristic
74C3×C3⋊Dic3C2411yescharacteristic
75C32×Dic3—24no—
76C2×C3×#41—92no—
77C2×#235—62no—
78C3×#135—98no—
79C2×C3×Q16—91no—
80C2×C3×SD16—92no—
81C2×C3×D8—91no—
82C22×C3×C8—91no—
83C23×C32—62no—
84C22×C3⋊D3C3×D411yescharacteristic
85C22×C3×D3—22no—
86C22×C3×D3—22no—
87C22×C3×D3—42no—
88C22×C3×D3—62no—
89C2×C32×C4—22no—
90C2×C32×C4—42no—
91C2×C32×C4—68no—
92C2×C32×C4—62no—
93C2×C32×C4C3×D411yescharacteristic
94C2×C3⋊Dic3C23×C311yescharacteristic
95C4×C3⋊D3C3×D412yes—
96C4×C3⋊D3C23×C312yes—
97C3×#67—64no—
98C3×#67—68no—
99C3×#67—64no—
100C2×C3×Dic3—22no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—42no—
103C2×C3×Dic3—62no—
104C3×D12—68no—
105C3×C4×D3—28no—
106C3×C4×D3—28no—
107C3×C4×D3—48no—
108C3×C4×D3—64no—
109C3×C4×D3—64no—
110C3×Dic6—64no—
111C3×Dic6—64no—
112#627—22no—
113#627C23×C314yes—
114#626—28no—
115#625—22no—
116#625C23×C314yes—
117#622C23×C312yes—
118#622C23×C312yes—
119C2×#135—91noSylow
120C2×C33#13512yescomplemented
121C2×C33C22×D411yescharacteristic
122C3×C3⋊D3—22no—
123C32×D3—44no—
124C3×#41—98no—
125C3×#41—184no—
126C2×C3×Q8—92no—
127C2×C3×D4—92no—
128C2×C3×D4—182no—
129C22×C3×C4—62no—
130C22×C3×C4—62no—
131C22×C3×C4—91no—
132C22×C3×C4—122no—
133C2×#67—62no—
134C2×#67—62no—
135#235—68no—
136#235—68no—
137C2×D12—62no—
138C2×C4×D3—22no—
139C2×C4×D3—62no—
140C2×Dic6—62no—
141C3×Q16—94no—
142C3×SD16—98no—
143C3×D8—94no—
144C2×C3×C8—94no—
145C2×C3×C8—91no—
146C2×C3×C8—91no—
147C22×C32—22no—
148C22×C32—42no—
149C22×C32—64no—
150C22×C32—124no—
151C22×C32C2×C3×D411yescharacteristic
152C2×C3⋊D3—22no—
153C2×C3⋊D3C2×C3×D412yes—
154C2×C3×D3—24no—
155C2×C3×D3—24no—
156C2×C3×D3—44no—
157C2×C3×D3—44no—
158C2×C3×D3—48no—
159C2×C3×D3—44no—
160C2×C3×D3—64no—
161C2×C3×D3—124no—
162C32×C4—24no—
163C32×C4—44no—
164C32×C4—64no—
165C32×C4C2×C3×D412yes—
166C3⋊Dic3C2×C3×D411yescharacteristic
167C3⋊Dic3C24×C311yescharacteristic
168C3×Dic3—24no—
169C3×Dic3—24no—
170C3×Dic3—44no—
171C3×Dic3—62no—
172C3×Dic3—62no—
173C2×#41—92no—
174#135—98no—
175C2×Q16—91no—
176C2×SD16—92no—
177C2×D8—91no—
178C22×C8—91no—
179C33C2×#13511yescharacteristic, complemented, Sylow
180C23×C3—62no—
181C23×C3—62no—
182C23×C3—91no—
183C23×C3—122no—
184C22×D3—22no—
185C22×D3—62no—
186C3×Q8—94no—
187C3×Q8—182no—
188C3×D4—94no—
189C3×D4—188no—
190C3×D4—182no—
191C2×C3×C4—22no—
192C2×C3×C4—42no—
193C2×C3×C4—68no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—68no—
197C2×C3×C4—91no—
198C2×C3×C4—92no—
199C2×C3×C4—92no—
200C2×C3×C4—128no—
201C2×C3×C4—122no—
202C2×C3×C4GO+(4,𝔽2)11yescharacteristic
203#67—64no—
204#67—68no—
205#67—64no—
206C2×Dic3—22no—
207C2×Dic3—62no—
208D12—68no—
209C4×D3—28no—
210C4×D3—64no—
211C4×D3—64no—
212Dic6—64no—
213Dic6—64no—
214C3×C8—92no—
215C3×C8—92no—
216C2×C32—22no—
217C2×C32—24no—
218C2×C32—44no—
219C2×C32—42no—
220C2×C32—124no—
221C2×C32C3×#13512yescomplemented
222C2×C32C22×C3×D411yescharacteristic
223C3⋊D3—22no—
224C3×D3—44no—
225C3×D3—48no—
226C3×D3—44no—
227C3×D3—124no—
228#41—98no—
229#41—184no—
230C2×Q8—92no—
231C2×D4—92no—
232C2×D4—182no—
233C22×C4—62no—
234C22×C4—91no—
235Q16—94no—
236SD16—98no—
237D8—94no—
238C2×C8—91no—
239C2×C8—91no—
240C2×C8—94no—
241C22×C3—22no—
242C22×C3—42no—
243C22×C3—64no—
244C22×C3—64no—
245C22×C3—92no—
246C22×C3—124no—
247C22×C3—124no—
248C22×C3—128no—
249C22×C3—124no—
250C22×C3—182no—
251C22×C3C2×GO+(4,𝔽2)11yescharacteristic
252C2×D3—24no—
253C2×D3—44no—
254C2×D3—64no—
255C2×D3—124no—
256C3×C4—24no—
257C3×C4—44no—
258C3×C4—64no—
259C3×C4—64no—
260C3×C4—91no—
261C3×C4—91no—
262C3×C4—124no—
263C3×C4C2×GO+(4,𝔽2)12yes—
264Dic3—24no—
265Dic3—62no—
266Dic3—62no—
267C32—22no—
268C32—42no—
269C32C2×C3×#13511yescharacteristic, complemented
270C23—62no—
271C23—91no—
272Q8—94no—
273Q8—182no—
274D4—94no—
275D4—188no—
276D4—182no—
277C2×C4—62no—
278C2×C4—68no—
279C2×C4—91no—
280C2×C4—92no—
281C2×C4—92no—
282C2×C4C3×GO+(4,𝔽2)11yescharacteristic
283C8—92no—
284C8—92no—
285C2×C3—22no—
286C2×C3—24no—
287C2×C3—42no—
288C2×C3—44no—
289C2×C3—124no—
290C2×C3—128no—
291C2×C3—124no—
292C2×C3—182no—
293C2×C3#6422312yesdirect product factor
294C2×C3C22×GO+(4,𝔽2)11yescharacteristic
295D3—44no—
296D3—124no—
297C22—64no—
298C22—92no—
299C22—124no—
300C22—182no—
301C22C2×C3×GO+(4,𝔽2)11yescharacteristic
302C4—64no—
303C4—91no—
304C4—91no—
305C4C2×C3×GO+(4,𝔽2)12yes—
306C3—22no—
307C3—42no—
308C3C2×#6422311yescharacteristic, direct product factor
309C2—124no—
310C2—182no—
311C2C3×#6422312yesdirect product factor
312C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypocenterC32C2×C3×#135
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3798C2semidirect
2. C2×C3×#3787C2semidirect
3. C2×C3×#3776C2semidirect
4. C2×C3×#3775C2semidirect
5. C2×C3×#3774C2semidirect
6. C3×#64223C2direct (N × H)
7. C2×#64223C3direct (N × H)
14. C3×#3798C22semidirect
17. C3×#3787C22semidirect
18. C3×#3776C22semidirect
19. C3×#3775C22semidirect
20. C3×#3774C22semidirect
22. C2×#3798C2×C3semidirect
23. C2×#3787C2×C3semidirect
24. C2×#3776C2×C3semidirect
25. C2×#3775C2×C3semidirect
26. C2×#3774C2×C3semidirect
27. #64223C2×C3direct (N × H)
60. #3798C22×C3semidirect
63. #3787C22×C3semidirect
64. #3776C22×C3semidirect
65. #3775C22×C3semidirect
66. #3774C22×C3semidirect
120. C2×C33#135semidirect
179. C33C2×#135semidirect
221. C2×C32C3×#135semidirect
269. C32C2×C3×#135semidirect
293. C2×C3#64223direct (N × H)
308. C3C2×#64223direct (N × H)
311. C2C3×#64223direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37982
C2×C3×#37872
C2×C3×#37762
C2×C3×#37752
C2×C3×#37742
C3×#642232
C2×#642233
C2×C3×#1359