Subgroups — C₃×#2544 #84408

Order 384.

704 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#394	C₂	1	4	yes	maximal, complemented
6	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#386	C₂	1	4	yes	maximal, complemented
9	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#2544	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×D₄	C₂²	1	4	yes	—
14	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₃×C₄	C₂²	1	2	yes	—
16	C₃×#127	—	2	4	no	—
17	C₃×#124	—	2	4	no	—
18	C₃×#124	—	2	4	no	—
19	C₃×C₄×D₄	—	2	4	no	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
27	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
30	C₃×#95	C₂²	1	2	yes	—
31	C₃×#95	C₂²	1	4	yes	—
32	C₂×#127	C₂×C₃	1	1	yes	characteristic
33	C₂×#124	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×#124	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
36	#394	C₂×C₃	1	4	yes	complemented
37	#391	C₂×C₃	1	1	yes	characteristic, complemented
38	#390	C₂×C₃	1	1	yes	characteristic, complemented
39	#386	C₂×C₃	1	4	yes	complemented
40	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂⁴×C₃	C₂³	1	2	yes	—
42	C₂⁴×C₃	C₂³	1	2	yes	—
43	C₂×C₃×Q₈	—	2	2	no	—
44	C₂×C₃×Q₈	—	2	2	no	—
45	C₂×C₃×Q₈	—	2	2	no	—
46	C₂×C₃×D₄	—	2	8	no	—
47	C₂×C₃×D₄	—	2	2	no	—
48	C₂×C₃×D₄	—	2	2	no	—
49	C₂×C₃×D₄	—	2	2	no	—
50	C₂×C₃×D₄	—	2	2	no	—
51	C₂×C₃×D₄	—	2	8	no	—
52	C₂×C₃×D₄	—	2	2	no	—
53	C₂×C₃×D₄	—	4	4	no	—
54	C₂×C₃×D₄	D₄	1	4	yes	—
55	C₂²×C₃×C₄	—	2	4	no	—
56	C₂²×C₃×C₄	—	4	4	no	—
57	C₂²×C₃×C₄	C₂³	1	2	yes	—
58	C₂²×C₃×C₄	C₂³	1	2	yes	—
59	C₂²×C₃×C₄	C₂³	1	2	yes	—
60	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂³	1	2	yes	—
63	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	2	4	no	—
66	C₃×C₂²⋊C₄	—	2	8	no	—
67	C₃×C₂²⋊C₄	—	2	4	no	—
68	C₃×C₂²⋊C₄	—	2	4	no	—
69	C₃×C₂²⋊C₄	—	2	4	no	—
70	C₃×C₂²⋊C₄	—	4	4	no	—
71	C₃×C₄²	—	2	2	no	—
72	C₃×C₄²	—	2	2	no	—
73	C₃×C₄²	D₄	1	4	yes	complemented
74	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
75	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
76	C₂²×D₄	C₂²×C₃	1	4	yes	—
77	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
78	C₂³×C₄	C₂²×C₃	1	2	yes	—
79	#127	—	2	4	no	—
80	#124	—	2	4	no	—
81	#124	—	2	4	no	—
82	C₄×D₄	—	2	4	no	—
83	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
85	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
86	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
88	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
89	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
90	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
91	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
92	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
93	#95	C₂²×C₃	1	2	yes	—
94	#95	C₂²×C₃	1	4	yes	—
95	C₂³×C₃	—	2	2	no	—
96	C₂³×C₃	—	2	2	no	—
97	C₂³×C₃	—	2	4	no	—
98	C₂³×C₃	—	2	4	no	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	—	2	2	no	—
101	C₂³×C₃	—	2	2	no	—
102	C₂³×C₃	—	2	4	no	—
103	C₂³×C₃	—	2	4	no	—
104	C₂³×C₃	C₂×D₄	1	4	yes	—
105	C₂³×C₃	C₂⁴	1	1	yes	characteristic
106	C₃×Q₈	—	4	4	no	—
107	C₃×D₄	—	2	8	no	—
108	C₃×D₄	—	4	16	no	—
109	C₃×D₄	—	4	4	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	4	no	—
122	C₂×C₃×C₄	—	4	8	no	—
123	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
124	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
125	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
126	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
127	C₂×C₃×C₄	#41	1	4	yes	—
128	C₂×C₃×C₄	#41	1	2	yes	—
129	C₂⁴	C₂³×C₃	1	2	yes	—
130	C₂⁴	C₂³×C₃	1	2	yes	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	2	2	no	—
134	C₂×D₄	—	2	8	no	—
135	C₂×D₄	—	2	2	no	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	2	8	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	—	4	4	no	—
142	C₂×D₄	C₃×D₄	1	4	yes	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	4	4	no	—
145	C₂²×C₄	C₂³×C₃	1	2	yes	—
146	C₂²×C₄	C₂³×C₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₃	1	2	yes	—
151	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₂²⋊C₄	—	2	8	no	—
155	C₂²⋊C₄	—	2	4	no	—
156	C₂²⋊C₄	—	2	4	no	—
157	C₂²⋊C₄	—	2	4	no	—
158	C₂²⋊C₄	—	4	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	C₃×D₄	1	4	yes	complemented
162	C₂²×C₃	—	2	4	no	—
163	C₂²×C₃	—	2	8	no	—
164	C₂²×C₃	—	4	2	no	—
165	C₂²×C₃	—	4	2	no	—
166	C₂²×C₃	—	4	4	no	—
167	C₂²×C₃	—	4	2	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	—	4	2	no	—
170	C₂²×C₃	—	4	4	no	—
171	C₂²×C₃	—	4	4	no	—
172	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
173	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
174	C₂²×C₃	C₂×#41	1	1	yes	characteristic
175	C₂²×C₃	C₂×#41	1	2	yes	—
176	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	2	4	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	—	4	2	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	#121	1	4	yes	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	4	no	—
187	C₂³	—	2	4	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	2	4	no	—
193	C₂³	C₂×C₃×D₄	1	4	yes	—
194	C₂³	C₂⁴×C₃	1	1	yes	characteristic
195	Q₈	—	4	4	no	—
196	D₄	—	2	8	no	—
197	D₄	—	4	16	no	—
198	D₄	—	4	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
213	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
214	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
215	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
216	C₂×C₄	C₃×#41	1	4	yes	—
217	C₂×C₄	C₃×#41	1	2	yes	—
218	C₂×C₃	—	4	4	no	—
219	C₂×C₃	—	8	2	no	—
220	C₂×C₃	C₂×#121	1	1	yes	characteristic
221	C₂×C₃	#532	1	1	yes	characteristic
222	C₂×C₃	#535	1	1	yes	characteristic
223	C₂×C₃	#546	1	2	yes	—
224	C₂×C₃	#559	1	2	yes	—
225	C₂²	—	2	4	no	—
226	C₂²	—	2	8	no	—
227	C₂²	—	4	4	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	4	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	4	no	—
235	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
236	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
237	C₂²	C₂×C₃×#41	1	1	yes	characteristic
238	C₂²	C₂×C₃×#41	1	2	yes	—
239	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	4	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	C₃×#121	1	4	yes	—
247	C₃	#2544	1	1	yes	characteristic, direct product factor, Sylow
248	C₂	—	4	4	no	—
249	C₂	—	8	2	no	—
250	C₂	C₂×C₃×#121	1	1	yes	characteristic
251	C₂	C₃×#532	1	1	yes	characteristic
252	C₂	C₃×#535	1	1	yes	characteristic
253	C₂	C₃×#546	1	2	yes	—
254	C₂	C₃×#559	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂×C₃×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#124	C₂	semidirect
3. C₂×C₃×#124	C₂	semidirect
4. C₂×C₃×C₄×D₄	C₂	semidirect
5. C₃×#394	C₂	semidirect
6. C₃×#391	C₂	semidirect
7. C₃×#390	C₂	semidirect
8. C₃×#386	C₂	semidirect
9. C₃×C₄×C₄⋊C₄	C₂	semidirect
10. #2544	C₃	direct (N × H)
33. C₂×#124	C₂×C₃	semidirect
34. C₂×#124	C₂×C₃	semidirect
35. C₂×C₄×D₄	C₂×C₃	semidirect
36. #394	C₂×C₃	semidirect
37. #391	C₂×C₃	semidirect
38. #390	C₂×C₃	semidirect
39. #386	C₂×C₃	semidirect
40. C₄×C₄⋊C₄	C₂×C₃	semidirect
73. C₃×C₄²	D₄	semidirect
161. C₄²	C₃×D₄	semidirect
247. C₃	#2544	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#127	2
C₂×C₃×#124	2
C₂×C₃×C₄×D₄	2
C₃×#394	2
C₃×#391	2
C₃×#390	2
C₃×#386	2
C₃×C₄×C₄⋊C₄	2
#2544	3

Subgroups — C3×#2544 #84408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2544 #84408