Subgroups — C₃×C₄×#124 #84167

Order 384.

652 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
4	C₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₃×#388	C₂	1	4	yes	maximal, complemented
7	C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
9	C₃×C₄³	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#124	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	C₂³×C₃×C₄	C₂²	1	2	yes	—
14	C₃×#124	C₄	1	8	yes	direct product factor
15	C₃×C₄×Q₈	—	2	4	no	—
16	C₃×C₄×D₄	—	2	4	no	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₂×C₃×C₄²	C₂²	1	2	yes	—
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂²	1	2	yes	—
26	C₂×C₃×C₄²	C₂²	1	2	yes	—
27	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
29	C₃×#95	C₂²	1	4	yes	—
30	C₃×#95	C₂²	1	4	yes	—
31	C₂×#124	C₂×C₃	1	1	yes	characteristic
32	C₂×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
34	#391	C₂×C₃	1	1	yes	characteristic, complemented
35	#390	C₂×C₃	1	1	yes	characteristic
36	#388	C₂×C₃	1	4	yes	complemented
37	#382	C₂×C₃	1	1	yes	characteristic, complemented
38	C₄×C₂²⋊C₄	C₂×C₃	1	4	yes	complemented
39	C₄³	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂⁴×C₃	C₂³	1	2	yes	—
41	C₂×C₃×Q₈	—	2	2	no	—
42	C₂×C₃×Q₈	—	2	2	no	—
43	C₂×C₃×Q₈	C₂×C₄	1	4	yes	complemented
44	C₂×C₃×D₄	—	2	2	no	—
45	C₂×C₃×D₄	—	2	2	no	—
46	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
47	C₂²×C₃×C₄	—	2	4	no	—
48	C₂²×C₃×C₄	—	2	8	no	—
49	C₂²×C₃×C₄	C₂³	1	2	yes	—
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	2	yes	—
52	C₂²×C₃×C₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂³	1	2	yes	—
55	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₃×C₄⋊C₄	—	2	2	no	—
58	C₃×C₄⋊C₄	—	2	4	no	—
59	C₃×C₄⋊C₄	—	2	2	no	—
60	C₃×C₂²⋊C₄	—	2	4	no	—
61	C₃×C₂²⋊C₄	—	2	8	no	—
62	C₃×C₂²⋊C₄	C₂×C₄	1	16	yes	complemented
63	C₃×C₄²	—	2	4	no	—
64	C₃×C₄²	—	2	4	no	—
65	C₃×C₄²	—	2	2	no	—
66	C₃×C₄²	—	2	4	no	—
67	C₃×C₄²	C₂×C₄	1	4	yes	complemented
68	C₃×C₄²	D₄	1	4	yes	—
69	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
70	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
71	C₂³×C₄	C₂²×C₃	1	2	yes	—
72	#124	C₃×C₄	1	8	yes	direct product factor
73	C₄×Q₈	—	2	4	no	—
74	C₄×D₄	—	2	4	no	—
75	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
80	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
81	C₂×C₄²	C₂²×C₃	1	2	yes	—
82	C₂×C₄²	C₂²×C₃	1	2	yes	—
83	C₂×C₄²	C₂²×C₃	1	2	yes	—
84	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
87	#95	C₂²×C₃	1	4	yes	—
88	#95	C₂²×C₃	1	4	yes	—
89	C₂³×C₃	—	2	2	no	—
90	C₂³×C₃	—	2	4	no	—
91	C₂³×C₃	—	2	4	no	—
92	C₂³×C₃	—	2	2	no	—
93	C₂³×C₃	C₂²×C₄	1	4	yes	—
94	C₂³×C₃	C₂⁴	1	1	yes	characteristic
95	C₃×Q₈	—	2	8	no	—
96	C₃×D₄	—	2	8	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	2	4	no	—
108	C₂×C₃×C₄	—	4	8	no	—
109	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
110	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
111	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
112	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
113	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
114	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
115	C₂×C₃×C₄	#41	1	4	yes	—
116	C₂×C₃×C₄	#41	1	2	yes	—
117	C₂×C₃×C₄	#41	1	4	yes	—
118	C₂⁴	C₂³×C₃	1	2	yes	—
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	—	2	2	no	—
121	C₂×Q₈	C₂×C₃×C₄	1	4	yes	complemented
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	8	no	—
127	C₂²×C₄	C₂³×C₃	1	2	yes	—
128	C₂²×C₄	C₂³×C₃	1	2	yes	—
129	C₂²×C₄	C₂³×C₃	1	2	yes	—
130	C₂²×C₄	C₂³×C₃	1	2	yes	—
131	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
135	C₄⋊C₄	—	2	2	no	—
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₂²⋊C₄	—	2	4	no	—
139	C₂²⋊C₄	—	2	8	no	—
140	C₂²⋊C₄	C₂×C₃×C₄	1	16	yes	complemented
141	C₄²	—	2	4	no	—
142	C₄²	—	2	4	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	4	no	—
145	C₄²	C₂×C₃×C₄	1	4	yes	complemented
146	C₄²	C₃×D₄	1	4	yes	—
147	C₂²×C₃	—	2	4	no	—
148	C₂²×C₃	—	2	8	no	—
149	C₂²×C₃	—	4	2	no	—
150	C₂²×C₃	—	4	4	no	—
151	C₂²×C₃	—	4	2	no	—
152	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
153	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
154	C₂²×C₃	C₂×#41	1	1	yes	characteristic
155	C₂²×C₃	C₂×#41	1	2	yes	—
156	C₂²×C₃	C₂×#41	1	2	yes	—
157	C₃×C₄	—	2	4	no	—
158	C₃×C₄	—	2	4	no	—
159	C₃×C₄	—	2	4	no	—
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	—	4	2	no	—
162	C₃×C₄	—	4	2	no	—
163	C₃×C₄	C₄×D₄	1	4	yes	—
164	C₃×C₄	#124	1	4	yes	direct product factor
165	C₂³	—	2	2	no	—
166	C₂³	—	2	4	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	2	2	no	—
169	C₂³	C₂²×C₃×C₄	1	4	yes	—
170	C₂³	C₂⁴×C₃	1	1	yes	characteristic
171	Q₈	—	2	8	no	—
172	D₄	—	2	8	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
186	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
187	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
188	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
189	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
190	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
191	C₂×C₄	C₃×#41	1	4	yes	—
192	C₂×C₄	C₃×#41	1	2	yes	—
193	C₂×C₄	C₃×#41	1	4	yes	—
194	C₂×C₃	—	4	4	no	—
195	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂×C₃	C₄×#41	1	2	yes	—
197	C₂×C₃	C₂×#124	1	1	yes	characteristic
198	C₂×C₃	#529	1	2	yes	—
199	C₂×C₃	#532	1	1	yes	characteristic
200	C₂²	—	2	4	no	—
201	C₂²	—	2	8	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
206	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
207	C₂²	C₂×C₃×#41	1	1	yes	characteristic
208	C₂²	C₂×C₃×#41	1	2	yes	—
209	C₂²	C₂×C₃×#41	1	2	yes	—
210	C₄	—	2	4	no	—
211	C₄	—	2	4	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	C₃×C₄×D₄	1	4	yes	—
217	C₄	C₃×#124	1	4	yes	direct product factor
218	C₃	C₄×#124	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	4	4	no	—
220	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
221	C₂	C₃×C₄×#41	1	2	yes	—
222	C₂	C₂×C₃×#124	1	1	yes	characteristic
223	C₂	C₃×#529	1	2	yes	—
224	C₂	C₃×#532	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
socle	C₂³×C₃	C₂⁴
prime radical	C₂²×C₃×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×Q₈	C₂	semidirect
4. C₃×#391	C₂	semidirect
6. C₃×#388	C₂	semidirect
7. C₃×#382	C₂	semidirect
8. C₃×C₄×C₂²⋊C₄	C₂	semidirect
9. C₃×C₄³	C₂	semidirect
10. C₄×#124	C₃	direct (N × H)
14. C₃×#124	C₄	direct (N × H)
32. C₂×C₄×Q₈	C₂×C₃	semidirect
34. #391	C₂×C₃	semidirect
36. #388	C₂×C₃	semidirect
37. #382	C₂×C₃	semidirect
38. C₄×C₂²⋊C₄	C₂×C₃	semidirect
39. C₄³	C₂×C₃	semidirect
43. C₂×C₃×Q₈	C₂×C₄	semidirect
62. C₃×C₂²⋊C₄	C₂×C₄	semidirect
67. C₃×C₄²	C₂×C₄	semidirect
72. #124	C₃×C₄	direct (N × H)
121. C₂×Q₈	C₂×C₃×C₄	semidirect
140. C₂²⋊C₄	C₂×C₃×C₄	semidirect
145. C₄²	C₂×C₃×C₄	semidirect
164. C₃×C₄	#124	direct (N × H)
217. C₄	C₃×#124	direct (N × H)
218. C₃	C₄×#124	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×#124	2
C₂×C₃×C₄×Q₈	2
C₂×C₃×C₄×D₄	2
C₃×#391	2
C₃×#390	2
C₃×#388	2
C₃×#382	2
C₃×C₄×C₂²⋊C₄	2
C₃×C₄³	2
C₄×#124	3

Subgroups — C3×C4×#124 #84167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#124 #84167