Subgroups — C₃×#2349 #84213

Order 384.

704 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₄	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#389	C₂	1	2	yes	maximal, complemented
4	C₃×#385	C₂	1	4	yes	maximal, complemented
5	C₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#380	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×#95	C₂	1	2	yes	maximal, complemented
9	#2349	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	4	yes	—
12	C₂³×C₃×C₄	C₂²	1	2	yes	—
13	C₃×C₄×D₄	—	2	8	no	—
14	C₃×C₄×D₄	C₄	1	8	yes	complemented
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₄²	C₂²	1	2	yes	—
22	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₃×#95	C₂²	1	1	yes	characteristic
24	C₃×#95	C₂²	1	2	yes	—
25	C₃×#95	C₂²	1	4	yes	—
26	C₃×#95	C₂²	1	1	yes	characteristic
27	C₃×#95	C₂²	1	2	yes	—
28	C₂×C₄×D₄	C₂×C₃	1	2	yes	—
29	C₂×C₄×D₄	C₂×C₃	1	1	yes	characteristic
30	#389	C₂×C₃	1	2	yes	complemented
31	#385	C₂×C₃	1	4	yes	complemented
32	#382	C₂×C₃	1	1	yes	characteristic, complemented
33	#380	C₂×C₃	1	2	yes	complemented
34	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×#95	C₂×C₃	1	2	yes	complemented
36	C₂⁴×C₃	C₂³	1	2	yes	—
37	C₂×C₃×D₄	C₂×C₄	1	4	yes	—
38	C₂×C₃×D₄	D₄	1	4	yes	—
39	C₂×C₃×D₄	Q₈	1	4	yes	—
40	C₂²×C₃×C₄	—	2	8	no	—
41	C₂²×C₃×C₄	—	2	8	no	—
42	C₂²×C₃×C₄	—	4	4	no	—
43	C₂²×C₃×C₄	—	4	4	no	—
44	C₂²×C₃×C₄	C₂×C₄	1	8	yes	complemented
45	C₂²×C₃×C₄	C₂³	1	4	yes	—
46	C₂²×C₃×C₄	C₂³	1	2	yes	—
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
49	C₂²×C₃×C₄	C₂³	1	2	yes	—
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₃×C₄⋊C₄	—	2	4	no	—
53	C₃×C₄⋊C₄	—	2	4	no	—
54	C₃×C₄⋊C₄	—	4	4	no	—
55	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
56	C₃×C₂²⋊C₄	—	2	8	no	—
57	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	complemented
58	C₃×C₄²	—	2	4	no	—
59	C₃×C₄²	C₂×C₄	1	4	yes	complemented
60	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
61	C₂³×C₄	C₂²×C₃	1	4	yes	—
62	C₂³×C₄	C₂²×C₃	1	2	yes	—
63	C₄×D₄	—	2	8	no	—
64	C₄×D₄	C₃×C₄	1	8	yes	complemented
65	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
67	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
68	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
70	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₄²	C₂²×C₃	1	2	yes	—
72	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
73	#95	C₂²×C₃	1	1	yes	characteristic
74	#95	C₂²×C₃	1	2	yes	—
75	#95	C₂²×C₃	1	4	yes	—
76	#95	C₂²×C₃	1	1	yes	characteristic
77	#95	C₂²×C₃	1	2	yes	—
78	C₂³×C₃	—	2	4	no	—
79	C₂³×C₃	—	2	4	no	—
80	C₂³×C₃	C₂²×C₄	1	4	yes	—
81	C₂³×C₃	C₂×D₄	1	4	yes	—
82	C₂³×C₃	C₂×Q₈	1	4	yes	—
83	C₂³×C₃	C₂⁴	1	1	yes	characteristic
84	C₃×D₄	C₄⋊C₄	1	16	yes	complemented
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	2	4	no	—
88	C₂×C₃×C₄	—	2	2	no	—
89	C₂×C₃×C₄	—	2	2	no	—
90	C₂×C₃×C₄	—	2	2	no	—
91	C₂×C₃×C₄	—	2	4	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	4	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	4	no	—
96	C₂×C₃×C₄	—	4	8	no	—
97	C₂×C₃×C₄	—	4	16	no	—
98	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
99	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
100	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
101	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
102	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
103	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
104	C₂×C₃×C₄	#41	1	4	yes	—
105	C₂⁴	C₂³×C₃	1	2	yes	—
106	C₂×D₄	C₂×C₃×C₄	1	4	yes	—
107	C₂×D₄	C₃×D₄	1	4	yes	—
108	C₂×D₄	C₃×Q₈	1	4	yes	—
109	C₂²×C₄	—	2	8	no	—
110	C₂²×C₄	—	2	8	no	—
111	C₂²×C₄	—	4	4	no	—
112	C₂²×C₄	—	4	4	no	—
113	C₂²×C₄	C₂×C₃×C₄	1	8	yes	complemented
114	C₂²×C₄	C₂³×C₃	1	4	yes	—
115	C₂²×C₄	C₂³×C₃	1	2	yes	—
116	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₃	1	2	yes	—
119	C₂²×C₄	C₂³×C₃	1	2	yes	—
120	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	4	4	no	—
124	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	complemented
125	C₂²⋊C₄	—	2	8	no	—
126	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
127	C₄²	—	2	4	no	—
128	C₄²	C₂×C₃×C₄	1	4	yes	complemented
129	C₂²×C₃	—	2	4	no	—
130	C₂²×C₃	—	2	4	no	—
131	C₂²×C₃	—	2	8	no	—
132	C₂²×C₃	—	2	8	no	—
133	C₂²×C₃	C₂×C₄⋊C₄	1	8	yes	—
134	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
135	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
136	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
137	C₂²×C₃	C₂×#41	1	2	yes	—
138	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
139	C₃×C₄	—	2	2	no	—
140	C₃×C₄	—	2	2	no	—
141	C₃×C₄	—	2	4	no	—
142	C₃×C₄	—	2	4	no	—
143	C₃×C₄	—	4	2	no	—
144	C₃×C₄	—	4	4	no	—
145	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
146	C₂³	—	2	4	no	—
147	C₂³	—	2	4	no	—
148	C₂³	C₂²×C₃×C₄	1	4	yes	—
149	C₂³	C₂×C₃×D₄	1	4	yes	—
150	C₂³	C₂×C₃×Q₈	1	4	yes	—
151	C₂³	C₂⁴×C₃	1	1	yes	characteristic
152	D₄	C₃×C₄⋊C₄	1	16	yes	complemented
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	8	no	—
165	C₂×C₄	—	4	16	no	—
166	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
167	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
168	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
169	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
170	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
171	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
172	C₂×C₄	C₃×#41	1	4	yes	—
173	C₂×C₃	—	2	8	no	—
174	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₃	C₄×#41	1	1	yes	characteristic
176	C₂×C₃	#518	1	1	yes	characteristic
177	C₂×C₃	#546	1	2	yes	—
178	C₂×C₃	#554	1	2	yes	—
179	C₂²	—	2	4	no	—
180	C₂²	—	2	4	no	—
181	C₂²	—	2	8	no	—
182	C₂²	—	2	8	no	—
183	C₂²	C₂×C₃×C₄⋊C₄	1	8	yes	—
184	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
185	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
186	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
187	C₂²	C₂×C₃×#41	1	2	yes	—
188	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	—	4	4	no	—
194	C₄	—	4	2	no	—
195	C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
196	C₃	#2349	1	1	yes	characteristic, direct product factor, Sylow
197	C₂	—	2	8	no	—
198	C₂	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
199	C₂	C₃×C₄×#41	1	1	yes	characteristic
200	C₂	C₃×#518	1	1	yes	characteristic
201	C₂	C₃×#546	1	2	yes	—
202	C₂	C₃×#554	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃×#389	C₂	semidirect
4. C₃×#385	C₂	semidirect
5. C₃×#382	C₂	semidirect
6. C₃×#380	C₂	semidirect
7. C₃×C₄×C₄⋊C₄	C₂	semidirect
8. C₂×C₃×#95	C₂	semidirect
9. #2349	C₃	direct (N × H)
14. C₃×C₄×D₄	C₄	semidirect
30. #389	C₂×C₃	semidirect
31. #385	C₂×C₃	semidirect
32. #382	C₂×C₃	semidirect
33. #380	C₂×C₃	semidirect
34. C₄×C₄⋊C₄	C₂×C₃	semidirect
35. C₂×#95	C₂×C₃	semidirect
44. C₂²×C₃×C₄	C₂×C₄	semidirect
55. C₃×C₄⋊C₄	C₂×C₄	semidirect
57. C₃×C₂²⋊C₄	C₂×C₄	semidirect
59. C₃×C₄²	C₂×C₄	semidirect
64. C₄×D₄	C₃×C₄	semidirect
84. C₃×D₄	C₄⋊C₄	semidirect
113. C₂²×C₄	C₂×C₃×C₄	semidirect
124. C₄⋊C₄	C₂×C₃×C₄	semidirect
126. C₂²⋊C₄	C₂×C₃×C₄	semidirect
128. C₄²	C₂×C₃×C₄	semidirect
152. D₄	C₃×C₄⋊C₄	semidirect
196. C₃	#2349	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×C₄×D₄	2
C₃×#389	2
C₃×#385	2
C₃×#382	2
C₃×#380	2
C₃×C₄×C₄⋊C₄	2
C₂×C₃×#95	2
#2349	3

Subgroups — C3×#2349 #84213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2349 #84213