Subgroups — C₃×#2484 #84348

Order 384.

512 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#400	C₂	1	2	yes	maximal, complemented
4	C₃×#398	C₂	1	2	yes	maximal, complemented
5	C₃×#387	C₂	1	4	yes	maximal, complemented
6	C₃×#385	C₂	1	2	yes	maximal
7	C₃×#381	C₂	1	1	yes	maximal, characteristic
8	C₃×#381	C₂	1	2	yes	maximal
9	#2484	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₃×#117	—	2	4	no	—
14	C₂×C₃×C₄⋊C₄	—	2	4	no	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₄²	C₂²	1	2	yes	—
23	C₃×#95	C₂²	1	2	yes	—
24	C₃×#95	C₂²	1	4	yes	—
25	C₃×#95	C₂²	1	4	yes	—
26	C₃×#95	C₂²	1	2	yes	—
27	C₃×#95	C₂²	1	4	yes	—
28	C₂×#117	C₂×C₃	1	1	yes	characteristic
29	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
30	#400	C₂×C₃	1	2	yes	complemented
31	#398	C₂×C₃	1	2	yes	complemented
32	#387	C₂×C₃	1	4	yes	complemented
33	#385	C₂×C₃	1	2	yes	—
34	#381	C₂×C₃	1	1	yes	characteristic
35	#381	C₂×C₃	1	2	yes	—
36	C₂⁴×C₃	C₂³	1	1	yes	characteristic
37	C₂²×C₃×C₄	—	2	2	no	—
38	C₂²×C₃×C₄	—	2	4	no	—
39	C₂²×C₃×C₄	—	2	4	no	—
40	C₂²×C₃×C₄	—	2	2	no	—
41	C₂²×C₃×C₄	—	2	4	no	—
42	C₂²×C₃×C₄	D₄	1	4	yes	—
43	C₂²×C₃×C₄	C₂³	1	4	yes	—
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₂²×C₃×C₄	C₂³	1	2	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	C₃×C₄⋊C₄	—	2	4	no	—
51	C₃×C₄⋊C₄	—	2	4	no	—
52	C₃×C₄⋊C₄	—	2	2	no	—
53	C₃×C₄⋊C₄	—	2	2	no	—
54	C₃×C₄⋊C₄	—	2	8	no	—
55	C₃×C₂²⋊C₄	—	2	4	no	—
56	C₃×C₂²⋊C₄	—	2	4	no	—
57	C₃×C₄²	—	2	4	no	—
58	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
59	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
60	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
61	#117	—	2	4	no	—
62	C₂×C₄⋊C₄	—	2	4	no	—
63	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
64	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
68	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
69	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
70	C₂×C₄²	C₂²×C₃	1	2	yes	—
71	#95	C₂²×C₃	1	2	yes	—
72	#95	C₂²×C₃	1	4	yes	—
73	#95	C₂²×C₃	1	4	yes	—
74	#95	C₂²×C₃	1	2	yes	—
75	#95	C₂²×C₃	1	4	yes	—
76	C₂³×C₃	—	2	2	no	—
77	C₂³×C₃	—	2	2	no	—
78	C₂³×C₃	C₂×D₄	1	2	yes	—
79	C₂³×C₃	#41	1	4	yes	—
80	C₂³×C₃	C₂⁴	1	1	yes	characteristic
81	C₂×C₃×C₄	—	2	8	no	—
82	C₂×C₃×C₄	—	2	8	no	—
83	C₂×C₃×C₄	—	2	2	no	—
84	C₂×C₃×C₄	—	2	2	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	2	2	no	—
88	C₂×C₃×C₄	—	2	2	no	—
89	C₂×C₃×C₄	—	2	2	no	—
90	C₂×C₃×C₄	—	2	4	no	—
91	C₂×C₃×C₄	—	2	4	no	—
92	C₂×C₃×C₄	—	2	4	no	—
93	C₂×C₃×C₄	—	2	4	no	—
94	C₂×C₃×C₄	—	2	4	no	—
95	C₂×C₃×C₄	—	2	4	no	—
96	C₂×C₃×C₄	—	4	4	no	—
97	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
98	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
99	C₂×C₃×C₄	#41	1	4	yes	—
100	C₂⁴	C₂³×C₃	1	1	yes	characteristic
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	2	4	no	—
103	C₂²×C₄	—	2	4	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	C₃×D₄	1	4	yes	—
107	C₂²×C₄	C₂³×C₃	1	4	yes	—
108	C₂²×C₄	C₂³×C₃	1	2	yes	—
109	C₂²×C₄	C₂³×C₃	1	2	yes	—
110	C₂²×C₄	C₂³×C₃	1	2	yes	—
111	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₂²×C₄	C₂³×C₃	1	2	yes	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	8	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₄²	—	2	4	no	—
122	C₂²×C₃	—	2	2	no	—
123	C₂²×C₃	—	2	2	no	—
124	C₂²×C₃	—	2	8	no	—
125	C₂²×C₃	#123	1	4	yes	—
126	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₃	C₂×#41	1	2	yes	—
128	C₂²×C₃	C₂×#41	1	2	yes	—
129	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
130	C₃×C₄	—	2	2	no	—
131	C₃×C₄	—	2	2	no	—
132	C₃×C₄	—	2	4	no	—
133	C₃×C₄	—	4	2	no	—
134	C₃×C₄	—	4	2	no	—
135	C₃×C₄	—	4	2	no	—
136	C₃×C₄	—	4	4	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	2	2	no	—
139	C₂³	C₂×C₃×D₄	1	2	yes	—
140	C₂³	C₃×#41	1	4	yes	—
141	C₂³	C₂⁴×C₃	1	1	yes	characteristic
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
159	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
160	C₂×C₄	C₃×#41	1	4	yes	—
161	C₂×C₃	—	2	4	no	—
162	C₂×C₃	C₂×#123	1	1	yes	characteristic
163	C₂×C₃	#525	1	1	yes	characteristic
164	C₂×C₃	#536	1	1	yes	characteristic
165	C₂×C₃	#552	1	4	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	8	no	—
169	C₂²	C₃×#123	1	4	yes	—
170	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
171	C₂²	C₂×C₃×#41	1	2	yes	—
172	C₂²	C₂×C₃×#41	1	2	yes	—
173	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
174	C₄	—	2	2	no	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	—	4	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	4	no	—
181	C₃	#2484	1	1	yes	characteristic, direct product factor, Sylow
182	C₂	—	2	4	no	—
183	C₂	C₂×C₃×#123	1	1	yes	characteristic
184	C₂	C₃×#525	1	1	yes	characteristic
185	C₂	C₃×#536	1	1	yes	characteristic
186	C₂	C₃×#552	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³
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
3. C₃×#400	C₂	semidirect
4. C₃×#398	C₂	semidirect
5. C₃×#387	C₂	semidirect
9. #2484	C₃	direct (N × H)
30. #400	C₂×C₃	semidirect
31. #398	C₂×C₃	semidirect
32. #387	C₂×C₃	semidirect
181. C₃	#2484	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×#117	2
C₂²×C₃×C₄⋊C₄	2
C₃×#400	2
C₃×#398	2
C₃×#387	2
C₃×#385	2
C₃×#381	2
#2484	3

Subgroups — C3×#2484 #84348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2484 #84348