Subgroups — C₃×#2336 #84200

Order 384.

428 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₃×#389	C₂	1	2	yes	maximal
3	C₃×#387	C₂	1	2	yes	maximal
4	C₃×#387	C₂	1	2	yes	maximal
5	C₃×#387	C₂	1	2	yes	maximal
6	C₃×#387	C₂	1	2	yes	maximal
7	C₃×#383	C₂	1	1	yes	maximal, characteristic
8	C₃×C₄×C₄⋊C₄	C₂	1	2	yes	maximal
9	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
10	#2336	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×#125	C₄	1	8	yes	complemented
12	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄²	C₂²	1	2	yes	—
20	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄²	C₂²	1	2	yes	—
23	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
24	C₃×#95	C₂²	1	2	yes	—
25	C₃×#95	C₂²	1	2	yes	—
26	C₃×#95	C₂²	1	2	yes	—
27	C₃×#95	C₂²	1	1	yes	characteristic
28	C₃×#95	C₂²	1	2	yes	—
29	C₃×#95	C₂²	1	1	yes	characteristic
30	C₃×#95	C₂²	1	2	yes	—
31	C₃×#95	C₂²	1	2	yes	—
32	C₃×#95	C₂²	1	2	yes	—
33	C₂×#125	C₂×C₃	1	1	yes	characteristic
34	#389	C₂×C₃	1	2	yes	—
35	#387	C₂×C₃	1	2	yes	—
36	#387	C₂×C₃	1	2	yes	—
37	#387	C₂×C₃	1	2	yes	—
38	#387	C₂×C₃	1	2	yes	—
39	#383	C₂×C₃	1	1	yes	characteristic
40	C₄×C₄⋊C₄	C₂×C₃	1	2	yes	—
41	C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂³	1	2	yes	—
43	C₂²×C₃×C₄	C₂³	1	2	yes	—
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
47	C₂²×C₃×C₄	C₂³	1	2	yes	—
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	C₃×C₄⋊C₄	—	2	4	no	—
52	C₃×C₄⋊C₄	—	2	4	no	—
53	C₃×C₄⋊C₄	—	4	2	no	—
54	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
55	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
56	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
57	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
58	C₃×C₄²	—	2	4	no	—
59	C₃×C₄²	—	4	1	no	—
60	C₃×C₄²	—	4	2	no	—
61	C₃×C₄²	—	4	1	no	—
62	C₃×C₄²	C₂×C₄	1	4	yes	—
63	#125	C₃×C₄	1	8	yes	complemented
64	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
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	2	yes	—
68	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₄²	C₂²×C₃	1	2	yes	—
72	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₄²	C₂²×C₃	1	2	yes	—
75	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
76	#95	C₂²×C₃	1	2	yes	—
77	#95	C₂²×C₃	1	2	yes	—
78	#95	C₂²×C₃	1	2	yes	—
79	#95	C₂²×C₃	1	1	yes	characteristic
80	#95	C₂²×C₃	1	2	yes	—
81	#95	C₂²×C₃	1	1	yes	characteristic
82	#95	C₂²×C₃	1	2	yes	—
83	#95	C₂²×C₃	1	2	yes	—
84	#95	C₂²×C₃	1	2	yes	—
85	C₂³×C₃	C₂⁴	1	1	yes	characteristic
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	1	no	—
90	C₂×C₃×C₄	—	2	2	no	—
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	2	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	C₂²×C₄	1	2	yes	—
106	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
107	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
108	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
109	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
110	C₂×C₃×C₄	#41	1	4	yes	—
111	C₂²×C₄	C₂³×C₃	1	2	yes	—
112	C₂²×C₄	C₂³×C₃	1	2	yes	—
113	C₂²×C₄	C₂³×C₃	1	2	yes	—
114	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
115	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
116	C₂²×C₄	C₂³×C₃	1	2	yes	—
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	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
124	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
125	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
126	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
127	C₄²	—	2	4	no	—
128	C₄²	—	4	1	no	—
129	C₄²	—	4	2	no	—
130	C₄²	—	4	1	no	—
131	C₄²	C₂×C₃×C₄	1	4	yes	—
132	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
133	C₂²×C₃	C₂×#41	1	2	yes	—
134	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
136	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
137	C₃×C₄	—	2	4	no	—
138	C₃×C₄	—	2	2	no	—
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	2	no	—
145	C₃×C₄	—	4	2	no	—
146	C₂³	C₂⁴×C₃	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	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	2	yes	—
169	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
170	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
171	C₂×C₄	C₃×#41	1	4	yes	—
172	C₂×C₃	C₄×#41	1	1	yes	characteristic
173	C₂×C₃	#519	1	1	yes	characteristic
174	C₂×C₃	#520	1	1	yes	characteristic
175	C₂×C₃	#539	1	2	yes	—
176	C₂×C₃	#541	1	2	yes	—
177	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
178	C₂²	C₂×C₃×#41	1	2	yes	—
179	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
180	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
181	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	2	4	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	2	no	—
191	C₃	#2336	1	1	yes	characteristic, direct product factor, Sylow
192	C₂	C₃×C₄×#41	1	1	yes	characteristic
193	C₂	C₃×#519	1	1	yes	characteristic
194	C₂	C₃×#520	1	1	yes	characteristic
195	C₂	C₃×#539	1	2	yes	—
196	C₂	C₃×#541	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
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
10. #2336	C₃	direct (N × H)
11. C₃×#125	C₄	semidirect
63. #125	C₃×C₄	semidirect
191. C₃	#2336	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×#125	2
C₃×#389	2
C₃×#387	2
C₃×#383	2
C₃×C₄×C₄⋊C₄	2
#2336	3

Subgroups — C3×#2336 #84200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2336 #84200