Subgroups — C₃×#2344 #84208

Order 384.

604 conjugacy classes of subgroups.

The subgroup lattice has 202 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	2	yes	maximal
2	C₂²×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×#389	C₂	1	2	yes	maximal, complemented
4	C₃×#387	C₂	1	4	yes	maximal, complemented
5	C₃×#381	C₂	1	2	yes	maximal
6	C₃×#380	C₂	1	1	yes	maximal, characteristic
7	C₃×C₄×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
9	#2344	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	2	yes	—
12	C₃×#117	—	2	8	no	—
13	C₂×C₃×C₄⋊C₄	C₄	1	8	yes	complemented
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄²	C₂²	1	2	yes	—
23	C₂×C₃×C₄²	C₂²	1	4	yes	—
24	C₃×#95	C₂²	1	2	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	2	yes	—
29	C₂²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
30	#389	C₂×C₃	1	2	yes	complemented
31	#387	C₂×C₃	1	4	yes	complemented
32	#381	C₂×C₃	1	2	yes	—
33	#380	C₂×C₃	1	1	yes	characteristic
34	C₄×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
35	C₄×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
36	C₂⁴×C₃	C₂³	1	1	yes	characteristic
37	C₂²×C₃×C₄	—	2	4	no	—
38	C₂²×C₃×C₄	—	2	4	no	—
39	C₂²×C₃×C₄	—	2	4	no	—
40	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
41	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
42	C₂²×C₃×C₄	C₂³	1	2	yes	—
43	C₂²×C₃×C₄	C₂³	1	4	yes	—
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₂²×C₃×C₄	C₂³	1	2	yes	—
46	C₂²×C₃×C₄	C₂³	1	2	yes	—
47	C₂²×C₃×C₄	C₂³	1	2	yes	—
48	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
49	C₃×C₄⋊C₄	—	2	8	no	—
50	C₃×C₄⋊C₄	—	2	4	no	—
51	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
52	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
53	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
54	C₃×C₂²⋊C₄	—	2	4	no	—
55	C₃×C₂²⋊C₄	—	2	2	no	—
56	C₃×C₂²⋊C₄	D₄	1	4	yes	—
57	C₃×C₄²	—	2	8	no	—
58	C₃×C₄²	—	2	4	no	—
59	C₂³×C₄	C₂²×C₃	1	2	yes	—
60	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
61	#117	—	2	8	no	—
62	C₂×C₄⋊C₄	C₃×C₄	1	8	yes	complemented
63	C₂×C₄⋊C₄	C₂²×C₃	1	4	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	2	yes	—
67	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
69	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₄²	C₂²×C₃	1	4	yes	—
72	C₂×C₄²	C₂²×C₃	1	2	yes	—
73	#95	C₂²×C₃	1	2	yes	—
74	#95	C₂²×C₃	1	4	yes	—
75	#95	C₂²×C₃	1	2	yes	—
76	#95	C₂²×C₃	1	4	yes	—
77	C₂³×C₃	—	2	2	no	—
78	C₂³×C₃	—	2	2	no	—
79	C₂³×C₃	C₂²×C₄	1	2	yes	—
80	C₂³×C₃	C₂×D₄	1	2	yes	—
81	C₂³×C₃	#41	1	2	yes	—
82	C₂³×C₃	C₂⁴	1	1	yes	characteristic
83	C₂×C₃×C₄	—	2	16	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	2	no	—
91	C₂×C₃×C₄	—	2	8	no	—
92	C₂×C₃×C₄	—	2	4	no	—
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	4	no	—
96	C₂×C₃×C₄	—	2	4	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
99	C₂×C₃×C₄	C₂²×C₄	1	4	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	4	yes	—
103	C₂×C₃×C₄	#41	1	4	yes	—
104	C₂⁴	C₂³×C₃	1	1	yes	characteristic
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	C₂×C₃×C₄	1	8	yes	—
109	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
110	C₂²×C₄	C₂³×C₃	1	2	yes	—
111	C₂²×C₄	C₂³×C₃	1	4	yes	—
112	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₂²×C₄	C₂³×C₃	1	2	yes	—
114	C₂²×C₄	C₂³×C₃	1	2	yes	—
115	C₂²×C₄	C₂³×C₃	1	2	yes	—
116	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
120	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
121	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
122	C₂²⋊C₄	—	2	4	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	C₃×D₄	1	4	yes	—
125	C₄²	—	2	8	no	—
126	C₄²	—	2	4	no	—
127	C₂²×C₃	—	2	2	no	—
128	C₂²×C₃	—	2	2	no	—
129	C₂²×C₃	—	2	4	no	—
130	C₂²×C₃	—	2	4	no	—
131	C₂²×C₃	C₄×D₄	1	4	yes	—
132	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
133	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
134	C₂²×C₃	C₂×#41	1	2	yes	—
135	C₂²×C₃	C₂×#41	1	1	yes	characteristic
136	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
137	C₃×C₄	—	2	2	no	—
138	C₃×C₄	—	2	4	no	—
139	C₃×C₄	—	2	4	no	—
140	C₃×C₄	—	2	4	no	—
141	C₃×C₄	—	4	4	no	—
142	C₃×C₄	—	4	2	no	—
143	C₃×C₄	#117	1	4	yes	complemented
144	C₂³	—	2	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	C₂²×C₃×C₄	1	2	yes	—
147	C₂³	C₂×C₃×D₄	1	2	yes	—
148	C₂³	C₃×#41	1	2	yes	—
149	C₂³	C₂⁴×C₃	1	1	yes	characteristic
150	C₂×C₄	—	2	16	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
166	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
167	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
168	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
169	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
170	C₂×C₄	C₃×#41	1	4	yes	—
171	C₂×C₃	—	2	4	no	—
172	C₂×C₃	C₂×#117	1	1	yes	characteristic
173	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×C₃	#519	1	1	yes	characteristic
175	C₂×C₃	#547	1	2	yes	—
176	C₂×C₃	#552	1	2	yes	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	4	no	—
180	C₂²	—	2	4	no	—
181	C₂²	C₃×C₄×D₄	1	4	yes	—
182	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
183	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
184	C₂²	C₂×C₃×#41	1	2	yes	—
185	C₂²	C₂×C₃×#41	1	1	yes	characteristic
186	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	4	4	no	—
192	C₄	—	4	2	no	—
193	C₄	C₃×#117	1	4	yes	complemented
194	C₃	#2344	1	1	yes	characteristic, direct product factor, Sylow
195	C₂	—	2	4	no	—
196	C₂	C₂×C₃×#117	1	1	yes	characteristic
197	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
198	C₂	C₃×#519	1	1	yes	characteristic
199	C₂	C₃×#547	1	2	yes	—
200	C₂	C₃×#552	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁴×C₃	C₂³
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×#387	C₂	semidirect
7. C₃×C₄×C₄⋊C₄	C₂	semidirect
9. #2344	C₃	direct (N × H)
13. C₂×C₃×C₄⋊C₄	C₄	semidirect
30. #389	C₂×C₃	semidirect
31. #387	C₂×C₃	semidirect
34. C₄×C₄⋊C₄	C₂×C₃	semidirect
62. C₂×C₄⋊C₄	C₃×C₄	semidirect
143. C₃×C₄	#117	semidirect
193. C₄	C₃×#117	semidirect
194. C₃	#2344	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×#389	2
C₃×#387	2
C₃×#381	2
C₃×#380	2
C₃×C₄×C₄⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
#2344	3

Subgroups — C3×#2344 #84208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2344 #84208