Subgroups — C₂×#4554 #71986

Order 384.

432 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#208	C₂	1	1	yes	maximal, characteristic
4	#4554	C₂	1	4	yes	maximal, direct product factor
5	C₂×#343	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂³×Dic₃	C₄	1	2	yes	—
8	C₂×C₃×M₁₆	—	2	2	no	—
9	C₂×C₃×M₁₆	—	2	1	no	—
10	C₂×C₃×M₁₆	C₄	1	4	yes	complemented
11	C₂×C₃×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₃×C₈	—	2	1	no	—
13	C₂×C₄⋊Dic₃	—	2	1	no	—
14	C₂×C₄⋊Dic₃	—	2	1	no	—
15	C₂×C₄⋊Dic₃	—	2	1	no	—
16	C₂×C₄⋊Dic₃	—	2	1	no	—
17	C₂×C₄⋊Dic₃	C₄	1	2	yes	—
18	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
19	C₂×#208	—	2	1	no	—
20	C₂×#208	—	2	2	no	—
21	C₂×#208	C₄	1	4	yes	complemented
22	C₂×#208	C₂²	1	2	yes	—
23	C₂²×C₃⋊C₈	—	2	1	no	—
24	C₂²×M₁₆	—	3	1	no	—
25	C₂²×M₁₆	D₃	1	1	yes	characteristic
26	C₂²×C₄⋊C₄	—	3	1	no	—
27	#343	—	3	4	no	—
28	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₄	D₄	1	2	yes	—
35	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
37	C₂²×C₃×C₄	Q₈	1	2	yes	—
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×Dic₃	—	2	2	no	—
40	C₂²×Dic₃	—	2	2	no	—
41	C₂²×Dic₃	—	4	4	no	—
42	C₂²×Dic₃	C₂×C₄	1	4	yes	—
43	C₃×M₁₆	—	2	4	no	—
44	C₃×M₁₆	—	2	2	no	—
45	C₃×M₁₆	C₂×C₄	1	4	yes	complemented
46	C₂×C₃×C₈	—	2	4	no	—
47	C₂×C₃×C₈	—	2	2	no	—
48	C₄⋊Dic₃	—	2	2	no	—
49	C₄⋊Dic₃	—	2	2	no	—
50	C₄⋊Dic₃	—	2	2	no	—
51	C₄⋊Dic₃	—	2	2	no	—
52	#208	—	2	4	no	—
53	#208	—	2	2	no	—
54	#208	C₂×C₄	1	4	yes	complemented
55	C₂×C₃⋊C₈	—	2	4	no	—
56	C₂×C₃⋊C₈	—	2	2	no	—
57	C₂³×C₄	—	3	2	no	—
58	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
59	C₂×M₁₆	—	2	2	no	—
60	C₂×M₁₆	—	2	1	no	—
61	C₂×M₁₆	—	3	4	no	—
62	C₂×M₁₆	—	3	2	no	—
63	C₂×M₁₆	—	6	1	no	—
64	C₂×M₁₆	—	6	2	no	—
65	C₂×M₁₆	Dic₃	1	4	yes	complemented
66	C₂×M₁₆	C₂×D₃	1	2	yes	—
67	C₂²×C₈	—	2	1	no	—
68	C₂²×C₈	—	6	1	no	—
69	C₂×C₄⋊C₄	—	3	2	no	—
70	C₂×C₄⋊C₄	—	3	2	no	—
71	C₂×C₄⋊C₄	—	6	1	no	—
72	C₂×C₄⋊C₄	—	6	1	no	—
73	C₂×C₄⋊C₄	—	6	1	no	—
74	C₂×C₄⋊C₄	—	6	1	no	—
75	C₂³×C₃	—	2	2	no	—
76	C₂³×C₃	—	2	2	no	—
77	C₂³×C₃	C₄²	1	2	yes	—
78	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂³×C₃	C₄⋊C₄	1	2	yes	—
81	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄	C₄²	1	2	yes	—
83	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
86	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
87	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
88	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
89	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
90	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
91	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
92	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
93	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
94	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
100	C₂×Dic₃	—	2	4	no	—
101	C₂×Dic₃	—	4	4	no	—
102	C₂×Dic₃	—	4	4	no	—
103	C₂×Dic₃	—	4	4	no	—
104	C₃×C₈	—	2	4	no	—
105	C₃⋊C₈	—	2	4	no	—
106	C₂⁴	C₄×D₃	1	1	yes	characteristic
107	C₂²×C₄	—	3	4	no	—
108	C₂²×C₄	—	6	2	no	—
109	C₂²×C₄	—	6	2	no	—
110	C₂²×C₄	—	12	4	no	—
111	C₂²×C₄	Dic₆	1	2	yes	—
112	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
113	C₂²×C₄	C₄×D₃	1	2	yes	—
114	C₂²×C₄	D₁₂	1	1	yes	characteristic
115	C₂²×C₄	D₁₂	1	1	yes	characteristic
116	C₂²×C₄	C₂×Dic₃	1	2	yes	—
117	C₂²×C₄	#67	1	1	yes	characteristic
118	C₂²×C₄	#67	1	1	yes	characteristic
119	C₂²×C₄	#67	1	2	yes	—
120	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
121	M₁₆	—	2	4	no	—
122	M₁₆	—	2	2	no	—
123	M₁₆	—	3	4	no	—
124	M₁₆	—	6	4	no	—
125	M₁₆	—	6	2	no	—
126	M₁₆	C₂×Dic₃	1	4	yes	complemented
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	6	4	no	—
130	C₂×C₈	—	6	2	no	—
131	C₄⋊C₄	—	6	2	no	—
132	C₄⋊C₄	—	6	2	no	—
133	C₄⋊C₄	—	6	2	no	—
134	C₄⋊C₄	—	6	2	no	—
135	C₂²×C₃	—	2	4	no	—
136	C₂²×C₃	—	2	4	no	—
137	C₂²×C₃	—	2	1	no	—
138	C₂²×C₃	—	2	1	no	—
139	C₂²×C₃	—	2	1	no	—
140	C₂²×C₃	—	2	1	no	—
141	C₂²×C₃	#95	1	2	yes	—
142	C₂²×C₃	#95	1	2	yes	—
143	C₂²×C₃	#100	1	2	yes	—
144	C₂²×C₃	#101	1	2	yes	—
145	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
146	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₃×C₄	#95	1	2	yes	—
149	C₃×C₄	#95	1	2	yes	—
150	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
154	Dic₃	—	4	4	no	—
155	C₂³	—	2	2	no	—
156	C₂³	—	2	2	no	—
157	C₂³	C₄×Dic₃	1	2	yes	—
158	C₂³	Dic₃⋊C₄	1	2	yes	—
159	C₂³	(C₂×D₃)⋊C₄	1	1	yes	characteristic
160	C₂³	(C₂×D₃)⋊C₄	1	1	yes	characteristic
161	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂×C₄	—	6	4	no	—
163	C₂×C₄	—	12	4	no	—
164	C₂×C₄	—	12	4	no	—
165	C₂×C₄	—	12	4	no	—
166	C₂×C₄	C₄×Dic₃	1	2	yes	—
167	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
168	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
169	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
170	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
171	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
172	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
175	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
176	C₂×C₄	#217	1	2	yes	—
177	C₂×C₄	#217	1	2	yes	—
178	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
180	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
181	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
182	C₂×C₄	C₂×#67	1	1	yes	characteristic
183	C₂×C₄	C₂×#67	1	1	yes	characteristic
184	C₈	—	2	4	no	—
185	C₈	—	6	4	no	—
186	C₂×C₃	—	2	2	no	—
187	C₂×C₃	—	2	1	no	—
188	C₂×C₃	—	2	1	no	—
189	C₂×C₃	#343	1	4	yes	complemented
190	C₂×C₃	C₂×#95	1	1	yes	characteristic
191	C₂×C₃	C₂×#100	1	1	yes	characteristic
192	C₂×C₃	C₂×#101	1	1	yes	characteristic
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	4	no	—
199	C₂²	#823	1	2	yes	—
200	C₂²	#824	1	2	yes	—
201	C₂²	C₃⋊#95	1	2	yes	—
202	C₂²	C₃⋊#95	1	2	yes	—
203	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
204	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
205	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
206	C₄	—	12	4	no	—
207	C₄	C₃⋊#95	1	2	yes	—
208	C₄	C₃⋊#95	1	2	yes	—
209	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
210	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
211	C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
212	C₄	C₂×#217	1	1	yes	characteristic
213	C₃	C₂×#343	1	1	yes	characteristic, complemented, Sylow
214	C₂	—	2	2	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	#4554	1	4	yes	direct product factor
218	C₂	C₂×#823	1	1	yes	characteristic
219	C₂	C₂×#824	1	1	yes	characteristic
220	C₂	C₂×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂³×C₃	C₂²⋊C₄
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	C₂×#343
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #4554	C₂	direct (N × H)
10. C₂×C₃×M₁₆	C₄	semidirect
21. C₂×#208	C₄	semidirect
45. C₃×M₁₆	C₂×C₄	semidirect
54. #208	C₂×C₄	semidirect
65. C₂×M₁₆	Dic₃	semidirect
126. M₁₆	C₂×Dic₃	semidirect
189. C₂×C₃	#343	semidirect
213. C₃	C₂×#343	semidirect
217. C₂	#4554	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×M₁₆	2
C₂²×C₄⋊Dic₃	2
C₂²×#208	2
#4554	2
C₂×#343	3

Subgroups — C2×#4554 #71986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4554 #71986