Subgroups — C₂×#4555 #71981

Order 384.

488 conjugacy classes of subgroups.

The subgroup lattice has 214 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	C₂×#926	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
4	#4555	C₂	1	4	yes	maximal, direct product factor
5	C₂×#339	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂³×Dic₃	—	2	1	no	—
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×#217	—	2	1	no	—
14	#926	—	2	1	no	—
15	#926	—	2	1	no	—
16	#926	C₄	1	2	yes	complemented
17	#926	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₃	C₄	1	2	yes	complemented
19	C₂×C₄×Dic₃	—	2	1	no	—
20	C₂×C₄×Dic₃	—	2	2	no	—
21	C₂×C₄×Dic₃	—	2	1	no	—
22	C₂×C₄×Dic₃	C₄	1	4	yes	complemented
23	C₂×C₄×Dic₃	C₂²	1	2	yes	—
24	C₂²×M₁₆	D₃	1	1	yes	characteristic
25	C₂×#117	—	3	1	no	—
26	C₂²×C₄²	—	3	1	no	—
27	#339	—	3	4	no	—
28	C₂⁴×C₃	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
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	8	no	—
40	C₂²×Dic₃	—	2	4	no	—
41	C₂²×Dic₃	—	2	2	no	—
42	C₂²×Dic₃	—	2	2	no	—
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	#217	—	2	2	no	—
49	#217	—	2	2	no	—
50	C₄⋊Dic₃	—	2	2	no	—
51	C₄⋊Dic₃	C₂×C₄	1	4	yes	complemented
52	C₄×Dic₃	—	2	4	no	—
53	C₄×Dic₃	—	2	2	no	—
54	C₄×Dic₃	—	2	2	no	—
55	C₄×Dic₃	—	2	2	no	—
56	C₄×Dic₃	C₂×C₄	1	4	yes	complemented
57	C₂³×C₄	—	6	1	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	Dic₃	1	4	yes	complemented
62	C₂×M₁₆	C₂×D₃	1	2	yes	—
63	C₂²×C₈	—	2	1	no	—
64	#117	—	3	2	no	—
65	#117	—	3	2	no	—
66	#117	—	6	1	no	—
67	#117	—	6	1	no	—
68	C₂×C₄⋊C₄	—	3	2	no	—
69	C₂×C₂²⋊C₄	—	6	1	no	—
70	C₂×C₄²	—	3	4	no	—
71	C₂×C₄²	—	3	2	no	—
72	C₂×C₄²	—	6	1	no	—
73	C₂×C₄²	—	6	2	no	—
74	C₂×C₄²	—	6	1	no	—
75	C₂³×C₃	—	2	4	no	—
76	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
78	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂³×C₃	C₄⋊C₄	1	2	yes	—
80	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄	C₄²	1	2	yes	—
82	C₂×C₃×C₄	C₄²	1	2	yes	—
83	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
84	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
87	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
88	C₂×C₃×C₄	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
99	C₂×Dic₃	—	2	16	no	—
100	C₂×Dic₃	—	2	8	no	—
101	C₂×Dic₃	—	2	4	no	—
102	C₂×Dic₃	—	2	4	no	—
103	C₂×Dic₃	—	4	4	no	—
104	C₃×C₈	—	2	4	no	—
105	C₂⁴	D₁₂	1	1	yes	characteristic
106	C₂²×C₄	—	6	8	no	—
107	C₂²×C₄	—	6	4	no	—
108	C₂²×C₄	—	6	2	no	—
109	C₂²×C₄	—	6	2	no	—
110	C₂²×C₄	Dic₆	1	2	yes	—
111	C₂²×C₄	C₄×D₃	1	2	yes	—
112	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
113	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
114	C₂²×C₄	D₁₂	1	1	yes	characteristic
115	C₂²×C₄	C₂×Dic₃	1	2	yes	—
116	C₂²×C₄	#67	1	2	yes	—
117	C₂²×C₄	#67	1	1	yes	characteristic
118	C₂²×C₄	#67	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
120	M₁₆	—	2	4	no	—
121	M₁₆	—	2	2	no	—
122	M₁₆	C₂×Dic₃	1	4	yes	complemented
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	2	2	no	—
125	C₄⋊C₄	—	3	4	no	—
126	C₄⋊C₄	—	6	2	no	—
127	C₂²⋊C₄	—	6	2	no	—
128	C₂²⋊C₄	—	6	2	no	—
129	C₄²	—	3	4	no	—
130	C₄²	—	6	4	no	—
131	C₄²	—	6	2	no	—
132	C₄²	—	6	2	no	—
133	C₄²	—	6	2	no	—
134	C₂²×C₃	—	2	4	no	—
135	C₂²×C₃	—	2	4	no	—
136	C₂²×C₃	—	2	2	no	—
137	C₂²×C₃	—	2	2	no	—
138	C₂²×C₃	#95	1	2	yes	—
139	C₂²×C₃	#95	1	2	yes	—
140	C₂²×C₃	#104	1	4	yes	—
141	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₃×C₄	#95	1	2	yes	—
145	C₃×C₄	#95	1	2	yes	—
146	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
147	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
150	Dic₃	—	2	8	no	—
151	Dic₃	—	4	4	no	—
152	C₂³	—	2	4	no	—
153	C₂³	C₄⋊Dic₃	1	2	yes	—
154	C₂³	(C₂×D₃)⋊C₄	1	1	yes	characteristic
155	C₂³	(C₂×D₃)⋊C₄	1	2	yes	—
156	C₂³	(C₂×D₃)⋊C₄	1	1	yes	characteristic
157	C₂³	C₂×D₁₂	1	1	yes	characteristic
158	C₂×C₄	—	6	8	no	—
159	C₂×C₄	—	6	16	no	—
160	C₂×C₄	—	6	4	no	—
161	C₂×C₄	—	6	4	no	—
162	C₂×C₄	—	12	4	no	—
163	C₂×C₄	C₄×Dic₃	1	2	yes	—
164	C₂×C₄	C₄×Dic₃	1	2	yes	—
165	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
166	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
167	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
168	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
169	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
170	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
171	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₂×C₄	#217	1	2	yes	—
174	C₂×C₄	#217	1	2	yes	—
175	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
179	C₂×C₄	C₂×#67	1	1	yes	characteristic
180	C₂×C₄	C₂×#67	1	1	yes	characteristic
181	C₈	—	2	4	no	—
182	C₂×C₃	—	2	1	no	—
183	C₂×C₃	—	2	2	no	—
184	C₂×C₃	—	2	1	no	—
185	C₂×C₃	#339	1	4	yes	complemented
186	C₂×C₃	C₂×#95	1	1	yes	characteristic
187	C₂×C₃	C₂×#104	1	2	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	#825	1	4	yes	—
193	C₂²	C₃⋊#95	1	2	yes	—
194	C₂²	C₃⋊#95	1	2	yes	—
195	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
196	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
197	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
198	C₄	—	6	8	no	—
199	C₄	—	12	4	no	—
200	C₄	C₃⋊#95	1	2	yes	—
201	C₄	C₃⋊#95	1	2	yes	—
202	C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
203	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
204	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
205	C₄	C₂×#217	1	1	yes	characteristic
206	C₃	C₂×#339	1	1	yes	characteristic, complemented, Sylow
207	C₂	—	2	1	no	—
208	C₂	—	2	2	no	—
209	C₂	—	2	1	no	—
210	C₂	#4555	1	4	yes	direct product factor
211	C₂	C₂×#825	1	2	yes	—
212	C₂	C₂×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#67
commutator	C₃×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂³×C₃	C₂²⋊C₄
prime radical	C₂⁴×C₃	D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	C₂×#339
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #4555	C₂	direct (N × H)
10. C₂×C₃×M₁₆	C₄	semidirect
16. #926	C₄	semidirect
18. C₂×C₄⋊Dic₃	C₄	semidirect
22. C₂×C₄×Dic₃	C₄	semidirect
45. C₃×M₁₆	C₂×C₄	semidirect
51. C₄⋊Dic₃	C₂×C₄	semidirect
56. C₄×Dic₃	C₂×C₄	semidirect
61. C₂×M₁₆	Dic₃	semidirect
122. M₁₆	C₂×Dic₃	semidirect
185. C₂×C₃	#339	semidirect
206. C₃	C₂×#339	semidirect
210. C₂	#4555	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
C₂×#926	2
C₂²×C₄×Dic₃	2
#4555	2
C₂×#339	3

Subgroups — C2×#4555 #71981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4555 #71981