Subgroups — C₂×#4550 #71964

Order 384.

528 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#4550	C₂	1	4	yes	maximal, direct product factor
5	C₂×#336	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂³×Dic₃	C₄	1	2	yes	—
8	C₂²×C₃×C₈	—	2	2	no	—
9	C₂²×C₃×C₈	—	2	2	no	—
10	C₂²×C₃×C₈	C₄	1	4	yes	complemented
11	C₂²×C₃×C₈	C₂²	1	2	yes	—
12	C₂×C₄×Dic₃	—	2	2	no	—
13	C₂×C₄×Dic₃	—	2	2	no	—
14	C₂×C₄×Dic₃	C₄	1	2	yes	—
15	C₂×C₄×Dic₃	C₂²	1	2	yes	—
16	C₂²×C₃⋊C₈	—	2	2	no	—
17	C₂²×C₃⋊C₈	—	2	2	no	—
18	C₂²×C₃⋊C₈	C₄	1	4	yes	complemented
19	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
20	C₂³×C₈	—	3	1	no	—
21	C₂³×C₈	D₃	1	1	yes	characteristic
22	C₂²×C₄²	—	3	1	no	—
23	#336	—	3	4	no	—
24	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄	D₄	1	2	yes	—
29	C₂²×C₃×C₄	D₄	1	2	yes	—
30	C₂²×C₃×C₄	D₄	1	2	yes	—
31	C₂²×C₃×C₄	Q₈	1	2	yes	—
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂²×Dic₃	—	2	4	no	—
34	C₂²×Dic₃	—	2	4	no	—
35	C₂²×Dic₃	C₈	1	8	yes	complemented
36	C₂²×Dic₃	C₂×C₄	1	4	yes	—
37	C₂×C₃×C₈	—	2	8	no	—
38	C₂×C₃×C₈	—	2	2	no	—
39	C₂×C₃×C₈	—	2	2	no	—
40	C₂×C₃×C₈	C₂×C₄	1	4	yes	complemented
41	C₄×Dic₃	—	2	4	no	—
42	C₄×Dic₃	—	2	2	no	—
43	C₄×Dic₃	—	2	2	no	—
44	C₂×C₃⋊C₈	—	2	8	no	—
45	C₂×C₃⋊C₈	—	2	2	no	—
46	C₂×C₃⋊C₈	—	2	2	no	—
47	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
48	C₂³×C₄	—	3	2	no	—
49	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₈	—	2	2	no	—
51	C₂²×C₈	—	2	2	no	—
52	C₂²×C₈	—	3	4	no	—
53	C₂²×C₈	—	3	2	no	—
54	C₂²×C₈	—	6	2	no	—
55	C₂²×C₈	—	6	2	no	—
56	C₂²×C₈	Dic₃	1	4	yes	complemented
57	C₂²×C₈	C₂×D₃	1	2	yes	—
58	C₂×C₄²	—	3	2	no	—
59	C₂×C₄²	—	3	2	no	—
60	C₂×C₄²	—	6	2	no	—
61	C₂×C₄²	—	6	2	no	—
62	C₂³×C₃	C₄²	1	2	yes	—
63	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
64	C₂³×C₃	C₄⋊C₄	1	2	yes	—
65	C₂³×C₃	C₂×C₈	1	4	yes	—
66	C₂³×C₃	M₁₆	1	4	yes	—
67	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄	C₄²	1	2	yes	—
69	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
70	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
71	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
72	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
73	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
74	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
75	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
81	C₂×Dic₃	—	2	16	no	—
82	C₂×Dic₃	—	2	4	no	—
83	C₂×Dic₃	—	2	4	no	—
84	C₂×Dic₃	C₂×C₈	1	8	yes	complemented
85	C₃×C₈	—	2	4	no	—
86	C₃⋊C₈	—	2	4	no	—
87	C₂⁴	C₄×D₃	1	1	yes	characteristic
88	C₂²×C₄	—	3	8	no	—
89	C₂²×C₄	—	3	4	no	—
90	C₂²×C₄	—	6	4	no	—
91	C₂²×C₄	—	6	4	no	—
92	C₂²×C₄	Dic₆	1	2	yes	—
93	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
94	C₂²×C₄	C₄×D₃	1	2	yes	—
95	C₂²×C₄	D₁₂	1	2	yes	—
96	C₂²×C₄	C₂×Dic₃	1	2	yes	—
97	C₂²×C₄	#67	1	2	yes	—
98	C₂²×C₄	#67	1	2	yes	—
99	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂×C₈	—	2	8	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	3	4	no	—
104	C₂×C₈	—	6	8	no	—
105	C₂×C₈	—	6	2	no	—
106	C₂×C₈	—	6	2	no	—
107	C₂×C₈	C₂×Dic₃	1	4	yes	complemented
108	C₄²	—	6	4	no	—
109	C₄²	—	6	2	no	—
110	C₄²	—	6	2	no	—
111	C₂²×C₃	#95	1	4	yes	—
112	C₂²×C₃	C₄×C₈	1	4	yes	—
113	C₂²×C₃	#97	1	4	yes	—
114	C₂²×C₃	C₂²⋊C₈	1	8	yes	—
115	C₂²×C₃	C₄⋊C₈	1	8	yes	—
116	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
117	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₃	C₂²×C₈	1	2	yes	—
120	C₂²×C₃	C₂×M₁₆	1	2	yes	—
121	C₃×C₄	#95	1	4	yes	—
122	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
126	Dic₃	—	2	8	no	—
127	C₂³	D₃×C₈	1	4	yes	—
128	C₂³	ΓL(1,𝔽₂₅)	1	4	yes	—
129	C₂³	C₄×Dic₃	1	2	yes	—
130	C₂³	Dic₃⋊C₄	1	2	yes	—
131	C₂³	(C₂×D₃)⋊C₄	1	2	yes	—
132	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
133	C₂×C₄	—	3	8	no	—
134	C₂×C₄	—	6	16	no	—
135	C₂×C₄	—	6	4	no	—
136	C₂×C₄	—	6	4	no	—
137	C₂×C₄	C₄×Dic₃	1	2	yes	—
138	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
139	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
140	C₂×C₄	C₄⋊Dic₃	1	4	yes	—
141	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
142	C₂×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
143	C₂×C₄	#217	1	4	yes	—
144	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
146	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
147	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
148	C₂×C₄	C₂×#67	1	1	yes	characteristic
149	C₂×C₄	C₂×#67	1	1	yes	characteristic
150	C₈	—	2	4	no	—
151	C₈	—	6	4	no	—
152	C₂×C₃	#336	1	8	yes	complemented
153	C₂×C₃	C₂×#95	1	1	yes	characteristic
154	C₂×C₃	C₂×C₄×C₈	1	1	yes	characteristic
155	C₂×C₃	C₂×#97	1	1	yes	characteristic
156	C₂×C₃	C₂×C₂²⋊C₈	1	2	yes	—
157	C₂×C₃	C₂×C₄⋊C₈	1	2	yes	—
158	C₂²	C₈×Dic₃	1	4	yes	—
159	C₂²	Dic₃⋊C₈	1	8	yes	—
160	C₂²	#815	1	4	yes	—
161	C₂²	(C₂×D₃)⋊C₈	1	8	yes	—
162	C₂²	C₃⋊#95	1	4	yes	—
163	C₂²	C₂×D₃×C₈	1	2	yes	—
164	C₂²	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
165	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
166	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
167	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
168	C₄	—	6	8	no	—
169	C₄	C₃⋊#95	1	4	yes	—
170	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
171	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
172	C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₄	C₂×#217	1	1	yes	characteristic
174	C₃	C₂×#336	1	1	yes	characteristic, complemented, Sylow
175	C₂	#4550	1	8	yes	direct product factor
176	C₂	C₂×C₈×Dic₃	1	1	yes	characteristic
177	C₂	C₂×Dic₃⋊C₈	1	2	yes	—
178	C₂	C₂×#815	1	1	yes	characteristic
179	C₂	C₂×(C₂×D₃)⋊C₈	1	2	yes	—
180	C₂	C₂×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂³×C₃×C₈	C₂
socle, prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂³×C₈	D₃
hypocenter	C₃	C₂×#336
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
4. #4550	C₂	direct (N × H)
10. C₂²×C₃×C₈	C₄	semidirect
18. C₂²×C₃⋊C₈	C₄	semidirect
35. C₂²×Dic₃	C₈	semidirect
40. C₂×C₃×C₈	C₂×C₄	semidirect
47. C₂×C₃⋊C₈	C₂×C₄	semidirect
56. C₂²×C₈	Dic₃	semidirect
84. C₂×Dic₃	C₂×C₈	semidirect
107. C₂×C₈	C₂×Dic₃	semidirect
152. C₂×C₃	#336	semidirect
174. C₃	C₂×#336	semidirect
175. C₂	#4550	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₈	2
C₂²×C₄×Dic₃	2
C₂³×C₃⋊C₈	2
#4550	2
C₂×#336	3

Subgroups — C2×#4550 #71964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4550 #71964