Subgroups — C₂×#4718 #78581

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	C₂²×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#830	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×Dic₃	C₂	1	2	yes	maximal, complemented
6	#4718	C₂	1	8	yes	maximal, direct product factor
7	C₂×#431	—	3	1	no	maximal, Sylow
8	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂³×Dic₃	C₄	1	1	yes	characteristic
10	C₂³×Dic₃	C₄	1	1	yes	characteristic
11	C₂²×C₃×C₈	C₂²	1	2	yes	—
12	C₂×C₄×Dic₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₃	C₄	1	4	yes	—
14	C₂×C₄×Dic₃	C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₃	C₂²	1	4	yes	—
16	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
18	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
19	C₃×C₂²⋊C₈	C₂²	1	4	yes	—
20	#830	C₂²	1	4	yes	—
21	Dic₃⋊C₈	C₂²	1	8	yes	complemented
22	C₈×Dic₃	C₂²	1	8	yes	complemented
23	C₂²×C₄²	—	3	1	no	—
24	#431	—	3	8	no	—
25	C₂×C₄⋊C₈	—	3	2	no	—
26	C₂×C₂²⋊C₈	—	3	1	no	—
27	C₂×C₂²⋊C₈	D₃	1	1	yes	characteristic
28	C₂×C₄×C₈	—	3	2	no	—
29	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄	—	2	4	no	—
31	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄	C₂³	1	2	yes	—
34	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃	—	2	4	no	—
36	C₂²×Dic₃	C₈	1	8	yes	complemented
37	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×Dic₃	C₂×C₄	1	4	yes	—
39	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₃	C₂×C₄	1	4	yes	—
42	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₈	—	2	4	no	—
44	C₂×C₃×C₈	C₂³	1	4	yes	—
45	C₄×Dic₃	C₂×C₄	1	4	yes	—
46	C₄×Dic₃	C₂×C₄	1	2	yes	—
47	C₄×Dic₃	C₂×C₄	1	4	yes	—
48	C₄×Dic₃	C₂×C₄	1	2	yes	—
49	C₄×Dic₃	C₂³	1	2	yes	—
50	C₄×Dic₃	C₂³	1	2	yes	—
51	C₂×C₃⋊C₈	—	2	4	no	—
52	C₂×C₃⋊C₈	C₂³	1	4	yes	—
53	C₂³×C₄	—	3	1	no	—
54	C₂³×C₄	—	3	1	no	—
55	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₈	—	3	2	no	—
57	C₂²×C₈	C₂×D₃	1	2	yes	—
58	C₂×C₄²	—	3	1	no	—
59	C₂×C₄²	—	3	4	no	—
60	C₂×C₄²	—	3	1	no	—
61	C₂×C₄²	—	3	4	no	—
62	C₂×C₄²	—	3	1	no	—
63	C₂×C₄²	—	3	1	no	—
64	C₄⋊C₈	—	3	8	no	—
65	C₂²⋊C₈	—	3	4	no	—
66	C₂²⋊C₈	C₂×D₃	1	4	yes	—
67	C₄×C₈	—	3	8	no	—
68	C₂³×C₃	—	2	2	no	—
69	C₂³×C₃	—	2	2	no	—
70	C₂³×C₃	C₂×C₈	1	4	yes	—
71	C₂³×C₃	C₂²×C₄	1	2	yes	—
72	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₄	—	2	2	no	—
74	C₂×C₃×C₄	—	2	4	no	—
75	C₂×C₃×C₄	—	2	4	no	—
76	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
77	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄	#41	1	4	yes	—
79	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₃	—	2	2	no	—
81	C₂×Dic₃	—	2	8	no	—
82	C₂×Dic₃	C₂×C₈	1	4	yes	—
83	C₂×Dic₃	C₂×C₈	1	4	yes	—
84	C₂×Dic₃	C₂×C₈	1	16	yes	complemented
85	C₂×Dic₃	M₁₆	1	4	yes	—
86	C₂×Dic₃	C₂²×C₄	1	2	yes	—
87	C₂×Dic₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₃	C₂²×C₄	1	2	yes	—
90	C₃×C₈	—	2	4	no	—
91	C₃⋊C₈	—	2	4	no	—
92	C₂⁴	C₄×D₃	1	1	yes	characteristic
93	C₂²×C₄	—	2	4	no	—
94	C₂²×C₄	—	3	8	no	—
95	C₂²×C₄	—	3	1	no	—
96	C₂²×C₄	—	3	1	no	—
97	C₂²×C₄	—	3	4	no	—
98	C₂²×C₄	—	3	1	no	—
99	C₂²×C₄	—	3	4	no	—
100	C₂²×C₄	—	3	1	no	—
101	C₂²×C₄	—	6	4	no	—
102	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
103	C₂²×C₄	C₄×D₃	1	2	yes	—
104	C₂²×C₄	C₂²×D₃	1	2	yes	—
105	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₈	—	2	4	no	—
107	C₂×C₈	—	3	4	no	—
108	C₂×C₈	—	6	4	no	—
109	C₂×C₈	C₂²×D₃	1	4	yes	—
110	C₄²	—	3	4	no	—
111	C₄²	—	3	2	no	—
112	C₄²	—	3	4	no	—
113	C₄²	—	3	2	no	—
114	C₄²	—	3	2	no	—
115	C₄²	—	3	2	no	—
116	C₂²×C₃	—	2	2	no	—
117	C₂²×C₃	—	2	2	no	—
118	C₂²×C₃	—	2	8	no	—
119	C₂²×C₃	#117	1	2	yes	—
120	C₂²×C₃	C₂²×C₈	1	4	yes	—
121	C₂²×C₃	C₂²×C₈	1	2	yes	—
122	C₂²×C₃	C₂×M₁₆	1	2	yes	—
123	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
124	C₃×C₄	—	2	2	no	—
125	C₃×C₄	#117	1	2	yes	—
126	C₃×C₄	C₂×#41	1	2	yes	—
127	Dic₃	—	2	2	no	—
128	Dic₃	C₂²×C₈	1	4	yes	—
129	Dic₃	C₂²×C₈	1	4	yes	—
130	Dic₃	C₂×M₁₆	1	4	yes	—
131	C₂³	—	2	2	no	—
132	C₂³	—	2	2	no	—
133	C₂³	D₃×C₈	1	4	yes	—
134	C₂³	C₂×C₄×D₃	1	2	yes	—
135	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	3	4	no	—
140	C₂×C₄	—	3	4	no	—
141	C₂×C₄	—	3	16	no	—
142	C₂×C₄	—	3	4	no	—
143	C₂×C₄	—	3	2	no	—
144	C₂×C₄	—	3	2	no	—
145	C₂×C₄	—	3	2	no	—
146	C₂×C₄	—	3	2	no	—
147	C₂×C₄	—	6	8	no	—
148	C₂×C₄	—	6	2	no	—
149	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
150	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂×C₄	#237	1	4	yes	—
152	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
153	C₈	—	2	4	no	—
154	C₈	—	6	4	no	—
155	C₂×C₃	—	2	4	no	—
156	C₂×C₃	#431	1	4	yes	complemented
157	C₂×C₃	C₂×#117	1	1	yes	characteristic
158	C₂×C₃	C₂³×C₈	1	1	yes	characteristic
159	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
160	C₂²	—	2	2	no	—
161	C₂²	—	2	8	no	—
162	C₂²	—	2	2	no	—
163	C₂²	#877	1	2	yes	—
164	C₂²	C₂×D₃×C₈	1	4	yes	—
165	C₂²	C₂×D₃×C₈	1	2	yes	—
166	C₂²	D₃×M₁₆	1	2	yes	—
167	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	3	4	no	—
170	C₄	—	3	4	no	—
171	C₄	—	3	4	no	—
172	C₄	—	6	2	no	—
173	C₄	#877	1	2	yes	—
174	C₄	C₂×#237	1	2	yes	—
175	C₃	C₂×#431	1	1	yes	characteristic, complemented, Sylow
176	C₂	—	2	4	no	—
177	C₂	#4718	1	4	yes	direct product factor
178	C₂	C₂×#877	1	1	yes	characteristic
179	C₂	C₂²×D₃×C₈	1	1	yes	characteristic
180	C₂	C₂×D₃×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂×C₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₂²⋊C₈	C₂
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₃
hypocenter	C₃	C₂×#431
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₃⋊C₈	C₂	semidirect
5. C₂×C₈×Dic₃	C₂	semidirect
6. #4718	C₂	direct (N × H)
21. Dic₃⋊C₈	C₂²	semidirect
22. C₈×Dic₃	C₂²	semidirect
36. C₂²×Dic₃	C₈	semidirect
84. C₂×Dic₃	C₂×C₈	semidirect
156. C₂×C₃	#431	semidirect
175. C₃	C₂×#431	semidirect
177. C₂	#4718	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×C₄×Dic₃	2
C₂×C₃×C₂²⋊C₈	2
C₂×#830	2
C₂×Dic₃⋊C₈	2
C₂×C₈×Dic₃	2
#4718	2
C₂×#431	3

Subgroups — C2×#4718 #78581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4718 #78581