Subgroups — C₂×D₃×C₄⋊C₈ #79009

Order 384.

640 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃×C₈	C₂	1	2	yes	maximal
2	C₂×C₄²×D₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₃×C₄⋊C₈	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₄⋊C₈	—	3	1	no	maximal, Sylow
8	C₂²×C₄×D₃	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₃	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₃	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₃	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₃	C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
16	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×D₃×C₈	—	2	8	no	—
18	C₂×D₃×C₈	C₂²	1	8	yes	—
19	C₄²×D₃	C₄	1	4	yes	—
20	C₄²×D₃	C₂²	1	4	yes	—
21	C₃×C₄⋊C₈	C₂²	1	4	yes	complemented
22	Dic₃⋊C₈	C₂²	1	8	yes	complemented
23	C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	complemented
24	C₂³×C₈	—	3	2	no	—
25	C₂²×C₄²	—	3	1	no	—
26	C₂×C₄⋊C₈	—	3	2	no	—
27	C₂×C₄⋊C₈	—	3	1	no	—
28	C₂×C₄⋊C₈	—	3	8	no	—
29	C₂×C₄⋊C₈	D₃	1	1	yes	characteristic, direct product factor
30	C₂³×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₃	—	2	4	no	—
38	C₂×C₄×D₃	C₈	1	8	yes	complemented
39	C₂×C₄×D₃	C₂×C₄	1	4	yes	—
40	C₂×C₄×D₃	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₃	C₂×C₄	1	4	yes	—
42	C₂×C₄×D₃	D₄	1	4	yes	—
43	C₂×C₄×D₃	Q₈	1	4	yes	—
44	C₂×C₄×D₃	C₂³	1	2	yes	—
45	C₂×C₃×C₈	—	2	4	no	—
46	C₂×C₃×C₈	C₂³	1	4	yes	—
47	C₃×C₄²	C₂×C₄	1	2	yes	—
48	C₃×C₄²	C₂³	1	2	yes	—
49	C₄×Dic₃	C₂×C₄	1	2	yes	—
50	C₄×Dic₃	C₂×C₄	1	4	yes	—
51	C₄×Dic₃	C₂×C₄	1	4	yes	—
52	C₄×Dic₃	C₂³	1	2	yes	—
53	C₂×C₃⋊C₈	—	2	4	no	—
54	C₂×C₃⋊C₈	C₂³	1	4	yes	—
55	D₃×C₈	—	2	16	no	—
56	C₂³×C₄	—	3	1	no	—
57	C₂³×C₄	—	3	1	no	—
58	C₂³×C₄	—	3	1	no	—
59	C₂²×C₈	—	3	2	no	—
60	C₂²×C₈	—	3	8	no	—
61	C₂²×C₈	—	6	8	no	—
62	C₂²×C₈	C₂×D₃	1	2	yes	—
63	C₂×C₄²	—	3	1	no	—
64	C₂×C₄²	—	3	1	no	—
65	C₂×C₄²	—	3	1	no	—
66	C₂×C₄²	—	3	4	no	—
67	C₂×C₄²	—	3	4	no	—
68	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
69	C₄⋊C₈	—	3	8	no	—
70	C₄⋊C₈	—	3	4	no	—
71	C₄⋊C₈	C₂×D₃	1	4	yes	direct product factor
72	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×D₃	C₄⋊C₄	1	4	yes	—
74	C₂²×D₃	C₂×C₈	1	4	yes	—
75	C₂²×D₃	M₁₆	1	4	yes	—
76	C₂²×D₃	C₂²×C₄	1	2	yes	—
77	C₂×C₃×C₄	—	2	2	no	—
78	C₂×C₃×C₄	C₂×C₈	1	4	yes	complemented
79	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
80	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
82	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
84	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
85	C₂×Dic₃	—	2	2	no	—
86	C₂×Dic₃	C₄⋊C₄	1	4	yes	—
87	C₂×Dic₃	C₂×C₈	1	4	yes	complemented
88	C₂×Dic₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₃	C₂²×C₄	1	2	yes	—
90	C₂×Dic₃	C₂²×C₄	1	2	yes	—
91	C₄×D₃	—	2	8	no	—
92	C₄×D₃	C₄⋊C₄	1	8	yes	—
93	C₄×D₃	C₂×C₈	1	16	yes	complemented
94	C₄×D₃	C₂×D₄	1	4	yes	—
95	C₄×D₃	C₂×Q₈	1	4	yes	—
96	C₃×C₈	—	2	4	no	—
97	C₃⋊C₈	—	2	4	no	—
98	C₂⁴	—	3	1	no	—
99	C₂²×C₄	—	3	1	no	—
100	C₂²×C₄	—	3	1	no	—
101	C₂²×C₄	—	3	8	no	—
102	C₂²×C₄	—	3	2	no	—
103	C₂²×C₄	—	3	4	no	—
104	C₂²×C₄	—	3	4	no	—
105	C₂²×C₄	—	3	4	no	—
106	C₂²×C₄	—	3	2	no	—
107	C₂²×C₄	—	3	1	no	—
108	C₂²×C₄	—	3	4	no	—
109	C₂²×C₄	—	6	4	no	—
110	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
111	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	3	4	no	—
115	C₂×C₈	—	6	16	no	—
116	C₂×C₈	—	6	4	no	—
117	C₂×C₈	C₂²×D₃	1	4	yes	—
118	C₄²	—	3	4	no	—
119	C₄²	—	3	2	no	—
120	C₄²	—	3	4	no	—
121	C₄²	—	3	2	no	—
122	C₄²	C₄×D₃	1	2	yes	—
123	C₄²	C₂²×D₃	1	2	yes	—
124	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂²×C₃	C₂²×C₈	1	2	yes	—
126	C₂²×C₃	C₂×M₁₆	1	2	yes	—
127	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×D₃	C₄⋊C₈	1	16	yes	direct product factor
129	C₂×D₃	C₂×C₄⋊C₄	1	4	yes	—
130	C₂×D₃	C₂²×C₈	1	4	yes	—
131	C₂×D₃	C₂×M₁₆	1	4	yes	—
132	C₃×C₄	—	2	2	no	—
133	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
134	C₃×C₄	C₂²×C₈	1	4	yes	complemented
135	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
136	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
137	Dic₃	—	2	2	no	—
138	Dic₃	C₂×C₄⋊C₄	1	4	yes	—
139	Dic₃	C₂²×C₈	1	4	yes	complemented
140	C₂³	—	3	4	no	—
141	C₂³	—	3	4	no	—
142	C₂³	—	3	4	no	—
143	C₂³	—	3	2	no	—
144	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	3	4	no	—
147	C₂×C₄	—	3	4	no	—
148	C₂×C₄	—	3	2	no	—
149	C₂×C₄	—	3	2	no	—
150	C₂×C₄	—	3	8	no	—
151	C₂×C₄	—	3	16	no	—
152	C₂×C₄	—	3	4	no	—
153	C₂×C₄	—	3	4	no	—
154	C₂×C₄	—	3	2	no	—
155	C₂×C₄	—	6	8	no	—
156	C₂×C₄	—	6	2	no	—
157	C₂×C₄	D₃×C₈	1	4	yes	complemented
158	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
161	C₂×C₄	D₃×D₄	1	2	yes	—
162	C₂×C₄	D₃×Q₈	1	2	yes	—
163	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
164	C₈	—	2	4	no	—
165	C₈	—	6	4	no	—
166	C₂×C₃	C₂×C₄⋊C₈	1	4	yes	complemented
167	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₃	C₂³×C₈	1	1	yes	characteristic
169	C₂×C₃	C₂²×M₁₆	1	1	yes	characteristic
170	D₃	C₂×C₄⋊C₈	1	8	yes	direct product factor
171	C₂²	—	3	4	no	—
172	C₂²	—	3	4	no	—
173	C₂²	—	3	4	no	—
174	C₂²	—	3	16	no	—
175	C₂²	D₃×C₄⋊C₄	1	2	yes	—
176	C₂²	C₂×D₃×C₈	1	2	yes	—
177	C₂²	D₃×M₁₆	1	2	yes	—
178	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	3	4	no	—
181	C₄	—	3	4	no	—
182	C₄	—	6	2	no	—
183	C₄	D₃×C₄⋊C₄	1	2	yes	—
184	C₄	C₂×D₃×C₈	1	4	yes	complemented
185	C₄	C₂×D₃×D₄	1	1	yes	characteristic
186	C₄	C₂×D₃×Q₈	1	1	yes	characteristic
187	C₃	C₂²×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
188	C₂	—	3	8	no	—
189	C₂	D₃×C₄⋊C₈	1	4	yes	direct product factor
190	C₂	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
191	C₂	C₂²×D₃×C₈	1	1	yes	characteristic
192	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×C₄⋊C₈	C₂
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂³×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₃
hypocenter	C₃	C₂²×C₄⋊C₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₄⋊C₈	C₂	semidirect
4. C₂×Dic₃⋊C₈	C₂	semidirect
5. C₂×C₄⋊(C₃⋊C₈)	C₂	semidirect
6. D₃×C₄⋊C₈	C₂	direct (N × H)
21. C₃×C₄⋊C₈	C₂²	semidirect
22. Dic₃⋊C₈	C₂²	semidirect
23. C₄⋊(C₃⋊C₈)	C₂²	semidirect
29. C₂×C₄⋊C₈	D₃	direct (N × H)
38. C₂×C₄×D₃	C₈	semidirect
71. C₄⋊C₈	C₂×D₃	direct (N × H)
78. C₂×C₃×C₄	C₂×C₈	semidirect
87. C₂×Dic₃	C₂×C₈	semidirect
93. C₄×D₃	C₂×C₈	semidirect
128. C₂×D₃	C₄⋊C₈	direct (N × H)
134. C₃×C₄	C₂²×C₈	semidirect
139. Dic₃	C₂²×C₈	semidirect
157. C₂×C₄	D₃×C₈	semidirect
166. C₂×C₃	C₂×C₄⋊C₈	semidirect
170. D₃	C₂×C₄⋊C₈	direct (N × H)
184. C₄	C₂×D₃×C₈	semidirect
187. C₃	C₂²×C₄⋊C₈	semidirect
189. C₂	D₃×C₄⋊C₈	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×D₃×C₈	2
C₂×C₄²×D₃	2
C₂×C₃×C₄⋊C₈	2
C₂×Dic₃⋊C₈	2
C₂×C₄⋊(C₃⋊C₈)	2
D₃×C₄⋊C₈	2
C₂²×C₄⋊C₈	3

Subgroups — C2×D3×C4⋊C8 #79009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×C₄⋊C₈ #79009