Subgroups — C₂×C₃×C₄×C₂²⋊C₄ #84132

Order 384.

1360 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₂²×C₃×C₄²	C₂	1	2	yes	maximal, complemented
4	C₃×C₄×C₂²⋊C₄	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#95	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₂²⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂⁵×C₃	C₂²	1	1	yes	characteristic
8	C₂³×C₃×C₄	—	2	8	no	—
9	C₂³×C₃×C₄	C₄	1	8	yes	complemented
10	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₂²	1	4	yes	—
12	C₂³×C₃×C₄	C₂²	1	4	yes	—
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₂²⋊C₄	C₄	1	16	yes	direct product factor
15	C₂×C₃×C₂²⋊C₄	C₂²	1	8	yes	—
16	C₂×C₃×C₄²	—	2	8	no	—
17	C₂×C₃×C₄²	C₂²	1	8	yes	complemented
18	C₃×#95	C₂²	1	8	yes	complemented
19	C₂⁴×C₄	C₂×C₃	1	1	yes	characteristic
20	C₂²×C₂²⋊C₄	C₂×C₃	1	2	yes	—
21	C₂²×C₄²	C₂×C₃	1	2	yes	complemented
22	C₄×C₂²⋊C₄	C₂×C₃	1	8	yes	direct product factor
23	C₂×#95	C₂×C₃	1	2	yes	complemented
24	C₂⁴×C₃	—	2	4	no	—
25	C₂⁴×C₃	—	2	4	no	—
26	C₂⁴×C₃	C₂×C₄	1	8	yes	—
27	C₂⁴×C₃	C₂×C₄	1	4	yes	—
28	C₂⁴×C₃	C₂³	1	2	yes	—
29	C₂⁴×C₃	C₂³	1	1	yes	characteristic
30	C₂²×C₃×C₄	—	2	8	no	—
31	C₂²×C₃×C₄	—	2	4	no	—
32	C₂²×C₃×C₄	—	2	8	no	—
33	C₂²×C₃×C₄	—	2	32	no	—
34	C₂²×C₃×C₄	—	2	16	no	—
35	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
36	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
37	C₂²×C₃×C₄	C₂×C₄	1	16	yes	complemented
38	C₂²×C₃×C₄	C₂×C₄	1	16	yes	complemented
39	C₂²×C₃×C₄	D₄	1	8	yes	—
40	C₂²×C₃×C₄	C₂³	1	2	yes	—
41	C₂²×C₃×C₄	C₂³	1	8	yes	—
42	C₂²×C₃×C₄	C₂³	1	2	yes	—
43	C₃×C₂²⋊C₄	C₂×C₄	1	32	yes	direct product factor
44	C₃×C₄²	—	2	16	no	—
45	C₂⁵	C₂²×C₃	1	1	yes	characteristic
46	C₂³×C₄	—	2	8	no	—
47	C₂³×C₄	C₃×C₄	1	8	yes	complemented
48	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
49	C₂³×C₄	C₂²×C₃	1	4	yes	—
50	C₂³×C₄	C₂²×C₃	1	4	yes	—
51	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₂²⋊C₄	C₃×C₄	1	16	yes	direct product factor
53	C₂×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
54	C₂×C₄²	—	2	8	no	—
55	C₂×C₄²	C₂²×C₃	1	8	yes	complemented
56	#95	C₂²×C₃	1	8	yes	complemented
57	C₂³×C₃	—	2	8	no	—
58	C₂³×C₃	—	2	8	no	—
59	C₂³×C₃	—	2	32	no	—
60	C₂³×C₃	—	2	4	no	—
61	C₂³×C₃	—	2	4	no	—
62	C₂³×C₃	C₄²	1	16	yes	complemented
63	C₂³×C₃	C₂²×C₄	1	8	yes	—
64	C₂³×C₃	C₂²×C₄	1	4	yes	—
65	C₂³×C₃	C₂²×C₄	1	2	yes	—
66	C₂³×C₃	C₂²×C₄	1	4	yes	—
67	C₂³×C₃	C₂×D₄	1	4	yes	—
68	C₂³×C₃	#41	1	4	yes	—
69	C₂³×C₃	C₂⁴	1	1	yes	characteristic
70	C₂×C₃×C₄	—	2	8	no	—
71	C₂×C₃×C₄	—	2	32	no	—
72	C₂×C₃×C₄	—	2	8	no	—
73	C₂×C₃×C₄	—	2	4	no	—
74	C₂×C₃×C₄	—	2	8	no	—
75	C₂×C₃×C₄	—	2	32	no	—
76	C₂×C₃×C₄	C₂²⋊C₄	1	16	yes	direct product factor
77	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
78	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
79	C₂×C₃×C₄	C₂²×C₄	1	16	yes	complemented
80	C₂×C₃×C₄	C₂×D₄	1	8	yes	—
81	C₂⁴	—	2	4	no	—
82	C₂⁴	—	2	4	no	—
83	C₂⁴	C₂×C₃×C₄	1	8	yes	—
84	C₂⁴	C₂×C₃×C₄	1	4	yes	—
85	C₂⁴	C₂³×C₃	1	2	yes	—
86	C₂⁴	C₂³×C₃	1	1	yes	characteristic
87	C₂²×C₄	—	2	8	no	—
88	C₂²×C₄	—	2	4	no	—
89	C₂²×C₄	—	2	8	no	—
90	C₂²×C₄	—	2	32	no	—
91	C₂²×C₄	—	2	16	no	—
92	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
93	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
94	C₂²×C₄	C₂×C₃×C₄	1	16	yes	complemented
95	C₂²×C₄	C₂×C₃×C₄	1	16	yes	complemented
96	C₂²×C₄	C₃×D₄	1	8	yes	—
97	C₂²×C₄	C₂³×C₃	1	2	yes	—
98	C₂²×C₄	C₂³×C₃	1	8	yes	—
99	C₂²×C₄	C₂³×C₃	1	2	yes	—
100	C₂²⋊C₄	C₂×C₃×C₄	1	32	yes	direct product factor
101	C₄²	—	2	16	no	—
102	C₂²×C₃	—	2	4	no	—
103	C₂²×C₃	—	2	4	no	—
104	C₂²×C₃	—	2	16	no	—
105	C₂²×C₃	—	2	32	no	—
106	C₂²×C₃	C₂×C₄²	1	4	yes	—
107	C₂²×C₃	C₂×C₄²	1	8	yes	complemented
108	C₂²×C₃	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂²×C₃	#117	1	4	yes	—
110	C₂²×C₃	C₄×D₄	1	16	yes	—
111	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
112	C₂²×C₃	C₂³×C₄	1	2	yes	—
113	C₂²×C₃	C₂²×D₄	1	2	yes	—
114	C₂²×C₃	C₂×#41	1	2	yes	—
115	C₃×C₄	—	2	4	no	—
116	C₃×C₄	—	2	16	no	—
117	C₃×C₄	C₂×C₂²⋊C₄	1	8	yes	direct product factor
118	C₂³	—	2	8	no	—
119	C₂³	—	2	8	no	—
120	C₂³	—	2	32	no	—
121	C₂³	—	2	4	no	—
122	C₂³	—	2	4	no	—
123	C₂³	C₃×C₄²	1	16	yes	complemented
124	C₂³	C₂²×C₃×C₄	1	8	yes	—
125	C₂³	C₂²×C₃×C₄	1	4	yes	—
126	C₂³	C₂²×C₃×C₄	1	2	yes	—
127	C₂³	C₂²×C₃×C₄	1	4	yes	—
128	C₂³	C₂×C₃×D₄	1	4	yes	—
129	C₂³	C₃×#41	1	4	yes	—
130	C₂³	C₂⁴×C₃	1	1	yes	characteristic
131	C₂×C₄	—	2	8	no	—
132	C₂×C₄	—	2	32	no	—
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	—	2	32	no	—
137	C₂×C₄	C₃×C₂²⋊C₄	1	16	yes	direct product factor
138	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
139	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
140	C₂×C₄	C₂²×C₃×C₄	1	16	yes	complemented
141	C₂×C₄	C₂×C₃×D₄	1	8	yes	—
142	C₂×C₃	—	2	8	no	—
143	C₂×C₃	C₄×C₂²⋊C₄	1	8	yes	direct product factor
144	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
145	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₃	C₂×#117	1	1	yes	characteristic
147	C₂×C₃	C₂×C₄×D₄	1	4	yes	—
148	C₂²	—	2	32	no	—
149	C₂²	—	2	4	no	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	16	no	—
152	C₂²	C₂×C₃×C₄²	1	4	yes	—
153	C₂²	C₂×C₃×C₄²	1	8	yes	complemented
154	C₂²	C₂×C₃×C₂²⋊C₄	1	4	yes	—
155	C₂²	C₃×#117	1	4	yes	—
156	C₂²	C₃×C₄×D₄	1	16	yes	—
157	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
158	C₂²	C₂³×C₃×C₄	1	2	yes	—
159	C₂²	C₂²×C₃×D₄	1	2	yes	—
160	C₂²	C₂×C₃×#41	1	2	yes	—
161	C₄	—	2	4	no	—
162	C₄	—	2	16	no	—
163	C₄	C₂×C₃×C₂²⋊C₄	1	8	yes	direct product factor
164	C₃	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
165	C₂	—	2	8	no	—
166	C₂	C₃×C₄×C₂²⋊C₄	1	8	yes	direct product factor
167	C₂	C₂²×C₃×C₄²	1	1	yes	characteristic
168	C₂	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₂	C₂×C₃×#117	1	1	yes	characteristic
170	C₂	C₂×C₃×C₄×D₄	1	4	yes	—

	subgroup	G/H
center	C₂³×C₃×C₄	C₂²
commutator	C₂	C₂²×C₃×C₄²
Frattini, cosocle	C₂³	C₂⁴×C₃
socle	C₂⁴×C₃	C₂³
prime radical	C₂⁵×C₃	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₃×C₄²	C₂	semidirect
4. C₃×C₄×C₂²⋊C₄	C₂	direct (N × H)
5. C₂×C₃×#95	C₂	semidirect
6. C₂×C₄×C₂²⋊C₄	C₃	direct (N × H)
9. C₂³×C₃×C₄	C₄	semidirect
14. C₂×C₃×C₂²⋊C₄	C₄	direct (N × H)
17. C₂×C₃×C₄²	C₂²	semidirect
18. C₃×#95	C₂²	semidirect
21. C₂²×C₄²	C₂×C₃	semidirect
22. C₄×C₂²⋊C₄	C₂×C₃	direct (N × H)
23. C₂×#95	C₂×C₃	semidirect
37. C₂²×C₃×C₄	C₂×C₄	semidirect
38. C₂²×C₃×C₄	C₂×C₄	semidirect
43. C₃×C₂²⋊C₄	C₂×C₄	direct (N × H)
47. C₂³×C₄	C₃×C₄	semidirect
52. C₂×C₂²⋊C₄	C₃×C₄	direct (N × H)
55. C₂×C₄²	C₂²×C₃	semidirect
56. #95	C₂²×C₃	semidirect
62. C₂³×C₃	C₄²	semidirect
76. C₂×C₃×C₄	C₂²⋊C₄	direct (N × H)
79. C₂×C₃×C₄	C₂²×C₄	semidirect
94. C₂²×C₄	C₂×C₃×C₄	semidirect
95. C₂²×C₄	C₂×C₃×C₄	semidirect
100. C₂²⋊C₄	C₂×C₃×C₄	direct (N × H)
107. C₂²×C₃	C₂×C₄²	semidirect
117. C₃×C₄	C₂×C₂²⋊C₄	direct (N × H)
123. C₂³	C₃×C₄²	semidirect
137. C₂×C₄	C₃×C₂²⋊C₄	direct (N × H)
140. C₂×C₄	C₂²×C₃×C₄	semidirect
143. C₂×C₃	C₄×C₂²⋊C₄	direct (N × H)
153. C₂²	C₂×C₃×C₄²	semidirect
163. C₄	C₂×C₃×C₂²⋊C₄	direct (N × H)
164. C₃	C₂×C₄×C₂²⋊C₄	direct (N × H)
166. C₂	C₃×C₄×C₂²⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₄	2
C₂²×C₃×C₂²⋊C₄	2
C₂²×C₃×C₄²	2
C₃×C₄×C₂²⋊C₄	2
C₂×C₃×#95	2
C₂×C₄×C₂²⋊C₄	3

Subgroups — C2×C3×C4×C22⋊C4 #84132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×C₂²⋊C₄ #84132