Subgroups — C₂×C₄×#217 #84024

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×#217	C₂	1	2	yes	maximal
3	C₂²×C₄×Dic₃	C₂	1	2	yes	maximal, complemented
4	C₄×#217	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃⋊#95	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₂²⋊C₄	—	3	1	no	maximal, 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	1	yes	characteristic
13	C₂³×Dic₃	C₂²	1	4	yes	—
14	C₂×#217	C₄	1	16	yes	direct product factor
15	C₂×#217	C₂²	1	8	yes	—
16	C₂×C₄×Dic₃	—	2	8	no	—
17	C₂×C₄×Dic₃	C₂²	1	8	yes	complemented
18	C₃⋊#95	C₂²	1	8	yes	complemented
19	C₂⁴×C₄	D₃	1	1	yes	characteristic
20	C₂²×C₂²⋊C₄	—	3	2	no	—
21	C₂²×C₄²	—	3	2	no	—
22	C₄×C₂²⋊C₄	—	3	8	no	—
23	C₂×#95	—	3	2	no	—
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	4	no	—
31	C₂²×C₃×C₄	—	2	32	no	—
32	C₂²×C₃×C₄	—	2	16	no	—
33	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
34	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
35	C₂²×C₃×C₄	C₂×C₄	1	16	yes	complemented
36	C₂²×C₃×C₄	D₄	1	8	yes	—
37	C₂²×C₃×C₄	C₂³	1	2	yes	—
38	C₂²×C₃×C₄	C₂³	1	2	yes	—
39	C₂²×Dic₃	—	2	8	no	—
40	C₂²×Dic₃	—	2	8	no	—
41	C₂²×Dic₃	C₂×C₄	1	16	yes	complemented
42	C₂²×Dic₃	C₂³	1	8	yes	—
43	#217	C₂×C₄	1	32	yes	direct product factor
44	C₄×Dic₃	—	2	16	no	—
45	C₂⁵	C₂×D₃	1	1	yes	characteristic
46	C₂³×C₄	—	2	8	no	—
47	C₂³×C₄	—	3	4	no	—
48	C₂³×C₄	Dic₃	1	8	yes	complemented
49	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
50	C₂³×C₄	C₂×D₃	1	4	yes	—
51	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
52	C₂×C₂²⋊C₄	—	3	16	no	—
53	C₂×C₂²⋊C₄	—	3	8	no	—
54	C₂×C₄²	—	3	8	no	—
55	C₂×C₄²	—	6	8	no	—
56	#95	—	3	8	no	—
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	4	yes	—
66	C₂³×C₃	C₂²×C₄	1	2	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	4	no	—
72	C₂×C₃×C₄	—	2	32	no	—
73	C₂×C₃×C₄	C₂²⋊C₄	1	16	yes	complemented
74	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
75	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
76	C₂×C₃×C₄	C₂×D₄	1	8	yes	—
77	C₂×Dic₃	—	2	32	no	—
78	C₂×Dic₃	—	2	8	no	—
79	C₂×Dic₃	—	2	8	no	—
80	C₂×Dic₃	C₂²×C₄	1	16	yes	complemented
81	C₂⁴	—	2	4	no	—
82	C₂⁴	—	2	4	no	—
83	C₂⁴	C₄×D₃	1	8	yes	—
84	C₂⁴	C₂×Dic₃	1	4	yes	—
85	C₂⁴	C₂²×D₃	1	2	yes	—
86	C₂⁴	C₂²×D₃	1	1	yes	characteristic
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	—	2	32	no	—
89	C₂²×C₄	—	2	16	no	—
90	C₂²×C₄	—	3	16	no	—
91	C₂²×C₄	—	3	8	no	—
92	C₂²×C₄	—	6	8	no	—
93	C₂²×C₄	—	6	8	no	—
94	C₂²×C₄	C₂×Dic₃	1	4	yes	—
95	C₂²×C₄	C₂×Dic₃	1	4	yes	—
96	C₂²×C₄	C₂×Dic₃	1	16	yes	complemented
97	C₂²×C₄	#67	1	8	yes	—
98	C₂²×C₄	C₂²×D₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₃	1	2	yes	—
100	C₂²⋊C₄	—	3	32	no	—
101	C₄²	—	6	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	2	yes	—
112	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
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	C₂×C₂²⋊C₄	1	8	yes	complemented
117	Dic₃	—	2	16	no	—
118	C₂³	—	2	8	no	—
119	C₂³	—	2	32	no	—
120	C₂³	—	2	8	no	—
121	C₂³	—	2	4	no	—
122	C₂³	—	2	4	no	—
123	C₂³	C₄×Dic₃	1	16	yes	complemented
124	C₂³	C₂×C₄×D₃	1	8	yes	—
125	C₂³	C₂×C₄×D₃	1	4	yes	—
126	C₂³	#235	1	4	yes	—
127	C₂³	C₂²×Dic₃	1	4	yes	—
128	C₂³	C₂²×Dic₃	1	2	yes	—
129	C₂³	C₂×#67	1	4	yes	—
130	C₂³	C₂³×D₃	1	1	yes	characteristic
131	C₂×C₄	—	2	8	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	32	no	—
134	C₂×C₄	—	3	16	no	—
135	C₂×C₄	—	6	32	no	—
136	C₂×C₄	—	6	8	no	—
137	C₂×C₄	—	6	8	no	—
138	C₂×C₄	#217	1	16	yes	direct product factor
139	C₂×C₄	C₂²×Dic₃	1	4	yes	—
140	C₂×C₄	C₂²×Dic₃	1	4	yes	—
141	C₂×C₄	C₂×#67	1	8	yes	—
142	C₂×C₃	—	2	8	no	—
143	C₂×C₃	C₄×C₂²⋊C₄	1	8	yes	complemented
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	16	no	—
151	C₂²	—	2	4	no	—
152	C₂²	C₂×C₄×Dic₃	1	4	yes	—
153	C₂²	C₂×C₄×Dic₃	1	8	yes	complemented
154	C₂²	#926	1	4	yes	—
155	C₂²	C₄×#67	1	16	yes	—
156	C₂²	C₂×#217	1	4	yes	—
157	C₂²	C₂²×C₄×D₃	1	2	yes	—
158	C₂²	C₂×#235	1	2	yes	—
159	C₂²	C₂³×Dic₃	1	1	yes	characteristic
160	C₂²	C₂²×#67	1	2	yes	—
161	C₄	—	2	4	no	—
162	C₄	—	6	16	no	—
163	C₄	C₂×#217	1	8	yes	direct product factor
164	C₃	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
165	C₂	—	2	8	no	—
166	C₂	C₄×#217	1	8	yes	direct product factor
167	C₂	C₂²×C₄×Dic₃	1	1	yes	characteristic
168	C₂	C₂×#926	1	1	yes	characteristic
169	C₂	C₂×C₄×#67	1	4	yes	—
170	C₂	C₂²×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₃
commutator	C₂×C₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₃
Fitting	C₂⁴×C₃×C₄	C₂
socle	C₂⁴×C₃	C₂³
prime radical	C₂⁵×C₃	C₂²
cosocle	C₂³×C₃	C₂⁴
hypercenter	C₂⁴×C₄	D₃
hypocenter	C₃	C₂×C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×Dic₃	C₂	semidirect
4. C₄×#217	C₂	direct (N × H)
5. C₂×C₃⋊#95	C₂	semidirect
9. C₂³×C₃×C₄	C₄	semidirect
14. C₂×#217	C₄	direct (N × H)
17. C₂×C₄×Dic₃	C₂²	semidirect
18. C₃⋊#95	C₂²	semidirect
35. C₂²×C₃×C₄	C₂×C₄	semidirect
41. C₂²×Dic₃	C₂×C₄	semidirect
43. #217	C₂×C₄	direct (N × H)
48. C₂³×C₄	Dic₃	semidirect
62. C₂³×C₃	C₄²	semidirect
73. C₂×C₃×C₄	C₂²⋊C₄	semidirect
80. C₂×Dic₃	C₂²×C₄	semidirect
96. C₂²×C₄	C₂×Dic₃	semidirect
107. C₂²×C₃	C₂×C₄²	semidirect
116. C₃×C₄	C₂×C₂²⋊C₄	semidirect
123. C₂³	C₄×Dic₃	semidirect
138. C₂×C₄	#217	direct (N × H)
143. C₂×C₃	C₄×C₂²⋊C₄	semidirect
153. C₂²	C₂×C₄×Dic₃	semidirect
163. C₄	C₂×#217	direct (N × H)
164. C₃	C₂×C₄×C₂²⋊C₄	semidirect
166. C₂	C₄×#217	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×#217	2
C₂²×C₄×Dic₃	2
C₄×#217	2
C₂×C₃⋊#95	2
C₂×C₄×C₂²⋊C₄	3

Subgroups — C2×C4×#217 #84024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#217 #84024