Subgroups — C₂×C₃×#541 #87337

Order 384.

1004 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	C₃×#541	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#126	C₂	1	4	yes	maximal
4	C₂×C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#125	C₂	1	4	yes	maximal, complemented
6	C₂×C₃×C₂²⋊Q₈	C₂	1	2	yes	maximal
7	C₂×C₃×C₄×Q₈	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
9	C₂×#541	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₈	C₂²	1	2	yes	—
11	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
13	C₃×#126	C₂²	1	32	yes	—
14	C₃×#125	C₂²	1	8	yes	complemented
15	C₃×#125	C₂²	1	32	yes	complemented
16	C₃×C₂²⋊Q₈	C₂²	1	16	yes	—
17	C₃×C₄×Q₈	C₂²	1	16	yes	complemented
18	C₃×#117	C₂²	1	8	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
22	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
23	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
24	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
27	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂²	1	4	yes	—
30	#541	C₂×C₃	1	16	yes	direct product factor
31	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×#126	C₂×C₃	1	4	yes	—
33	C₂×#125	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×#125	C₂×C₃	1	4	yes	complemented
35	C₂×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
36	C₂×C₄×Q₈	C₂×C₃	1	2	yes	complemented
37	C₂×#117	C₂×C₃	1	1	yes	characteristic
38	C₂⁴×C₃	C₂³	1	1	yes	characteristic
39	C₂×C₃×Q₈	—	2	8	no	—
40	C₂×C₃×Q₈	C₂³	1	8	yes	—
41	C₂²×C₃×C₄	—	2	4	no	—
42	C₂²×C₃×C₄	C₂³	1	4	yes	—
43	C₂²×C₃×C₄	C₂³	1	2	yes	—
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₂²×C₃×C₄	C₂³	1	4	yes	—
46	C₂²×C₃×C₄	C₂³	1	4	yes	—
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
49	C₃×C₄⋊C₄	C₂³	1	8	yes	—
50	C₃×C₄⋊C₄	C₂³	1	8	yes	—
51	C₃×C₄⋊C₄	C₂³	1	16	yes	—
52	C₃×C₄⋊C₄	C₂³	1	16	yes	—
53	C₃×C₄⋊C₄	C₂³	1	16	yes	—
54	C₃×C₄⋊C₄	C₂³	1	16	yes	—
55	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
56	C₃×C₂²⋊C₄	C₂³	1	16	yes	—
57	C₃×C₄²	C₂³	1	4	yes	—
58	C₃×C₄²	C₂³	1	4	yes	—
59	C₃×C₄²	C₂³	1	16	yes	—
60	C₂²×Q₈	C₂²×C₃	1	2	yes	—
61	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
62	C₄⋊Q₈	C₂²×C₃	1	8	yes	complemented
63	#126	C₂²×C₃	1	32	yes	—
64	#125	C₂²×C₃	1	8	yes	complemented
65	#125	C₂²×C₃	1	32	yes	complemented
66	C₂²⋊Q₈	C₂²×C₃	1	16	yes	—
67	C₄×Q₈	C₂²×C₃	1	16	yes	complemented
68	#117	C₂²×C₃	1	8	yes	—
69	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
72	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
73	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
74	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
75	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
76	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
77	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₄²	C₂²×C₃	1	4	yes	—
80	C₂³×C₃	—	2	2	no	—
81	C₂³×C₃	—	2	4	no	—
82	C₂³×C₃	C₂⁴	1	2	yes	—
83	C₂³×C₃	C₂⁴	1	1	yes	characteristic
84	C₃×Q₈	—	2	16	no	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	8	no	—
87	C₂×C₃×C₄	—	2	4	no	—
88	C₂×C₃×C₄	—	2	4	no	—
89	C₂×C₃×C₄	—	2	8	no	—
90	C₂×C₃×C₄	—	2	8	no	—
91	C₂×C₃×C₄	#41	1	4	yes	—
92	C₂×C₃×C₄	C₂⁴	1	4	yes	—
93	C₂×C₃×C₄	C₂⁴	1	4	yes	—
94	C₂×C₃×C₄	C₂⁴	1	8	yes	—
95	C₂×C₃×C₄	C₂⁴	1	8	yes	—
96	C₂×C₃×C₄	C₂⁴	1	2	yes	—
97	C₂×C₃×C₄	C₂⁴	1	2	yes	—
98	C₂⁴	C₂³×C₃	1	1	yes	characteristic
99	C₂×Q₈	—	2	8	no	—
100	C₂×Q₈	C₂³×C₃	1	8	yes	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	C₂³×C₃	1	2	yes	—
103	C₂²×C₄	C₂³×C₃	1	2	yes	—
104	C₂²×C₄	C₂³×C₃	1	4	yes	—
105	C₂²×C₄	C₂³×C₃	1	4	yes	—
106	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
107	C₂²×C₄	C₂³×C₃	1	4	yes	—
108	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
109	C₄⋊C₄	C₂³×C₃	1	8	yes	—
110	C₄⋊C₄	C₂³×C₃	1	8	yes	—
111	C₄⋊C₄	C₂³×C₃	1	16	yes	—
112	C₄⋊C₄	C₂³×C₃	1	16	yes	—
113	C₄⋊C₄	C₂³×C₃	1	16	yes	—
114	C₄⋊C₄	C₂³×C₃	1	16	yes	—
115	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
116	C₂²⋊C₄	C₂³×C₃	1	16	yes	—
117	C₄²	C₂³×C₃	1	4	yes	—
118	C₄²	C₂³×C₃	1	4	yes	—
119	C₄²	C₂³×C₃	1	16	yes	—
120	C₂²×C₃	—	2	2	no	—
121	C₂²×C₃	—	2	4	no	—
122	C₂²×C₃	—	4	4	no	—
123	C₂²×C₃	C₂×#41	1	2	yes	—
124	C₂²×C₃	2₋¹⁺⁴	1	4	yes	—
125	C₂²×C₃	C₂⁵	1	1	yes	characteristic
126	C₃×C₄	—	2	2	no	—
127	C₃×C₄	—	2	8	no	—
128	C₃×C₄	—	2	4	no	—
129	C₃×C₄	—	2	4	no	—
130	C₃×C₄	—	2	8	no	—
131	C₃×C₄	C₂×#41	1	4	yes	—
132	C₂³	—	2	2	no	—
133	C₂³	—	2	4	no	—
134	C₂³	C₂⁴×C₃	1	2	yes	—
135	C₂³	C₂⁴×C₃	1	1	yes	characteristic
136	Q₈	—	2	16	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	C₃×#41	1	4	yes	—
144	C₂×C₄	C₂⁴×C₃	1	4	yes	—
145	C₂×C₄	C₂⁴×C₃	1	4	yes	—
146	C₂×C₄	C₂⁴×C₃	1	8	yes	—
147	C₂×C₄	C₂⁴×C₃	1	8	yes	—
148	C₂×C₄	C₂⁴×C₃	1	2	yes	—
149	C₂×C₄	C₂⁴×C₃	1	2	yes	—
150	C₂×C₃	—	4	2	no	—
151	C₂×C₃	#541	1	4	yes	direct product factor
152	C₂×C₃	C₂²×#41	1	1	yes	characteristic
153	C₂×C₃	C₂×2₋¹⁺⁴	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂×C₃×#41	1	2	yes	—
158	C₂²	C₃×2₋¹⁺⁴	1	4	yes	—
159	C₂²	C₂⁵×C₃	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	8	no	—
162	C₄	—	2	4	no	—
163	C₄	—	2	4	no	—
164	C₄	—	2	8	no	—
165	C₄	C₂×C₃×#41	1	4	yes	—
166	C₃	C₂×#541	1	1	yes	characteristic, direct product factor, Sylow
167	C₂	—	4	2	no	—
168	C₂	C₃×#541	1	4	yes	direct product factor
169	C₂	C₂²×C₃×#41	1	1	yes	characteristic
170	C₂	C₂×C₃×2₋¹⁺⁴	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃×#541	C₂	direct (N × H)
2. C₂×C₃×C₄⋊Q₈	C₂	semidirect
4. C₂×C₃×#125	C₂	semidirect
5. C₂×C₃×#125	C₂	semidirect
7. C₂×C₃×C₄×Q₈	C₂	semidirect
9. C₂×#541	C₃	direct (N × H)
12. C₃×C₄⋊Q₈	C₂²	semidirect
14. C₃×#125	C₂²	semidirect
15. C₃×#125	C₂²	semidirect
17. C₃×C₄×Q₈	C₂²	semidirect
30. #541	C₂×C₃	direct (N × H)
31. C₂×C₄⋊Q₈	C₂×C₃	semidirect
33. C₂×#125	C₂×C₃	semidirect
34. C₂×#125	C₂×C₃	semidirect
36. C₂×C₄×Q₈	C₂×C₃	semidirect
62. C₄⋊Q₈	C₂²×C₃	semidirect
64. #125	C₂²×C₃	semidirect
65. #125	C₂²×C₃	semidirect
67. C₄×Q₈	C₂²×C₃	semidirect
151. C₂×C₃	#541	direct (N × H)
166. C₃	C₂×#541	direct (N × H)
168. C₂	C₃×#541	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#541	2
C₂×C₃×C₄⋊Q₈	2
C₂×C₃×#126	2
C₂×C₃×#125	2
C₂×C₃×C₂²⋊Q₈	2
C₂×C₃×C₄×Q₈	2
C₂×C₃×#117	2
C₂×#541	3

Subgroups — C2×C3×#541 #87337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#541 #87337