Subgroups — C₂²×C₃×#137 #87463

Order 384.

1464 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#137	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×Q₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃×SD₁₆	C₂	1	2	yes	maximal, complemented
6	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#137	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂×C₃×#41	—	2	6	no	—
9	C₂×C₃×#41	C₂²	1	12	yes	—
10	C₂²×C₃×Q₈	—	2	6	no	—
11	C₂²×C₃×Q₈	—	2	1	no	—
12	C₂²×C₃×Q₈	C₂²	1	2	yes	—
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Q₈	C₂²	1	12	yes	complemented
15	C₂²×C₃×D₄	—	2	1	no	—
16	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
17	C₂³×C₃×C₄	—	2	1	no	—
18	C₂³×C₃×C₄	—	2	1	no	—
19	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
20	C₃×#137	C₂²	1	64	yes	direct product factor
21	C₂×C₃×Q₁₆	C₂²	1	24	yes	complemented
22	C₂×C₃×SD₁₆	C₂²	1	24	yes	complemented
23	C₂×C₃×M₁₆	C₂²	1	12	yes	complemented
24	C₂²×C₃×C₈	C₂²	1	2	yes	—
25	C₂²×#41	C₂×C₃	1	1	yes	characteristic
26	C₂³×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×#137	C₂×C₃	1	24	yes	direct product factor
28	C₂²×Q₁₆	C₂×C₃	1	2	yes	complemented
29	C₂²×SD₁₆	C₂×C₃	1	2	yes	complemented
30	C₂²×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂⁴×C₃	—	2	1	no	—
32	C₂⁴×C₃	D₄	1	1	yes	characteristic
33	C₃×#41	—	2	24	no	—
34	C₃×#41	C₂³	1	16	yes	—
35	C₂×C₃×Q₈	—	2	6	no	—
36	C₂×C₃×Q₈	—	2	12	no	—
37	C₂×C₃×Q₈	—	2	24	no	—
38	C₂×C₃×Q₈	—	2	3	no	—
39	C₂×C₃×Q₈	C₂³	1	12	yes	—
40	C₂×C₃×Q₈	C₂³	1	6	yes	—
41	C₂×C₃×Q₈	C₂³	1	16	yes	complemented
42	C₂×C₃×D₄	—	2	12	no	—
43	C₂×C₃×D₄	—	2	3	no	—
44	C₂×C₃×D₄	C₂³	1	6	yes	—
45	C₂²×C₃×C₄	—	2	12	no	—
46	C₂²×C₃×C₄	—	2	1	no	—
47	C₂²×C₃×C₄	—	2	2	no	—
48	C₂²×C₃×C₄	—	2	12	no	—
49	C₂²×C₃×C₄	D₄	1	6	yes	—
50	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
51	C₂²×C₃×C₄	C₂³	1	6	yes	—
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₃×Q₁₆	C₂³	1	32	yes	complemented
54	C₃×SD₁₆	C₂³	1	32	yes	complemented
55	C₃×M₁₆	C₂³	1	16	yes	complemented
56	C₂×C₃×C₈	C₂³	1	12	yes	—
57	C₂×#41	—	2	6	no	—
58	C₂×#41	C₂²×C₃	1	12	yes	—
59	C₂²×Q₈	—	2	1	no	—
60	C₂²×Q₈	—	2	6	no	—
61	C₂²×Q₈	C₂²×C₃	1	2	yes	—
62	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
63	C₂²×Q₈	C₂²×C₃	1	12	yes	complemented
64	C₂²×D₄	—	2	1	no	—
65	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
66	C₂³×C₄	—	2	1	no	—
67	C₂³×C₄	—	2	1	no	—
68	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
69	#137	C₂²×C₃	1	64	yes	direct product factor
70	C₂×Q₁₆	C₂²×C₃	1	24	yes	complemented
71	C₂×SD₁₆	C₂²×C₃	1	24	yes	complemented
72	C₂×M₁₆	C₂²×C₃	1	12	yes	complemented
73	C₂²×C₈	C₂²×C₃	1	2	yes	—
74	C₂³×C₃	—	2	4	no	—
75	C₂³×C₃	—	2	6	no	—
76	C₂³×C₃	—	4	4	no	—
77	C₂³×C₃	C₂×D₄	1	6	yes	—
78	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
79	C₃×Q₈	—	2	12	no	—
80	C₃×Q₈	—	2	16	no	—
81	C₃×Q₈	—	2	6	no	—
82	C₃×Q₈	C₂⁴	1	8	yes	—
83	C₃×Q₈	C₂⁴	1	4	yes	—
84	C₃×D₄	—	2	16	no	—
85	C₃×D₄	—	2	6	no	—
86	C₃×D₄	C₂⁴	1	4	yes	—
87	C₂×C₃×C₄	—	2	6	no	—
88	C₂×C₃×C₄	—	2	12	no	—
89	C₂×C₃×C₄	—	2	16	no	—
90	C₂×C₃×C₄	—	2	16	no	—
91	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
92	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
93	C₂×C₃×C₄	C₂×D₄	1	12	yes	—
94	C₂×C₃×C₄	C₂⁴	1	3	yes	—
95	C₂×C₃×C₄	C₂⁴	1	4	yes	—
96	C₃×C₈	C₂⁴	1	8	yes	—
97	C₂⁴	—	2	1	no	—
98	C₂⁴	C₃×D₄	1	1	yes	characteristic
99	#41	—	2	24	no	—
100	#41	C₂³×C₃	1	16	yes	—
101	C₂×Q₈	—	2	6	no	—
102	C₂×Q₈	—	2	12	no	—
103	C₂×Q₈	—	2	24	no	—
104	C₂×Q₈	—	2	3	no	—
105	C₂×Q₈	C₂³×C₃	1	12	yes	—
106	C₂×Q₈	C₂³×C₃	1	6	yes	—
107	C₂×Q₈	C₂³×C₃	1	16	yes	complemented
108	C₂×D₄	—	2	12	no	—
109	C₂×D₄	—	2	3	no	—
110	C₂×D₄	C₂³×C₃	1	6	yes	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	2	12	no	—
114	C₂²×C₄	—	2	12	no	—
115	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
116	C₂²×C₄	C₃×D₄	1	6	yes	—
117	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₃	1	6	yes	—
119	Q₁₆	C₂³×C₃	1	32	yes	complemented
120	SD₁₆	C₂³×C₃	1	32	yes	complemented
121	M₁₆	C₂³×C₃	1	16	yes	complemented
122	C₂×C₈	C₂³×C₃	1	12	yes	—
123	C₂²×C₃	—	2	12	no	—
124	C₂²×C₃	—	2	4	no	—
125	C₂²×C₃	—	4	12	no	—
126	C₂²×C₃	#137	1	4	yes	direct product factor
127	C₂²×C₃	C₂²×D₄	1	4	yes	—
128	C₂²×C₃	C₂²×D₄	1	3	yes	—
129	C₃×C₄	—	2	4	no	—
130	C₃×C₄	—	2	8	no	—
131	C₃×C₄	C₂²×D₄	1	4	yes	—
132	C₃×C₄	C₂²×D₄	1	3	yes	—
133	C₃×C₄	C₂⁵	1	1	yes	characteristic
134	C₂³	—	2	4	no	—
135	C₂³	—	2	6	no	—
136	C₂³	—	4	4	no	—
137	C₂³	C₂×C₃×D₄	1	6	yes	—
138	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
139	Q₈	—	2	12	no	—
140	Q₈	—	2	16	no	—
141	Q₈	—	2	6	no	—
142	Q₈	C₂⁴×C₃	1	8	yes	—
143	Q₈	C₂⁴×C₃	1	4	yes	—
144	D₄	—	2	16	no	—
145	D₄	—	2	6	no	—
146	D₄	C₂⁴×C₃	1	4	yes	—
147	C₂×C₄	—	2	6	no	—
148	C₂×C₄	—	2	12	no	—
149	C₂×C₄	—	2	16	no	—
150	C₂×C₄	—	2	16	no	—
151	C₂×C₄	C₂×C₃×D₄	1	6	yes	—
152	C₂×C₄	C₂×C₃×D₄	1	3	yes	—
153	C₂×C₄	C₂×C₃×D₄	1	12	yes	—
154	C₂×C₄	C₂⁴×C₃	1	3	yes	—
155	C₂×C₄	C₂⁴×C₃	1	4	yes	—
156	C₈	C₂⁴×C₃	1	8	yes	—
157	C₂×C₃	—	2	4	no	—
158	C₂×C₃	—	4	4	no	—
159	C₂×C₃	C₂×#137	1	6	yes	direct product factor
160	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
161	C₂²	—	2	12	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	4	12	no	—
164	C₂²	C₃×#137	1	4	yes	direct product factor
165	C₂²	C₂²×C₃×D₄	1	4	yes	—
166	C₂²	C₂²×C₃×D₄	1	3	yes	—
167	C₄	—	2	4	no	—
168	C₄	—	2	8	no	—
169	C₄	C₂²×C₃×D₄	1	4	yes	—
170	C₄	C₂²×C₃×D₄	1	3	yes	—
171	C₄	C₂⁵×C₃	1	1	yes	characteristic
172	C₃	C₂²×#137	1	1	yes	characteristic, direct product factor, Sylow
173	C₂	—	2	4	no	—
174	C₂	—	4	4	no	—
175	C₂	C₂×C₃×#137	1	6	yes	direct product factor
176	C₂	C₂³×C₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₃
prime radical	C₂²×C₃×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×Q₈	C₂	semidirect
3. C₂×C₃×#137	C₂	direct (N × H)
4. C₂²×C₃×Q₁₆	C₂	semidirect
5. C₂²×C₃×SD₁₆	C₂	semidirect
6. C₂²×C₃×M₁₆	C₂	semidirect
7. C₂²×#137	C₃	direct (N × H)
14. C₂²×C₃×Q₈	C₂²	semidirect
20. C₃×#137	C₂²	direct (N × H)
21. C₂×C₃×Q₁₆	C₂²	semidirect
22. C₂×C₃×SD₁₆	C₂²	semidirect
23. C₂×C₃×M₁₆	C₂²	semidirect
26. C₂³×Q₈	C₂×C₃	semidirect
27. C₂×#137	C₂×C₃	direct (N × H)
28. C₂²×Q₁₆	C₂×C₃	semidirect
29. C₂²×SD₁₆	C₂×C₃	semidirect
30. C₂²×M₁₆	C₂×C₃	semidirect
41. C₂×C₃×Q₈	C₂³	semidirect
53. C₃×Q₁₆	C₂³	semidirect
54. C₃×SD₁₆	C₂³	semidirect
55. C₃×M₁₆	C₂³	semidirect
63. C₂²×Q₈	C₂²×C₃	semidirect
69. #137	C₂²×C₃	direct (N × H)
70. C₂×Q₁₆	C₂²×C₃	semidirect
71. C₂×SD₁₆	C₂²×C₃	semidirect
72. C₂×M₁₆	C₂²×C₃	semidirect
107. C₂×Q₈	C₂³×C₃	semidirect
119. Q₁₆	C₂³×C₃	semidirect
120. SD₁₆	C₂³×C₃	semidirect
121. M₁₆	C₂³×C₃	semidirect
126. C₂²×C₃	#137	direct (N × H)
159. C₂×C₃	C₂×#137	direct (N × H)
164. C₂²	C₃×#137	direct (N × H)
172. C₃	C₂²×#137	direct (N × H)
175. C₂	C₂×C₃×#137	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×C₃×#41	2
C₂³×C₃×Q₈	2
C₂×C₃×#137	2
C₂²×C₃×Q₁₆	2
C₂²×C₃×SD₁₆	2
C₂²×C₃×M₁₆	2
C₂²×#137	3

Subgroups — C22×C3×#137 #87463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#137 #87463