Subgroups — C₂²×C₃×#135 #87461

Order 384.

1504 conjugacy classes of subgroups.

The subgroup lattice has 152 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	2	yes	maximal, complemented
2	C₂×C₃×#135	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#135	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂×C₃×#41	—	2	12	no	—
9	C₂×C₃×#41	C₂²	1	24	yes	complemented
10	C₂²×C₃×Q₈	C₂²	1	2	yes	—
11	C₂²×C₃×D₄	—	2	2	no	—
12	C₂²×C₃×D₄	C₂²	1	2	yes	—
13	C₂³×C₃×C₄	—	2	2	no	—
14	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
15	C₃×#135	C₂²	1	64	yes	direct product factor
16	C₂×C₃×Q₁₆	C₂²	1	12	yes	complemented
17	C₂×C₃×SD₁₆	C₂²	1	24	yes	complemented
18	C₂×C₃×D₈	C₂²	1	12	yes	complemented
19	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
20	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
21	C₂²×C₃×C₈	C₂²	1	12	yes	complemented
22	C₂²×#41	C₂×C₃	1	2	yes	complemented
23	C₂×#135	C₂×C₃	1	24	yes	direct product factor
24	C₂²×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂²×SD₁₆	C₂×C₃	1	2	yes	complemented
26	C₂²×D₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂³×C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂⁴×C₃	—	2	2	no	—
29	C₂⁴×C₃	D₄	1	1	yes	characteristic
30	C₃×#41	—	2	48	no	—
31	C₃×#41	C₂³	1	32	yes	complemented
32	C₂×C₃×Q₈	—	2	6	no	—
33	C₂×C₃×Q₈	C₂³	1	12	yes	—
34	C₂×C₃×D₄	—	2	24	no	—
35	C₂×C₃×D₄	—	2	6	no	—
36	C₂×C₃×D₄	C₂³	1	12	yes	—
37	C₂²×C₃×C₄	—	2	2	no	—
38	C₂²×C₃×C₄	—	2	24	no	—
39	C₂²×C₃×C₄	D₄	1	6	yes	—
40	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
41	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂³	1	6	yes	—
43	C₃×Q₁₆	C₂³	1	16	yes	complemented
44	C₃×SD₁₆	C₂³	1	32	yes	complemented
45	C₃×D₈	C₂³	1	16	yes	complemented
46	C₂×C₃×C₈	C₂³	1	6	yes	—
47	C₂×C₃×C₈	C₂³	1	6	yes	—
48	C₂×C₃×C₈	C₂³	1	16	yes	complemented
49	C₂×#41	—	2	12	no	—
50	C₂×#41	C₂²×C₃	1	24	yes	complemented
51	C₂²×Q₈	C₂²×C₃	1	2	yes	—
52	C₂²×D₄	—	2	2	no	—
53	C₂²×D₄	C₂²×C₃	1	2	yes	—
54	C₂³×C₄	—	2	2	no	—
55	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
56	#135	C₂²×C₃	1	64	yes	direct product factor
57	C₂×Q₁₆	C₂²×C₃	1	12	yes	complemented
58	C₂×SD₁₆	C₂²×C₃	1	24	yes	complemented
59	C₂×D₈	C₂²×C₃	1	12	yes	complemented
60	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₈	C₂²×C₃	1	12	yes	complemented
63	C₂³×C₃	—	2	4	no	—
64	C₂³×C₃	—	2	12	no	—
65	C₂³×C₃	—	4	8	no	—
66	C₂³×C₃	C₂×D₄	1	6	yes	—
67	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
68	C₃×Q₈	—	2	12	no	—
69	C₃×Q₈	C₂⁴	1	8	yes	—
70	C₃×D₄	—	2	32	no	—
71	C₃×D₄	—	2	12	no	—
72	C₃×D₄	C₂⁴	1	8	yes	—
73	C₂×C₃×C₄	—	2	12	no	—
74	C₂×C₃×C₄	—	2	32	no	—
75	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
76	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
77	C₂×C₃×C₄	C₂×D₄	1	12	yes	—
78	C₂×C₃×C₄	C₂⁴	1	3	yes	—
79	C₂×C₃×C₄	C₂⁴	1	4	yes	—
80	C₃×C₈	C₂⁴	1	4	yes	—
81	C₃×C₈	C₂⁴	1	4	yes	—
82	C₂⁴	—	2	2	no	—
83	C₂⁴	C₃×D₄	1	1	yes	characteristic
84	#41	—	2	48	no	—
85	#41	C₂³×C₃	1	32	yes	complemented
86	C₂×Q₈	—	2	6	no	—
87	C₂×Q₈	C₂³×C₃	1	12	yes	—
88	C₂×D₄	—	2	24	no	—
89	C₂×D₄	—	2	6	no	—
90	C₂×D₄	C₂³×C₃	1	12	yes	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	—	2	24	no	—
93	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
94	C₂²×C₄	C₃×D₄	1	6	yes	—
95	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
96	C₂²×C₄	C₂³×C₃	1	6	yes	—
97	Q₁₆	C₂³×C₃	1	16	yes	complemented
98	SD₁₆	C₂³×C₃	1	32	yes	complemented
99	D₈	C₂³×C₃	1	16	yes	complemented
100	C₂×C₈	C₂³×C₃	1	6	yes	—
101	C₂×C₈	C₂³×C₃	1	6	yes	—
102	C₂×C₈	C₂³×C₃	1	16	yes	complemented
103	C₂²×C₃	—	2	12	no	—
104	C₂²×C₃	—	2	8	no	—
105	C₂²×C₃	—	4	24	no	—
106	C₂²×C₃	#135	1	4	yes	direct product factor
107	C₂²×C₃	C₂²×D₄	1	4	yes	—
108	C₂²×C₃	C₂²×D₄	1	3	yes	—
109	C₃×C₄	—	2	8	no	—
110	C₃×C₄	C₂²×D₄	1	4	yes	—
111	C₃×C₄	C₂²×D₄	1	3	yes	—
112	C₃×C₄	C₂⁵	1	1	yes	characteristic
113	C₂³	—	2	4	no	—
114	C₂³	—	2	12	no	—
115	C₂³	—	4	8	no	—
116	C₂³	C₂×C₃×D₄	1	6	yes	—
117	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
118	Q₈	—	2	12	no	—
119	Q₈	C₂⁴×C₃	1	8	yes	—
120	D₄	—	2	32	no	—
121	D₄	—	2	12	no	—
122	D₄	C₂⁴×C₃	1	8	yes	—
123	C₂×C₄	—	2	12	no	—
124	C₂×C₄	—	2	32	no	—
125	C₂×C₄	C₂×C₃×D₄	1	6	yes	—
126	C₂×C₄	C₂×C₃×D₄	1	3	yes	—
127	C₂×C₄	C₂×C₃×D₄	1	12	yes	—
128	C₂×C₄	C₂⁴×C₃	1	3	yes	—
129	C₂×C₄	C₂⁴×C₃	1	4	yes	—
130	C₈	C₂⁴×C₃	1	4	yes	—
131	C₈	C₂⁴×C₃	1	4	yes	—
132	C₂×C₃	—	2	4	no	—
133	C₂×C₃	—	4	8	no	—
134	C₂×C₃	C₂×#135	1	6	yes	direct product factor
135	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
136	C₂²	—	2	12	no	—
137	C₂²	—	2	8	no	—
138	C₂²	—	4	24	no	—
139	C₂²	C₃×#135	1	4	yes	direct product factor
140	C₂²	C₂²×C₃×D₄	1	4	yes	—
141	C₂²	C₂²×C₃×D₄	1	3	yes	—
142	C₄	—	2	8	no	—
143	C₄	C₂²×C₃×D₄	1	4	yes	—
144	C₄	C₂²×C₃×D₄	1	3	yes	—
145	C₄	C₂⁵×C₃	1	1	yes	characteristic
146	C₃	C₂²×#135	1	1	yes	characteristic, direct product factor, Sylow
147	C₂	—	2	4	no	—
148	C₂	—	4	8	no	—
149	C₂	C₂×C₃×#135	1	6	yes	direct product factor
150	C₂	C₂³×C₃×D₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#41	C₂	semidirect
2. C₂×C₃×#135	C₂	direct (N × H)
3. C₂²×C₃×Q₁₆	C₂	semidirect
4. C₂²×C₃×SD₁₆	C₂	semidirect
5. C₂²×C₃×D₈	C₂	semidirect
6. C₂³×C₃×C₈	C₂	semidirect
7. C₂²×#135	C₃	direct (N × H)
9. C₂×C₃×#41	C₂²	semidirect
15. C₃×#135	C₂²	direct (N × H)
16. C₂×C₃×Q₁₆	C₂²	semidirect
17. C₂×C₃×SD₁₆	C₂²	semidirect
18. C₂×C₃×D₈	C₂²	semidirect
21. C₂²×C₃×C₈	C₂²	semidirect
22. C₂²×#41	C₂×C₃	semidirect
23. C₂×#135	C₂×C₃	direct (N × H)
24. C₂²×Q₁₆	C₂×C₃	semidirect
25. C₂²×SD₁₆	C₂×C₃	semidirect
26. C₂²×D₈	C₂×C₃	semidirect
27. C₂³×C₈	C₂×C₃	semidirect
31. C₃×#41	C₂³	semidirect
43. C₃×Q₁₆	C₂³	semidirect
44. C₃×SD₁₆	C₂³	semidirect
45. C₃×D₈	C₂³	semidirect
48. C₂×C₃×C₈	C₂³	semidirect
50. C₂×#41	C₂²×C₃	semidirect
56. #135	C₂²×C₃	direct (N × H)
57. C₂×Q₁₆	C₂²×C₃	semidirect
58. C₂×SD₁₆	C₂²×C₃	semidirect
59. C₂×D₈	C₂²×C₃	semidirect
62. C₂²×C₈	C₂²×C₃	semidirect
85. #41	C₂³×C₃	semidirect
97. Q₁₆	C₂³×C₃	semidirect
98. SD₁₆	C₂³×C₃	semidirect
99. D₈	C₂³×C₃	semidirect
102. C₂×C₈	C₂³×C₃	semidirect
106. C₂²×C₃	#135	direct (N × H)
134. C₂×C₃	C₂×#135	direct (N × H)
139. C₂²	C₃×#135	direct (N × H)
146. C₃	C₂²×#135	direct (N × H)
149. C₂	C₂×C₃×#135	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
C₂×C₃×#135	2
C₂²×C₃×Q₁₆	2
C₂²×C₃×SD₁₆	2
C₂²×C₃×D₈	2
C₂³×C₃×C₈	2
C₂²×#135	3

Subgroups — C22×C3×#135 #87461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#135 #87461