Finite Group Databasebeta

← D₃×#135 #5767

Subgroups — D₃×#135 #5767

Order 192.

266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×#41	C₂	1	2	yes	maximal, complemented
2	C₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	#950	C₂	1	2	yes	maximal, complemented
4	#919	C₂	1	1	yes	maximal, characteristic, complemented
5	D₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#916	C₂	1	2	yes	maximal, complemented
7	D₃×SD₁₆	C₂	1	2	yes	maximal, complemented
8	#912	C₂	1	1	yes	maximal, characteristic, complemented
9	D₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#904	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	—	3	1	no	maximal, Sylow
13	C₃×#41	C₂²	1	2	yes	complemented
14	#239	C₂²	1	2	yes	complemented
15	D₃×Q₈	C₂²	1	2	yes	—
16	#237	—	2	2	no	—
17	#237	C₂²	1	2	yes	complemented
18	D₃×D₄	—	2	2	no	—
19	D₃×D₄	C₂²	1	2	yes	—
20	#235	—	2	2	no	—
21	#235	C₂²	1	2	yes	complemented
22	C₂×C₄×D₃	—	2	2	no	—
23	C₂×C₄×D₃	C₂²	1	1	yes	characteristic
24	C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
25	C₃×SD₁₆	C₂²	1	2	yes	complemented
26	C₃×D₈	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₃×C₈	C₂²	1	1	yes	characteristic, complemented
28	#216	C₂²	1	2	yes	complemented
29	#215	C₂²	1	2	yes	complemented
30	#214	C₂²	1	2	yes	complemented
31	#213	C₂²	1	2	yes	complemented
32	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	Dic₁₂	C₂²	1	1	yes	characteristic, complemented
34	D₂₄	C₂²	1	1	yes	characteristic, complemented
35	#204	C₂²	1	2	yes	complemented
36	D₃×C₈	C₂²	1	1	yes	characteristic
37	D₃×C₈	C₂²	1	1	yes	characteristic
38	D₃×C₈	C₂²	1	1	yes	characteristic, complemented
39	D₃×C₈	C₂²	1	1	yes	characteristic, complemented
40	C₂×#41	—	3	2	no	—
41	#135	—	3	2	no	—
42	#135	—	3	1	no	—
43	#135	—	3	2	no	—
44	#135	—	3	1	no	—
45	#135	—	3	1	no	—
46	#135	D₃	1	1	yes	characteristic, direct product factor
47	C₂×Q₁₆	—	3	1	no	—
48	C₂×SD₁₆	—	3	2	no	—
49	C₂×D₈	—	3	1	no	—
50	C₂²×C₈	—	3	1	no	—
51	C₂²×D₃	—	2	2	no	—
52	C₂²×D₃	D₄	1	1	yes	characteristic
53	C₃×Q₈	C₂³	1	2	yes	—
54	C₃×D₄	—	2	2	no	—
55	C₃×D₄	C₂³	1	2	yes	—
56	C₂×C₃×C₄	—	2	2	no	—
57	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
58	#67	—	2	2	no	—
59	#67	—	2	2	no	—
60	#67	—	2	2	no	—
61	C₂×Dic₃	—	2	2	no	—
62	C₂×Dic₃	D₄	1	1	yes	characteristic
63	D₁₂	—	2	2	no	—
64	D₁₂	C₂³	1	2	yes	—
65	C₄×D₃	—	2	2	no	—
66	C₄×D₃	—	2	2	no	—
67	C₄×D₃	—	2	2	no	—
68	C₄×D₃	D₄	1	1	yes	characteristic
69	C₄×D₃	D₄	1	1	yes	characteristic
70	C₄×D₃	C₂³	1	1	yes	characteristic
71	C₄×D₃	C₂³	1	1	yes	characteristic
72	Dic₆	—	2	2	no	—
73	Dic₆	C₂³	1	2	yes	—
74	C₃×C₈	C₂³	1	1	yes	characteristic
75	C₃×C₈	C₂³	1	1	yes	characteristic
76	C₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₃⋊C₈	C₂³	1	1	yes	characteristic
78	#41	—	3	2	no	—
79	#41	—	3	2	no	—
80	#41	—	3	2	no	—
81	#41	—	6	2	no	—
82	#41	—	6	2	no	—
83	#41	C₂×D₃	1	2	yes	complemented
84	C₂×Q₈	—	3	2	no	—
85	C₂×D₄	—	3	2	no	—
86	C₂×D₄	—	6	2	no	—
87	C₂²×C₄	—	3	1	no	—
88	C₂²×C₄	—	6	2	no	—
89	Q₁₆	—	3	2	no	—
90	Q₁₆	—	3	1	no	—
91	Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
92	SD₁₆	—	3	2	no	—
93	SD₁₆	—	3	2	no	—
94	SD₁₆	—	3	2	no	—
95	SD₁₆	C₂×D₃	1	2	yes	complemented
96	D₈	—	3	2	no	—
97	D₈	—	3	1	no	—
98	D₈	C₂×D₃	1	1	yes	characteristic, complemented
99	C₂×C₈	—	3	1	no	—
100	C₂×C₈	—	3	1	no	—
101	C₂×C₈	—	3	1	no	—
102	C₂×C₈	—	3	1	no	—
103	C₂×C₈	—	3	1	no	—
104	C₂×C₈	C₂×D₃	1	1	yes	characteristic, complemented
105	C₂²×C₃	—	2	2	no	—
106	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
107	C₂×D₃	—	2	2	no	—
108	C₂×D₃	—	2	2	no	—
109	C₂×D₃	—	4	4	no	—
110	C₂×D₃	C₂×D₄	1	1	yes	characteristic
111	C₂×D₃	C₂×D₄	1	1	yes	characteristic
112	C₃×C₄	—	2	2	no	—
113	C₃×C₄	C₂×D₄	1	1	yes	characteristic
114	C₃×C₄	C₂⁴	1	1	yes	characteristic
115	Dic₃	—	2	2	no	—
116	Dic₃	C₂×D₄	1	1	yes	characteristic
117	Dic₃	C₂×D₄	1	1	yes	characteristic
118	C₂³	—	3	1	no	—
119	C₂³	—	6	2	no	—
120	Q₈	—	3	2	no	—
121	Q₈	—	6	2	no	—
122	Q₈	C₂²×D₃	1	2	yes	—
123	D₄	—	2	2	no	—
124	D₄	—	3	2	no	—
125	D₄	—	6	2	no	—
126	D₄	—	6	2	no	—
127	D₄	—	6	2	no	—
128	D₄	—	6	2	no	—
129	D₄	C₂²×D₃	1	2	yes	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	3	1	no	—
132	C₂×C₄	—	3	1	no	—
133	C₂×C₄	—	3	1	no	—
134	C₂×C₄	—	3	1	no	—
135	C₂×C₄	—	3	1	no	—
136	C₂×C₄	—	6	2	no	—
137	C₂×C₄	—	6	2	no	—
138	C₂×C₄	—	6	2	no	—
139	C₂×C₄	—	6	2	no	—
140	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
141	C₈	—	3	1	no	—
142	C₈	—	3	1	no	—
143	C₈	C₂²×D₃	1	1	yes	characteristic
144	C₈	C₂²×D₃	1	1	yes	characteristic
145	C₂×C₃	—	2	1	no	—
146	C₂×C₃	—	4	2	no	—
147	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
148	D₃	—	2	1	no	—
149	D₃	—	4	2	no	—
150	D₃	#135	1	2	yes	direct product factor
151	C₂²	—	2	2	no	—
152	C₂²	—	3	1	no	—
153	C₂²	—	3	1	no	—
154	C₂²	—	6	2	no	—
155	C₂²	—	6	2	no	—
156	C₂²	—	12	4	no	—
157	C₂²	D₃×D₄	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	3	1	no	—
160	C₄	—	3	1	no	—
161	C₄	—	6	2	no	—
162	C₄	D₃×D₄	1	1	yes	characteristic
163	C₄	C₂³×D₃	1	1	yes	characteristic
164	C₃	C₂×#135	1	1	yes	characteristic, complemented, Sylow
165	C₂	—	2	1	no	—
166	C₂	—	3	2	no	—
167	C₂	—	4	2	no	—
168	C₂	—	6	1	no	—
169	C₂	—	12	2	no	—
170	C₂	C₂×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁴
Frattini	C₄	C₂³×D₃
Fitting	C₃×#135	C₂
socle	C₂×C₃	C₂²×D₄
hypercenter	#135	D₃
hypocenter	C₃	C₂×#135
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. D₃×#41	C₂	semidirect
2. C₃×#135	C₂	semidirect
3. #950	C₂	semidirect
4. #919	C₂	semidirect
5. D₃×Q₁₆	C₂	semidirect
6. #916	C₂	semidirect
7. D₃×SD₁₆	C₂	semidirect
8. #912	C₂	semidirect
9. D₃×D₈	C₂	semidirect
10. #904	C₂	semidirect
11. C₂×D₃×C₈	C₂	semidirect
13. C₃×#41	C₂²	semidirect
14. #239	C₂²	semidirect
17. #237	C₂²	semidirect
21. #235	C₂²	semidirect
24. C₃×Q₁₆	C₂²	semidirect
25. C₃×SD₁₆	C₂²	semidirect
26. C₃×D₈	C₂²	semidirect
27. C₂×C₃×C₈	C₂²	semidirect
28. #216	C₂²	semidirect
29. #215	C₂²	semidirect
30. #214	C₂²	semidirect
31. #213	C₂²	semidirect
32. C₂×C₃⋊C₈	C₂²	semidirect
33. Dic₁₂	C₂²	semidirect
34. D₂₄	C₂²	semidirect
35. #204	C₂²	semidirect
38. D₃×C₈	C₂²	semidirect
39. D₃×C₈	C₂²	semidirect
46. #135	D₃	direct (N × H)
83. #41	C₂×D₃	semidirect
91. Q₁₆	C₂×D₃	semidirect
95. SD₁₆	C₂×D₃	semidirect
98. D₈	C₂×D₃	semidirect
104. C₂×C₈	C₂×D₃	semidirect
150. D₃	#135	direct (N × H)
164. C₃	C₂×#135	semidirect

Maximal subgroups

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

subgroup	index
D₃×#41	2
C₃×#135	2
#950	2
#919	2
D₃×Q₁₆	2
#916	2
D₃×SD₁₆	2
#912	2
D₃×D₈	2
#904	2
C₂×D₃×C₈	2
C₂×#135	3