Subgroups — C₃×C₄×#135 #84803

Order 384.

500 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#41	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×Q₁₆	C₂	1	2	yes	maximal, complemented
4	C₃×C₄×SD₁₆	C₂	1	4	yes	maximal, complemented
5	C₃×C₄×D₈	C₂	1	2	yes	maximal, complemented
6	C₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#416	C₂	1	2	yes	maximal, complemented
8	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#135	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#41	C₂²	1	2	yes	—
11	C₃×#135	C₄	1	8	yes	direct product factor
12	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
14	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×Q₈	C₂²	1	4	yes	—
18	C₃×C₄×D₄	—	2	4	no	—
19	C₃×C₄×D₄	C₂²	1	4	yes	—
20	C₃×#117	—	2	2	no	—
21	C₃×#117	C₂²	1	2	yes	—
22	C₂×C₃×C₄²	—	2	2	no	—
23	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
24	C₃×#107	C₂²	1	2	yes	—
25	C₃×#106	C₂²	1	2	yes	—
26	C₃×#103	C₂²	1	4	yes	—
27	C₃×#102	C₂²	1	4	yes	—
28	C₃×C₄×C₈	C₂²	1	2	yes	—
29	C₃×C₄×C₈	C₂²	1	2	yes	—
30	C₂×#135	C₂×C₃	1	1	yes	characteristic
31	C₄×#41	C₂×C₃	1	2	yes	complemented
32	C₄×Q₁₆	C₂×C₃	1	2	yes	complemented
33	C₄×SD₁₆	C₂×C₃	1	4	yes	complemented
34	C₄×D₈	C₂×C₃	1	2	yes	complemented
35	#427	C₂×C₃	1	1	yes	characteristic, complemented
36	#416	C₂×C₃	1	2	yes	complemented
37	C₂×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
38	C₃×#41	—	2	4	no	—
39	C₃×#41	C₂×C₄	1	8	yes	complemented
40	C₂×C₃×Q₈	C₂³	1	2	yes	—
41	C₂×C₃×D₄	—	2	2	no	—
42	C₂×C₃×D₄	C₂³	1	2	yes	—
43	C₂²×C₃×C₄	—	2	2	no	—
44	C₂²×C₃×C₄	—	2	4	no	—
45	C₂²×C₃×C₄	D₄	1	2	yes	—
46	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
47	C₃×Q₁₆	C₂×C₄	1	4	yes	complemented
48	C₃×SD₁₆	C₂×C₄	1	8	yes	complemented
49	C₃×D₈	C₂×C₄	1	4	yes	complemented
50	C₂×C₃×C₈	—	2	2	no	—
51	C₂×C₃×C₈	C₂×C₄	1	4	yes	complemented
52	C₂×C₃×C₈	C₂³	1	2	yes	—
53	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
55	C₃×C₄⋊C₄	—	2	4	no	—
56	C₃×C₄⋊C₄	C₂³	1	4	yes	—
57	C₃×C₂²⋊C₄	—	2	2	no	—
58	C₃×C₂²⋊C₄	—	2	4	no	—
59	C₃×C₄²	—	2	2	no	—
60	C₃×C₄²	—	2	4	no	—
61	C₃×C₄²	D₄	1	1	yes	characteristic
62	C₃×C₄²	D₄	1	1	yes	characteristic
63	C₃×C₄²	C₂³	1	2	yes	—
64	C₂×#41	C₂²×C₃	1	2	yes	—
65	#135	C₃×C₄	1	8	yes	direct product factor
66	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
67	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
68	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
69	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
70	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
71	C₄×Q₈	C₂²×C₃	1	4	yes	—
72	C₄×D₄	—	2	4	no	—
73	C₄×D₄	C₂²×C₃	1	4	yes	—
74	#117	—	2	2	no	—
75	#117	C₂²×C₃	1	2	yes	—
76	C₂×C₄²	—	2	2	no	—
77	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
78	#107	C₂²×C₃	1	2	yes	—
79	#106	C₂²×C₃	1	2	yes	—
80	#103	C₂²×C₃	1	4	yes	—
81	#102	C₂²×C₃	1	4	yes	—
82	C₄×C₈	C₂²×C₃	1	2	yes	—
83	C₄×C₈	C₂²×C₃	1	2	yes	—
84	C₂³×C₃	—	2	2	no	—
85	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
86	C₃×Q₈	—	2	2	no	—
87	C₃×Q₈	C₂²×C₄	1	4	yes	—
88	C₃×D₄	—	2	8	no	—
89	C₃×D₄	—	2	2	no	—
90	C₃×D₄	C₂²×C₄	1	4	yes	—
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	8	no	—
93	C₂×C₃×C₄	—	2	4	no	—
94	C₂×C₃×C₄	—	2	4	no	—
95	C₂×C₃×C₄	—	4	8	no	—
96	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
97	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
98	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
100	C₂×C₃×C₄	#41	1	2	yes	—
101	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
102	C₃×C₈	—	2	2	no	—
103	C₃×C₈	C₂²×C₄	1	2	yes	—
104	C₃×C₈	C₂²×C₄	1	2	yes	—
105	#41	—	2	4	no	—
106	#41	C₂×C₃×C₄	1	8	yes	complemented
107	C₂×Q₈	C₂³×C₃	1	2	yes	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	C₂³×C₃	1	2	yes	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	C₃×D₄	1	2	yes	—
113	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
114	Q₁₆	C₂×C₃×C₄	1	4	yes	complemented
115	SD₁₆	C₂×C₃×C₄	1	8	yes	complemented
116	D₈	C₂×C₃×C₄	1	4	yes	complemented
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	C₂×C₃×C₄	1	4	yes	complemented
119	C₂×C₈	C₂³×C₃	1	2	yes	—
120	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
121	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	C₂³×C₃	1	4	yes	—
124	C₂²⋊C₄	—	2	2	no	—
125	C₂²⋊C₄	—	2	4	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	4	no	—
128	C₄²	C₃×D₄	1	1	yes	characteristic
129	C₄²	C₃×D₄	1	1	yes	characteristic
130	C₄²	C₂³×C₃	1	2	yes	—
131	C₂²×C₃	—	2	2	no	—
132	C₂²×C₃	—	2	4	no	—
133	C₂²×C₃	—	4	4	no	—
134	C₂²×C₃	C₄×D₄	1	2	yes	—
135	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
136	C₃×C₄	—	2	4	no	—
137	C₃×C₄	—	2	2	no	—
138	C₃×C₄	—	4	4	no	—
139	C₃×C₄	C₄×D₄	1	2	yes	—
140	C₃×C₄	#135	1	4	yes	direct product factor
141	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
142	C₃×C₄	C₂×#41	1	1	yes	characteristic
143	C₂³	—	2	2	no	—
144	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	Q₈	C₂²×C₃×C₄	1	4	yes	—
147	D₄	—	2	8	no	—
148	D₄	—	2	2	no	—
149	D₄	C₂²×C₃×C₄	1	4	yes	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	8	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	8	no	—
155	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
156	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
157	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
158	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
159	C₂×C₄	C₃×#41	1	2	yes	—
160	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	C₂²×C₃×C₄	1	2	yes	—
163	C₈	C₂²×C₃×C₄	1	2	yes	—
164	C₂×C₃	—	2	2	no	—
165	C₂×C₃	—	4	4	no	—
166	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂×C₃	C₂×#135	1	2	yes	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	4	no	—
170	C₂²	—	4	4	no	—
171	C₂²	C₃×C₄×D₄	1	2	yes	—
172	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	4	no	—
175	C₄	—	4	4	no	—
176	C₄	C₃×C₄×D₄	1	2	yes	—
177	C₄	C₃×#135	1	4	yes	direct product factor
178	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
179	C₄	C₂×C₃×#41	1	1	yes	characteristic
180	C₃	C₄×#135	1	1	yes	characteristic, direct product factor, Sylow
181	C₂	—	2	2	no	—
182	C₂	—	4	4	no	—
183	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic
184	C₂	C₂×C₃×#135	1	2	yes	—

	subgroup	G/H
center	C₃×C₄²	D₄
commutator	C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×C₃×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄² ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×#41	C₂	semidirect
3. C₃×C₄×Q₁₆	C₂	semidirect
4. C₃×C₄×SD₁₆	C₂	semidirect
5. C₃×C₄×D₈	C₂	semidirect
6. C₃×#427	C₂	semidirect
7. C₃×#416	C₂	semidirect
8. C₂×C₃×C₄×C₈	C₂	semidirect
9. C₄×#135	C₃	direct (N × H)
11. C₃×#135	C₄	direct (N × H)
31. C₄×#41	C₂×C₃	semidirect
32. C₄×Q₁₆	C₂×C₃	semidirect
33. C₄×SD₁₆	C₂×C₃	semidirect
34. C₄×D₈	C₂×C₃	semidirect
35. #427	C₂×C₃	semidirect
36. #416	C₂×C₃	semidirect
37. C₂×C₄×C₈	C₂×C₃	semidirect
39. C₃×#41	C₂×C₄	semidirect
47. C₃×Q₁₆	C₂×C₄	semidirect
48. C₃×SD₁₆	C₂×C₄	semidirect
49. C₃×D₈	C₂×C₄	semidirect
51. C₂×C₃×C₈	C₂×C₄	semidirect
65. #135	C₃×C₄	direct (N × H)
106. #41	C₂×C₃×C₄	semidirect
114. Q₁₆	C₂×C₃×C₄	semidirect
115. SD₁₆	C₂×C₃×C₄	semidirect
116. D₈	C₂×C₃×C₄	semidirect
118. C₂×C₈	C₂×C₃×C₄	semidirect
140. C₃×C₄	#135	direct (N × H)
177. C₄	C₃×#135	direct (N × H)
180. C₃	C₄×#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
C₂×C₃×#135	2
C₃×C₄×#41	2
C₃×C₄×Q₁₆	2
C₃×C₄×SD₁₆	2
C₃×C₄×D₈	2
C₃×#427	2
C₃×#416	2
C₂×C₃×C₄×C₈	2
C₄×#135	3

Subgroups — C3×C4×#135 #84803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#135 #84803