Subgroups — C₃×#3165 #85029

Order 384.

380 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	C₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#501	C₂	1	4	yes	maximal, complemented
4	C₃×#500	C₂	1	2	yes	maximal, complemented
5	C₃×#498	C₂	1	2	yes	maximal, complemented
6	C₃×#428	C₂	1	4	yes	maximal
7	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
8	#3165	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₈	—	2	1	no	—
10	C₂×C₃×M₁₆	C₂²	1	2	yes	—
11	C₃×C₄⋊Q₈	—	2	1	no	—
12	C₃×C₄⋊Q₈	—	2	1	no	—
13	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃×#125	C₂²	1	2	yes	—
18	C₃×C₂²⋊Q₈	—	2	1	no	—
19	C₃×C₂²⋊Q₈	—	2	1	no	—
20	C₃×C₄×Q₈	—	2	1	no	—
21	C₃×C₄×Q₈	—	2	1	no	—
22	C₃×#117	C₂²	1	2	yes	—
23	C₂×C₃×C₄⋊C₄	—	2	1	no	—
24	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
26	C₃×#107	C₂²	1	8	yes	—
27	C₃×#106	C₂²	1	8	yes	—
28	C₃×#97	C₂²	1	2	yes	—
29	C₃×C₄×C₈	C₂²	1	2	yes	—
30	#533	C₂×C₃	1	1	yes	characteristic
31	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
32	#501	C₂×C₃	1	4	yes	complemented
33	#500	C₂×C₃	1	2	yes	complemented
34	#498	C₂×C₃	1	2	yes	complemented
35	#428	C₂×C₃	1	4	yes	—
36	C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
37	C₂×C₃×Q₈	—	2	1	no	—
38	C₂×C₃×Q₈	—	2	1	no	—
39	C₂×C₃×Q₈	—	2	2	no	—
40	C₂×C₃×Q₈	—	2	2	no	—
41	C₂×C₃×Q₈	—	4	4	no	—
42	C₂²×C₃×C₄	—	2	2	no	—
43	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
44	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
45	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₃×M₁₆	Q₈	1	8	yes	complemented
47	C₂×C₃×C₈	C₂³	1	4	yes	—
48	C₃×C₄⋊C₄	—	2	2	no	—
49	C₃×C₄⋊C₄	—	2	2	no	—
50	C₃×C₄⋊C₄	—	2	2	no	—
51	C₃×C₄⋊C₄	—	2	2	no	—
52	C₃×C₄⋊C₄	—	2	2	no	—
53	C₃×C₄⋊C₄	—	2	1	no	—
54	C₃×C₄⋊C₄	—	2	1	no	—
55	C₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₃×C₄⋊C₄	C₂³	1	4	yes	—
57	C₃×C₄⋊C₄	C₂³	1	2	yes	—
58	C₃×C₂²⋊C₄	—	2	2	no	—
59	C₃×C₄²	—	2	2	no	—
60	C₃×C₄²	D₄	1	1	yes	characteristic
61	C₃×C₄²	D₄	1	1	yes	characteristic
62	C₃×C₄²	C₂³	1	1	yes	characteristic
63	C₃×C₄²	C₂³	1	1	yes	characteristic
64	C₂²×Q₈	—	2	1	no	—
65	C₂×M₁₆	C₂²×C₃	1	2	yes	—
66	C₄⋊Q₈	—	2	1	no	—
67	C₄⋊Q₈	—	2	1	no	—
68	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
69	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
70	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
71	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
72	#125	C₂²×C₃	1	2	yes	—
73	C₂²⋊Q₈	—	2	1	no	—
74	C₂²⋊Q₈	—	2	1	no	—
75	C₄×Q₈	—	2	1	no	—
76	C₄×Q₈	—	2	1	no	—
77	#117	C₂²×C₃	1	2	yes	—
78	C₂×C₄⋊C₄	—	2	1	no	—
79	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
81	#107	C₂²×C₃	1	8	yes	—
82	#106	C₂²×C₃	1	8	yes	—
83	#97	C₂²×C₃	1	2	yes	—
84	C₄×C₈	C₂²×C₃	1	2	yes	—
85	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
86	C₃×Q₈	—	4	4	no	—
87	C₃×Q₈	—	4	2	no	—
88	C₃×Q₈	—	4	4	no	—
89	C₃×Q₈	—	4	2	no	—
90	C₂×C₃×C₄	—	2	2	no	—
91	C₂×C₃×C₄	—	2	2	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	2	4	no	—
95	C₂×C₃×C₄	—	4	4	no	—
96	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
101	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
102	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
103	C₃×C₈	C₂×Q₈	1	8	yes	complemented
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	4	4	no	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
111	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
112	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
113	M₁₆	C₃×Q₈	1	8	yes	complemented
114	C₂×C₈	C₂³×C₃	1	4	yes	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	C₂³×C₃	1	2	yes	—
123	C₄⋊C₄	C₂³×C₃	1	4	yes	—
124	C₄⋊C₄	C₂³×C₃	1	2	yes	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	C₃×D₄	1	1	yes	characteristic
128	C₄²	C₃×D₄	1	1	yes	characteristic
129	C₄²	C₂³×C₃	1	1	yes	characteristic
130	C₄²	C₂³×C₃	1	1	yes	characteristic
131	C₂²×C₃	—	2	1	no	—
132	C₂²×C₃	—	2	1	no	—
133	C₂²×C₃	C₄⋊Q₈	1	2	yes	—
134	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
135	C₃×C₄	—	2	1	no	—
136	C₃×C₄	—	2	1	no	—
137	C₃×C₄	—	4	4	no	—
138	C₃×C₄	—	4	2	no	—
139	C₃×C₄	—	4	2	no	—
140	C₃×C₄	C₄⋊Q₈	1	2	yes	—
141	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
142	C₃×C₄	#137	1	2	yes	—
143	C₃×C₄	C₂²×Q₈	1	2	yes	—
144	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
145	Q₈	—	4	4	no	—
146	Q₈	—	4	4	no	—
147	Q₈	—	4	2	no	—
148	Q₈	—	4	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
156	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
158	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
159	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
160	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
161	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
162	C₈	C₂×C₃×Q₈	1	8	yes	complemented
163	C₂×C₃	—	2	2	no	—
164	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
165	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×C₃	C₂×#137	1	1	yes	characteristic
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	C₃×C₄⋊Q₈	1	2	yes	—
170	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	4	4	no	—
174	C₄	—	4	2	no	—
175	C₄	—	4	2	no	—
176	C₄	C₃×C₄⋊Q₈	1	2	yes	—
177	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
178	C₄	C₃×#137	1	2	yes	—
179	C₄	C₂²×C₃×Q₈	1	2	yes	—
180	C₃	#3165	1	1	yes	characteristic, direct product factor, Sylow
181	C₂	—	2	2	no	—
182	C₂	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
183	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂	C₂×C₃×#137	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃×#501	C₂	semidirect
4. C₃×#500	C₂	semidirect
5. C₃×#498	C₂	semidirect
8. #3165	C₃	direct (N × H)
32. #501	C₂×C₃	semidirect
33. #500	C₂×C₃	semidirect
34. #498	C₂×C₃	semidirect
46. C₃×M₁₆	Q₈	semidirect
103. C₃×C₈	C₂×Q₈	semidirect
113. M₁₆	C₃×Q₈	semidirect
162. C₈	C₂×C₃×Q₈	semidirect
180. C₃	#3165	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×#533	2
C₂×C₃×C₄⋊Q₈	2
C₃×#501	2
C₃×#500	2
C₃×#498	2
C₃×#428	2
C₃×C₄×M₁₆	2
#3165	3

Subgroups — C3×#3165 #85029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3165 #85029