Subgroups — C₃×#3229 #85093

Order 384.

360 conjugacy classes of subgroups.

The subgroup lattice has 184 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×#498	C₂	1	2	yes	maximal, complemented
4	C₃×#487	C₂	1	2	yes	maximal, complemented
5	C₃×#483	C₂	1	4	yes	maximal
6	C₃×#475	C₂	1	4	yes	maximal, complemented
7	C₃×#431	C₂	1	1	yes	maximal, characteristic
8	#3229	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₈	—	2	1	no	—
10	C₃×C₄⋊Q₈	—	2	1	no	—
11	C₃×C₄⋊Q₈	—	2	1	no	—
12	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
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×#125	—	2	1	no	—
17	C₃×C₂²⋊Q₈	—	2	1	no	—
18	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₃×C₄×Q₈	—	2	1	no	—
20	C₃×C₄×Q₈	C₂²	1	2	yes	—
21	C₃×#117	—	2	1	no	—
22	C₂×C₃×C₄⋊C₄	—	2	1	no	—
23	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₃×#106	C₂²	1	8	yes	—
26	C₃×C₄⋊C₈	C₂²	1	2	yes	—
27	C₃×#103	C₂²	1	8	yes	—
28	C₃×C₂²⋊C₈	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	#498	C₂×C₃	1	2	yes	complemented
33	#487	C₂×C₃	1	2	yes	complemented
34	#483	C₂×C₃	1	4	yes	—
35	#475	C₂×C₃	1	4	yes	complemented
36	#431	C₂×C₃	1	1	yes	characteristic
37	C₂×C₃×Q₈	—	2	1	no	—
38	C₂×C₃×Q₈	—	2	2	no	—
39	C₂×C₃×Q₈	—	2	2	no	—
40	C₂×C₃×Q₈	—	4	4	no	—
41	C₂×C₃×Q₈	C₂³	1	2	yes	—
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	C₂×C₃×C₈	C₂³	1	4	yes	—
47	C₃×C₄⋊C₄	—	2	2	no	—
48	C₃×C₄⋊C₄	—	2	2	no	—
49	C₃×C₄⋊C₄	—	2	2	no	—
50	C₃×C₄⋊C₄	—	2	1	no	—
51	C₃×C₄⋊C₄	—	2	1	no	—
52	C₃×C₄⋊C₄	—	2	1	no	—
53	C₃×C₄⋊C₄	—	2	2	no	—
54	C₃×C₄⋊C₄	—	2	2	no	—
55	C₃×C₄⋊C₄	C₂³	1	4	yes	—
56	C₃×C₄⋊C₄	C₂³	1	2	yes	—
57	C₃×C₂²⋊C₄	—	2	2	no	—
58	C₃×C₄²	—	2	2	no	—
59	C₃×C₄²	D₄	1	1	yes	characteristic
60	C₃×C₄²	D₄	1	1	yes	characteristic
61	C₃×C₄²	C₂³	1	1	yes	characteristic
62	C₃×C₄²	C₂³	1	1	yes	characteristic
63	C₂²×Q₈	—	2	1	no	—
64	C₄⋊Q₈	—	2	1	no	—
65	C₄⋊Q₈	—	2	1	no	—
66	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
68	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
69	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
70	#125	—	2	1	no	—
71	C₂²⋊Q₈	—	2	1	no	—
72	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
73	C₄×Q₈	—	2	1	no	—
74	C₄×Q₈	C₂²×C₃	1	2	yes	—
75	#117	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
79	#106	C₂²×C₃	1	8	yes	—
80	C₄⋊C₈	C₂²×C₃	1	2	yes	—
81	#103	C₂²×C₃	1	8	yes	—
82	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
83	C₄×C₈	C₂²×C₃	1	2	yes	—
84	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
85	C₃×Q₈	—	2	4	no	—
86	C₃×Q₈	—	4	4	no	—
87	C₃×Q₈	—	4	4	no	—
88	C₃×Q₈	—	4	2	no	—
89	C₂×C₃×C₄	—	2	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	4	no	—
94	C₂×C₃×C₄	—	4	4	no	—
95	C₂×C₃×C₄	SD₁₆	1	4	yes	—
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	C₂⁴	1	1	yes	characteristic
102	C₃×C₈	—	2	4	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	4	4	no	—
107	C₂×Q₈	C₂³×C₃	1	2	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
110	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
111	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂×C₈	C₂³×C₃	1	4	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	C₂³×C₃	1	4	yes	—
122	C₄⋊C₄	C₂³×C₃	1	2	yes	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	C₃×D₄	1	1	yes	characteristic
126	C₄²	C₃×D₄	1	1	yes	characteristic
127	C₄²	C₂³×C₃	1	1	yes	characteristic
128	C₄²	C₂³×C₃	1	1	yes	characteristic
129	C₂²×C₃	—	2	1	no	—
130	C₂²×C₃	—	2	1	no	—
131	C₂²×C₃	C₂×SD₁₆	1	2	yes	—
132	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
133	C₃×C₄	—	2	1	no	—
134	C₃×C₄	—	2	1	no	—
135	C₃×C₄	—	4	2	no	—
136	C₃×C₄	—	4	4	no	—
137	C₃×C₄	—	4	2	no	—
138	C₃×C₄	C₂×SD₁₆	1	2	yes	—
139	C₃×C₄	C₂×SD₁₆	1	2	yes	—
140	C₃×C₄	#137	1	2	yes	—
141	C₃×C₄	2₋¹⁺⁴	1	2	yes	—
142	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
143	Q₈	—	2	4	no	—
144	Q₈	—	4	4	no	—
145	Q₈	—	4	4	no	—
146	Q₈	—	4	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	C₃×SD₁₆	1	4	yes	—
154	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
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	C₂⁴×C₃	1	1	yes	characteristic
160	C₈	—	2	4	no	—
161	C₂×C₃	—	2	2	no	—
162	C₂×C₃	#536	1	1	yes	characteristic
163	C₂×C₃	C₂²×SD₁₆	1	1	yes	characteristic
164	C₂×C₃	C₂×#137	1	1	yes	characteristic
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	C₂×C₃×SD₁₆	1	2	yes	—
168	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	4	4	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	2	no	—
174	C₄	C₂×C₃×SD₁₆	1	2	yes	—
175	C₄	C₂×C₃×SD₁₆	1	2	yes	—
176	C₄	C₃×#137	1	2	yes	—
177	C₄	C₃×2₋¹⁺⁴	1	2	yes	—
178	C₃	#3229	1	1	yes	characteristic, direct product factor, Sylow
179	C₂	—	2	2	no	—
180	C₂	C₃×#536	1	1	yes	characteristic
181	C₂	C₂²×C₃×SD₁₆	1	1	yes	characteristic
182	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×#498	C₂	semidirect
4. C₃×#487	C₂	semidirect
6. C₃×#475	C₂	semidirect
8. #3229	C₃	direct (N × H)
32. #498	C₂×C₃	semidirect
33. #487	C₂×C₃	semidirect
35. #475	C₂×C₃	semidirect
178. C₃	#3229	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×#498	2
C₃×#487	2
C₃×#483	2
C₃×#475	2
C₃×#431	2
#3229	3

Subgroups — C3×#3229 #85093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3229 #85093