Subgroups — C₃×#2821 #84685

Order 384.

430 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#126	C₂	1	1	yes	maximal, characteristic
3	C₃×#400	C₂	1	2	yes	maximal, complemented
4	C₃×#399	C₂	1	2	yes	maximal
5	C₃×#398	C₂	1	2	yes	maximal, complemented
6	C₃×#397	C₂	1	2	yes	maximal
7	C₃×#395	C₂	1	2	yes	maximal, complemented
8	C₃×#393	C₂	1	2	yes	maximal
9	C₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
10	#2821	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	2	yes	—
12	C₃×C₄⋊Q₈	—	2	4	no	—
13	C₃×#126	—	2	4	no	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
25	C₃×#95	C₂²	1	2	yes	—
26	C₃×#95	C₂²	1	2	yes	—
27	C₃×#95	C₂²	1	2	yes	—
28	C₃×#95	C₂²	1	4	yes	—
29	C₃×#95	C₂²	1	4	yes	—
30	C₂×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#126	C₂×C₃	1	1	yes	characteristic
32	#400	C₂×C₃	1	2	yes	complemented
33	#399	C₂×C₃	1	2	yes	—
34	#398	C₂×C₃	1	2	yes	complemented
35	#397	C₂×C₃	1	2	yes	—
36	#395	C₂×C₃	1	2	yes	complemented
37	#393	C₂×C₃	1	2	yes	—
38	#383	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂⁴×C₃	C₂³	1	1	yes	characteristic
40	C₂×C₃×Q₈	—	2	4	no	—
41	C₂×C₃×Q₈	—	4	4	no	—
42	C₂²×C₃×C₄	C₂³	1	4	yes	—
43	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₂²×C₃×C₄	C₂³	1	4	yes	—
46	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
47	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
49	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
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	4	no	—
54	C₃×C₄⋊C₄	—	2	4	no	—
55	C₃×C₄⋊C₄	—	4	4	no	—
56	C₃×C₂²⋊C₄	—	2	2	no	—
57	C₃×C₂²⋊C₄	—	2	2	no	—
58	C₃×C₂²⋊C₄	—	2	2	no	—
59	C₃×C₂²⋊C₄	—	4	4	no	—
60	C₃×C₄²	D₄	1	4	yes	—
61	C₂²×Q₈	C₂²×C₃	1	2	yes	—
62	C₄⋊Q₈	—	2	4	no	—
63	#126	—	2	4	no	—
64	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
68	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
69	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
70	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
74	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
75	#95	C₂²×C₃	1	2	yes	—
76	#95	C₂²×C₃	1	2	yes	—
77	#95	C₂²×C₃	1	2	yes	—
78	#95	C₂²×C₃	1	4	yes	—
79	#95	C₂²×C₃	1	4	yes	—
80	C₂³×C₃	—	2	1	no	—
81	C₂³×C₃	—	2	2	no	—
82	C₂³×C₃	—	2	2	no	—
83	C₂³×C₃	—	2	2	no	—
84	C₂³×C₃	C₂⁴	1	1	yes	characteristic
85	C₃×Q₈	—	4	8	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	2	2	no	—
88	C₂×C₃×C₄	—	2	2	no	—
89	C₂×C₃×C₄	—	2	1	no	—
90	C₂×C₃×C₄	—	2	1	no	—
91	C₂×C₃×C₄	—	2	1	no	—
92	C₂×C₃×C₄	—	2	4	no	—
93	C₂×C₃×C₄	—	2	4	no	—
94	C₂×C₃×C₄	—	2	4	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	4	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	4	no	—
100	C₂×C₃×C₄	—	2	4	no	—
101	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
102	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
103	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
104	C₂⁴	C₂³×C₃	1	1	yes	characteristic
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	4	4	no	—
107	C₂²×C₄	C₂³×C₃	1	4	yes	—
108	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
109	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₂²×C₄	C₂³×C₃	1	4	yes	—
111	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
114	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	4	4	no	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	4	4	no	—
125	C₄²	C₃×D₄	1	4	yes	—
126	C₂²×C₃	—	4	1	no	—
127	C₂²×C₃	—	4	1	no	—
128	C₂²×C₃	—	4	1	no	—
129	C₂²×C₃	—	4	4	no	—
130	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
131	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
132	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
133	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
134	C₃×C₄	—	2	2	no	—
135	C₃×C₄	—	2	2	no	—
136	C₃×C₄	—	2	2	no	—
137	C₃×C₄	—	4	1	no	—
138	C₃×C₄	—	4	4	no	—
139	C₃×C₄	—	4	1	no	—
140	C₃×C₄	—	4	1	no	—
141	C₃×C₄	—	4	4	no	—
142	C₂³	—	2	1	no	—
143	C₂³	—	2	2	no	—
144	C₂³	—	2	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	C₂⁴×C₃	1	1	yes	characteristic
147	Q₈	—	4	8	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
164	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
165	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
166	C₂×C₃	—	8	1	no	—
167	C₂×C₃	#535	1	1	yes	characteristic
168	C₂×C₃	#536	1	1	yes	characteristic
169	C₂×C₃	#536	1	1	yes	characteristic
170	C₂×C₃	#563	1	4	yes	—
171	C₂²	—	4	4	no	—
172	C₂²	—	4	1	no	—
173	C₂²	—	4	1	no	—
174	C₂²	—	4	1	no	—
175	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
176	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
177	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
178	C₂²	C₃×2₋¹⁺⁴	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	4	no	—
183	C₄	—	4	4	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	1	no	—
187	C₃	#2821	1	1	yes	characteristic, direct product factor, Sylow
188	C₂	—	8	1	no	—
189	C₂	C₃×#535	1	1	yes	characteristic
190	C₂	C₃×#536	1	1	yes	characteristic
191	C₂	C₃×#536	1	1	yes	characteristic
192	C₂	C₃×#563	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄⋊Q₈	C₂	semidirect
3. C₃×#400	C₂	semidirect
5. C₃×#398	C₂	semidirect
7. C₃×#395	C₂	semidirect
9. C₃×#383	C₂	semidirect
10. #2821	C₃	direct (N × H)
30. C₂×C₄⋊Q₈	C₂×C₃	semidirect
32. #400	C₂×C₃	semidirect
34. #398	C₂×C₃	semidirect
36. #395	C₂×C₃	semidirect
38. #383	C₂×C₃	semidirect
187. C₃	#2821	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×C₄⋊Q₈	2
C₂×C₃×#126	2
C₃×#400	2
C₃×#399	2
C₃×#398	2
C₃×#397	2
C₃×#395	2
C₃×#393	2
C₃×#383	2
#2821	3

Subgroups — C3×#2821 #84685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2821 #84685