Subgroups — C₃×#2805 #84669

Order 384.

368 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#400	C₂	1	2	yes	maximal
2	C₃×#400	C₂	1	2	yes	maximal
3	C₃×#395	C₂	1	2	yes	maximal
4	C₃×#391	C₂	1	1	yes	maximal, characteristic
5	C₃×#389	C₂	1	1	yes	maximal, characteristic
6	C₃×#389	C₂	1	1	yes	maximal, characteristic
7	C₃×#389	C₂	1	2	yes	maximal
8	C₃×#387	C₂	1	2	yes	maximal
9	C₃×#387	C₂	1	2	yes	maximal
10	#2805	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄²	C₂²	1	2	yes	—
22	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₃×#95	C₂²	1	2	yes	—
24	C₃×#95	C₂²	1	2	yes	—
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	2	yes	—
29	C₃×#95	C₂²	1	2	yes	—
30	C₃×#95	C₂²	1	2	yes	—
31	C₃×#95	C₂²	1	2	yes	—
32	#400	C₂×C₃	1	2	yes	—
33	#400	C₂×C₃	1	2	yes	—
34	#395	C₂×C₃	1	2	yes	—
35	#391	C₂×C₃	1	1	yes	characteristic
36	#389	C₂×C₃	1	1	yes	characteristic
37	#389	C₂×C₃	1	1	yes	characteristic
38	#389	C₂×C₃	1	2	yes	—
39	#387	C₂×C₃	1	2	yes	—
40	#387	C₂×C₃	1	2	yes	—
41	C₂×C₃×Q₈	—	2	2	no	—
42	C₂×C₃×Q₈	—	4	1	no	—
43	C₂×C₃×Q₈	—	4	1	no	—
44	C₂²×C₃×C₄	C₂³	1	2	yes	—
45	C₂²×C₃×C₄	C₂³	1	2	yes	—
46	C₂²×C₃×C₄	C₂³	1	2	yes	—
47	C₂²×C₃×C₄	C₂³	1	2	yes	—
48	C₂²×C₃×C₄	C₂³	1	2	yes	—
49	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
53	C₃×C₄⋊C₄	—	2	4	no	—
54	C₃×C₄⋊C₄	—	2	2	no	—
55	C₃×C₄⋊C₄	—	2	4	no	—
56	C₃×C₄⋊C₄	—	4	2	no	—
57	C₃×C₄⋊C₄	—	4	1	no	—
58	C₃×C₄⋊C₄	—	4	1	no	—
59	C₃×C₄⋊C₄	—	4	1	no	—
60	C₃×C₄⋊C₄	—	4	2	no	—
61	C₃×C₄⋊C₄	Q₈	1	4	yes	complemented
62	C₃×C₄²	—	4	1	no	—
63	C₃×C₄²	—	4	2	no	—
64	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
65	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
67	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
70	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
72	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₄²	C₂²×C₃	1	2	yes	—
75	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
76	#95	C₂²×C₃	1	2	yes	—
77	#95	C₂²×C₃	1	2	yes	—
78	#95	C₂²×C₃	1	2	yes	—
79	#95	C₂²×C₃	1	2	yes	—
80	#95	C₂²×C₃	1	2	yes	—
81	#95	C₂²×C₃	1	2	yes	—
82	#95	C₂²×C₃	1	2	yes	—
83	#95	C₂²×C₃	1	2	yes	—
84	#95	C₂²×C₃	1	2	yes	—
85	C₂³×C₃	C₂⁴	1	1	yes	characteristic
86	C₃×Q₈	—	4	4	no	—
87	C₂×C₃×C₄	—	2	2	no	—
88	C₂×C₃×C₄	—	2	1	no	—
89	C₂×C₃×C₄	—	2	1	no	—
90	C₂×C₃×C₄	—	2	2	no	—
91	C₂×C₃×C₄	—	2	1	no	—
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	—	2	1	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
111	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
112	C₂×C₃×C₄	#41	1	2	yes	—
113	C₂×C₃×C₄	#41	1	2	yes	—
114	C₂×Q₈	—	2	2	no	—
115	C₂×Q₈	—	4	1	no	—
116	C₂×Q₈	—	4	1	no	—
117	C₂²×C₄	C₂³×C₃	1	2	yes	—
118	C₂²×C₄	C₂³×C₃	1	2	yes	—
119	C₂²×C₄	C₂³×C₃	1	2	yes	—
120	C₂²×C₄	C₂³×C₃	1	2	yes	—
121	C₂²×C₄	C₂³×C₃	1	2	yes	—
122	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
123	C₂²×C₄	C₂³×C₃	1	2	yes	—
124	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
125	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	1	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	C₃×Q₈	1	4	yes	complemented
135	C₄²	—	4	1	no	—
136	C₄²	—	4	2	no	—
137	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
138	C₂²×C₃	C₂×#41	1	1	yes	characteristic
139	C₂²×C₃	C₂×#41	1	1	yes	characteristic
140	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂²×C₃	2₋¹⁺⁴	1	1	yes	characteristic
144	C₃×C₄	—	2	2	no	—
145	C₃×C₄	—	2	2	no	—
146	C₃×C₄	—	2	2	no	—
147	C₃×C₄	—	2	4	no	—
148	C₃×C₄	—	4	2	no	—
149	C₃×C₄	—	4	2	no	—
150	C₃×C₄	—	4	2	no	—
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	—	4	2	no	—
153	C₂³	C₂⁴×C₃	1	1	yes	characteristic
154	Q₈	—	4	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
179	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
180	C₂×C₄	C₃×#41	1	2	yes	—
181	C₂×C₄	C₃×#41	1	2	yes	—
182	C₂×C₃	#538	1	1	yes	characteristic
183	C₂×C₃	#539	1	1	yes	characteristic
184	C₂×C₃	#544	1	1	yes	characteristic
185	C₂×C₃	#554	1	1	yes	characteristic
186	C₂×C₃	#556	1	1	yes	characteristic
187	C₂×C₃	#557	1	1	yes	characteristic
188	C₂×C₃	#563	1	1	yes	characteristic
189	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
190	C₂²	C₂×C₃×#41	1	1	yes	characteristic
191	C₂²	C₂×C₃×#41	1	1	yes	characteristic
192	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
193	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
194	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
195	C₂²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	4	no	—
200	C₄	—	4	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₃	#2805	1	1	yes	characteristic, direct product factor, Sylow
206	C₂	C₃×#538	1	1	yes	characteristic
207	C₂	C₃×#539	1	1	yes	characteristic
208	C₂	C₃×#544	1	1	yes	characteristic
209	C₂	C₃×#554	1	1	yes	characteristic
210	C₂	C₃×#556	1	1	yes	characteristic
211	C₂	C₃×#557	1	1	yes	characteristic
212	C₂	C₃×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
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
10. #2805	C₃	direct (N × H)
61. C₃×C₄⋊C₄	Q₈	semidirect
134. C₄⋊C₄	C₃×Q₈	semidirect
205. C₃	#2805	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×#400	2
C₃×#395	2
C₃×#391	2
C₃×#389	2
C₃×#387	2
#2805	3

Subgroups — C3×#2805 #84669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2805 #84669