Subgroups — C₃×#5721 #10595818

Order 576.

434 conjugacy classes of subgroups.

The subgroup lattice has 215 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×Dic₃	C₂	1	2	yes	maximal
3	C₃×#944	C₂	1	2	yes	maximal
4	C₃×#890	C₂	1	2	yes	maximal
5	C₃×#889	C₂	1	4	yes	maximal
6	C₃×#887	C₂	1	2	yes	maximal
7	C₃×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
8	C₃×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
9	C₃×#557	—	3	1	no	maximal
10	#5721	C₃	1	1	yes	maximal, characteristic, direct product factor
11	C₂×C₃²×Q₈	C₂²	1	2	yes	—
12	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
14	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃²×C₄²	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
18	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
20	C₃×Dic₃⋊C₄	C₂²	1	4	yes	—
21	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
22	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
23	C₃×Dic₃⋊C₄	C₂²	1	4	yes	—
24	C₃×C₄×Dic₃	C₂²	1	4	yes	—
25	C₃×C₄×Dic₃	C₂²	1	2	yes	—
26	C₃×C₄×Dic₃	C₂²	1	2	yes	—
27	C₃×C₄⋊Q₈	—	2	1	no	—
28	C₃×C₄⋊Q₈	—	3	2	no	—
29	C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
30	C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
31	C₃×#125	—	3	4	no	—
32	C₃×#125	—	3	2	no	—
33	C₃×C₄×Q₈	—	3	2	no	—
34	C₃×C₄×Q₈	—	3	2	no	—
35	C₃×C₄×Q₈	—	3	1	no	—
36	C₃×C₄×Q₈	—	3	1	no	—
37	Q₈×Dic₃	C₂×C₃	1	2	yes	—
38	#944	C₂×C₃	1	2	yes	—
39	#890	C₂×C₃	1	2	yes	—
40	#889	C₂×C₃	1	4	yes	—
41	#887	C₂×C₃	1	2	yes	—
42	C₄×Dic₆	C₂×C₃	1	1	yes	characteristic
43	C₄×Dic₆	C₂×C₃	1	1	yes	characteristic
44	C₃²×Q₈	—	2	4	no	—
45	C₂×C₃²×C₄	C₂³	1	4	yes	—
46	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
49	C₂×C₃×Dic₃	C₂³	1	2	yes	—
50	C₂×C₃×Dic₃	C₂³	1	2	yes	—
51	C₂×C₃×Dic₃	C₂³	1	2	yes	—
52	C₂×C₃×Dic₃	C₂³	1	2	yes	—
53	C₃×Dic₆	—	2	2	no	—
54	C₃×Dic₆	Q₈	1	4	yes	complemented
55	#557	—	3	1	no	Sylow
56	C₂×C₃×Q₈	—	2	2	no	—
57	C₂×C₃×Q₈	—	3	1	no	—
58	C₂×C₃×Q₈	—	3	1	no	—
59	C₂×C₃×Q₈	C₂×D₃	1	2	yes	—
60	C₂×C₃×Q₈	C₂²×C₃	1	2	yes	—
61	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
62	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
63	C₃×C₄⋊C₄	—	2	2	no	—
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	3	2	no	—
66	C₃×C₄⋊C₄	—	3	2	no	—
67	C₃×C₄⋊C₄	—	3	4	no	—
68	C₃×C₄⋊C₄	—	3	4	no	—
69	C₃×C₄⋊C₄	—	3	1	no	—
70	C₃×C₄⋊C₄	—	3	4	no	—
71	C₃×C₄⋊C₄	—	3	1	no	—
72	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
73	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
74	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
75	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
76	C₃×C₄²	—	2	1	no	—
77	C₃×C₄²	—	3	4	no	—
78	C₃×C₄²	—	3	2	no	—
79	C₃×C₄²	—	3	2	no	—
80	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
81	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
82	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
83	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
84	C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
85	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
86	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
87	Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
88	Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
89	C₄×Dic₃	C₂²×C₃	1	2	yes	—
90	C₄×Dic₃	C₂²×C₃	1	2	yes	—
91	C₄×Dic₃	C₂²×C₃	1	4	yes	—
92	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
93	C₃²×C₄	—	2	1	no	—
94	C₃²×C₄	—	2	4	no	—
95	C₃²×C₄	C₂×Q₈	1	2	yes	—
96	C₃²×C₄	#41	1	2	yes	—
97	C₃×Dic₃	—	2	2	no	—
98	C₃×Dic₃	—	2	2	no	—
99	C₃×Dic₃	—	2	2	no	—
100	C₃×Dic₃	C₂×Q₈	1	4	yes	—
101	C₄⋊Q₈	—	3	2	no	—
102	C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
103	#125	—	3	4	no	—
104	#125	—	3	2	no	—
105	C₄×Q₈	—	3	2	no	—
106	C₄×Q₈	—	3	2	no	—
107	C₄×Q₈	—	3	1	no	—
108	C₄×Q₈	—	3	1	no	—
109	C₃×Q₈	—	2	4	no	—
110	C₃×Q₈	—	2	4	no	—
111	C₃×Q₈	—	3	4	no	—
112	C₃×Q₈	—	4	4	no	—
113	C₃×Q₈	—	6	2	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	4	no	—
118	C₂×C₃×C₄	—	3	2	no	—
119	C₂×C₃×C₄	—	3	2	no	—
120	C₂×C₃×C₄	—	3	2	no	—
121	C₂×C₃×C₄	—	3	2	no	—
122	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
123	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂³×C₃	1	4	yes	—
127	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
130	C₂×Dic₃	C₂³×C₃	1	2	yes	—
131	C₂×Dic₃	C₂³×C₃	1	2	yes	—
132	C₂×Dic₃	C₂³×C₃	1	2	yes	—
133	C₂×Dic₃	C₂³×C₃	1	2	yes	—
134	Dic₆	—	2	2	no	—
135	Dic₆	C₃×Q₈	1	4	yes	complemented
136	C₂×C₃²	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₃²	C₂×#41	1	1	yes	characteristic
138	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
139	C₂×Q₈	—	3	1	no	—
140	C₂×Q₈	—	3	1	no	—
141	C₂×Q₈	C₂×C₃×D₃	1	2	yes	—
142	C₄⋊C₄	—	3	1	no	—
143	C₄⋊C₄	—	3	2	no	—
144	C₄⋊C₄	—	3	2	no	—
145	C₄⋊C₄	—	3	4	no	—
146	C₄⋊C₄	—	3	1	no	—
147	C₄⋊C₄	—	3	4	no	—
148	C₄⋊C₄	—	3	4	no	—
149	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
150	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
151	C₄²	—	3	4	no	—
152	C₄²	—	3	2	no	—
153	C₄²	—	3	2	no	—
154	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
155	C₂²×C₃	—	2	1	no	—
156	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
157	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	—	2	4	no	—
160	C₃×C₄	—	2	1	no	—
161	C₃×C₄	—	2	4	no	—
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	—	2	2	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	3	4	no	—
166	C₃×C₄	—	4	4	no	—
167	C₃×C₄	—	6	2	no	—
168	C₃×C₄	—	6	2	no	—
169	C₃×C₄	—	6	2	no	—
170	C₃×C₄	#237	1	2	yes	—
171	C₃×C₄	D₃×Q₈	1	2	yes	—
172	C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
173	C₃×C₄	C₃×#41	1	2	yes	—
174	Dic₃	—	2	2	no	—
175	Dic₃	—	2	2	no	—
176	Dic₃	—	2	2	no	—
177	Dic₃	C₂×C₃×Q₈	1	4	yes	—
178	C₃²	#557	1	1	yes	characteristic, complemented, Sylow
179	Q₈	—	2	4	no	—
180	Q₈	—	3	4	no	—
181	Q₈	—	6	2	no	—
182	C₂×C₄	—	3	2	no	—
183	C₂×C₄	—	3	2	no	—
184	C₂×C₄	—	3	2	no	—
185	C₂×C₄	—	3	2	no	—
186	C₂×C₄	C₂²×C₃×D₃	1	4	yes	—
187	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
188	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
189	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
190	C₂×C₃	—	2	1	no	—
191	C₂×C₃	—	2	1	no	—
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	C₂×#237	1	1	yes	characteristic
194	C₂×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
195	C₂×C₃	#1007	1	1	yes	characteristic
196	C₂×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
197	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
198	C₂×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
199	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	—	2	1	no	—
202	C₄	—	3	4	no	—
203	C₄	—	6	2	no	—
204	C₄	—	6	2	no	—
205	C₄	—	6	2	no	—
206	C₄	C₃×#237	1	2	yes	—
207	C₄	C₃×D₃×Q₈	1	2	yes	—
208	C₃	—	2	1	no	—
209	C₃	#5721	1	1	yes	characteristic, direct product factor
210	C₃	C₃×#557	1	1	yes	characteristic, complemented
211	C₂	C₂×C₃×#237	1	1	yes	characteristic
212	C₂	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
213	C₂	C₃×#1007	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting	C₃²×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃²	C₂⁴
hypercenter	C₃×C₄⋊Q₈	D₃
hypocenter	C₃	C₃×#557
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
10. #5721	C₃	direct (N × H)
54. C₃×Dic₆	Q₈	semidirect
135. Dic₆	C₃×Q₈	semidirect
178. C₃²	#557	semidirect
209. C₃	#5721	direct (N × H)
210. C₃	C₃×#557	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×C₄⋊Q₈	2
C₃×Q₈×Dic₃	2
C₃×#944	2
C₃×#890	2
C₃×#889	2
C₃×#887	2
C₃×C₄×Dic₆	2
C₃×#557	3
#5721	3

Subgroups — C3×#5721 #10595818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5721 #10595818