Subgroups — C₃×#5579 #10595676

Order 576.

491 conjugacy classes of subgroups.

The subgroup lattice has 238 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, complemented
2	C₃×#946	C₂	1	2	yes	maximal, complemented
3	C₃×#896	C₂	1	4	yes	maximal, complemented
4	C₃×D₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×D₁₂	C₂	1	2	yes	maximal
6	C₃×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
7	C₃×#870	C₂	1	2	yes	maximal, complemented
8	C₃×#869	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#554	—	3	1	no	maximal
10	#5579	C₃	1	1	yes	maximal, characteristic, direct product factor
11	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
14	C₂×C₃×C₄×D₃	C₂²	1	4	yes	—
15	C₂×C₃×Dic₆	C₂²	1	2	yes	—
16	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃²×C₄²	C₂²	1	1	yes	characteristic
19	C₃²×C₄²	C₂²	1	2	yes	—
20	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
21	C₃×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
22	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
23	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₃×Dic₃⋊C₄	C₂²	1	4	yes	—
26	C₃×Dic₃⋊C₄	C₂²	1	4	yes	—
27	C₃×C₄⋊Q₈	—	3	1	no	—
28	C₃×#125	—	3	2	no	—
29	C₃×C₂²⋊Q₈	—	3	2	no	—
30	C₃×C₂²⋊Q₈	—	3	4	no	—
31	C₃×C₄×Q₈	—	2	1	no	—
32	C₃×C₄×Q₈	D₃	1	1	yes	characteristic, complemented
33	C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×C₄×D₄	—	3	2	no	—
35	C₃×C₄×D₄	—	3	1	no	—
36	C₂×C₃×C₄⋊C₄	—	3	2	no	—
37	#946	C₂×C₃	1	2	yes	complemented
38	#896	C₂×C₃	1	4	yes	complemented
39	D₃×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
40	C₄×D₁₂	C₂×C₃	1	2	yes	—
41	C₄×D₁₂	C₂×C₃	1	1	yes	characteristic
42	#870	C₂×C₃	1	2	yes	complemented
43	#869	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂²×C₃×D₃	C₂³	1	2	yes	—
45	C₃²×Q₈	—	2	2	no	—
46	C₂×C₃²×C₄	C₂³	1	2	yes	—
47	C₂×C₃²×C₄	C₂³	1	2	yes	—
48	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
51	C₂×C₃×Dic₃	C₂³	1	2	yes	—
52	C₂×C₃×Dic₃	C₂³	1	4	yes	—
53	C₃×D₁₂	Q₈	1	4	yes	complemented
54	C₃×C₄×D₃	—	2	4	no	—
55	C₃×C₄×D₃	—	4	4	no	—
56	C₃×Dic₆	—	4	2	no	—
57	#554	—	3	1	no	Sylow
58	C₂×C₃×Q₈	—	2	1	no	—
59	C₂×C₃×Q₈	—	3	2	no	—
60	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₃×D₄	—	3	1	no	—
63	C₂²×C₃×C₄	—	3	2	no	—
64	C₂²×C₃×C₄	—	3	4	no	—
65	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
67	C₂×C₄×D₃	C₂²×C₃	1	4	yes	—
68	C₂×Dic₆	C₂²×C₃	1	2	yes	—
69	C₃×C₄⋊C₄	—	2	1	no	—
70	C₃×C₄⋊C₄	—	2	2	no	—
71	C₃×C₄⋊C₄	—	3	4	no	—
72	C₃×C₄⋊C₄	—	3	4	no	—
73	C₃×C₄⋊C₄	—	3	2	no	—
74	C₃×C₄⋊C₄	—	3	1	no	—
75	C₃×C₄⋊C₄	—	3	2	no	—
76	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
77	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
78	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
80	C₃×C₂²⋊C₄	—	3	2	no	—
81	C₃×C₂²⋊C₄	—	3	4	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	—	2	1	no	—
84	C₃×C₄²	C₂×D₃	1	2	yes	—
85	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
86	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
87	C₃×C₄²	C₂²×C₃	1	2	yes	—
88	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
89	(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
90	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
91	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
92	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
93	Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
94	Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
95	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
96	C₂×C₃×D₃	—	2	2	no	—
97	C₂×C₃×D₃	—	2	2	no	—
98	C₂×C₃×D₃	C₂×Q₈	1	4	yes	—
99	C₃²×C₄	—	2	2	no	—
100	C₃²×C₄	—	2	1	no	—
101	C₃²×C₄	—	2	2	no	—
102	C₃²×C₄	C₂×Q₈	1	2	yes	—
103	C₃²×C₄	#41	1	2	yes	—
104	C₃×Dic₃	—	2	2	no	—
105	C₃×Dic₃	—	2	4	no	—
106	C₄⋊Q₈	—	3	1	no	—
107	#125	—	3	2	no	—
108	C₂²⋊Q₈	—	3	2	no	—
109	C₂²⋊Q₈	—	3	4	no	—
110	C₄×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
111	C₄×D₄	—	3	2	no	—
112	C₄×D₄	—	3	1	no	—
113	C₂×C₄⋊C₄	—	3	2	no	—
114	C₂³×C₃	—	3	2	no	—
115	C₂²×D₃	C₂³×C₃	1	2	yes	—
116	C₃×Q₈	—	2	4	no	—
117	C₃×Q₈	—	2	2	no	—
118	C₃×Q₈	—	2	2	no	—
119	C₃×Q₈	—	12	2	no	—
120	C₃×D₄	—	3	4	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	3	2	no	—
127	C₂×C₃×C₄	—	3	4	no	—
128	C₂×C₃×C₄	—	6	4	no	—
129	C₂×C₃×C₄	—	12	4	no	—
130	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
131	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
133	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
136	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
137	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
140	C₂×Dic₃	C₂³×C₃	1	2	yes	—
141	C₂×Dic₃	C₂³×C₃	1	4	yes	—
142	D₁₂	C₃×Q₈	1	4	yes	complemented
143	C₄×D₃	—	2	4	no	—
144	C₄×D₃	—	4	4	no	—
145	Dic₆	—	4	2	no	—
146	C₂×C₃²	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₃²	C₂×#41	1	1	yes	characteristic
148	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
149	C₃×D₃	—	2	4	no	—
150	C₂×Q₈	—	3	2	no	—
151	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
152	C₂×D₄	—	3	1	no	—
153	C₂²×C₄	—	3	2	no	—
154	C₂²×C₄	—	3	4	no	—
155	C₄⋊C₄	—	3	4	no	—
156	C₄⋊C₄	—	3	2	no	—
157	C₄⋊C₄	—	3	1	no	—
158	C₄⋊C₄	—	3	2	no	—
159	C₄⋊C₄	—	3	4	no	—
160	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
161	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
162	C₂²⋊C₄	—	3	2	no	—
163	C₂²⋊C₄	—	3	4	no	—
164	C₄²	C₂×C₃×D₃	1	2	yes	—
165	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
166	C₂²×C₃	—	2	1	no	—
167	C₂²×C₃	—	3	4	no	—
168	C₂²×C₃	—	6	2	no	—
169	C₂²×C₃	—	6	2	no	—
170	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
171	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
172	C₂×D₃	—	2	2	no	—
173	C₂×D₃	—	2	2	no	—
174	C₂×D₃	C₂×C₃×Q₈	1	4	yes	—
175	C₃×C₄	—	2	4	no	—
176	C₃×C₄	—	2	2	no	—
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	4	2	no	—
185	C₃×C₄	—	4	1	no	—
186	C₃×C₄	—	6	2	no	—
187	C₃×C₄	—	6	4	no	—
188	C₃×C₄	#235	1	2	yes	—
189	C₃×C₄	D₃×Q₈	1	2	yes	—
190	C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
191	C₃×C₄	C₃×#41	1	2	yes	—
192	Dic₃	—	2	2	no	—
193	Dic₃	—	2	4	no	—
194	C₃²	#554	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	3	2	no	—
196	Q₈	—	2	2	no	—
197	Q₈	—	12	2	no	—
198	D₄	—	3	4	no	—
199	C₂×C₄	—	3	2	no	—
200	C₂×C₄	—	3	4	no	—
201	C₂×C₄	—	6	4	no	—
202	C₂×C₄	—	12	4	no	—
203	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
204	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
206	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
207	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	—	6	4	no	—
212	C₂×C₃	C₂×#235	1	1	yes	characteristic
213	C₂×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
214	C₂×C₃	#1009	1	1	yes	characteristic
215	C₂×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
216	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
217	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
218	D₃	—	2	4	no	—
219	C₂²	—	3	4	no	—
220	C₂²	—	6	2	no	—
221	C₂²	—	6	2	no	—
222	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	6	2	no	—
227	C₄	—	6	4	no	—
228	C₄	C₃×#235	1	2	yes	—
229	C₄	C₃×D₃×Q₈	1	2	yes	—
230	C₃	—	2	1	no	—
231	C₃	#5579	1	1	yes	characteristic, direct product factor
232	C₃	C₃×#554	1	1	yes	characteristic, complemented
233	C₂	—	6	4	no	—
234	C₂	C₂×C₃×#235	1	1	yes	characteristic
235	C₂	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
236	C₂	C₃×#1009	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	C₂²×C₃²	C₂⁴
prime radical	C₂×C₃×D₁₂	C₂²
hypercenter	C₃×C₄×Q₈	D₃
hypocenter	C₃	C₃×#554
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
1. C₃²×C₄×Q₈	C₂	semidirect
2. C₃×#946	C₂	semidirect
3. C₃×#896	C₂	semidirect
4. C₃×D₃×C₄⋊C₄	C₂	semidirect
7. C₃×#870	C₂	semidirect
8. C₃×#869	C₂	semidirect
10. #5579	C₃	direct (N × H)
32. C₃×C₄×Q₈	D₃	semidirect
33. C₃×C₄×Q₈	C₂×C₃	semidirect
37. #946	C₂×C₃	semidirect
38. #896	C₂×C₃	semidirect
39. D₃×C₄⋊C₄	C₂×C₃	semidirect
42. #870	C₂×C₃	semidirect
43. #869	C₂×C₃	semidirect
53. C₃×D₁₂	Q₈	semidirect
110. C₄×Q₈	C₃×D₃	semidirect
142. D₁₂	C₃×Q₈	semidirect
194. C₃²	#554	semidirect
231. C₃	#5579	direct (N × H)
232. C₃	C₃×#554	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×#946	2
C₃×#896	2
C₃×D₃×C₄⋊C₄	2
C₃×C₄×D₁₂	2
C₃×#870	2
C₃×#869	2
C₃×#554	3
#5579	3

Subgroups — C3×#5579 #10595676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5579 #10595676