Subgroups — C₃×#83231 #11807953

Order 1152.

993 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#5229	C₂	1	4	yes	maximal
3	C₃×#4980	C₂	1	8	yes	maximal
4	C₃×#4937	C₂	1	2	yes	maximal
5	C₃×#2381	—	3	1	no	maximal
6	#83231	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×Q₈	C₂²	1	2	yes	—
8	C₃²×C₄⋊Q₈	C₄	1	8	yes	complemented
9	C₂×C₃²×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
12	C₂×C₃×C₄×Dic₃	C₂²	1	4	yes	—
13	C₃×C₃⋊#95	C₂²	1	4	yes	—
14	C₃×C₃⋊#95	C₂²	1	8	yes	—
15	C₃×C₃⋊#95	C₂²	1	8	yes	—
16	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
17	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
18	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
19	C₃×#391	—	3	4	no	—
20	C₃×#387	—	3	8	no	—
21	C₃×#383	—	3	2	no	—
22	#5229	C₂×C₃	1	4	yes	—
23	#4980	C₂×C₃	1	8	yes	—
24	#4937	C₂×C₃	1	2	yes	—
25	C₂×C₃²×Q₈	—	4	2	no	—
26	C₂×C₃²×Q₈	—	4	2	no	—
27	C₂×C₃²×Q₈	C₂×C₄	1	8	yes	—
28	C₂²×C₃²×C₄	C₂³	1	4	yes	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂³	1	2	yes	—
31	C₂²×C₃×Dic₃	C₂³	1	8	yes	—
32	C₃²×C₄⋊C₄	C₂×C₄	1	16	yes	—
33	C₃²×C₄²	C₂×C₄	1	4	yes	—
34	C₃×C₄⋊Dic₃	—	4	4	no	—
35	C₃×C₄×Dic₃	—	4	4	no	—
36	#2381	—	3	1	no	Sylow
37	C₂²×C₃×Q₈	—	2	2	no	—
38	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
39	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
40	C₃×C₄⋊Q₈	—	2	8	no	—
41	C₃×C₄⋊Q₈	Dic₃	1	8	yes	complemented
42	C₃×C₄⋊Q₈	C₃×C₄	1	8	yes	complemented
43	C₂×C₃×C₄⋊C₄	—	2	4	no	—
44	C₂×C₃×C₄⋊C₄	—	3	4	no	—
45	C₂×C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
46	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
47	C₂×C₃×C₄²	—	2	1	no	—
48	C₂×C₃×C₄²	—	3	4	no	—
49	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
52	C₂×C₄×Dic₃	C₂²×C₃	1	4	yes	—
53	C₃×#95	—	3	8	no	—
54	C₃×#95	—	3	4	no	—
55	C₃×#95	—	3	8	no	—
56	C₃⋊#95	C₂²×C₃	1	4	yes	—
57	C₃⋊#95	C₂²×C₃	1	8	yes	—
58	C₃⋊#95	C₂²×C₃	1	8	yes	—
59	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
60	C₃²×Q₈	—	4	8	no	—
61	C₂×C₃²×C₄	—	2	2	no	—
62	C₂×C₃²×C₄	—	2	4	no	—
63	C₂×C₃²×C₄	—	2	4	no	—
64	C₂×C₃²×C₄	—	2	4	no	—
65	C₂×C₃²×C₄	C₂²×C₄	1	4	yes	—
66	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
67	C₂×C₃²×C₄	C₂²×C₄	1	8	yes	—
68	C₂×C₃×Dic₃	—	2	8	no	—
69	C₂×C₃×Dic₃	—	2	16	no	—
70	C₂×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
71	#391	—	3	4	no	—
72	#387	—	3	8	no	—
73	#383	—	3	2	no	—
74	C₂×C₃×Q₈	—	2	8	no	—
75	C₂×C₃×Q₈	—	4	2	no	—
76	C₂×C₃×Q₈	—	4	2	no	—
77	C₂×C₃×Q₈	—	4	2	no	—
78	C₂×C₃×Q₈	—	4	2	no	—
79	C₂×C₃×Q₈	—	4	4	no	—
80	C₂×C₃×Q₈	—	4	4	no	—
81	C₂×C₃×Q₈	C₂×Dic₃	1	8	yes	—
82	C₂×C₃×Q₈	C₂×C₃×C₄	1	8	yes	—
83	C₂²×C₃×C₄	—	2	2	no	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	2	4	no	—
86	C₂²×C₃×C₄	—	3	8	no	—
87	C₂²×C₃×C₄	C₂²×D₃	1	2	yes	—
88	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
90	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
91	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
93	C₂²×Dic₃	C₂³×C₃	1	8	yes	—
94	C₃×C₄⋊C₄	—	2	16	no	—
95	C₃×C₄⋊C₄	—	12	4	no	—
96	C₃×C₄⋊C₄	C₂×Dic₃	1	16	yes	—
97	C₃×C₄⋊C₄	C₂×C₃×C₄	1	16	yes	—
98	C₃×C₄²	—	2	4	no	—
99	C₃×C₄²	—	12	4	no	—
100	C₃×C₄²	C₂×Dic₃	1	4	yes	—
101	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
102	C₄⋊Dic₃	—	4	4	no	—
103	C₄×Dic₃	—	4	4	no	—
104	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₃²	H(5,𝔽₂)	1	2	yes	—
106	C₂²×C₃²	2₋¹⁺⁴	1	4	yes	—
107	C₃²×C₄	—	2	2	no	—
108	C₃²×C₄	—	2	4	no	—
109	C₃²×C₄	—	2	8	no	—
110	C₃×Dic₃	—	4	8	no	—
111	C₂²×Q₈	C₂×C₃×D₃	1	2	yes	—
112	C₄⋊Q₈	C₃×Dic₃	1	8	yes	complemented
113	C₂×C₄⋊C₄	—	3	4	no	—
114	C₂×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
115	C₂×C₄²	—	3	4	no	—
116	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
117	#95	—	3	8	no	—
118	#95	—	3	4	no	—
119	#95	—	3	8	no	—
120	C₂³×C₃	—	2	1	no	—
121	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
122	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
123	C₃×Q₈	—	4	8	no	—
124	C₃×Q₈	—	4	8	no	—
125	C₃×Q₈	—	4	16	no	—
126	C₂×C₃×C₄	—	2	8	no	—
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	—	2	4	no	—
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	4	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	2	2	no	—
138	C₂×C₃×C₄	—	2	8	no	—
139	C₂×C₃×C₄	—	4	4	no	—
140	C₂×C₃×C₄	—	4	4	no	—
141	C₂×C₃×C₄	—	6	8	no	—
142	C₂×C₃×C₄	—	6	16	no	—
143	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
144	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
145	C₂×C₃×C₄	C₂²×Dic₃	1	8	yes	—
146	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
147	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
148	C₂×C₃×C₄	C₂²×C₃×C₄	1	8	yes	—
149	C₂×Dic₃	—	2	8	no	—
150	C₂×Dic₃	—	2	16	no	—
151	C₂×C₃²	#518	1	1	yes	characteristic
152	C₂×C₃²	#519	1	2	yes	—
153	C₂×C₃²	#563	1	4	yes	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	C₂×C₃×Dic₃	1	8	yes	—
157	C₂²×C₄	—	3	8	no	—
158	C₂²×C₄	C₂²×C₃×D₃	1	2	yes	—
159	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
160	C₂²×C₄	C₂²×C₃×D₃	1	4	yes	—
161	C₄⋊C₄	—	12	4	no	—
162	C₄⋊C₄	C₂×C₃×Dic₃	1	16	yes	—
163	C₄²	—	12	4	no	—
164	C₄²	C₂×C₃×Dic₃	1	4	yes	—
165	C₂²×C₃	—	2	2	no	—
166	C₂²×C₃	—	2	4	no	—
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	#1004	1	2	yes	—
169	C₂²×C₃	#1007	1	4	yes	—
170	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
171	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
172	C₂²×C₃	C₃×H(5,𝔽₂)	1	2	yes	—
173	C₂²×C₃	C₃×2₋¹⁺⁴	1	4	yes	—
174	C₃×C₄	—	2	4	no	—
175	C₃×C₄	—	2	8	no	—
176	C₃×C₄	—	2	2	no	—
177	C₃×C₄	—	2	8	no	—
178	C₃×C₄	—	2	4	no	—
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	2	8	no	—
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	—	4	8	no	—
183	C₃×C₄	—	12	8	no	—
184	Dic₃	—	4	8	no	—
185	C₃²	#2381	1	1	yes	characteristic, complemented, Sylow
186	C₂³	C₂³×C₃×D₃	1	1	yes	characteristic
187	Q₈	—	4	8	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	6	8	no	—
193	C₂×C₄	—	6	16	no	—
194	C₂×C₄	C₂²×C₃×Dic₃	1	4	yes	—
195	C₂×C₄	C₂²×C₃×Dic₃	1	2	yes	—
196	C₂×C₄	C₂²×C₃×Dic₃	1	8	yes	—
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	—	2	2	no	—
199	C₂×C₃	—	2	4	no	—
200	C₂×C₃	#5735	1	4	yes	—
201	C₂×C₃	#5798	1	1	yes	characteristic
202	C₂×C₃	#5812	1	2	yes	—
203	C₂×C₃	C₃×#518	1	1	yes	characteristic
204	C₂×C₃	C₃×#519	1	2	yes	—
205	C₂×C₃	C₃×#563	1	4	yes	—
206	C₂²	C₃×#1004	1	2	yes	—
207	C₂²	C₃×#1007	1	4	yes	—
208	C₂²	C₂³×C₃×Dic₃	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	2	8	no	—
211	C₄	—	2	4	no	—
212	C₄	—	12	8	no	—
213	C₃	—	2	1	no	—
214	C₃	#83231	1	1	yes	characteristic, direct product factor
215	C₃	C₃×#2381	1	1	yes	characteristic, complemented
216	C₂	C₃×#5735	1	4	yes	—
217	C₂	C₃×#5798	1	1	yes	characteristic
218	C₂	C₃×#5812	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₃×C₄
Frattini	C₂³	C₂³×C₃×D₃
Fitting	C₂×C₃²×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃²	C₂⁴
cosocle	C₂³×C₃	C₂⁴×C₃
hypercenter	C₂×C₃×C₄⋊Q₈	D₃
hypocenter	C₃	C₃×#2381
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. #83231	C₃	direct (N × H)
8. C₃²×C₄⋊Q₈	C₄	semidirect
41. C₃×C₄⋊Q₈	Dic₃	semidirect
42. C₃×C₄⋊Q₈	C₃×C₄	semidirect
112. C₄⋊Q₈	C₃×Dic₃	semidirect
185. C₃²	#2381	semidirect
214. C₃	#83231	direct (N × H)
215. C₃	C₃×#2381	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄⋊Q₈	2
C₃×#5229	2
C₃×#4980	2
C₃×#4937	2
C₃×#2381	3
#83231	3

Subgroups — C3×#83231 #11807953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#83231 #11807953