Subgroups — C₃×#70949 #11798023

Order 1152.

303 conjugacy classes of subgroups.

The subgroup lattice has 219 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊(C₃⋊C₁₆)	C₂	1	2	yes	maximal
3	C₃×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	C₃×#4461	C₂	1	2	yes	maximal
5	C₃×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₃×#2183	—	3	1	no	maximal
7	#70949	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃²×C₄×Q₈	C₄	1	1	yes	characteristic
9	C₃²×C₄⋊C₈	C₄	1	2	yes	—
10	C₃²×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	2	yes	—
13	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
15	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₃×C₈×Q₈	—	2	1	no	—
17	C₃×C₈×Q₈	D₃	1	1	yes	characteristic
18	C₃×C₈×Q₈	C₂×C₃	1	1	yes	characteristic
19	C₃×C₄⋊C₁₆	—	3	1	no	—
20	C₃×C₄⋊C₁₆	—	3	2	no	—
21	C₃×#346	—	3	2	no	—
22	C₃×C₄×C₁₆	—	3	1	no	—
23	C₄⋊(C₃⋊C₁₆)	C₂×C₃	1	2	yes	—
24	C₄⋊(C₃⋊C₁₆)	C₂×C₃	1	1	yes	characteristic
25	#4461	C₂×C₃	1	2	yes	—
26	C₄×C₃⋊C₁₆	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃²×Q₈	C₈	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
31	C₃²×C₄⋊C₄	C₈	1	2	yes	—
32	C₃²×C₄⋊C₄	C₈	1	1	yes	characteristic
33	C₃²×C₄²	C₂×C₄	1	2	yes	—
34	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₃×C₃⋊C₁₆	—	2	1	no	—
36	C₃×C₃⋊C₁₆	—	2	2	no	—
37	C₃×C₃⋊C₁₆	Q₈	1	2	yes	complemented
38	#2183	—	3	1	no	Sylow
39	C₃×C₄×Q₈	—	2	1	no	—
40	C₃×C₄×Q₈	Dic₃	1	1	yes	characteristic
41	C₃×C₄×Q₈	C₃×C₄	1	1	yes	characteristic
42	C₂×C₃×C₁₆	—	3	1	no	—
43	C₂×C₃×C₁₆	—	3	1	no	—
44	C₂×C₃×C₁₆	—	3	2	no	—
45	C₃×C₄⋊C₈	—	2	2	no	—
46	C₃×C₄⋊C₈	—	2	1	no	—
47	C₃×C₄⋊C₈	Dic₃	1	2	yes	—
48	C₃×C₄⋊C₈	Dic₃	1	1	yes	characteristic
49	C₃×C₄⋊C₈	C₃×C₄	1	2	yes	—
50	C₃×C₄⋊C₈	C₃×C₄	1	1	yes	characteristic
51	C₃×C₄×C₈	—	2	1	no	—
52	C₃×C₄×C₈	—	2	2	no	—
53	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
55	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
57	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃⋊C₁₆	C₂²×C₃	1	2	yes	—
59	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
60	C₃²×Q₈	—	2	2	no	—
61	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
62	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
64	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₃²×C₈	—	2	1	no	—
67	C₃²×C₈	—	2	2	no	—
68	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₃²×C₈	#41	1	1	yes	characteristic
70	C₈×Q₈	C₃×D₃	1	1	yes	characteristic
71	C₄⋊C₁₆	—	3	1	no	—
72	C₄⋊C₁₆	—	3	2	no	—
73	#346	—	3	2	no	—
74	C₄×C₁₆	—	3	1	no	—
75	C₂×C₃×Q₈	—	2	1	no	—
76	C₂×C₃×Q₈	C₃⋊C₈	1	1	yes	characteristic
77	C₂×C₃×Q₈	C₃×C₈	1	1	yes	characteristic
78	C₂×C₃×C₈	—	2	1	no	—
79	C₂×C₃×C₈	—	2	2	no	—
80	C₂×C₃×C₈	—	2	1	no	—
81	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
82	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
83	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
85	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
86	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
87	C₃×C₄⋊C₄	—	2	2	no	—
88	C₃×C₄⋊C₄	—	2	1	no	—
89	C₃×C₄⋊C₄	C₃⋊C₈	1	2	yes	—
90	C₃×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
91	C₃×C₄⋊C₄	C₃×C₈	1	2	yes	—
92	C₃×C₄⋊C₄	C₃×C₈	1	1	yes	characteristic
93	C₃×C₄²	—	2	2	no	—
94	C₃×C₄²	—	2	1	no	—
95	C₃×C₄²	C₂×Dic₃	1	2	yes	—
96	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
97	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
98	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
99	C₃×C₁₆	—	3	2	no	—
100	C₃×C₁₆	—	6	1	no	—
101	C₃×C₁₆	—	6	2	no	—
102	C₃⋊C₁₆	—	2	1	no	—
103	C₃⋊C₁₆	—	2	2	no	—
104	C₃⋊C₁₆	C₃×Q₈	1	2	yes	complemented
105	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
106	C₃²×C₄	—	2	2	no	—
107	C₃²×C₄	—	2	2	no	—
108	C₃²×C₄	—	2	1	no	—
109	C₃²×C₄	M₃₂	1	2	yes	—
110	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
111	C₃²×C₄	#131	1	1	yes	characteristic
112	C₄×Q₈	C₃×Dic₃	1	1	yes	characteristic
113	C₂×C₁₆	—	3	1	no	—
114	C₂×C₁₆	—	3	1	no	—
115	C₂×C₁₆	—	3	2	no	—
116	C₄⋊C₈	C₃×Dic₃	1	2	yes	—
117	C₄⋊C₈	C₃×Dic₃	1	1	yes	characteristic
118	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
119	C₄×C₈	C₂×C₃×D₃	1	2	yes	—
120	C₃×Q₈	—	2	4	no	—
121	C₃×Q₈	—	2	2	no	—
122	C₃×Q₈	—	2	2	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
129	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
131	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
133	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
135	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
138	C₃×C₈	—	2	1	no	—
139	C₃×C₈	—	2	2	no	—
140	C₃×C₈	—	2	1	no	—
141	C₃×C₈	—	2	2	no	—
142	C₃×C₈	—	2	1	no	—
143	C₃×C₈	—	2	1	no	—
144	C₃×C₈	—	4	1	no	—
145	C₃×C₈	—	4	2	no	—
146	C₃×C₈	D₃×Q₈	1	1	yes	characteristic
147	C₃×C₈	#239	1	1	yes	characteristic
148	C₃×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
149	C₃×C₈	C₃×#41	1	1	yes	characteristic
150	C₂×C₃²	C₈×Q₈	1	1	yes	characteristic
151	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
152	C₂×C₃²	#504	1	1	yes	characteristic
153	C₂×Q₈	C₃×C₃⋊C₈	1	1	yes	characteristic
154	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
155	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
156	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
157	C₄⋊C₄	C₃×C₃⋊C₈	1	2	yes	—
158	C₄⋊C₄	C₃×C₃⋊C₈	1	1	yes	characteristic
159	C₄²	C₂×C₃×Dic₃	1	2	yes	—
160	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
161	C₁₆	—	3	2	no	—
162	C₁₆	—	6	1	no	—
163	C₁₆	—	6	2	no	—
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
166	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
167	C₃×C₄	—	2	4	no	—
168	C₃×C₄	—	2	2	no	—
169	C₃×C₄	—	2	2	no	—
170	C₃×C₄	—	2	1	no	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	2	1	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	2	2	no	—
177	C₃×C₄	—	4	2	no	—
178	C₃×C₄	—	4	1	no	—
179	C₃×C₄	#812	1	2	yes	—
180	C₃×C₄	C₃×M₃₂	1	2	yes	—
181	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
182	C₃×C₄	#948	1	1	yes	characteristic
183	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
184	C₃×C₄	C₃×#131	1	1	yes	characteristic
185	C₃²	#2183	1	1	yes	characteristic, complemented, Sylow
186	Q₈	—	2	2	no	—
187	C₂×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
188	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
189	C₂×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
190	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	2	no	—
194	C₈	C₃×D₃×Q₈	1	1	yes	characteristic
195	C₈	C₃×#239	1	1	yes	characteristic
196	C₂×C₃	—	2	1	no	—
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	—	2	1	no	—
199	C₂×C₃	Q₈×C₃⋊C₈	1	1	yes	characteristic
200	C₂×C₃	C₂×#812	1	1	yes	characteristic
201	C₂×C₃	#5140	1	1	yes	characteristic
202	C₂×C₃	C₃×C₈×Q₈	1	1	yes	characteristic
203	C₂×C₃	C₂×C₃×M₃₂	1	1	yes	characteristic
204	C₂×C₃	C₃×#504	1	1	yes	characteristic
205	C₂²	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	1	no	—
209	C₄	C₃×#812	1	2	yes	—
210	C₄	C₃×Q₈×Dic₃	1	1	yes	characteristic
211	C₄	C₃×#948	1	1	yes	characteristic
212	C₃	—	2	1	no	—
213	C₃	#70949	1	1	yes	characteristic, direct product factor
214	C₃	C₃×#2183	1	1	yes	characteristic, complemented
215	C₂	C₃×Q₈×C₃⋊C₈	1	1	yes	characteristic
216	C₂	C₂×C₃×#812	1	1	yes	characteristic
217	C₂	C₃×#5140	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×C₈×Q₈	C₂
socle, prime radical	C₂²×C₃²	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×C₈×Q₈	D₃
hypocenter	C₃	C₃×#2183
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₈×Q₈

Complements

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

N	H ≅ G/N	type
7. #70949	C₃	direct (N × H)
37. C₃×C₃⋊C₁₆	Q₈	semidirect
104. C₃⋊C₁₆	C₃×Q₈	semidirect
185. C₃²	#2183	semidirect
213. C₃	#70949	direct (N × H)
214. C₃	C₃×#2183	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₈×Q₈	2
C₃×C₄⋊(C₃⋊C₁₆)	2
C₃×#4461	2
C₃×C₄×C₃⋊C₁₆	2
C₃×#2183	3
#70949	3

Subgroups — C3×#70949 #11798023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70949 #11798023