Subgroups — C₃²×#1279 #11768328

Order 1152.

624 conjugacy classes of subgroups.

The subgroup lattice has 213 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×#1279	C₃	1	4	yes	maximal, direct product factor
5	C₂²×C₃²×C₈	—	2	1	no	—
6	C₂²×C₃²×C₈	—	2	1	no	—
7	C₂×C₃²×C₄⋊C₄	C₄	1	2	yes	—
8	C₂×C₃²×C₄²	—	2	1	no	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₃²×C₄⋊C₈	—	2	1	no	—
11	C₃²×C₄⋊C₈	C₄	1	2	yes	complemented
12	C₃²×C₄×C₈	—	2	1	no	—
13	C₃²×C₄×C₈	C₄	1	2	yes	complemented
14	C₃²×#95	—	2	1	no	—
15	C₂×C₃×C₄⋊C₈	C₂×C₃	1	4	yes	—
16	C₂×C₃×C₄×C₈	C₂×C₃	1	4	yes	—
17	C₃×C₄×C₄⋊C₄	C₂×C₃	1	4	yes	—
18	C₂²×C₃²×C₄	—	2	2	no	—
19	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
21	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
22	C₂×C₃²×C₈	—	2	4	no	—
23	C₂×C₃²×C₈	—	2	2	no	—
24	C₂×C₃²×C₈	—	2	2	no	—
25	C₂×C₃²×C₈	—	4	1	no	—
26	C₂×C₃²×C₈	—	4	1	no	—
27	C₃²×C₄⋊C₄	—	2	2	no	—
28	C₃²×C₄⋊C₄	C₈	1	4	yes	complemented
29	C₃²×C₄²	—	4	1	no	—
30	C₃²×C₄²	—	4	1	no	—
31	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
33	C₃²×C₄²	D₄	1	1	yes	characteristic
34	C₃²×C₄²	Q₈	1	1	yes	characteristic
35	#1279	C₃²	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²×C₃×C₈	—	2	4	no	—
37	C₂²×C₃×C₈	—	2	4	no	—
38	C₂×C₃×C₄⋊C₄	C₃×C₄	1	8	yes	—
39	C₂×C₃×C₄²	—	2	4	no	—
40	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
41	C₃×C₄⋊C₈	—	2	4	no	—
42	C₃×C₄⋊C₈	C₃×C₄	1	8	yes	complemented
43	C₃×C₄×C₈	—	2	4	no	—
44	C₃×C₄×C₈	C₃×C₄	1	8	yes	complemented
45	C₃×#95	—	2	4	no	—
46	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	—	2	2	no	—
48	C₂×C₃²×C₄	—	2	4	no	—
49	C₂×C₃²×C₄	—	4	2	no	—
50	C₂×C₃²×C₄	—	4	2	no	—
51	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
58	C₂×C₃²×C₄	M₁₆	1	2	yes	—
59	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
60	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
61	C₂×C₃²×C₄	SD₁₆	1	1	yes	characteristic
62	C₂×C₃²×C₄	Q₁₆	1	1	yes	characteristic
63	C₃²×C₈	—	2	4	no	—
64	C₃²×C₈	—	4	2	no	—
65	C₂×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
67	C₄×C₄⋊C₄	C₂×C₃²	1	1	yes	characteristic
68	C₂²×C₃×C₄	—	2	8	no	—
69	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
70	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
71	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
72	C₂×C₃×C₈	—	2	16	no	—
73	C₂×C₃×C₈	—	2	8	no	—
74	C₂×C₃×C₈	—	2	8	no	—
75	C₂×C₃×C₈	—	4	4	no	—
76	C₂×C₃×C₈	—	4	4	no	—
77	C₃×C₄⋊C₄	—	2	8	no	—
78	C₃×C₄⋊C₄	C₃×C₈	1	16	yes	complemented
79	C₃×C₄²	—	4	4	no	—
80	C₃×C₄²	—	4	4	no	—
81	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
82	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
83	C₃×C₄²	C₃×D₄	1	4	yes	—
84	C₃×C₄²	C₃×Q₈	1	4	yes	—
85	C₂²×C₃²	#95	1	1	yes	characteristic
86	C₂²×C₃²	C₂²⋊C₈	1	2	yes	—
87	C₂²×C₃²	#102	1	1	yes	characteristic
88	C₂²×C₃²	#103	1	1	yes	characteristic
89	C₂²×C₃²	#104	1	2	yes	—
90	C₃²×C₄	—	2	1	no	—
91	C₃²×C₄	—	2	1	no	—
92	C₃²×C₄	—	4	4	no	—
93	C₃²×C₄	C₄×C₈	1	1	yes	characteristic
94	C₃²×C₄	#97	1	1	yes	characteristic
95	C₃²×C₄	#102	1	1	yes	characteristic
96	C₃²×C₄	#103	1	1	yes	characteristic
97	C₃²×C₄	C₄⋊C₈	1	2	yes	—
98	C₃²×C₄	#106	1	1	yes	characteristic
99	C₃²×C₄	#107	1	1	yes	characteristic
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	—	2	1	no	—
102	C₂×C₄⋊C₄	C₃²×C₄	1	2	yes	—
103	C₂×C₄²	—	2	1	no	—
104	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
105	C₄⋊C₈	—	2	1	no	—
106	C₄⋊C₈	C₃²×C₄	1	2	yes	complemented
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	C₃²×C₄	1	2	yes	complemented
109	#95	—	2	1	no	—
110	C₂³×C₃	C₃×C₂²⋊C₄	1	4	yes	—
111	C₂×C₃×C₄	—	2	8	no	—
112	C₂×C₃×C₄	—	2	16	no	—
113	C₂×C₃×C₄	—	4	8	no	—
114	C₂×C₃×C₄	—	4	8	no	—
115	C₂×C₃×C₄	C₃×C₄²	1	4	yes	—
116	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
117	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
118	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
119	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
120	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
121	C₂×C₃×C₄	C₂×C₃×C₈	1	8	yes	—
122	C₂×C₃×C₄	C₃×M₁₆	1	8	yes	—
123	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
124	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
125	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
126	C₂×C₃×C₄	C₃×Q₁₆	1	4	yes	—
127	C₃×C₈	—	2	16	no	—
128	C₃×C₈	—	4	8	no	—
129	C₂×C₃²	D₄⋊C₈	1	2	yes	—
130	C₂×C₃²	Q₈⋊C₈	1	2	yes	—
131	C₂×C₃²	#336	1	1	yes	characteristic
132	C₂×C₃²	#339	1	1	yes	characteristic
133	C₂×C₃²	#340	1	1	yes	characteristic
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
136	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
137	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	4	1	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	C₃²×C₈	1	4	yes	complemented
145	C₄²	—	4	1	no	—
146	C₄²	—	4	1	no	—
147	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
148	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
149	C₄²	C₃²×D₄	1	1	yes	characteristic
150	C₄²	C₃²×Q₈	1	1	yes	characteristic
151	C₂²×C₃	C₃×#95	1	4	yes	—
152	C₂²×C₃	C₃×C₂²⋊C₈	1	8	yes	—
153	C₂²×C₃	C₃×#102	1	4	yes	—
154	C₂²×C₃	C₃×#103	1	4	yes	—
155	C₂²×C₃	C₃×#104	1	8	yes	—
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	—	2	4	no	—
158	C₃×C₄	—	4	16	no	—
159	C₃×C₄	C₃×C₄×C₈	1	4	yes	—
160	C₃×C₄	C₃×#97	1	4	yes	—
161	C₃×C₄	C₃×#102	1	4	yes	—
162	C₃×C₄	C₃×#103	1	4	yes	—
163	C₃×C₄	C₃×C₄⋊C₈	1	8	yes	—
164	C₃×C₄	C₃×#106	1	4	yes	—
165	C₃×C₄	C₃×#107	1	4	yes	—
166	C₃²	#1279	1	1	yes	characteristic, direct product factor, Sylow
167	C₂³	C₃²×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	C₃²×C₄²	1	1	yes	characteristic
173	C₂×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄	C₂×C₃²×C₈	1	2	yes	—
179	C₂×C₄	C₃²×M₁₆	1	2	yes	—
180	C₂×C₄	C₃²×D₈	1	1	yes	characteristic
181	C₂×C₄	C₃²×SD₁₆	1	1	yes	characteristic
182	C₂×C₄	C₃²×SD₁₆	1	1	yes	characteristic
183	C₂×C₄	C₃²×Q₁₆	1	1	yes	characteristic
184	C₈	—	2	4	no	—
185	C₈	—	4	2	no	—
186	C₂×C₃	C₃×D₄⋊C₈	1	8	yes	—
187	C₂×C₃	C₃×Q₈⋊C₈	1	8	yes	—
188	C₂×C₃	C₃×#336	1	4	yes	—
189	C₂×C₃	C₃×#339	1	4	yes	—
190	C₂×C₃	C₃×#340	1	4	yes	—
191	C₂²	C₃²×#95	1	1	yes	characteristic
192	C₂²	C₃²×C₂²⋊C₈	1	2	yes	—
193	C₂²	C₃²×#102	1	1	yes	characteristic
194	C₂²	C₃²×#103	1	1	yes	characteristic
195	C₂²	C₃²×#104	1	2	yes	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	4	no	—
199	C₄	C₃²×C₄×C₈	1	1	yes	characteristic
200	C₄	C₃²×#97	1	1	yes	characteristic
201	C₄	C₃²×#102	1	1	yes	characteristic
202	C₄	C₃²×#103	1	1	yes	characteristic
203	C₄	C₃²×C₄⋊C₈	1	2	yes	—
204	C₄	C₃²×#106	1	1	yes	characteristic
205	C₄	C₃²×#107	1	1	yes	characteristic
206	C₃	C₃×#1279	1	4	yes	direct product factor
207	C₂	C₃²×D₄⋊C₈	1	2	yes	—
208	C₂	C₃²×Q₈⋊C₈	1	2	yes	—
209	C₂	C₃²×#336	1	1	yes	characteristic
210	C₂	C₃²×#339	1	1	yes	characteristic
211	C₂	C₃²×#340	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²×C₄	D₄
commutator	C₄	C₃²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂²×C₃²
socle, prime radical	C₂³×C₃²	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃²×C₄ ◁ C₂×C₃²×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×#1279	C₃	direct (N × H)
11. C₃²×C₄⋊C₈	C₄	semidirect
13. C₃²×C₄×C₈	C₄	semidirect
28. C₃²×C₄⋊C₄	C₈	semidirect
35. #1279	C₃²	direct (N × H)
42. C₃×C₄⋊C₈	C₃×C₄	semidirect
44. C₃×C₄×C₈	C₃×C₄	semidirect
78. C₃×C₄⋊C₄	C₃×C₈	semidirect
106. C₄⋊C₈	C₃²×C₄	semidirect
108. C₄×C₈	C₃²×C₄	semidirect
144. C₄⋊C₄	C₃²×C₈	semidirect
166. C₃²	#1279	direct (N × H)
206. C₃	C₃×#1279	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄⋊C₈	2
C₂×C₃²×C₄×C₈	2
C₃²×C₄×C₄⋊C₄	2
C₃×#1279	3

Subgroups — C32×#1279 #11768328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1279 #11768328