Subgroups — C₃²×#1278 #11768327

Order 1152.

516 conjugacy classes of subgroups.

The subgroup lattice has 171 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#431	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×#1278	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×M₁₆	C₄	1	2	yes	—
6	C₂²×C₃²×C₈	—	2	1	no	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₃²×C₄⋊C₈	C₄	1	2	yes	—
9	C₃²×C₂²⋊C₈	—	2	1	no	—
10	C₃²×#97	—	2	1	no	—
11	C₃²×C₄×C₈	—	2	1	no	—
12	C₃²×C₄×C₈	—	2	1	no	—
13	C₃²×C₄×C₈	—	2	1	no	—
14	C₃²×C₄×C₈	C₄	1	2	yes	—
15	C₃×#431	C₂×C₃	1	4	yes	—
16	C₃×C₄×M₁₆	C₂×C₃	1	4	yes	—
17	C₂×C₃×C₄×C₈	C₂×C₃	1	4	yes	—
18	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
21	C₃²×M₁₆	—	2	2	no	—
22	C₃²×M₁₆	C₈	1	4	yes	complemented
23	C₂×C₃²×C₈	—	2	4	no	—
24	C₂×C₃²×C₈	—	2	2	no	—
25	C₂×C₃²×C₈	—	2	2	no	—
26	C₂×C₃²×C₈	—	2	2	no	—
27	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃²×C₄²	D₄	1	1	yes	characteristic
30	C₃²×C₄²	D₄	1	1	yes	characteristic
31	#1278	C₃²	1	1	yes	characteristic, direct product factor, Sylow
32	C₂×C₃×M₁₆	C₃×C₄	1	8	yes	—
33	C₂²×C₃×C₈	—	2	4	no	—
34	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
35	C₃×C₄⋊C₈	C₃×C₄	1	8	yes	—
36	C₃×C₂²⋊C₈	—	2	4	no	—
37	C₃×#97	—	2	4	no	—
38	C₃×C₄×C₈	—	2	4	no	—
39	C₃×C₄×C₈	—	2	4	no	—
40	C₃×C₄×C₈	—	2	4	no	—
41	C₃×C₄×C₈	C₃×C₄	1	8	yes	—
42	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₃²×C₄	—	2	2	no	—
44	C₂×C₃²×C₄	—	2	2	no	—
45	C₂×C₃²×C₄	C₄²	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
52	C₂×C₃²×C₄	M₁₆	1	2	yes	—
53	C₃²×C₈	—	2	4	no	—
54	C₃²×C₈	—	2	4	no	—
55	C₃²×C₈	—	4	2	no	—
56	#431	C₂×C₃²	1	1	yes	characteristic
57	C₄×M₁₆	C₂×C₃²	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
60	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
61	C₂²×C₃×C₄	C₃×Q₈	1	4	yes	—
62	C₃×M₁₆	—	2	8	no	—
63	C₃×M₁₆	C₃×C₈	1	16	yes	complemented
64	C₂×C₃×C₈	—	2	16	no	—
65	C₂×C₃×C₈	—	2	8	no	—
66	C₂×C₃×C₈	—	2	8	no	—
67	C₂×C₃×C₈	—	2	8	no	—
68	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
69	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
70	C₃×C₄²	C₃×D₄	1	4	yes	—
71	C₃×C₄²	C₃×D₄	1	4	yes	—
72	C₂²×C₃²	—	2	1	no	—
73	C₂²×C₃²	—	2	1	no	—
74	C₂²×C₃²	#95	1	1	yes	characteristic
75	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
76	C₃²×C₄	—	2	1	no	—
77	C₃²×C₄	—	2	1	no	—
78	C₃²×C₄	C₄×C₈	1	1	yes	characteristic
79	C₃²×C₄	#97	1	1	yes	characteristic
80	C₃²×C₄	C₂²⋊C₈	1	2	yes	—
81	C₃²×C₄	#104	1	2	yes	—
82	C₃²×C₄	#108	1	2	yes	—
83	C₂×M₁₆	C₃²×C₄	1	2	yes	—
84	C₂²×C₈	—	2	1	no	—
85	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
86	C₄⋊C₈	C₃²×C₄	1	2	yes	—
87	C₂²⋊C₈	—	2	1	no	—
88	#97	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	C₃²×C₄	1	2	yes	—
93	C₂³×C₃	C₃×C₄⋊C₄	1	4	yes	—
94	C₂×C₃×C₄	—	2	8	no	—
95	C₂×C₃×C₄	—	2	8	no	—
96	C₂×C₃×C₄	C₃×C₄²	1	4	yes	—
97	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
98	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
99	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
100	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
101	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
102	C₂×C₃×C₄	C₂×C₃×C₈	1	8	yes	—
103	C₂×C₃×C₄	C₃×M₁₆	1	8	yes	—
104	C₃×C₈	—	2	16	no	—
105	C₃×C₈	—	2	16	no	—
106	C₃×C₈	—	4	8	no	—
107	C₂×C₃²	—	2	2	no	—
108	C₂×C₃²	#336	1	1	yes	characteristic
109	C₂×C₃²	#339	1	1	yes	characteristic
110	C₂×C₃²	#341	1	1	yes	characteristic
111	C₂²×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
112	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
113	C₂²×C₄	C₃²×Q₈	1	1	yes	characteristic
114	M₁₆	—	2	2	no	—
115	M₁₆	C₃²×C₈	1	4	yes	complemented
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
121	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
122	C₄²	C₃²×D₄	1	1	yes	characteristic
123	C₄²	C₃²×D₄	1	1	yes	characteristic
124	C₂²×C₃	—	2	4	no	—
125	C₂²×C₃	—	2	4	no	—
126	C₂²×C₃	C₃×#95	1	4	yes	—
127	C₂²×C₃	C₃×C₄⋊C₈	1	8	yes	—
128	C₃×C₄	—	2	4	no	—
129	C₃×C₄	—	2	4	no	—
130	C₃×C₄	C₃×C₄×C₈	1	4	yes	—
131	C₃×C₄	C₃×#97	1	4	yes	—
132	C₃×C₄	C₃×C₂²⋊C₈	1	8	yes	—
133	C₃×C₄	C₃×#104	1	8	yes	—
134	C₃×C₄	C₃×#108	1	8	yes	—
135	C₃²	#1278	1	1	yes	characteristic, direct product factor, Sylow
136	C₂³	C₃²×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₃²×C₄²	1	1	yes	characteristic
140	C₂×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄	C₂×C₃²×C₈	1	2	yes	—
146	C₂×C₄	C₃²×M₁₆	1	2	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	2	4	no	—
149	C₈	—	4	2	no	—
150	C₂×C₃	—	2	8	no	—
151	C₂×C₃	C₃×#336	1	4	yes	—
152	C₂×C₃	C₃×#339	1	4	yes	—
153	C₂×C₃	C₃×#341	1	4	yes	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	C₃²×#95	1	1	yes	characteristic
157	C₂²	C₃²×C₄⋊C₈	1	2	yes	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	C₃²×C₄×C₈	1	1	yes	characteristic
161	C₄	C₃²×#97	1	1	yes	characteristic
162	C₄	C₃²×C₂²⋊C₈	1	2	yes	—
163	C₄	C₃²×#104	1	2	yes	—
164	C₄	C₃²×#108	1	2	yes	—
165	C₃	C₃×#1278	1	4	yes	direct product factor
166	C₂	—	2	2	no	—
167	C₂	C₃²×#336	1	1	yes	characteristic
168	C₂	C₃²×#339	1	1	yes	characteristic
169	C₂	C₃²×#341	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄²	D₄
commutator	C₄	C₃²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂²×C₃²
socle	C₂²×C₃²	#95
prime radical	C₂³×C₃²	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄² ◁ C₂×C₃²×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×#1278	C₃	direct (N × H)
22. C₃²×M₁₆	C₈	semidirect
31. #1278	C₃²	direct (N × H)
63. C₃×M₁₆	C₃×C₈	semidirect
115. M₁₆	C₃²×C₈	semidirect
135. C₃²	#1278	direct (N × H)
165. C₃	C₃×#1278	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#431	2
C₃²×C₄×M₁₆	2
C₂×C₃²×C₄×C₈	2
C₃×#1278	3

Subgroups — C32×#1278 #11768327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1278 #11768327