Subgroups — C₃²×#1564 #11768613

Order 1152.

540 conjugacy classes of subgroups.

The subgroup lattice has 183 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	2	yes	maximal
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃²×#335	C₂	1	2	yes	maximal, complemented
4	C₃²×#334	C₂	1	2	yes	maximal, complemented
5	C₃×#1564	C₃	1	4	yes	maximal, direct product factor
6	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₃²×C₄⋊C₈	C₂²	1	4	yes	—
10	C₃²×C₂²⋊C₈	—	2	2	no	—
11	C₃²×C₄×C₈	—	2	2	no	—
12	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×#431	C₂×C₃	1	8	yes	—
17	C₂×C₃×C₄×C₈	C₂×C₃	1	4	yes	—
18	C₃×#335	C₂×C₃	1	8	yes	complemented
19	C₃×#334	C₂×C₃	1	8	yes	complemented
20	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
22	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
23	C₂×C₃²×C₈	—	2	2	no	—
24	C₂×C₃²×C₈	—	2	4	no	—
25	C₂×C₃²×C₈	C₈	1	4	yes	complemented
26	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃²×C₄²	D₄	1	1	yes	characteristic
32	C₃²×C₄²	Q₈	1	1	yes	characteristic
33	C₃²×C₄²	C₂³	1	1	yes	characteristic
34	#1564	C₃²	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²×C₃×C₈	C₃×C₄	1	4	yes	—
36	C₂²×C₃×C₈	C₃×C₄	1	4	yes	—
37	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
38	C₃×C₄⋊C₈	C₂²×C₃	1	16	yes	—
39	C₃×C₂²⋊C₈	—	2	8	no	—
40	C₃×C₄×C₈	—	2	8	no	—
41	C₃×C₄×C₈	C₃×C₄	1	4	yes	—
42	C₃×C₄×C₈	C₃×C₄	1	4	yes	—
43	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
44	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
45	C₂³×C₃²	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	—	2	2	no	—
47	C₂×C₃²×C₄	—	2	2	no	—
48	C₂×C₃²×C₄	C₄⋊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	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
56	C₃²×C₈	—	2	1	no	—
57	C₃²×C₈	—	2	1	no	—
58	C₃²×C₈	—	4	4	no	—
59	C₃²×C₈	C₂×C₈	1	2	yes	—
60	C₃²×C₈	C₂×C₈	1	2	yes	—
61	#431	C₂×C₃²	1	2	yes	—
62	C₂×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
63	#335	C₂×C₃²	1	2	yes	complemented
64	#334	C₂×C₃²	1	2	yes	complemented
65	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
66	C₂²×C₃×C₄	C₃×D₄	1	4	yes	—
67	C₂²×C₃×C₄	C₃×Q₈	1	4	yes	—
68	C₂×C₃×C₈	—	2	8	no	—
69	C₂×C₃×C₈	—	2	16	no	—
70	C₂×C₃×C₈	C₃×C₈	1	16	yes	complemented
71	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
72	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
73	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
74	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
75	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
76	C₃×C₄²	C₃×D₄	1	4	yes	—
77	C₃×C₄²	C₃×Q₈	1	4	yes	—
78	C₃×C₄²	C₂³×C₃	1	4	yes	—
79	C₂²×C₃²	—	2	1	no	—
80	C₂²×C₃²	—	2	1	no	—
81	C₂²×C₃²	C₄⋊C₈	1	2	yes	—
82	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₃²×C₄	—	2	1	no	—
84	C₃²×C₄	—	2	1	no	—
85	C₃²×C₄	C₄⋊C₈	1	2	yes	—
86	C₃²×C₄	#108	1	2	yes	—
87	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
88	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
89	C₃²×C₄	#135	1	2	yes	—
90	C₂²×C₈	C₃²×C₄	1	1	yes	characteristic
91	C₂²×C₈	C₃²×C₄	1	1	yes	characteristic
92	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
93	C₄⋊C₈	C₂²×C₃²	1	4	yes	—
94	C₂²⋊C₈	—	2	2	no	—
95	C₄×C₈	—	2	2	no	—
96	C₄×C₈	C₃²×C₄	1	1	yes	characteristic
97	C₄×C₈	C₃²×C₄	1	1	yes	characteristic
98	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
99	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
100	C₂³×C₃	C₃×C₄⋊C₄	1	4	yes	—
101	C₂×C₃×C₄	—	2	8	no	—
102	C₂×C₃×C₄	—	2	8	no	—
103	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
104	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
105	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
106	C₂×C₃×C₄	C₂×C₃×C₈	1	8	yes	—
107	C₂×C₃×C₄	C₃×M₁₆	1	8	yes	—
108	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
109	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
110	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
111	C₃×C₈	—	2	4	no	—
112	C₃×C₈	—	2	4	no	—
113	C₃×C₈	—	4	16	no	—
114	C₃×C₈	C₂×C₃×C₈	1	8	yes	—
115	C₃×C₈	C₂×C₃×C₈	1	8	yes	—
116	C₂×C₃²	—	2	2	no	—
117	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂×C₃²	#427	1	1	yes	characteristic
119	C₂×C₃²	C₂×#108	1	1	yes	characteristic
120	C₂²×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
121	C₂²×C₄	C₃²×D₄	1	1	yes	characteristic
122	C₂²×C₄	C₃²×Q₈	1	1	yes	characteristic
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	C₃²×C₈	1	4	yes	complemented
126	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
127	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
128	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
129	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
130	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
131	C₄²	C₃²×D₄	1	1	yes	characteristic
132	C₄²	C₃²×Q₈	1	1	yes	characteristic
133	C₄²	C₂³×C₃²	1	1	yes	characteristic
134	C₂²×C₃	—	2	4	no	—
135	C₂²×C₃	—	2	4	no	—
136	C₂²×C₃	C₃×C₄⋊C₈	1	8	yes	—
137	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	4	yes	—
138	C₃×C₄	—	2	4	no	—
139	C₃×C₄	—	2	4	no	—
140	C₃×C₄	C₃×C₄⋊C₈	1	8	yes	—
141	C₃×C₄	C₃×#108	1	8	yes	—
142	C₃×C₄	C₂²×C₃×C₈	1	4	yes	—
143	C₃×C₄	C₂×C₃×M₁₆	1	4	yes	—
144	C₃×C₄	C₃×#135	1	8	yes	—
145	C₃²	#1564	1	1	yes	characteristic, direct product factor, Sylow
146	C₂³	C₃²×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₂×C₃²×C₈	1	2	yes	—
153	C₂×C₄	C₃²×M₁₆	1	2	yes	—
154	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
155	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
156	C₂×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	4	4	no	—
160	C₈	C₂×C₃²×C₈	1	2	yes	—
161	C₈	C₂×C₃²×C₈	1	2	yes	—
162	C₂×C₃	—	2	8	no	—
163	C₂×C₃	C₂×C₃×C₄⋊C₈	1	4	yes	—
164	C₂×C₃	C₃×#427	1	4	yes	—
165	C₂×C₃	C₂×C₃×#108	1	4	yes	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	C₃²×C₄⋊C₈	1	2	yes	—
169	C₂²	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	C₃²×C₄⋊C₈	1	2	yes	—
173	C₄	C₃²×#108	1	2	yes	—
174	C₄	C₂²×C₃²×C₈	1	1	yes	characteristic
175	C₄	C₂×C₃²×M₁₆	1	1	yes	characteristic
176	C₄	C₃²×#135	1	2	yes	—
177	C₃	C₃×#1564	1	4	yes	direct product factor
178	C₂	—	2	2	no	—
179	C₂	C₂×C₃²×C₄⋊C₈	1	1	yes	characteristic
180	C₂	C₃²×#427	1	1	yes	characteristic
181	C₂	C₂×C₃²×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄²	D₄
commutator	C₄	C₂²×C₃²×C₈
Frattini, cosocle	C₄²	C₂³×C₃²
socle	C₂²×C₃²	C₂×C₄⋊C₄
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
3. C₃²×#335	C₂	semidirect
4. C₃²×#334	C₂	semidirect
5. C₃×#1564	C₃	direct (N × H)
18. C₃×#335	C₂×C₃	semidirect
19. C₃×#334	C₂×C₃	semidirect
25. C₂×C₃²×C₈	C₈	semidirect
34. #1564	C₃²	direct (N × H)
63. #335	C₂×C₃²	semidirect
64. #334	C₂×C₃²	semidirect
70. C₂×C₃×C₈	C₃×C₈	semidirect
125. C₂×C₈	C₃²×C₈	semidirect
145. C₃²	#1564	direct (N × H)
177. C₃	C₃×#1564	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
C₂×C₃²×C₄×C₈	2
C₃²×#335	2
C₃²×#334	2
C₃×#1564	3

Subgroups — C32×#1564 #11768613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1564 #11768613