Subgroups — C₃²×#1810 #11768859

Order 1152.

876 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#108	C₂	1	2	yes	maximal, complemented
3	C₃²×#407	C₂	1	2	yes	maximal
4	C₃²×#341	C₂	1	2	yes	maximal, complemented
5	C₃×#1810	C₃	1	4	yes	maximal, direct product factor
6	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃²×M₁₆	C₂²	1	4	yes	—
8	C₂²×C₃²×C₈	—	2	4	no	—
9	C₂²×C₃²×C₈	C₄	1	4	yes	—
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
12	C₃²×#108	—	2	4	no	—
13	C₃²×C₂²⋊C₈	—	2	4	no	—
14	C₂³×C₃×C₈	C₂×C₃	1	4	yes	—
15	C₂×C₃×#108	C₂×C₃	1	8	yes	complemented
16	C₃×#407	C₂×C₃	1	8	yes	—
17	C₃×#341	C₂×C₃	1	8	yes	complemented
18	C₂⁴×C₃²	Q₈	1	1	yes	characteristic
19	C₂²×C₃²×C₄	—	2	2	no	—
20	C₂²×C₃²×C₄	—	2	2	no	—
21	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
22	C₂²×C₃²×C₄	D₄	1	2	yes	—
23	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
24	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₃²×M₁₆	—	2	4	no	—
26	C₃²×M₁₆	—	2	4	no	—
27	C₂×C₃²×C₈	—	2	2	no	—
28	C₂×C₃²×C₈	—	2	8	no	—
29	C₂×C₃²×C₈	—	2	4	no	—
30	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃²×C₈	D₄	1	4	yes	—
33	#1810	C₃²	1	1	yes	characteristic, direct product factor, Sylow
34	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
35	C₂×C₃×M₁₆	C₂²×C₃	1	16	yes	—
36	C₂²×C₃×C₈	—	2	16	no	—
37	C₂²×C₃×C₈	C₃×C₄	1	16	yes	—
38	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
39	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
40	C₃×#108	—	2	16	no	—
41	C₃×C₂²⋊C₈	—	2	16	no	—
42	C₂³×C₃²	—	2	4	no	—
43	C₂³×C₃²	—	2	2	no	—
44	C₂³×C₃²	C₄⋊C₄	1	2	yes	—
45	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₃²×C₄	—	2	2	no	—
47	C₂×C₃²×C₄	—	2	2	no	—
48	C₂×C₃²×C₄	—	2	2	no	—
49	C₂×C₃²×C₄	—	2	4	no	—
50	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
51	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
53	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	#41	1	2	yes	—
55	C₃²×C₈	—	2	2	no	—
56	C₃²×C₈	—	4	4	no	—
57	C₃²×C₈	C₂²⋊C₄	1	4	yes	—
58	C₂³×C₈	C₂×C₃²	1	1	yes	characteristic
59	C₂×#108	C₂×C₃²	1	2	yes	complemented
60	#407	C₂×C₃²	1	2	yes	—
61	#341	C₂×C₃²	1	2	yes	complemented
62	C₂⁴×C₃	C₃×Q₈	1	4	yes	—
63	C₂²×C₃×C₄	—	2	8	no	—
64	C₂²×C₃×C₄	—	2	8	no	—
65	C₂²×C₃×C₄	C₂×C₃×C₄	1	8	yes	—
66	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
67	C₂²×C₃×C₄	C₃×Q₈	1	4	yes	—
68	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
69	C₃×M₁₆	—	2	16	no	—
70	C₃×M₁₆	—	2	16	no	—
71	C₂×C₃×C₈	—	2	8	no	—
72	C₂×C₃×C₈	—	2	32	no	—
73	C₂×C₃×C₈	—	2	16	no	—
74	C₂×C₃×C₈	C₂×C₃×C₄	1	8	yes	—
75	C₂×C₃×C₈	C₂×C₃×C₄	1	8	yes	—
76	C₂×C₃×C₈	C₃×D₄	1	16	yes	—
77	C₂²×C₃²	—	2	8	no	—
78	C₂²×C₃²	—	2	2	no	—
79	C₂²×C₃²	—	2	2	no	—
80	C₂²×C₃²	—	2	2	no	—
81	C₂²×C₃²	#108	1	4	yes	complemented
82	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
84	C₃²×C₄	—	2	2	no	—
85	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₃²×C₄	#117	1	1	yes	characteristic
87	C₃²×C₄	#121	1	2	yes	—
88	C₂³×C₄	C₂²×C₃²	1	1	yes	characteristic
89	C₂×M₁₆	C₂²×C₃²	1	4	yes	—
90	C₂²×C₈	—	2	4	no	—
91	C₂²×C₈	C₃²×C₄	1	4	yes	—
92	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
94	#108	—	2	4	no	—
95	C₂²⋊C₈	—	2	4	no	—
96	C₂³×C₃	—	2	16	no	—
97	C₂³×C₃	—	2	8	no	—
98	C₂³×C₃	C₃×C₄⋊C₄	1	8	yes	—
99	C₂³×C₃	C₂×C₃×Q₈	1	4	yes	—
100	C₂×C₃×C₄	—	2	8	no	—
101	C₂×C₃×C₄	—	2	8	no	—
102	C₂×C₃×C₄	—	2	8	no	—
103	C₂×C₃×C₄	—	2	16	no	—
104	C₂×C₃×C₄	C₃×C₄⋊C₄	1	8	yes	—
105	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
106	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
107	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
108	C₂×C₃×C₄	C₃×#41	1	8	yes	—
109	C₃×C₈	—	2	8	no	—
110	C₃×C₈	—	4	16	no	—
111	C₃×C₈	C₃×C₂²⋊C₄	1	16	yes	—
112	C₂×C₃²	—	2	4	no	—
113	C₂×C₃²	—	2	2	no	—
114	C₂×C₃²	#380	1	1	yes	characteristic
115	C₂×C₃²	C₂×#108	1	2	yes	—
116	C₂⁴	C₃²×Q₈	1	1	yes	characteristic
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	C₂×C₃²×C₄	1	2	yes	—
120	C₂²×C₄	C₃²×D₄	1	2	yes	—
121	C₂²×C₄	C₃²×Q₈	1	1	yes	characteristic
122	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
123	M₁₆	—	2	4	no	—
124	M₁₆	—	2	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	C₂×C₃²×C₄	1	2	yes	—
129	C₂×C₈	C₂×C₃²×C₄	1	2	yes	—
130	C₂×C₈	C₃²×D₄	1	4	yes	—
131	C₂²×C₃	—	2	32	no	—
132	C₂²×C₃	—	2	8	no	—
133	C₂²×C₃	—	2	8	no	—
134	C₂²×C₃	—	2	8	no	—
135	C₂²×C₃	C₃×#108	1	16	yes	complemented
136	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	4	yes	—
137	C₂²×C₃	C₃×C₂²⋊Q₈	1	8	yes	—
138	C₃×C₄	—	2	8	no	—
139	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	4	yes	—
140	C₃×C₄	C₃×#117	1	4	yes	—
141	C₃×C₄	C₃×#121	1	8	yes	—
142	C₃²	#1810	1	1	yes	characteristic, direct product factor, Sylow
143	C₂³	—	2	2	no	—
144	C₂³	—	2	4	no	—
145	C₂³	C₃²×C₄⋊C₄	1	2	yes	—
146	C₂³	C₂×C₃²×Q₈	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	C₃²×C₄⋊C₄	1	2	yes	—
152	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
153	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
154	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
155	C₂×C₄	C₃²×#41	1	2	yes	—
156	C₈	—	2	2	no	—
157	C₈	—	4	4	no	—
158	C₈	C₃²×C₂²⋊C₄	1	4	yes	—
159	C₂×C₃	—	2	16	no	—
160	C₂×C₃	—	2	8	no	—
161	C₂×C₃	C₃×#380	1	4	yes	—
162	C₂×C₃	C₂×C₃×#108	1	8	yes	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	C₃²×#108	1	4	yes	complemented
168	C₂²	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
169	C₂²	C₃²×C₂²⋊Q₈	1	2	yes	—
170	C₄	—	2	2	no	—
171	C₄	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
172	C₄	C₃²×#117	1	1	yes	characteristic
173	C₄	C₃²×#121	1	2	yes	—
174	C₃	C₃×#1810	1	4	yes	direct product factor
175	C₂	—	2	2	no	—
176	C₂	—	2	4	no	—
177	C₂	C₃²×#380	1	1	yes	characteristic
178	C₂	C₂×C₃²×#108	1	2	yes	—

	subgroup	G/H
center	C₂×C₃²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃²	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂³×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×#108	C₂	semidirect
4. C₃²×#341	C₂	semidirect
5. C₃×#1810	C₃	direct (N × H)
15. C₂×C₃×#108	C₂×C₃	semidirect
17. C₃×#341	C₂×C₃	semidirect
33. #1810	C₃²	direct (N × H)
59. C₂×#108	C₂×C₃²	semidirect
61. #341	C₂×C₃²	semidirect
81. C₂²×C₃²	#108	semidirect
135. C₂²×C₃	C₃×#108	semidirect
142. C₃²	#1810	direct (N × H)
167. C₂²	C₃²×#108	semidirect
174. C₃	C₃×#1810	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃²×C₈	2
C₂×C₃²×#108	2
C₃²×#407	2
C₃²×#341	2
C₃×#1810	3

Subgroups — C32×#1810 #11768859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1810 #11768859