Subgroups — #11785957

Order 1152.

780 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×C₄³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	2	yes	maximal
3	#10589390	C₂	1	4	yes	maximal
4	#69392	—	3	4	no	maximal
5	C₂×C₃²×C₄²	C₄	1	2	yes	—
6	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
7	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
8	C₂×C₃²×C₄²	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	4	yes	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	4	no	—
12	C₃×C₄³	D₃	1	4	yes	—
13	C₂×C₄⋊(C₃⋊C₈)	—	3	8	no	—
14	#4524	—	3	16	no	—
15	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
17	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
18	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
20	C₃²×C₄²	—	2	4	no	—
21	C₃²×C₄²	—	2	2	no	—
22	C₃²×C₄²	—	2	4	no	—
23	C₃²×C₄²	C₈	1	4	yes	complemented
24	C₃²×C₄²	D₄	1	2	yes	—
25	C₃²×C₄²	Q₈	1	2	yes	—
26	C₂×C₃⋊(C₃⋊C₈)	—	2	4	no	—
27	C₂×C₃⋊(C₃⋊C₈)	—	2	8	no	—
28	#1831	—	9	1	no	Sylow
29	C₂×C₃×C₄²	Dic₃	1	8	yes	—
30	C₂×C₃×C₄²	Dic₃	1	4	yes	—
31	C₂×C₃×C₄²	Dic₃	1	4	yes	—
32	C₂×C₃×C₄²	C₂×D₃	1	8	yes	—
33	C₂×C₃×C₄²	C₂×D₃	1	4	yes	—
34	C₂²×C₃⋊C₈	—	3	16	no	—
35	C₄⋊(C₃⋊C₈)	—	6	16	no	—
36	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₃²×C₄	—	2	2	no	—
38	C₂×C₃²×C₄	—	2	2	no	—
39	C₂×C₃²×C₄	—	2	2	no	—
40	C₂×C₃²×C₄	—	2	2	no	—
41	C₂×C₃²×C₄	—	2	2	no	—
42	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
43	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
44	C₂×C₃²×C₄	C₂×C₈	1	4	yes	—
45	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
46	C₂×C₃²×C₄	M₁₆	1	4	yes	—
47	C₂×C₃²×C₄	M₁₆	1	2	yes	—
48	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₃²×C₄	#41	1	4	yes	—
51	C₃⋊(C₃⋊C₈)	—	4	4	no	—
52	C₂×C₄⋊C₈	—	9	2	no	—
53	C₄³	C₃⋊D₃	1	1	yes	characteristic
54	#336	—	9	4	no	—
55	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
56	C₂²×C₃×C₄	C₂×Dic₃	1	8	yes	—
57	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
58	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
59	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
60	C₃×C₄²	—	2	16	no	—
61	C₃×C₄²	—	2	8	no	—
62	C₃×C₄²	—	2	16	no	—
63	C₃×C₄²	C₃⋊C₈	1	16	yes	complemented
64	C₃×C₄²	Dic₆	1	8	yes	—
65	C₃×C₄²	D₁₂	1	8	yes	—
66	C₂×C₃⋊C₈	—	6	16	no	—
67	C₂×C₃⋊C₈	—	6	32	no	—
68	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₃²	#117	1	2	yes	—
70	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
71	C₂²×C₃²	C₂×M₁₆	1	1	yes	characteristic
72	C₂²×C₃²	C₂×M₁₆	1	2	yes	—
73	C₃²×C₄	—	2	2	no	—
74	C₃²×C₄	—	2	4	no	—
75	C₃²×C₄	—	2	4	no	—
76	C₃²×C₄	C₄⋊C₈	1	4	yes	—
77	C₃²×C₄	#124	1	2	yes	—
78	C₃²×C₄	#125	1	2	yes	—
79	C₂²×C₈	—	9	4	no	—
80	C₂×C₄²	C₃⋊Dic₃	1	2	yes	—
81	C₂×C₄²	C₃⋊Dic₃	1	1	yes	characteristic
82	C₂×C₄²	C₃⋊Dic₃	1	1	yes	characteristic
83	C₂×C₄²	C₂×C₃⋊D₃	1	2	yes	—
84	C₂×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
85	C₄⋊C₈	—	18	4	no	—
86	C₂³×C₃	C₂²×Dic₃	1	4	yes	—
87	C₂×C₃×C₄	—	2	8	no	—
88	C₂×C₃×C₄	—	2	8	no	—
89	C₂×C₃×C₄	—	2	8	no	—
90	C₂×C₃×C₄	—	2	8	no	—
91	C₂×C₃×C₄	—	2	8	no	—
92	C₂×C₃×C₄	C₂×C₃⋊C₈	1	16	yes	—
93	C₂×C₃×C₄	C₂×C₃⋊C₈	1	8	yes	—
94	C₂×C₃×C₄	#208	1	16	yes	—
95	C₂×C₃×C₄	#208	1	8	yes	—
96	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
97	C₂×C₃×C₄	C₄⋊Dic₃	1	8	yes	—
98	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
99	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
100	C₂×C₃×C₄	#235	1	16	yes	—
101	C₃⋊C₈	—	12	16	no	—
102	C₂×C₃²	#382	1	1	yes	characteristic
103	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₂×C₃²	#423	1	1	yes	characteristic
105	C₂×C₃²	#431	1	2	yes	—
106	C₂×C₃²	#432	1	2	yes	—
107	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
108	C₂²×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
109	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
110	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
111	C₂²×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
112	C₂×C₈	—	18	4	no	—
113	C₂×C₈	—	18	8	no	—
114	C₄²	—	2	4	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	4	no	—
117	C₄²	C₃⋊(C₃⋊C₈)	1	4	yes	complemented
118	C₄²	#634	1	2	yes	—
119	C₄²	C₄⋊(C₃⋊D₃)	1	2	yes	—
120	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
121	C₂²×C₃	C₂×#208	1	4	yes	—
122	C₂²×C₃	C₂×#208	1	8	yes	—
123	C₂²×C₃	C₂×C₄⋊Dic₃	1	4	yes	—
124	C₂²×C₃	#926	1	8	yes	—
125	C₃×C₄	—	2	8	no	—
126	C₃×C₄	—	2	16	no	—
127	C₃×C₄	—	2	16	no	—
128	C₃×C₄	C₄⋊(C₃⋊C₈)	1	16	yes	—
129	C₃×C₄	#870	1	8	yes	—
130	C₃×C₄	#875	1	8	yes	—
131	C₃²	#1831	1	1	yes	characteristic, complemented, Sylow
132	C₂³	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
139	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
140	C₂×C₄	#3748	1	4	yes	—
141	C₂×C₄	#3748	1	2	yes	—
142	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
143	C₂×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
144	C₂×C₄	C₂×#634	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
146	C₂×C₄	#3828	1	4	yes	—
147	C₈	—	36	4	no	—
148	C₂×C₃	C₂×C₄⋊(C₃⋊C₈)	1	4	yes	—
149	C₂×C₃	#4924	1	4	yes	—
150	C₂×C₃	#4925	1	8	yes	—
151	C₂×C₃	#4926	1	8	yes	—
152	C₂×C₃	#4936	1	4	yes	—
153	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
154	C₂²	C₂×#3748	1	1	yes	characteristic
155	C₂²	C₂×#3748	1	2	yes	—
156	C₂²	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
157	C₂²	#64135	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	4	no	—
160	C₄	—	2	4	no	—
161	C₄	C₄⋊(C₃⋊(C₃⋊C₈))	1	4	yes	—
162	C₄	#64079	1	2	yes	—
163	C₄	#64084	1	2	yes	—
164	C₃	#69392	1	4	yes	complemented
165	C₂	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
166	C₂	#10592623	1	1	yes	characteristic
167	C₂	#10592624	1	2	yes	—
168	C₂	#10592625	1	2	yes	—
169	C₂	#10592635	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂²×C₈
Fitting	C₃²×C₄³	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₄³	C₃⋊D₃
hypocenter	C₃²	#1831
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
23. C₃²×C₄²	C₈	semidirect
63. C₃×C₄²	C₃⋊C₈	semidirect
117. C₄²	C₃⋊(C₃⋊C₈)	semidirect
131. C₃²	#1831	semidirect
164. C₃	#69392	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄³	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	2
#10589390	2
#69392	3