Subgroups — #72225815

Order 1536.

604 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	#10711613	C₂	1	4	yes	maximal
2	#10688988	C₂	1	1	yes	maximal, characteristic
3	#10688954	C₂	1	1	yes	maximal, characteristic
4	C₃×C₈×M₃₂	C₂	1	1	yes	maximal, characteristic
5	#135758	—	3	1	no	maximal, Sylow
6	C₃×C₄×M₃₂	C₂²	1	2	yes	—
7	C₂×C₃×C₈²	C₂²	1	1	yes	characteristic
8	C₂×#4513	—	2	8	no	—
9	#69792	C₂²	1	2	yes	—
10	#69764	—	2	2	no	—
11	#69764	C₂²	1	2	yes	—
12	#69730	—	2	2	no	—
13	C₃×#1311	C₄	1	2	yes	complemented
14	C₃×C₈×C₁₆	C₄	1	2	yes	complemented
15	#12903	—	3	4	no	—
16	C₈×M₃₂	D₃	1	1	yes	characteristic
17	#8822	—	3	1	no	—
18	#8813	—	3	1	no	—
19	C₂×C₃×M₃₂	D₄	1	4	yes	—
20	C₂×C₃×C₄×C₈	D₄	1	2	yes	—
21	C₂×C₃×C₄×C₈	C₂³	1	1	yes	characteristic
22	C₂×#819	—	2	4	no	—
23	C₂×#817	—	2	4	no	—
24	C₂×#817	—	4	4	no	—
25	#4935	—	2	2	no	—
26	#4924	—	2	2	no	—
27	C₃×#346	Q₈	1	4	yes	—
28	C₃×C₄×C₁₆	C₂×C₄	1	4	yes	—
29	C₃×C₈²	C₂×C₄	1	1	yes	characteristic
30	C₃×C₈²	C₂×C₄	1	1	yes	characteristic
31	C₃×C₈²	Q₈	1	2	yes	—
32	#4513	—	4	16	no	—
33	#4457	—	4	4	no	—
34	C₂×#365	—	6	8	no	—
35	C₄×M₃₂	C₂×D₃	1	2	yes	—
36	#1848	—	3	2	no	—
37	#1844	—	3	2	no	—
38	#1844	—	6	2	no	—
39	#1566	—	6	2	no	—
40	C₂×C₈²	C₂×D₃	1	1	yes	characteristic
41	#1311	Dic₃	1	2	yes	complemented
42	C₈×C₁₆	Dic₃	1	2	yes	complemented
43	C₂²×C₃×C₈	SD₁₆	1	4	yes	—
44	C₂²×C₃×C₈	C₂×D₄	1	2	yes	—
45	C₂×C₃×C₄²	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄⋊Dic₃	—	4	2	no	—
47	C₂×C₄⋊Dic₃	—	4	4	no	—
48	C₂×#208	—	4	2	no	—
49	C₃×M₃₂	SD₁₆	1	16	yes	—
50	C₂×C₃×C₁₆	C₄⋊C₄	1	8	yes	—
51	C₃×C₄×C₈	C₄⋊C₄	1	2	yes	—
52	C₃×C₄×C₈	C₄⋊C₄	1	2	yes	—
53	C₃×C₄×C₈	SD₁₆	1	4	yes	—
54	C₃×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂×Q₈	1	2	yes	—
57	#819	—	4	8	no	—
58	#817	—	4	8	no	—
59	#817	—	8	8	no	—
60	C₄⋊(C₃⋊C₈)	—	4	4	no	—
61	C₂×M₃₂	D₁₂	1	4	yes	—
62	C₂×#108	—	6	4	no	—
63	C₂×#106	—	6	4	no	—
64	C₂×#106	—	12	4	no	—
65	#423	—	6	2	no	—
66	C₂×C₄×C₈	D₁₂	1	2	yes	—
67	C₂×C₄×C₈	C₂²×D₃	1	1	yes	characteristic
68	#384	—	6	2	no	—
69	#365	—	12	16	no	—
70	#346	Dic₆	1	4	yes	—
71	C₄×C₁₆	C₂×Dic₃	1	4	yes	—
72	#334	—	12	4	no	—
73	C₈²	Dic₆	1	2	yes	—
74	C₈²	C₂×Dic₃	1	1	yes	characteristic
75	C₈²	C₂×Dic₃	1	1	yes	characteristic
76	C₂²×C₃×C₄	#127	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂×SD₁₆	1	2	yes	—
78	C₂²×Dic₃	—	8	2	no	—
79	C₂×C₃×C₈	—	2	8	no	—
80	C₂×C₃×C₈	#106	1	4	yes	—
81	C₂×C₃×C₈	#106	1	4	yes	—
82	C₂×C₃×C₈	C₂×C₄⋊C₄	1	2	yes	—
83	C₂×C₃×C₈	C₄⋊Q₈	1	2	yes	—
84	C₂×C₃×C₈	C₂×SD₁₆	1	8	yes	—
85	C₃×C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₃×C₄²	C₄⋊Q₈	1	1	yes	characteristic
87	C₃×C₄²	C₂×SD₁₆	1	2	yes	—
88	C₄⋊Dic₃	—	8	8	no	—
89	C₄⋊Dic₃	—	8	4	no	—
90	#208	—	8	4	no	—
91	C₂×C₃⋊C₈	—	8	2	no	—
92	C₃×C₁₆	#106	1	16	yes	complemented
93	C₂×M₁₆	—	12	2	no	—
94	C₂²×C₈	#204	1	4	yes	—
95	C₂²×C₈	C₂×D₁₂	1	2	yes	—
96	C₂×C₄⋊C₄	—	12	2	no	—
97	C₂×C₄⋊C₄	—	12	4	no	—
98	C₂×C₄²	C₂×D₁₂	1	1	yes	characteristic
99	M₃₂	#204	1	16	yes	—
100	C₂×C₁₆	C₄⋊Dic₃	1	8	yes	—
101	#108	—	12	8	no	—
102	#106	—	12	8	no	—
103	#106	—	24	8	no	—
104	C₄⋊C₈	—	12	4	no	—
105	C₄×C₈	#204	1	4	yes	—
106	C₄×C₈	C₄⋊Dic₃	1	2	yes	—
107	C₄×C₈	C₄⋊Dic₃	1	2	yes	—
108	C₄×C₈	C₂×Dic₆	1	2	yes	—
109	C₄×C₈	C₂×Dic₆	1	1	yes	characteristic
110	C₄×C₈	C₂²×Dic₃	1	1	yes	characteristic
111	C₂³×C₃	#492	1	1	yes	characteristic
112	C₂×C₃×C₄	—	2	4	no	—
113	C₂×C₃×C₄	#384	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂×#106	1	2	yes	—
115	C₂×C₃×C₄	#492	1	4	yes	—
116	C₂×C₃×C₄	#492	1	1	yes	characteristic
117	C₂×C₃×C₄	#498	1	2	yes	—
118	C₂×Dic₃	—	16	4	no	—
119	C₂×Dic₃	—	16	2	no	—
120	C₃×C₈	—	2	4	no	—
121	C₃×C₈	#365	1	8	yes	complemented
122	C₃×C₈	C₂×#106	1	4	yes	—
123	C₃×C₈	#498	1	4	yes	—
124	C₃⋊C₈	—	16	2	no	—
125	C₂²×C₄	—	24	2	no	—
126	C₂²×C₄	#874	1	1	yes	characteristic
127	C₂²×C₄	C₂×#204	1	2	yes	—
128	M₁₆	—	24	4	no	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	—	24	2	no	—
131	C₂×C₈	#817	1	4	yes	—
132	C₂×C₈	#817	1	4	yes	—
133	C₂×C₈	#869	1	2	yes	—
134	C₂×C₈	C₂×#204	1	8	yes	—
135	C₂×C₈	C₂×C₄⋊Dic₃	1	2	yes	—
136	C₄⋊C₄	—	24	8	no	—
137	C₄⋊C₄	—	24	4	no	—
138	C₄²	#869	1	1	yes	characteristic
139	C₄²	C₂×#204	1	2	yes	—
140	C₄²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
141	C₁₆	#817	1	16	yes	complemented
142	C₂²×C₃	—	2	2	no	—
143	C₂²×C₃	#1705	1	2	yes	—
144	C₂²×C₃	#1844	1	1	yes	characteristic
145	C₃×C₄	—	2	2	no	—
146	C₃×C₄	#1705	1	2	yes	—
147	C₃×C₄	#1844	1	2	yes	—
148	C₃×C₄	C₂×#365	1	4	yes	—
149	Dic₃	—	32	2	no	—
150	C₂³	#4693	1	1	yes	characteristic
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	48	4	no	—
153	C₂×C₄	—	48	2	no	—
154	C₂×C₄	#4680	1	2	yes	—
155	C₂×C₄	#4693	1	4	yes	—
156	C₂×C₄	#4693	1	1	yes	characteristic
157	C₂×C₄	#4935	1	1	yes	characteristic
158	C₂×C₄	C₂×#817	1	2	yes	—
159	C₈	—	2	4	no	—
160	C₈	—	48	2	no	—
161	C₈	#4513	1	8	yes	complemented
162	C₈	#4680	1	4	yes	—
163	C₈	C₂×#817	1	4	yes	—
164	C₂×C₃	—	2	2	no	—
165	C₂×C₃	#8813	1	1	yes	characteristic
166	C₂×C₃	#12903	1	2	yes	—
167	C₂²	—	2	2	no	—
168	C₂²	#68057	1	2	yes	—
169	C₂²	#69764	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	96	2	no	—
172	C₄	#68057	1	2	yes	—
173	C₄	#69764	1	2	yes	—
174	C₄	C₂×#4513	1	4	yes	—
175	C₃	#135758	1	1	yes	characteristic, complemented, Sylow
176	C₂	—	2	2	no	—
177	C₂	#10688954	1	1	yes	characteristic
178	C₂	#10711613	1	2	yes	—

	subgroup	G/H
center	C₂²	#69764
commutator	C₃×C₄×C₈	C₂²×C₄
Frattini	C₂×C₄×C₈	C₂²×D₃
Fitting	C₃×C₈×M₃₂	C₂
socle	C₂²×C₃	#1844
prime radical	C₂³×C₃	#492
cosocle	C₂×C₃×C₄×C₈	C₂³
hypercenter	C₈×M₃₂	D₃
hypocenter	C₃	#135758
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₈×M₃₂

Complements

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

N	H ≅ G/N	type
13. C₃×#1311	C₄	semidirect
14. C₃×C₈×C₁₆	C₄	semidirect
41. #1311	Dic₃	semidirect
42. C₈×C₁₆	Dic₃	semidirect
92. C₃×C₁₆	#106	semidirect
121. C₃×C₈	#365	semidirect
141. C₁₆	#817	semidirect
161. C₈	#4513	semidirect
175. C₃	#135758	semidirect

Maximal subgroups

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

subgroup	index
#10711613	2
#10688988	2
#10688954	2
C₃×C₈×M₃₂	2
#135758	3