Subgroups — #72225854

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	#10711612	C₂	1	4	yes	maximal
2	C₈×#812	C₂	1	1	yes	maximal, characteristic
3	#10688988	C₂	1	1	yes	maximal, characteristic
4	C₃×#8813	C₂	1	1	yes	maximal, characteristic
5	#135758	—	3	1	no	maximal, Sylow
6	C₃×#1844	—	2	2	no	—
7	C₃×#1844	C₂²	1	2	yes	—
8	C₂×C₃×C₈²	C₂²	1	1	yes	characteristic
9	C₂×#4487	—	2	8	no	—
10	#69792	C₂²	1	2	yes	—
11	C₄×#812	C₂²	1	2	yes	—
12	#69730	—	2	2	no	—
13	#67437	C₄	1	2	yes	complemented
14	C₈×C₃⋊C₁₆	C₄	1	2	yes	complemented
15	#12903	—	3	4	no	—
16	C₈×M₃₂	—	3	1	no	—
17	#8822	—	3	1	no	—
18	#8813	D₃	1	1	yes	characteristic
19	C₂×C₃×#106	—	2	4	no	—
20	C₂×C₃×#106	—	4	4	no	—
21	C₂×C₃×C₄×C₈	D₄	1	2	yes	—
22	C₂×C₃×C₄×C₈	C₂³	1	1	yes	characteristic
23	C₃×#384	—	2	2	no	—
24	C₂×#819	—	2	4	no	—
25	C₂×#812	D₄	1	4	yes	—
26	#4924	—	2	2	no	—
27	C₃×C₈²	C₂×C₄	1	1	yes	characteristic
28	C₃×C₈²	C₂×C₄	1	1	yes	characteristic
29	C₃×C₈²	Q₈	1	2	yes	—
30	#4487	—	4	16	no	—
31	#4461	Q₈	1	4	yes	—
32	C₄×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
33	#4457	—	4	4	no	—
34	C₂×#365	—	6	8	no	—
35	C₄×M₃₂	—	3	2	no	—
36	#1848	—	3	2	no	—
37	#1844	—	2	2	no	—
38	#1844	C₂×D₃	1	2	yes	—
39	#1566	—	6	2	no	—
40	C₂×C₈²	C₂×D₃	1	1	yes	characteristic
41	#1311	—	3	2	no	—
42	C₈×C₁₆	—	3	2	no	—
43	C₂²×C₃×C₈	SD₁₆	1	4	yes	—
44	C₂²×C₃×C₈	C₂×D₄	1	2	yes	—
45	C₂×C₃×C₄⋊C₄	—	4	2	no	—
46	C₂×C₃×C₄⋊C₄	—	4	4	no	—
47	C₂×C₃×C₄²	C₂×D₄	1	1	yes	characteristic
48	C₂×#208	—	4	2	no	—
49	C₃×#106	—	4	8	no	—
50	C₃×#106	—	8	8	no	—
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	#812	SD₁₆	1	16	yes	—
59	C₂×C₃⋊C₁₆	C₄⋊C₄	1	8	yes	—
60	C₄⋊(C₃⋊C₈)	—	4	4	no	—
61	C₂×M₃₂	—	3	4	no	—
62	C₂×#108	—	6	4	no	—
63	C₂×#106	—	2	4	no	—
64	C₂×#106	—	4	4	no	—
65	#423	—	6	2	no	—
66	C₂×C₄×C₈	#67	1	2	yes	—
67	C₂×C₄×C₈	C₂²×D₃	1	1	yes	characteristic
68	#384	—	2	2	no	—
69	#365	—	12	16	no	—
70	#346	—	3	4	no	—
71	C₄×C₁₆	—	3	4	no	—
72	#334	—	12	4	no	—
73	C₈²	Dic₆	1	2	yes	—
74	C₈²	C₄×D₃	1	1	yes	characteristic
75	C₈²	C₄×D₃	1	1	yes	characteristic
76	C₂²×C₃×C₄	—	8	2	no	—
77	C₂²×C₃×C₄	#127	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂×SD₁₆	1	2	yes	—
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×C₄⋊C₄	—	8	8	no	—
86	C₃×C₄⋊C₄	—	8	4	no	—
87	C₃×C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₃×C₄²	C₄⋊Q₈	1	1	yes	characteristic
89	C₃×C₄²	C₂×SD₁₆	1	2	yes	—
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	#214	1	4	yes	—
95	C₂²×C₈	D₃×D₄	1	2	yes	—
96	C₂×C₄⋊C₄	—	4	4	no	—
97	C₂×C₄⋊C₄	—	4	2	no	—
98	C₂×C₄²	C₂×#67	1	1	yes	characteristic
99	M₃₂	—	3	16	no	—
100	C₂×C₁₆	—	3	8	no	—
101	#108	—	12	8	no	—
102	#106	—	4	8	no	—
103	#106	—	8	8	no	—
104	C₄⋊C₈	—	12	4	no	—
105	C₄×C₈	Dic₃⋊C₄	1	2	yes	—
106	C₄×C₈	Dic₃⋊C₄	1	2	yes	—
107	C₄×C₈	#215	1	4	yes	—
108	C₄×C₈	C₂×Dic₆	1	1	yes	characteristic
109	C₄×C₈	C₂×C₄×D₃	1	1	yes	characteristic
110	C₄×C₈	D₃×Q₈	1	2	yes	—
111	C₂³×C₃	#492	1	1	yes	characteristic
112	C₂×C₃×C₄	—	2	4	no	—
113	C₂×C₃×C₄	—	16	4	no	—
114	C₂×C₃×C₄	—	16	2	no	—
115	C₂×C₃×C₄	#384	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×#106	1	2	yes	—
117	C₂×C₃×C₄	#492	1	4	yes	—
118	C₂×C₃×C₄	#492	1	1	yes	characteristic
119	C₂×C₃×C₄	#498	1	2	yes	—
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	—	8	2	no	—
126	C₂²×C₄	C₂×#214	1	2	yes	—
127	C₂²×C₄	#940	1	1	yes	characteristic
128	M₁₆	—	24	4	no	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	—	24	2	no	—
131	C₂×C₈	#808	1	4	yes	—
132	C₂×C₈	#808	1	4	yes	—
133	C₂×C₈	#888	1	2	yes	—
134	C₂×C₈	D₃×C₄⋊C₄	1	2	yes	—
135	C₂×C₈	D₃×SD₁₆	1	8	yes	—
136	C₄⋊C₄	—	8	8	no	—
137	C₄⋊C₄	—	8	4	no	—
138	C₄²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
139	C₄²	C₂×#215	1	2	yes	—
140	C₄²	#944	1	1	yes	characteristic
141	C₁₆	—	3	16	no	—
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	—	32	2	no	—
147	C₃×C₄	#1705	1	2	yes	—
148	C₃×C₄	#1844	1	2	yes	—
149	C₃×C₄	C₂×#365	1	4	yes	—
150	C₂³	#5076	1	1	yes	characteristic
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	16	4	no	—
153	C₂×C₄	—	16	2	no	—
154	C₂×C₄	C₂×#808	1	2	yes	—
155	C₂×C₄	#4975	1	1	yes	characteristic
156	C₂×C₄	#5080	1	2	yes	—
157	C₂×C₄	#5085	1	1	yes	characteristic
158	C₂×C₄	#5175	1	4	yes	—
159	C₈	—	2	4	no	—
160	C₈	—	48	2	no	—
161	C₈	#4487	1	8	yes	complemented
162	C₈	#4855	1	4	yes	—
163	C₈	D₃×#106	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	#69576	1	1	yes	characteristic
169	C₂²	#71254	1	2	yes	—
170	C₄	—	2	2	no	—
171	C₄	—	32	2	no	—
172	C₄	C₂×#4487	1	4	yes	—
173	C₄	#70554	1	2	yes	—
174	C₄	#71332	1	2	yes	—
175	C₃	#135758	1	1	yes	characteristic, complemented, Sylow
176	C₂	—	2	2	no	—
177	C₂	#10688953	1	1	yes	characteristic
178	C₂	#10711612	1	2	yes	—

	subgroup	G/H
center	C₂²	#69576
commutator	C₃×C₄×C₈	C₂²×C₄
Frattini	C₂×C₄×C₈	C₂²×D₃
Fitting	C₃×#8813	C₂
socle	C₂²×C₃	#1844
prime radical	C₂³×C₃	#492
cosocle	C₂×C₃×C₄×C₈	C₂³
hypercenter	#8813	D₃
hypocenter	C₃	#135758
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #8813

Complements

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

N	H ≅ G/N	type
13. #67437	C₄	semidirect
14. C₈×C₃⋊C₁₆	C₄	semidirect
92. C₃⋊C₁₆	#106	semidirect
121. C₃×C₈	#365	semidirect
161. C₈	#4487	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
#10711612	2
C₈×#812	2
#10688988	2
C₃×#8813	2
#135758	3