Subgroups — #11797322

Order 1152.

456 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	#10589415	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	1	2	yes	maximal, complemented
4	#10589327	C₂	1	2	yes	maximal, complemented
5	#69745	—	3	4	no	maximal
6	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
9	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
10	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	4	yes	—
14	C₂×C₃×C₄×C₈	D₃	1	4	yes	—
15	#4549	—	3	8	no	—
16	C₄⋊(C₃⋊C₁₆)	—	3	8	no	—
17	#4461	—	3	8	no	—
18	C₂²×C₃²×C₄	C₈	1	1	yes	characteristic
19	C₂²×C₃²×C₄	C₈	1	1	yes	characteristic
20	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃²×C₈	—	2	2	no	—
22	C₂×C₃²×C₈	—	2	2	no	—
23	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
27	C₃²×C₄²	C₈	1	1	yes	characteristic
28	C₃²×C₄²	C₈	1	1	yes	characteristic
29	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₃⋊(C₃⋊C₁₆)	—	2	4	no	—
32	#2163	—	9	1	no	Sylow
33	C₂²×C₃×C₈	Dic₃	1	4	yes	—
34	C₂²×C₃×C₈	C₂×D₃	1	4	yes	—
35	C₂×C₃×C₄²	Dic₃	1	4	yes	—
36	C₃×C₄×C₈	Dic₃	1	4	yes	—
37	C₃×C₄×C₈	Dic₃	1	4	yes	—
38	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
39	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
40	C₂×C₃⋊C₁₆	—	3	16	no	—
41	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
42	C₂×C₃²×C₄	—	2	2	no	—
43	C₂×C₃²×C₄	—	2	2	no	—
44	C₂×C₃²×C₄	—	2	1	no	—
45	C₂×C₃²×C₄	—	2	1	no	—
46	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₃²×C₈	—	2	1	no	—
53	C₃²×C₈	—	2	1	no	—
54	C₃²×C₈	M₁₆	1	2	yes	—
55	C₃²×C₈	#41	1	2	yes	—
56	C₂×C₄×C₈	C₃⋊D₃	1	1	yes	characteristic
57	C₄⋊C₁₆	—	9	2	no	—
58	C₂²⋊C₁₆	—	9	2	no	—
59	#346	—	9	2	no	—
60	C₂²×C₃×C₄	C₃⋊C₈	1	4	yes	—
61	C₂²×C₃×C₄	C₃⋊C₈	1	4	yes	—
62	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
63	C₂×C₃×C₈	—	2	8	no	—
64	C₂×C₃×C₈	—	2	8	no	—
65	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
66	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
67	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
68	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
69	C₃×C₄²	C₃⋊C₈	1	4	yes	—
70	C₃×C₄²	C₃⋊C₈	1	4	yes	—
71	C₃×C₄²	C₂×Dic₃	1	4	yes	—
72	C₃×C₄²	C₂×Dic₃	1	4	yes	—
73	C₃⋊C₁₆	—	6	16	no	—
74	C₂²×C₃²	—	2	1	no	—
75	C₂²×C₃²	—	2	1	no	—
76	C₂²×C₃²	M₃₂	1	2	yes	—
77	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
78	C₃²×C₄	—	2	1	no	—
79	C₃²×C₄	—	2	1	no	—
80	C₃²×C₄	—	2	1	no	—
81	C₃²×C₄	—	2	1	no	—
82	C₃²×C₄	M₃₂	1	2	yes	—
83	C₃²×C₄	#117	1	1	yes	characteristic
84	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
85	C₂²×C₈	C₃⋊Dic₃	1	1	yes	characteristic
86	C₂²×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
87	C₂×C₄²	C₃⋊Dic₃	1	1	yes	characteristic
88	C₂×C₁₆	—	9	4	no	—
89	C₄×C₈	C₃⋊Dic₃	1	1	yes	characteristic
90	C₄×C₈	C₃⋊Dic₃	1	1	yes	characteristic
91	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
92	C₄×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
93	C₂³×C₃	C₂×C₃⋊C₈	1	4	yes	—
94	C₂×C₃×C₄	—	2	8	no	—
95	C₂×C₃×C₄	—	2	8	no	—
96	C₂×C₃×C₄	—	2	4	no	—
97	C₂×C₃×C₄	—	2	4	no	—
98	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
99	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
100	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
101	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
102	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
103	C₂×C₃×C₄	C₂²×Dic₃	1	4	yes	—
104	C₃×C₈	—	2	4	no	—
105	C₃×C₈	—	2	4	no	—
106	C₃×C₈	#208	1	8	yes	—
107	C₃×C₈	#235	1	8	yes	—
108	C₂×C₃²	—	2	2	no	—
109	C₂×C₃²	#431	1	1	yes	characteristic
110	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
111	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
112	C₂²×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
113	C₂²×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
118	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
119	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
120	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
121	C₄²	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
122	C₄²	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
123	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
124	C₄²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
125	C₁₆	—	18	4	no	—
126	C₂²×C₃	—	2	4	no	—
127	C₂²×C₃	—	2	4	no	—
128	C₂²×C₃	#812	1	8	yes	—
129	C₂²×C₃	C₂²×C₃⋊C₈	1	4	yes	—
130	C₃×C₄	—	2	4	no	—
131	C₃×C₄	—	2	4	no	—
132	C₃×C₄	—	2	4	no	—
133	C₃×C₄	—	2	4	no	—
134	C₃×C₄	#812	1	8	yes	—
135	C₃×C₄	C₂×#208	1	4	yes	—
136	C₃×C₄	#926	1	4	yes	—
137	C₃²	#2163	1	1	yes	characteristic, complemented, Sylow
138	C₂³	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
144	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
145	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
146	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
147	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
148	C₂×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	#3748	1	2	yes	—
152	C₈	#3828	1	2	yes	—
153	C₂×C₃	—	2	8	no	—
154	C₂×C₃	#4925	1	4	yes	—
155	C₂×C₃	C₂×#812	1	4	yes	—
156	C₂×C₃	C₂×#812	1	4	yes	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	#63639	1	2	yes	—
160	C₂²	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	#63639	1	2	yes	—
166	C₄	C₂×#3748	1	1	yes	characteristic
167	C₄	#64135	1	1	yes	characteristic
168	C₃	#69745	1	4	yes	complemented
169	C₂	—	2	2	no	—
170	C₂	#10592624	1	1	yes	characteristic
171	C₂	C₂×#63639	1	1	yes	characteristic
172	C₂	C₂×#63639	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×C₃⋊D₃
commutator, socle	C₂²×C₃²	C₂²×C₈
Fitting	C₂×C₃²×C₄×C₈	C₂
prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂×C₄×C₈	C₃⋊D₃
hypocenter	C₃²	#2163
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃⋊(C₃⋊C₁₆))	C₂	semidirect
4. #10589327	C₂	semidirect
137. C₃²	#2163	semidirect
168. C₃	#69745	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
#10589415	2
C₄⋊(C₃⋊(C₃⋊C₁₆))	2
#10589327	2
#69745	3