Subgroups — #29833662

Order 15616.

494 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447728	C₂	1	6	yes	maximal, complemented
2	Q₈×#11758185	C₂	1	2	yes	maximal, complemented
3	#21447622	C₂	1	3	yes	maximal
4	#21447618	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11758185	C₂²	1	6	yes	—
7	C₄⋊#11758185	C₂²	1	6	yes	—
8	C₂²⋊#11758185	C₂²	1	1	yes	characteristic
9	C₂²⋊#11758185	C₂²	1	3	yes	—
10	#16839538	C₂²	1	6	yes	—
11	C₄×#11758185	C₂²	1	6	yes	—
12	C₂×C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₆₁	C₄	1	3	yes	—
14	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
15	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₈×Q₈×C₆₁	C₂²	1	3	yes	—
17	C₂×C₄⋊C₈×C₆₁	C₄	1	3	yes	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
19	C₂×#11758185	C₂³	1	2	yes	—
20	C₂×#11758185	C₂³	1	6	yes	—
21	C₂²×Q₈×C₆₁	C₈	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
23	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
24	C₄×Q₈×C₆₁	C₈	1	3	yes	—
25	C₄×Q₈×C₆₁	C₈	1	1	yes	characteristic
26	C₄×Q₈×C₆₁	C₂×C₄	1	3	yes	—
27	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₆₁	C₈	1	3	yes	—
29	C₂×C₄²×C₆₁	C₂×C₄	1	3	yes	—
30	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
31	C₄⋊C₈×C₆₁	C₂×C₄	1	6	yes	—
32	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
34	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₆₁	C₂³	1	3	yes	—
36	C₄×C₈×C₆₁	C₂³	1	3	yes	—
37	C₂×Q₈×C₆₁	—	2	4	no	—
38	C₂×Q₈×C₆₁	C₂×C₈	1	3	yes	—
39	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₆₁	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	—	4	3	no	—
45	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₆₁	C₂×C₈	1	6	yes	—
50	C₄⋊C₄×C₆₁	C₂×C₈	1	3	yes	—
51	C₄⋊C₄×C₆₁	C₂×C₈	1	3	yes	—
52	C₄²×C₆₁	C₂×C₈	1	3	yes	—
53	C₄²×C₆₁	C₂×C₈	1	3	yes	—
54	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
55	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
56	#11758185	—	2	2	no	—
57	#11758185	—	2	6	no	—
58	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
59	Q₈×C₆₁	—	2	6	no	—
60	Q₈×C₆₁	M₃₂	1	4	yes	—
61	C₂×C₄×C₆₁	—	2	6	no	—
62	C₂×C₄×C₆₁	—	2	1	no	—
63	C₂×C₄×C₆₁	—	2	1	no	—
64	C₂×C₄×C₆₁	—	2	6	no	—
65	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
71	C₈×C₆₁	—	2	1	no	—
72	C₈×C₆₁	—	2	3	no	—
73	C₈×C₆₁	—	2	3	no	—
74	C₈×C₆₁	2₋¹⁺⁴	1	2	yes	—
75	#33498	—	61	1	no	maximal, Sylow
76	C₂²×C₆₁	—	2	1	no	—
77	C₂²×C₆₁	—	2	1	no	—
78	C₂²×C₆₁	#504	1	2	yes	—
79	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₆₁	—	2	3	no	—
81	C₄×C₆₁	—	2	3	no	—
82	C₄×C₆₁	—	2	3	no	—
83	C₄×C₆₁	—	2	1	no	—
84	C₄×C₆₁	C₂×M₃₂	1	6	yes	—
85	C₄×C₆₁	#519	1	1	yes	characteristic
86	C₄×C₆₁	#568	1	1	yes	characteristic
87	C₂×C₈×Q₈	D₆₁	1	1	yes	characteristic
88	#2183	—	61	6	no	—
89	Q₈×C₁₆	—	61	2	no	—
90	#2163	—	61	3	no	—
91	#2162	—	61	3	no	—
92	C₂×C₆₁	—	2	2	no	—
93	C₂×C₆₁	#2982	1	1	yes	characteristic
94	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
95	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
96	C₂×C₄×Q₈	Dic₆₁	1	1	yes	characteristic
97	C₈×Q₈	Dic₆₁	1	3	yes	—
98	C₈×Q₈	Dic₆₁	1	1	yes	characteristic
99	C₈×Q₈	C₂×D₆₁	1	1	yes	characteristic
100	C₈×Q₈	C₂×D₆₁	1	3	yes	—
101	C₂×C₄⋊C₈	Dic₆₁	1	3	yes	—
102	C₂×C₄×C₈	C₂×D₆₁	1	3	yes	—
103	C₄⋊C₁₆	—	61	6	no	—
104	C₄⋊C₁₆	—	61	6	no	—
105	C₂²⋊C₁₆	—	61	1	no	—
106	C₂²⋊C₁₆	—	61	3	no	—
107	#346	—	61	6	no	—
108	C₄×C₁₆	—	61	6	no	—
109	C₆₁	#33498	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	#92418	1	1	yes	characteristic
111	C₂²×C₈	C₂×Dic₆₁	1	3	yes	—
112	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
113	C₄×Q₈	#92418	1	3	yes	—
114	C₄×Q₈	#92418	1	1	yes	characteristic
115	C₄×Q₈	C₂×Dic₆₁	1	3	yes	—
116	C₄×Q₈	C₂×Dic₆₁	1	1	yes	characteristic
117	C₂×C₄⋊C₄	#92418	1	3	yes	—
118	C₂×C₄²	C₂×Dic₆₁	1	3	yes	—
119	C₂×C₁₆	—	61	2	no	—
120	C₂×C₁₆	—	61	6	no	—
121	C₄⋊C₈	C₂×Dic₆₁	1	3	yes	—
122	C₄⋊C₈	C₂×Dic₆₁	1	6	yes	—
123	C₄⋊C₈	C₂×Dic₆₁	1	3	yes	—
124	C₄×C₈	C₂×Dic₆₁	1	3	yes	—
125	C₄×C₈	C₂×Dic₆₁	1	3	yes	—
126	C₄×C₈	C₂²×D₆₁	1	3	yes	—
127	C₄×C₈	C₂²×D₆₁	1	3	yes	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	C₂×#92418	1	3	yes	—
130	C₂×Q₈	C₂×#92418	1	1	yes	characteristic
131	C₂²×C₄	C₂×#92418	1	3	yes	—
132	C₂²×C₄	C₂×#92418	1	3	yes	—
133	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	4	3	no	—
136	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₆₁	1	3	yes	—
138	C₂×C₈	C₂²×Dic₆₁	1	3	yes	—
139	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
140	C₄⋊C₄	C₂×#92418	1	6	yes	—
141	C₄⋊C₄	C₂×#92418	1	3	yes	—
142	C₄⋊C₄	C₂×#92418	1	3	yes	—
143	C₄²	C₂×#92418	1	3	yes	—
144	C₄²	C₂×#92418	1	3	yes	—
145	C₄²	C₂²×Dic₆₁	1	3	yes	—
146	C₄²	C₂²×Dic₆₁	1	3	yes	—
147	C₁₆	—	122	2	no	—
148	C₁₆	—	122	6	no	—
149	C₂³	C₂²×#92418	1	1	yes	characteristic
150	Q₈	—	2	6	no	—
151	Q₈	#14519717	1	4	yes	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	6	no	—
156	C₂×C₄	C₂²×#92418	1	1	yes	characteristic
157	C₂×C₄	C₂²×#92418	1	3	yes	—
158	C₂×C₄	C₂²×#92418	1	3	yes	—
159	C₂×C₄	C₂²×#92418	1	3	yes	—
160	C₂×C₄	C₂²×#92418	1	3	yes	—
161	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	3	no	—
164	C₈	—	2	3	no	—
165	C₈	#14519825	1	2	yes	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	#16840179	1	2	yes	—
169	C₂²	C₂³×#92418	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	3	no	—
172	C₄	—	2	3	no	—
173	C₄	—	2	3	no	—
174	C₄	C₂×#14519717	1	6	yes	—
175	C₄	#16840240	1	1	yes	characteristic
176	C₄	#16840529	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	#21454900	1	1	yes	characteristic
179	C₂	C₂²×#14519717	1	1	yes	characteristic
180	C₂	C₂×#16840179	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₆₁	C₂
prime radical	C₂³×C₆₁	C₂²×C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₆₁
hypocenter	C₆₁	#33498
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #21447728	C₂	semidirect
2. Q₈×#11758185	C₂	semidirect
109. C₆₁	#33498	semidirect

Maximal subgroups

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

subgroup	index
#21447728	2
Q₈×#11758185	2
#21447622	2
#21447618	2
C₂×C₈×Q₈×C₆₁	2
#33498	61