Subgroups — #72234417

Order 8192.

1676 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	#72234340	C₂	1	3	yes	maximal
2	#72234312	—	2	6	no	—
3	#72234306	C₄	1	6	yes	—
4	#72234287	—	2	6	no	—
5	#72234287	—	2	6	no	—
6	#72234271	C₂²	1	1	yes	characteristic
7	#72234260	D₄	1	3	yes	—
8	#72234260	Q₈	1	1	yes	characteristic
9	C₈²×C₁₆	C₂×C₄	1	3	yes	—
10	C₄×C₈×C₃₂	C₈	1	12	yes	—
11	C₄²×C₃₂	—	2	12	no	—
12	C₂×C₈×C₃₂	—	2	12	no	—
13	C₂×C₈×C₃₂	C₁₆	1	24	yes	—
14	C₄×C₈×C₁₆	C₂²⋊C₄	1	3	yes	—
15	C₄×C₈×C₁₆	C₄⋊C₄	1	3	yes	—
16	C₄×C₈×C₁₆	C₂×C₈	1	6	yes	—
17	C₄×C₈×C₁₆	M₁₆	1	6	yes	—
18	C₈³	C₄²	1	1	yes	characteristic
19	#93989	—	4	12	no	—
20	#93989	—	4	12	no	—
21	#92895	—	4	3	no	—
22	#92895	—	4	1	no	—
23	C₂×C₄×C₃₂	—	2	24	no	—
24	C₂×C₄×C₃₂	—	4	12	no	—
25	C₂×C₄×C₃₂	—	4	12	no	—
26	C₄²×C₁₆	C₂²⋊C₈	1	6	yes	—
27	C₄²×C₁₆	C₄⋊C₈	1	6	yes	—
28	C₂×C₈×C₁₆	—	2	6	no	—
29	C₂×C₈×C₁₆	—	2	3	no	—
30	C₂×C₈×C₁₆	—	2	3	no	—
31	C₂×C₈×C₁₆	—	2	12	no	—
32	C₂×C₈×C₁₆	C₂×C₁₆	1	12	yes	—
33	C₂×C₈×C₁₆	M₃₂	1	12	yes	—
34	C₄×C₈²	#95	1	1	yes	characteristic
35	C₄×C₈²	C₄×C₈	1	3	yes	—
36	C₄×C₈²	#97	1	3	yes	—
37	C₈×C₃₂	—	2	24	no	—
38	C₈×C₃₂	—	4	24	no	—
39	C₈×C₃₂	C₃₂	1	48	yes	complemented
40	C₂²×C₃₂	—	4	24	no	—
41	C₂×C₄×C₁₆	—	2	6	no	—
42	C₂×C₄×C₁₆	—	2	6	no	—
43	C₂×C₄×C₁₆	—	2	6	no	—
44	C₂×C₄×C₁₆	—	2	6	no	—
45	C₂×C₄×C₁₆	C₂²⋊C₁₆	1	12	yes	—
46	C₂×C₄×C₁₆	C₄⋊C₁₆	1	12	yes	—
47	C₄²×C₈	C₈²	1	1	yes	characteristic
48	C₄²×C₈	#322	1	3	yes	—
49	C₄²×C₈	#336	1	3	yes	—
50	C₂×C₈²	H(3,ℤ/4)	1	3	yes	—
51	C₂×C₈²	#338	1	1	yes	characteristic
52	C₂×C₈²	C₄×C₁₆	1	6	yes	—
53	C₂×C₈²	#346	1	6	yes	—
54	C₂×C₈²	#347	1	12	yes	—
55	C₄×C₃₂	—	2	48	no	—
56	C₄×C₃₂	—	4	24	no	—
57	C₄×C₃₂	—	8	24	no	—
58	C₄×C₃₂	—	8	24	no	—
59	C₈×C₁₆	—	2	24	no	—
60	C₈×C₁₆	—	8	3	no	—
61	C₈×C₁₆	—	8	6	no	—
62	C₈×C₁₆	—	8	3	no	—
63	C₈×C₁₆	—	8	12	no	—
64	C₈×C₁₆	C₂×C₃₂	1	24	yes	—
65	C₈×C₁₆	M₆₄	1	24	yes	—
66	C₂²×C₁₆	—	2	12	no	—
67	C₂²×C₁₆	—	2	12	no	—
68	C₂×C₄×C₈	#1274	1	3	yes	—
69	C₂×C₄×C₈	#1275	1	3	yes	—
70	C₂×C₄×C₈	C₈×C₁₆	1	3	yes	—
71	C₂×C₄×C₈	#1311	1	3	yes	—
72	C₂×C₄×C₈	#1312	1	6	yes	—
73	C₂×C₄×C₈	#1313	1	12	yes	—
74	C₂×C₄×C₈	#1374	1	6	yes	—
75	C₂×C₄×C₈	#1375	1	6	yes	—
76	C₄³	#1273	1	1	yes	characteristic
77	C₂×C₃₂	—	4	48	no	—
78	C₂×C₃₂	—	8	24	no	—
79	C₂×C₃₂	—	8	24	no	—
80	C₄×C₁₆	—	2	12	no	—
81	C₄×C₁₆	—	2	12	no	—
82	C₄×C₁₆	—	4	12	no	—
83	C₄×C₁₆	—	4	6	no	—
84	C₄×C₁₆	—	4	6	no	—
85	C₄×C₁₆	—	4	12	no	—
86	C₄×C₁₆	—	4	12	no	—
87	C₄×C₁₆	C₂²⋊C₃₂	1	24	yes	—
88	C₄×C₁₆	C₄⋊C₃₂	1	24	yes	—
89	C₈²	—	4	3	no	—
90	C₈²	—	4	1	no	—
91	C₈²	—	4	12	no	—
92	C₈²	C₄×C₃₂	1	12	yes	—
93	C₈²	#1397	1	12	yes	—
94	C₈²	#1398	1	24	yes	—
95	C₂²×C₈	C₁₆²	1	1	yes	characteristic
96	C₂²×C₈	#7052	1	3	yes	—
97	C₂²×C₈	#7053	1	12	yes	—
98	C₂²×C₈	#7142	1	6	yes	—
99	C₂²×C₈	#7145	1	6	yes	—
100	C₂×C₄²	#7014	1	1	yes	characteristic
101	C₂×C₄²	#7136	1	3	yes	—
102	C₂×C₄²	#7137	1	3	yes	—
103	C₂×C₁₆	—	2	24	no	—
104	C₂×C₁₆	—	2	24	no	—
105	C₂×C₁₆	—	4	12	no	—
106	C₂×C₁₆	—	4	12	no	—
107	C₂×C₁₆	—	4	12	no	—
108	C₂×C₁₆	—	4	12	no	—
109	C₄×C₈	—	2	12	no	—
110	C₄×C₈	—	2	24	no	—
111	C₄×C₈	—	2	6	no	—
112	C₄×C₈	—	2	3	no	—
113	C₄×C₈	—	2	3	no	—
114	C₄×C₈	C₈×C₃₂	1	6	yes	—
115	C₄×C₈	#7329	1	6	yes	—
116	C₄×C₈	#7330	1	12	yes	—
117	C₄×C₈	#7331	1	12	yes	—
118	C₄×C₈	#7332	1	12	yes	—
119	C₄×C₈	#7465	1	12	yes	—
120	C₄×C₈	#7466	1	12	yes	—
121	C₃₂	—	8	48	no	—
122	C₂²×C₄	#92895	1	1	yes	characteristic
123	C₂²×C₄	#92896	1	3	yes	—
124	C₂²×C₄	#93249	1	3	yes	—
125	C₂×C₈	—	2	24	no	—
126	C₂×C₈	—	2	6	no	—
127	C₂×C₈	—	2	6	no	—
128	C₂×C₈	—	2	6	no	—
129	C₂×C₈	—	2	6	no	—
130	C₂×C₈	C₁₆×C₃₂	1	3	yes	—
131	C₂×C₈	#93664	1	3	yes	—
132	C₂×C₈	#93665	1	6	yes	—
133	C₂×C₈	#93666	1	12	yes	—
134	C₂×C₈	#93667	1	24	yes	—
135	C₂×C₈	#93998	1	12	yes	—
136	C₂×C₈	#94005	1	12	yes	—
137	C₄²	H(3,ℤ/8)	1	3	yes	—
138	C₄²	#93255	1	1	yes	characteristic
139	C₄²	#93989	1	6	yes	—
140	C₄²	#93990	1	6	yes	—
141	C₄²	#93995	1	12	yes	—
142	C₁₆	—	4	24	no	—
143	C₁₆	—	4	24	no	—
144	C₂³	#72210714	1	1	yes	characteristic
145	C₂×C₄	#72210629	1	3	yes	—
146	C₂×C₄	#72210635	1	3	yes	—
147	C₂×C₄	#72210682	1	6	yes	—
148	C₂×C₄	#72234264	1	12	yes	—
149	C₂×C₄	#72234282	1	6	yes	—
150	C₂×C₄	#72234299	1	6	yes	—
151	C₂×C₄	#72234310	1	3	yes	—
152	C₂×C₄	#72234311	1	3	yes	—
153	C₈	—	2	12	no	—
154	C₈	—	2	12	no	—
155	C₈	—	2	24	no	—
156	C₈	C₃₂²	1	1	yes	characteristic
157	C₈	#72225038	1	3	yes	—
158	C₈	#72225071	1	12	yes	—
159	C₂²	#72210719	1	1	yes	characteristic
160	C₂²	#72234265	1	3	yes	—
161	C₂²	#72234312	1	3	yes	—
162	C₄	#72234283	1	6	yes	—
163	C₄	#72234284	1	6	yes	—
164	C₄	#72234301	1	12	yes	—
165	C₄	#72234410	1	1	yes	characteristic
166	C₄	#72234411	1	3	yes	—
167	C₂	#72234313	1	3	yes	—
168	C₂	#72234412	1	1	yes	characteristic
169	C₂	#72234414	1	3	yes	—

	subgroup	G/H
center	C₄²×C₈	C₈²
commutator	C₈	C₃₂²
Frattini, cosocle	#72234271	C₂²
socle, prime radical	C₂³	#72210714
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₄²×C₈ ◁ G
LECS	G ▷ #72234271 ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
39. C₈×C₃₂	C₃₂	semidirect

Maximal subgroups

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

subgroup	index
#72234340	2