Subgroups — #17337

Order 256.

399 conjugacy classes of subgroups.

The subgroup lattice has 181 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2965	C₂	1	1	yes	maximal, characteristic
2	#2965	C₂	1	1	yes	maximal, characteristic
3	C₄×#131	C₂	1	1	yes	maximal, characteristic
4	#1455	C₂	1	2	yes	maximal, complemented
5	#1453	C₂	1	2	yes	maximal, complemented
6	#1452	C₂	1	1	yes	maximal, characteristic, complemented
7	#1451	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×M₁₆	C₂	1	2	yes	maximal, complemented
10	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#322	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131	C₄	1	2	yes	—
13	C₄×#41	C₂²	1	1	yes	characteristic
14	#446	C₄	1	2	yes	—
15	#446	C₄	1	2	yes	—
16	#436	C₄	1	2	yes	—
17	#436	C₄	1	2	yes	—
18	#435	C₄	1	4	yes	—
19	#435	C₄	1	4	yes	—
20	#431	C₂²	1	1	yes	characteristic
21	#431	C₂²	1	1	yes	characteristic
22	#431	C₂²	1	2	yes	—
23	#431	C₂²	1	2	yes	—
24	#405	C₄	1	4	yes	—
25	#405	C₄	1	2	yes	—
26	C₄×M₁₆	C₂²	1	1	yes	characteristic
27	C₄×M₁₆	C₂²	1	2	yes	—
28	C₄×M₁₆	C₂²	1	2	yes	—
29	C₄×M₁₆	C₂²	1	2	yes	—
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂²	1	2	yes	—
32	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
34	#322	C₂²	1	1	yes	characteristic
35	#322	C₂²	1	1	yes	characteristic
36	#322	C₂²	1	1	yes	characteristic
37	#322	C₂²	1	1	yes	characteristic
38	#322	C₂²	1	2	yes	—
39	#322	C₂²	1	2	yes	—
40	#322	C₂²	1	2	yes	—
41	#322	C₂²	1	2	yes	—
42	C₈²	C₂²	1	2	yes	—
43	C₈²	C₂²	1	2	yes	—
44	C₂×#41	C₂×C₄	1	1	yes	characteristic
45	#131	—	2	8	no	—
46	C₂×M₁₆	C₂×C₄	1	4	yes	—
47	C₂×M₁₆	C₂×C₄	1	4	yes	—
48	C₂×M₁₆	C₂×C₄	1	4	yes	—
49	C₂×M₁₆	C₂×C₄	1	2	yes	—
50	C₂²×C₈	C₂×C₄	1	2	yes	—
51	C₂²×C₈	C₂×C₄	1	4	yes	—
52	C₂²×C₈	C₂×C₄	1	2	yes	—
53	C₂²×C₈	C₂×C₄	1	2	yes	—
54	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
56	C₄×D₄	C₂×C₄	1	2	yes	—
57	C₄×D₄	C₂×C₄	1	2	yes	—
58	C₄×D₄	C₂×C₄	1	1	yes	characteristic
59	C₄×D₄	C₂×C₄	1	1	yes	characteristic
60	#117	C₂×C₄	1	2	yes	—
61	#117	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²	C₂³	1	1	yes	characteristic
63	C₂×C₄²	C₂³	1	2	yes	—
64	C₄⋊C₈	C₂×C₄	1	4	yes	—
65	C₄⋊C₈	C₂×C₄	1	4	yes	—
66	C₄⋊C₈	C₂×C₄	1	2	yes	—
67	C₄⋊C₈	C₂×C₄	1	2	yes	—
68	C₂²⋊C₈	C₂×C₄	1	2	yes	—
69	C₂²⋊C₈	C₂×C₄	1	2	yes	—
70	C₂²⋊C₈	C₂×C₄	1	4	yes	—
71	C₂²⋊C₈	C₂×C₄	1	4	yes	—
72	#97	C₂×C₄	1	4	yes	—
73	#97	C₂×C₄	1	2	yes	—
74	#97	C₂×C₄	1	4	yes	—
75	#97	C₂×C₄	1	4	yes	—
76	C₄×C₈	C₂×C₄	1	2	yes	—
77	C₄×C₈	C₂×C₄	1	4	yes	—
78	C₄×C₈	C₂×C₄	1	2	yes	—
79	C₄×C₈	C₂×C₄	1	2	yes	—
80	C₄×C₈	C₂³	1	2	yes	—
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₈	C₂³	1	1	yes	characteristic
83	C₄×C₈	C₂³	1	2	yes	—
84	C₄×C₈	C₂³	1	2	yes	—
85	C₄×C₈	C₂³	1	1	yes	characteristic
86	C₄×C₈	C₂³	1	1	yes	characteristic
87	C₄×C₈	C₂³	1	1	yes	characteristic
88	C₄×C₈	C₂³	1	1	yes	characteristic
89	#41	—	2	4	no	—
90	C₂×Q₈	C₄²	1	1	yes	characteristic
91	C₂×D₄	C₄²	1	2	yes	—
92	C₂×D₄	C₄²	1	1	yes	characteristic
93	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄	C₂²×C₄	1	2	yes	—
96	C₂²×C₄	C₂²×C₄	1	2	yes	—
97	C₂²×C₄	C₂²×C₄	1	2	yes	—
98	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
99	M₁₆	—	2	8	no	—
100	M₁₆	—	2	8	no	—
101	M₁₆	—	4	4	no	—
102	M₁₆	M₁₆	1	8	yes	complemented
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	2	8	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₈	C₂²×C₄	1	2	yes	—
110	C₂×C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₈	C₂²×C₄	1	2	yes	—
112	C₂×C₈	C₂²×C₄	1	4	yes	—
113	C₂×C₈	C₂²×C₄	1	4	yes	—
114	C₂×C₈	C₂²×C₄	1	4	yes	—
115	C₂×C₈	C₂²×C₄	1	2	yes	—
116	C₄⋊C₄	C₄²	1	2	yes	—
117	C₄⋊C₄	C₄²	1	2	yes	—
118	C₄⋊C₄	C₄²	1	1	yes	characteristic
119	C₄⋊C₄	C₄²	1	1	yes	characteristic
120	C₂²⋊C₄	C₄²	1	2	yes	—
121	C₂²⋊C₄	C₄²	1	2	yes	—
122	C₂²⋊C₄	C₄²	1	2	yes	—
123	C₄²	C₂²×C₄	1	1	yes	characteristic
124	C₄²	C₂²×C₄	1	2	yes	—
125	C₄²	C₂²×C₄	1	1	yes	characteristic
126	C₄²	C₂²×C₄	1	1	yes	characteristic
127	C₄²	C₂²×C₄	1	2	yes	—
128	C₄²	C₂²×C₄	1	2	yes	—
129	C₄²	C₂⁴	1	1	yes	characteristic
130	C₂³	C₂×C₄²	1	2	yes	—
131	C₂³	C₂×C₄²	1	1	yes	characteristic
132	Q₈	—	2	2	no	—
133	D₄	—	2	4	no	—
134	D₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄²	1	2	yes	—
142	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄²	1	2	yes	—
144	C₂×C₄	C₂×C₄²	1	2	yes	—
145	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
146	C₂×C₄	C₂×M₁₆	1	4	yes	—
147	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	2	4	no	—
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₈	—	2	4	no	—
157	C₈	C₂×M₁₆	1	8	yes	—
158	C₈	#131	1	4	yes	complemented
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	C₄×M₁₆	1	2	yes	—
165	C₂²	C₂²×C₄²	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	2	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	2	no	—
171	C₄	C₄×M₁₆	1	2	yes	—
172	C₄	C₂²×M₁₆	1	2	yes	—
173	C₄	C₂×#131	1	2	yes	—
174	C₄	#568	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	4	2	no	—
177	C₂	C₂×C₄×M₁₆	1	1	yes	characteristic
178	C₂	C₄×#131	1	1	yes	characteristic
179	C₂	#2875	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₄²	C₂⁴
commutator, socle	C₂²	C₂²×C₄²
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1455	C₂	semidirect
5. #1453	C₂	semidirect
6. #1452	C₂	semidirect
7. #1451	C₂	semidirect
8. C₈×M₁₆	C₂	semidirect
9. C₈×M₁₆	C₂	semidirect
10. C₈×M₁₆	C₂	semidirect
11. C₂×#322	C₂	semidirect
102. M₁₆	M₁₆	semidirect
158. C₈	#131	semidirect

Maximal subgroups

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

subgroup	index
#2965	2
C₄×#131	2
#1455	2
#1453	2
#1452	2
#1451	2
C₈×M₁₆	2
C₂×#322	2