Subgroups — #33433

Order 256.

341 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#2929	C₂	1	1	yes	maximal, characteristic
4	#2169	C₂	1	4	yes	maximal, complemented
5	#2168	C₂	1	4	yes	maximal, complemented
6	#2151	C₂	1	1	yes	maximal, characteristic, complemented
7	#2113	C₂	1	1	yes	maximal, characteristic, complemented
8	#2112	C₂	1	1	yes	maximal, characteristic, complemented
9	#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#131	C₂²	1	1	yes	characteristic
11	C₂³×C₈	C₂²	1	1	yes	characteristic
12	#525	C₄	1	1	yes	characteristic
13	#504	—	2	4	no	—
14	C₂×M₃₂	—	2	4	no	—
15	C₂×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×M₃₂	C₂²	1	1	yes	characteristic
17	C₂×M₃₂	C₂²	1	2	yes	—
18	C₂×M₃₂	C₂²	1	2	yes	—
19	C₂×M₃₂	C₂²	1	2	yes	complemented
20	C₂²×C₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆	C₂²	1	2	yes	complemented
23	#435	C₄	1	4	yes	—
24	C₈×D₄	C₂²	1	4	yes	—
25	#424	C₄	1	1	yes	characteristic
26	#408	C₄	1	1	yes	characteristic
27	#407	C₄	1	1	yes	characteristic
28	#405	C₂²	1	1	yes	characteristic
29	C₄⋊C₁₆	C₂²	1	4	yes	complemented
30	C₂²⋊C₁₆	C₂²	1	4	yes	—
31	C₂²⋊C₁₆	C₂²	1	4	yes	complemented
32	#346	C₂²	1	2	yes	complemented
33	C₄×C₁₆	C₂²	1	2	yes	complemented
34	C₂×#41	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
36	#131	—	2	2	no	—
37	#131	—	2	2	no	—
38	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆	C₂×C₄	1	2	yes	—
40	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈	—	2	2	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂×C₄	1	2	yes	—
45	C₂²×C₈	C₂³	1	2	yes	—
46	C₂²×C₈	C₂³	1	2	yes	—
47	C₂²×C₈	C₂³	1	1	yes	characteristic
48	#123	C₈	1	2	yes	—
49	C₂²⋊Q₈	C₈	1	2	yes	—
50	#121	C₈	1	2	yes	—
51	#120	C₈	1	2	yes	—
52	C₄×D₄	C₂×C₄	1	4	yes	—
53	#117	C₂×C₄	1	1	yes	characteristic
54	M₃₂	—	2	2	no	—
55	M₃₂	—	2	2	no	—
56	M₃₂	—	2	4	no	—
57	M₃₂	—	2	4	no	—
58	M₃₂	—	2	4	no	—
59	C₂×C₁₆	—	2	2	no	—
60	C₂×C₁₆	—	2	4	no	—
61	C₂×C₁₆	D₄	1	4	yes	complemented
62	C₂×C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₆	C₂³	1	2	yes	—
64	C₂×C₁₆	C₂³	1	2	yes	—
65	C₂×C₁₆	C₂³	1	1	yes	characteristic
66	C₂×C₁₆	C₂³	1	1	yes	characteristic
67	C₄⋊C₈	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈	C₂×C₄	1	4	yes	—
70	#97	C₂×C₄	1	2	yes	—
71	C₄×C₈	C₂³	1	2	yes	—
72	C₂⁴	C₂×C₈	1	1	yes	characteristic
73	#41	—	2	2	no	—
74	#41	—	2	2	no	—
75	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
76	C₂×D₄	C₂×C₈	1	2	yes	—
77	C₂×D₄	C₂×C₈	1	1	yes	characteristic
78	C₂×D₄	C₂×C₈	1	4	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	C₂×C₈	1	2	yes	—
82	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄	C₂²×C₄	1	2	yes	—
84	C₂²×C₄	C₂²×C₄	1	2	yes	—
85	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₈	#41	1	2	yes	—
101	C₂×C₈	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄	C₂×C₈	1	2	yes	—
103	C₄⋊C₄	C₂×C₈	1	4	yes	—
104	C₂²⋊C₄	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄	C₂×C₈	1	4	yes	—
106	C₂²⋊C₄	C₂×C₈	1	4	yes	—
107	C₄²	C₂²×C₄	1	2	yes	—
108	C₁₆	—	2	2	no	—
109	C₁₆	—	2	2	no	—
110	C₁₆	—	2	2	no	—
111	C₁₆	C₂×D₄	1	2	yes	—
112	C₁₆	C₂×D₄	1	2	yes	—
113	C₂³	—	2	2	no	—
114	C₂³	—	4	2	no	—
115	C₂³	C₂²×C₈	1	2	yes	—
116	C₂³	C₂²×C₈	1	2	yes	—
117	C₂³	C₂²×C₈	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	D₄	—	4	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	C₄×D₄	1	2	yes	—
129	C₂×C₄	C₂²×C₈	1	4	yes	—
130	C₂×C₄	C₂²×C₈	1	2	yes	—
131	C₂×C₄	C₂²×C₈	1	2	yes	—
132	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄	#131	1	2	yes	—
134	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	2	2	no	—
138	C₈	C₄×D₄	1	2	yes	—
139	C₈	C₂²×D₄	1	1	yes	characteristic
140	C₈	C₂×#41	1	1	yes	characteristic
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	2	4	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	4	no	—
148	C₂²	C₈×D₄	1	2	yes	—
149	C₂²	C₂³×C₈	1	1	yes	characteristic
150	C₄	—	2	4	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	C₈×D₄	1	2	yes	—
154	C₄	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄	C₂×#131	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	—	4	2	no	—
158	C₂	—	4	2	no	—
159	C₂	C₂×C₈×D₄	1	1	yes	characteristic
160	C₂	#3407	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	#525	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#504	C₂	semidirect
2. C₂²×M₃₂	C₂	semidirect
4. #2169	C₂	semidirect
5. #2168	C₂	semidirect
6. #2151	C₂	semidirect
7. #2113	C₂	semidirect
8. #2112	C₂	semidirect
9. #2108	C₂	semidirect
19. C₂×M₃₂	C₂²	semidirect
22. C₂²×C₁₆	C₂²	semidirect
29. C₄⋊C₁₆	C₂²	semidirect
31. C₂²⋊C₁₆	C₂²	semidirect
32. #346	C₂²	semidirect
33. C₄×C₁₆	C₂²	semidirect
61. C₂×C₁₆	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504	2
C₂²×M₃₂	2
#2929	2
#2169	2
#2168	2
#2151	2
#2113	2
#2112	2
#2108	2