Subgroups — #33476

Order 256.

249 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2960	C₂	1	1	yes	maximal, characteristic
2	#2183	C₂	1	2	yes	maximal, complemented
3	#2183	C₂	1	4	yes	maximal, complemented
4	Q₈×C₁₆	C₂	1	2	yes	maximal, complemented
5	#2151	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	#2108	C₂	1	2	yes	maximal
8	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
9	#533	C₄	1	1	yes	characteristic
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆	C₂²	1	2	yes	—
13	#446	C₄	1	2	yes	—
14	#446	C₄	1	2	yes	—
15	C₈×Q₈	C₂²	1	2	yes	—
16	C₈×Q₈	C₂²	1	2	yes	—
17	#424	C₄	1	2	yes	—
18	#423	C₄	1	1	yes	characteristic
19	#405	C₂²	1	2	yes	—
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₄⋊C₁₆	C₂²	1	4	yes	—
25	C₄⋊C₁₆	C₂²	1	4	yes	—
26	#346	C₂²	1	2	yes	—
27	#346	C₂²	1	4	yes	—
28	C₄×C₁₆	C₂²	1	2	yes	—
29	C₄×C₁₆	C₂²	1	4	yes	—
30	C₂×M₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈	C₈	1	1	yes	characteristic
34	C₄⋊Q₈	C₈	1	1	yes	characteristic
35	#125	C₈	1	2	yes	—
36	C₂²⋊Q₈	C₈	1	2	yes	—
37	C₂²⋊Q₈	C₈	1	2	yes	—
38	C₄×Q₈	C₂×C₄	1	2	yes	—
39	C₄×Q₈	C₂×C₄	1	2	yes	—
40	#117	C₂×C₄	1	2	yes	—
41	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
42	M₃₂	—	2	2	no	—
43	M₃₂	Q₈	1	4	yes	complemented
44	C₂×C₁₆	—	2	4	no	—
45	C₂×C₁₆	C₂³	1	2	yes	—
46	C₂×C₁₆	C₂³	1	2	yes	—
47	C₂×C₁₆	C₂³	1	2	yes	—
48	C₂×C₁₆	C₂³	1	2	yes	—
49	C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₄⋊C₈	C₂×C₄	1	2	yes	—
51	C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₄⋊C₈	C₂×C₄	1	4	yes	—
53	#97	C₂×C₄	1	4	yes	—
54	C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈	C₂³	1	1	yes	characteristic
57	C₄×C₈	C₂³	1	4	yes	—
58	C₄×C₈	C₂³	1	1	yes	characteristic
59	C₂×Q₈	C₂×C₈	1	2	yes	—
60	C₂×Q₈	C₂×C₈	1	2	yes	—
61	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
64	M₁₆	—	2	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈	C₂²×C₄	1	4	yes	—
69	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈	C₂×Q₈	1	2	yes	—
71	C₂×C₈	#41	1	2	yes	—
72	C₂×C₈	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄	C₂×C₈	1	2	yes	—
74	C₄⋊C₄	C₂×C₈	1	2	yes	—
75	C₄⋊C₄	C₂×C₈	1	4	yes	—
76	C₄⋊C₄	C₂×C₈	1	4	yes	—
77	C₄⋊C₄	C₂×C₈	1	2	yes	—
78	C₄⋊C₄	C₂×C₈	1	2	yes	—
79	C₂²⋊C₄	C₂×C₈	1	4	yes	—
80	C₄²	C₂×C₈	1	1	yes	characteristic
81	C₄²	C₂×C₈	1	1	yes	characteristic
82	C₄²	C₂²×C₄	1	1	yes	characteristic
83	C₄²	C₂²×C₄	1	1	yes	characteristic
84	C₄²	C₂²×C₄	1	4	yes	—
85	C₁₆	—	2	2	no	—
86	C₁₆	—	2	2	no	—
87	C₁₆	—	2	2	no	—
88	C₁₆	C₂×Q₈	1	4	yes	—
89	C₂³	C₂²×C₈	1	1	yes	characteristic
90	Q₈	—	2	4	no	—
91	Q₈	—	4	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Q₈	1	2	yes	—
95	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄	C₂²×C₈	1	4	yes	—
99	C₂×C₄	C₂²×C₈	1	4	yes	—
100	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄	#131	1	2	yes	—
103	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
104	C₈	—	2	1	no	—
105	C₈	—	2	1	no	—
106	C₈	—	2	4	no	—
107	C₈	C₄×Q₈	1	2	yes	—
108	C₈	C₂²×Q₈	1	1	yes	characteristic
109	C₈	C₂×#41	1	1	yes	characteristic
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	C₈×Q₈	1	2	yes	—
113	C₂²	C₂³×C₈	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	2	4	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	4	no	—
119	C₄	C₈×Q₈	1	2	yes	—
120	C₄	#504	1	2	yes	—
121	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
122	C₄	C₂×#131	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	C₂×C₈×Q₈	1	1	yes	characteristic
125	C₂	C₂×#504	1	1	yes	characteristic
126	C₂	#3407	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	C₂³	C₂²×C₈
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
2. #2183	C₂	semidirect
3. #2183	C₂	semidirect
4. Q₈×C₁₆	C₂	semidirect
43. M₃₂	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#2960	2
#2183	2
Q₈×C₁₆	2
#2151	2
C₂×C₄⋊C₁₆	2
#2108	2
C₄×M₃₂	2