Subgroups — #33421

Order 256.

291 conjugacy classes of subgroups.

The subgroup lattice has 133 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#131	C₂	1	1	yes	maximal, characteristic
2	#2164	C₂	1	4	yes	maximal, complemented
3	#2163	C₂	1	4	yes	maximal, complemented
4	#2151	C₂	1	2	yes	maximal, complemented
5	#2113	C₂	1	2	yes	maximal
6	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#346	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131	C₄	1	1	yes	characteristic
9	C₂×#131	C₂²	1	1	yes	characteristic
10	C₄×#41	C₄	1	1	yes	characteristic
11	C₂×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₆	C₂²	1	2	yes	—
13	#405	C₄	1	2	yes	—
14	#405	C₄	1	1	yes	characteristic
15	#405	C₂²	1	1	yes	characteristic
16	#405	C₂²	1	2	yes	—
17	C₄×M₁₆	C₄	1	1	yes	characteristic
18	C₄×M₁₆	C₄	1	2	yes	—
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂²	1	2	yes	—
21	C₄⋊C₁₆	C₂²	1	8	yes	—
22	C₂²⋊C₁₆	C₂²	1	8	yes	—
23	#346	C₂²	1	2	yes	—
24	#346	C₂²	1	2	yes	—
25	#346	C₂²	1	2	yes	—
26	C₄×C₁₆	C₂²	1	2	yes	—
27	C₂×#41	C₂×C₄	1	1	yes	characteristic
28	#131	—	2	4	no	—
29	#131	—	2	4	no	—
30	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆	C₂×C₄	1	2	yes	—
32	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆	C₂×C₄	1	2	yes	—
34	C₂²×C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈	C₂³	1	2	yes	—
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	C₄×Q₈	C₈	1	2	yes	—
39	C₄×D₄	C₈	1	4	yes	—
40	C₄×D₄	C₈	1	2	yes	—
41	#117	C₂×C₄	1	2	yes	—
42	#117	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²	C₂×C₄	1	2	yes	—
44	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
45	M₃₂	—	2	4	no	—
46	C₂×C₁₆	—	2	4	no	—
47	C₂×C₁₆	C₂³	1	4	yes	—
48	C₂×C₁₆	C₂³	1	4	yes	—
49	#97	C₂×C₄	1	2	yes	—
50	#97	C₂×C₄	1	2	yes	—
51	#97	C₂×C₄	1	2	yes	—
52	C₄×C₈	C₂×C₄	1	2	yes	—
53	C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈	C₂×C₄	1	2	yes	—
55	C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈	C₂³	1	1	yes	characteristic
57	C₄×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₈	C₂³	1	2	yes	—
59	#41	—	2	2	no	—
60	#41	—	2	2	no	—
61	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
62	C₂×D₄	C₂×C₈	1	2	yes	—
63	C₂×D₄	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄	C₂×C₈	1	4	yes	—
65	C₂²×C₄	C₂×C₈	1	2	yes	—
66	C₂²×C₄	C₂²×C₄	1	2	yes	—
67	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	4	no	—
71	M₁₆	—	2	4	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	M₁₆	1	4	yes	—
75	C₂×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈	#41	1	4	yes	—
81	C₂×C₈	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄	C₂×C₈	1	4	yes	—
83	C₄⋊C₄	C₂×C₈	1	2	yes	—
84	C₂²⋊C₄	C₂×C₈	1	4	yes	—
85	C₂²⋊C₄	C₂×C₈	1	2	yes	—
86	C₄²	C₂×C₈	1	4	yes	—
87	C₄²	C₂×C₈	1	2	yes	—
88	C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₄²	C₂²×C₄	1	2	yes	—
91	C₁₆	—	2	4	no	—
92	C₁₆	—	2	4	no	—
93	C₂³	C₂²×C₈	1	2	yes	—
94	C₂³	C₂²×C₈	1	1	yes	characteristic
95	Q₈	—	2	2	no	—
96	D₄	—	2	4	no	—
97	D₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	#117	1	2	yes	—
103	C₂×C₄	C₂²×C₈	1	2	yes	—
104	C₂×C₄	C₂²×C₈	1	2	yes	—
105	C₂×C₄	C₂²×C₈	1	4	yes	—
106	C₂×C₄	C₂²×C₈	1	2	yes	—
107	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄	C₂×M₁₆	1	2	yes	—
109	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	#117	1	2	yes	—
113	C₈	C₂×M₁₆	1	2	yes	—
114	C₈	C₂×M₁₆	1	2	yes	—
115	C₈	C₂×#41	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	#431	1	2	yes	—
120	C₂²	C₂³×C₈	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	#431	1	2	yes	—
126	C₄	C₂×#117	1	1	yes	characteristic
127	C₄	C₂²×M₁₆	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	C₂×#431	1	1	yes	characteristic
131	C₂	#3407	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	C₂×#41	C₂×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
2. #2164	C₂	semidirect
3. #2163	C₂	semidirect
4. #2151	C₂	semidirect
6. C₄×M₃₂	C₂	semidirect
7. C₂×#346	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×#131	2
#2164	2
#2163	2
#2151	2
#2113	2
C₄×M₃₂	2
C₂×#346	2