Subgroups — #17798

Order 256.

333 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2989	C₂	1	2	yes	maximal
2	C₄×#131	C₂	1	1	yes	maximal, characteristic
3	#1569	C₂	1	4	yes	maximal, complemented
4	#1568	C₂	1	2	yes	maximal, complemented
5	#1566	C₂	1	2	yes	maximal, complemented
6	#1565	C₂	1	2	yes	maximal, complemented
7	#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#334	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131	C₄	1	1	yes	characteristic
10	C₂×#131	C₄	1	1	yes	characteristic
11	C₄×#41	C₂²	1	1	yes	characteristic
12	#446	—	2	2	no	—
13	#436	—	2	2	no	—
14	#435	—	2	4	no	—
15	#432	C₂²	1	4	yes	—
16	#431	C₂²	1	2	yes	—
17	#423	C₂²	1	4	yes	—
18	C₂×C₄⋊C₈	C₂²	1	2	yes	—
19	#405	C₄	1	1	yes	characteristic
20	#405	C₄	1	2	yes	—
21	#405	C₄	1	2	yes	—
22	#405	C₄	1	1	yes	characteristic
23	C₄×M₁₆	C₂²	1	1	yes	characteristic
24	C₄×M₁₆	C₂²	1	2	yes	—
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂²	1	2	yes	—
27	#335	C₂²	1	2	yes	—
28	#335	C₂²	1	4	yes	—
29	#334	C₂²	1	2	yes	—
30	#334	C₂²	1	2	yes	—
31	#334	C₂²	1	1	yes	characteristic
32	#334	C₂²	1	1	yes	characteristic
33	#334	C₂²	1	4	yes	—
34	C₂×#41	C₂×C₄	1	1	yes	characteristic
35	#131	—	2	4	no	—
36	#131	—	2	4	no	—
37	C₂×M₁₆	—	2	4	no	—
38	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆	C₂×C₄	1	2	yes	—
40	C₂×M₁₆	C₂×C₄	1	2	yes	—
41	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
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	C₂×C₄	1	2	yes	—
46	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×Q₈	D₄	1	1	yes	characteristic
48	C₄×Q₈	Q₈	1	1	yes	characteristic
49	C₄×D₄	D₄	1	2	yes	—
50	C₄×D₄	D₄	1	1	yes	characteristic
51	C₄×D₄	Q₈	1	2	yes	—
52	C₄×D₄	Q₈	1	1	yes	characteristic
53	#117	C₂×C₄	1	2	yes	—
54	#117	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	C₂×C₄²	C₂³	1	2	yes	—
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	2	4	no	—
59	C₄⋊C₈	C₂³	1	4	yes	—
60	C₄⋊C₈	C₂³	1	2	yes	—
61	C₄⋊C₈	C₂³	1	2	yes	—
62	C₂²⋊C₈	—	2	2	no	—
63	C₂²⋊C₈	—	2	4	no	—
64	#97	—	2	4	no	—
65	#97	C₂×C₄	1	2	yes	—
66	#97	C₂×C₄	1	2	yes	—
67	#97	C₂×C₄	1	2	yes	—
68	C₄×C₈	—	2	2	no	—
69	C₄×C₈	C₂×C₄	1	2	yes	—
70	C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈	C₂×C₄	1	2	yes	—
73	C₄×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₈	C₂³	1	1	yes	characteristic
75	C₄×C₈	C₂³	1	2	yes	—
76	#41	—	2	2	no	—
77	#41	—	2	2	no	—
78	C₂×Q₈	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₄	C₄⋊C₄	1	2	yes	—
80	C₂×D₄	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄	C₂²×C₄	1	2	yes	—
82	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄	1	2	yes	—
85	C₂²×C₄	C₂×Q₈	1	2	yes	—
86	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	4	no	—
89	M₁₆	—	2	4	no	—
90	M₁₆	—	8	4	no	—
91	M₁₆	M₁₆	1	4	yes	complemented
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	8	2	no	—
99	C₂×C₈	M₁₆	1	4	yes	complemented
100	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈	C₂²×C₄	1	2	yes	—
103	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
107	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
108	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
109	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
110	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
111	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
112	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
113	C₄²	C₂²×C₄	1	1	yes	characteristic
114	C₄²	C₂²×C₄	1	2	yes	—
115	C₄²	C₂×D₄	1	1	yes	characteristic
116	C₄²	C₂×D₄	1	2	yes	—
117	C₄²	C₂×Q₈	1	2	yes	—
118	C₄²	C₂×Q₈	1	1	yes	characteristic
119	C₄²	C₂⁴	1	1	yes	characteristic
120	C₂³	C₂×C₄⋊C₄	1	2	yes	—
121	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
122	Q₈	—	2	2	no	—
123	D₄	—	2	4	no	—
124	D₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₂×M₁₆	1	2	yes	—
138	C₂×C₄	C₂×M₁₆	1	2	yes	—
139	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₈	—	4	2	no	—
146	C₈	—	4	4	no	—
147	C₈	—	4	2	no	—
148	C₈	C₂×M₁₆	1	2	yes	—
149	C₈	C₂×M₁₆	1	4	yes	—
150	C₈	C₂×M₁₆	1	2	yes	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#423	1	2	yes	—
157	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	2	no	—
164	C₄	#423	1	2	yes	—
165	C₄	C₂²×M₁₆	1	1	yes	characteristic
166	C₄	C₂²×M₁₆	1	1	yes	characteristic
167	C₄	#577	1	2	yes	—
168	C₂	—	2	2	no	—
169	C₂	—	4	2	no	—
170	C₂	C₂×#423	1	1	yes	characteristic
171	C₂	#2912	1	1	yes	characteristic
172	C₂	#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂²×C₄⋊C₄
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1569	C₂	semidirect
4. #1568	C₂	semidirect
5. #1566	C₂	semidirect
6. #1565	C₂	semidirect
7. #1564	C₂	semidirect
8. C₂×#334	C₂	semidirect
91. M₁₆	M₁₆	semidirect
99. C₂×C₈	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2989	2
C₄×#131	2
#1569	2
#1568	2
#1566	2
#1565	2
#1564	2
C₂×#334	2