Subgroups — #17328

Order 256.

375 conjugacy classes of subgroups.

The subgroup lattice has 178 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	C₈×#41	C₂	1	1	yes	maximal, characteristic
3	#2922	C₂	1	1	yes	maximal, characteristic
4	#1454	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	1	yes	maximal, characteristic, complemented
8	#1450	C₂	1	2	yes	maximal, complemented
9	C₈×M₁₆	C₂	1	2	yes	maximal, complemented
10	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#41	C₂²	1	1	yes	characteristic
13	#446	C₄	1	2	yes	—
14	C₈×Q₈	C₄	1	2	yes	—
15	#436	C₄	1	2	yes	—
16	#435	C₄	1	4	yes	—
17	C₈×D₄	C₄	1	2	yes	—
18	C₈×D₄	C₄	1	4	yes	—
19	#433	C₄	1	4	yes	—
20	#431	C₂²	1	2	yes	—
21	#431	C₂²	1	2	yes	—
22	#431	C₂²	1	1	yes	characteristic
23	#431	C₂²	1	2	yes	—
24	#431	C₂²	1	1	yes	characteristic
25	#431	C₂²	1	2	yes	—
26	#424	C₄	1	2	yes	—
27	#408	C₄	1	2	yes	—
28	C₄×M₁₆	C₂²	1	1	yes	characteristic
29	C₄×M₁₆	C₂²	1	2	yes	—
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂²	1	2	yes	—
34	#322	C₂²	1	2	yes	—
35	#322	C₂²	1	2	yes	—
36	#322	C₂²	1	2	yes	—
37	#322	C₂²	1	2	yes	—
38	#322	C₂²	1	2	yes	—
39	C₈²	C₂²	1	1	yes	characteristic
40	C₈²	C₂²	1	1	yes	characteristic
41	C₈²	C₂²	1	1	yes	characteristic
42	C₈²	C₂²	1	1	yes	characteristic
43	C₈²	C₂²	1	2	yes	—
44	C₂×#41	C₂×C₄	1	1	yes	characteristic
45	C₂×M₁₆	C₂×C₄	1	2	yes	—
46	C₂×M₁₆	C₂×C₄	1	4	yes	—
47	C₂²×C₈	C₂×C₄	1	2	yes	—
48	C₂²×C₈	C₂×C₄	1	2	yes	—
49	C₂²×C₈	C₂×C₄	1	4	yes	—
50	C₂²×C₈	C₂×C₄	1	2	yes	—
51	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
52	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
53	C₄×D₄	C₂×C₄	1	2	yes	—
54	C₄×D₄	C₂×C₄	1	1	yes	characteristic
55	C₄×D₄	C₂×C₄	1	2	yes	—
56	C₄×D₄	C₂×C₄	1	1	yes	characteristic
57	#117	C₂×C₄	1	1	yes	characteristic
58	#117	C₂×C₄	1	2	yes	—
59	C₂×C₄²	C₂³	1	1	yes	characteristic
60	C₂×C₄²	C₂³	1	2	yes	—
61	C₄⋊C₈	C₂×C₄	1	4	yes	—
62	C₄⋊C₈	C₂×C₄	1	4	yes	—
63	C₄⋊C₈	C₂×C₄	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	C₂²⋊C₈	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈	C₂×C₄	1	2	yes	—
70	C₂²⋊C₈	C₂×C₄	1	4	yes	—
71	C₂²⋊C₈	C₂×C₄	1	2	yes	—
72	C₂²⋊C₈	C₂×C₄	1	4	yes	—
73	#97	C₂×C₄	1	2	yes	—
74	#97	C₂×C₄	1	4	yes	—
75	C₄×C₈	C₂×C₄	1	2	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	C₂³	1	1	yes	characteristic
80	C₄×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₈	C₂³	1	1	yes	characteristic
83	C₄×C₈	C₂³	1	1	yes	characteristic
84	C₄×C₈	C₂³	1	2	yes	—
85	C₄×C₈	C₂³	1	2	yes	—
86	C₄×C₈	C₂³	1	2	yes	—
87	C₄×C₈	C₂³	1	1	yes	characteristic
88	#41	—	2	4	no	—
89	C₂×Q₈	C₄²	1	1	yes	characteristic
90	C₂×D₄	C₄²	1	2	yes	—
91	C₂×D₄	C₄²	1	1	yes	characteristic
92	C₂²×C₄	C₂²×C₄	1	2	yes	—
93	C₂²×C₄	C₂²×C₄	1	2	yes	—
94	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄	C₂²×C₄	1	2	yes	—
97	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
98	M₁₆	—	2	4	no	—
99	M₁₆	—	2	8	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	8	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₈	C₂²×C₄	1	4	yes	—
108	C₂×C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₈	C₂²×C₄	1	4	yes	—
110	C₂×C₈	C₂²×C₄	1	4	yes	—
111	C₂×C₈	C₂²×C₄	1	2	yes	—
112	C₂×C₈	C₂²×C₄	1	2	yes	—
113	C₄⋊C₄	C₄²	1	2	yes	—
114	C₄⋊C₄	C₄²	1	2	yes	—
115	C₄⋊C₄	C₄²	1	1	yes	characteristic
116	C₄⋊C₄	C₄²	1	1	yes	characteristic
117	C₂²⋊C₄	C₄²	1	2	yes	—
118	C₂²⋊C₄	C₄²	1	2	yes	—
119	C₂²⋊C₄	C₄²	1	2	yes	—
120	C₄²	C₂²×C₄	1	1	yes	characteristic
121	C₄²	C₂²×C₄	1	2	yes	—
122	C₄²	C₂²×C₄	1	1	yes	characteristic
123	C₄²	C₂²×C₄	1	1	yes	characteristic
124	C₄²	C₂²×C₄	1	2	yes	—
125	C₄²	C₂²×C₄	1	2	yes	—
126	C₄²	C₂⁴	1	1	yes	characteristic
127	C₂³	C₂×C₄²	1	2	yes	—
128	C₂³	C₂×C₄²	1	1	yes	characteristic
129	Q₈	—	2	2	no	—
130	D₄	—	2	4	no	—
131	D₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄²	1	2	yes	—
140	C₂×C₄	C₂×C₄²	1	2	yes	—
141	C₂×C₄	C₂×C₄²	1	2	yes	—
142	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄	#131	1	4	yes	—
144	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	2	2	no	—
149	C₈	—	2	2	no	—
150	C₈	—	2	4	no	—
151	C₈	—	2	2	no	—
152	C₈	—	2	4	no	—
153	C₈	—	2	4	no	—
154	C₈	#131	1	4	yes	—
155	C₈	#131	1	4	yes	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	2	2	no	—
161	C₂²	#405	1	2	yes	—
162	C₂²	C₂²×C₄²	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	2	no	—
167	C₄	—	2	2	no	—
168	C₄	#405	1	2	yes	—
169	C₄	C₂×#131	1	2	yes	—
170	C₄	C₂×#131	1	2	yes	—
171	C₄	C₂×#131	1	2	yes	—
172	C₂	—	2	2	no	—
173	C₂	—	4	2	no	—
174	C₂	C₂×#405	1	1	yes	characteristic
175	C₂	C₄×#131	1	1	yes	characteristic
176	C₂	C₄×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂³
commutator, socle	C₂²	C₂²×C₄²
Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1454	C₂	semidirect
5. #1453	C₂	semidirect
6. #1452	C₂	semidirect
7. #1451	C₂	semidirect
8. #1450	C₂	semidirect
9. C₈×M₁₆	C₂	semidirect
10. C₈×M₁₆	C₂	semidirect
11. C₂×C₈²	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2965	2
C₈×#41	2
#2922	2
#1454	2
#1453	2
#1452	2
#1451	2
#1450	2
C₈×M₁₆	2
C₂×C₈²	2