Subgroups — C₂×#1285 #7947

Order 256.

373 conjugacy classes of subgroups.

The subgroup lattice has 195 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×#432	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#1285	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₄⋊C₄	—	2	1	no	—
6	C₂²×C₄²	C₄	1	2	yes	—
7	C₂²×C₄²	C₂²	1	1	yes	characteristic
8	#432	—	2	1	no	—
9	#432	—	2	1	no	—
10	#432	—	2	1	no	—
11	#432	—	2	1	no	—
12	#432	C₄	1	2	yes	complemented
13	#432	C₄	1	2	yes	complemented
14	#432	C₂²	1	2	yes	—
15	#432	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈	—	2	1	no	—
17	C₂×C₄⋊C₈	C₄	1	2	yes	complemented
18	C₂×C₂²⋊C₈	—	2	1	no	—
19	C₂×C₂²⋊C₈	—	2	1	no	—
20	C₂×#97	—	2	1	no	—
21	C₂×#97	C₄	1	2	yes	complemented
22	C₄×C₄⋊C₄	—	2	1	no	—
23	C₄×C₄⋊C₄	—	2	1	no	—
24	C₄×C₄⋊C₄	C₄	1	2	yes	—
25	C₄×C₄⋊C₄	C₂²	1	2	yes	—
26	C₂×#95	—	2	1	no	—
27	C₂³×C₄	—	2	2	no	—
28	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄	D₄	1	1	yes	characteristic
30	C₂³×C₄	D₄	1	1	yes	characteristic
31	C₂²×C₈	—	2	2	no	—
32	C₂²×C₈	—	2	2	no	—
33	C₂×C₄⋊C₄	—	2	8	no	—
34	C₂×C₄⋊C₄	—	2	2	no	—
35	C₂×C₄⋊C₄	—	2	2	no	—
36	C₂×C₄²	—	2	2	no	—
37	C₂×C₄²	—	4	4	no	—
38	C₂×C₄²	C₂×C₄	1	4	yes	—
39	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²	D₄	1	1	yes	characteristic
45	C₂×C₄²	D₄	1	1	yes	characteristic
46	C₂×C₄²	D₄	1	1	yes	characteristic
47	C₂×C₄²	D₄	1	1	yes	characteristic
48	C₂×C₄²	Q₈	1	1	yes	characteristic
49	C₂×C₄²	Q₈	1	1	yes	characteristic
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	C₄⋊C₈	—	2	2	no	—
52	C₄⋊C₈	—	2	2	no	—
53	C₄⋊C₈	—	2	2	no	—
54	C₄⋊C₈	C₂×C₄	1	4	yes	complemented
55	C₂²⋊C₈	—	2	2	no	—
56	C₂²⋊C₈	—	2	2	no	—
57	C₂²⋊C₈	—	2	2	no	—
58	C₂²⋊C₈	—	2	2	no	—
59	#97	—	2	2	no	—
60	#97	—	2	2	no	—
61	#97	—	2	2	no	—
62	#97	C₂×C₄	1	4	yes	complemented
63	#95	—	2	2	no	—
64	#95	—	2	2	no	—
65	C₂⁴	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	2	4	no	—
69	C₂²×C₄	—	2	8	no	—
70	C₂²×C₄	—	2	4	no	—
71	C₂²×C₄	—	4	2	no	—
72	C₂²×C₄	—	4	4	no	—
73	C₂²×C₄	—	4	4	no	—
74	C₂²×C₄	C₄²	1	1	yes	characteristic
75	C₂²×C₄	C₄²	1	1	yes	characteristic
76	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	4	4	no	—
95	C₂×C₈	—	4	4	no	—
96	C₄⋊C₄	—	2	8	no	—
97	C₄⋊C₄	—	2	8	no	—
98	C₄²	—	4	4	no	—
99	C₄²	—	4	4	no	—
100	C₄²	C₄²	1	1	yes	characteristic
101	C₄²	C₄²	1	1	yes	characteristic
102	C₄²	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²	C₂²⋊C₄	1	1	yes	characteristic
106	C₄²	C₄⋊C₄	1	1	yes	characteristic
107	C₄²	C₄⋊C₄	1	1	yes	characteristic
108	C₄²	C₄⋊C₄	1	1	yes	characteristic
109	C₄²	C₄⋊C₄	1	1	yes	characteristic
110	C₄²	C₄⋊C₄	1	1	yes	characteristic
111	C₄²	C₄⋊C₄	1	1	yes	characteristic
112	C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₄²	C₂²×C₄	1	1	yes	characteristic
114	C₄²	C₂×D₄	1	1	yes	characteristic
115	C₄²	C₂×Q₈	1	1	yes	characteristic
116	C₂³	—	2	2	no	—
117	C₂³	—	2	2	no	—
118	C₂³	#95	1	1	yes	characteristic
119	C₂³	#95	1	1	yes	characteristic
120	C₂³	#100	1	2	yes	—
121	C₂³	#101	1	2	yes	—
122	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	16	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	#95	1	1	yes	characteristic
147	C₂×C₄	#95	1	1	yes	characteristic
148	C₂×C₄	#95	1	1	yes	characteristic
149	C₂×C₄	#95	1	1	yes	characteristic
150	C₂×C₄	#95	1	1	yes	characteristic
151	C₂×C₄	#95	1	1	yes	characteristic
152	C₂×C₄	#104	1	4	yes	—
153	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
154	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₈	—	4	4	no	—
160	C₈	—	4	4	no	—
161	C₂²	—	2	4	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	H(3,ℤ/4)	1	2	yes	—
168	C₂²	#339	1	2	yes	—
169	C₂²	#343	1	2	yes	—
170	C₂²	#343	1	2	yes	—
171	C₂²	C₂×#95	1	1	yes	characteristic
172	C₂²	C₂×#100	1	1	yes	characteristic
173	C₂²	C₂×#101	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	8	no	—
185	C₄	#339	1	2	yes	—
186	C₄	C₂×#104	1	2	yes	—
187	C₂	—	2	1	no	—
188	C₂	—	2	2	no	—
189	C₂	—	2	1	no	—
190	C₂	#1285	1	4	yes	direct product factor
191	C₂	C₂×H(3,ℤ/4)	1	1	yes	characteristic
192	C₂	C₂×#339	1	1	yes	characteristic
193	C₂	C₂×#343	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#95
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1285	C₂	direct (N × H)
12. #432	C₄	semidirect
13. #432	C₄	semidirect
17. C₂×C₄⋊C₈	C₄	semidirect
21. C₂×#97	C₄	semidirect
54. C₄⋊C₈	C₂×C₄	semidirect
62. #97	C₂×C₄	semidirect
190. C₂	#1285	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#432	2
C₂×C₄×C₄⋊C₄	2
#1285	2

Subgroups — C2×#1285 #7947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1285 #7947