Subgroups — #7938

Order 256.

330 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2920	C₂	1	1	yes	maximal, characteristic
2	C₂×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#1283	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆	C₄	1	2	yes	—
6	C₂²×M₁₆	C₄	1	2	yes	—
7	C₂²×C₄²	C₂²	1	1	yes	characteristic
8	#432	—	2	1	no	—
9	#432	C₄	1	2	yes	—
10	#431	—	2	1	no	—
11	#431	C₂²	1	2	yes	—
12	#423	—	2	1	no	—
13	#423	—	2	1	no	—
14	#423	—	2	1	no	—
15	#423	—	2	1	no	—
16	#423	—	2	1	no	—
17	C₂×C₄⋊C₈	C₄	1	2	yes	—
18	C₂×C₄⋊C₈	C₂²	1	2	yes	—
19	#407	C₄	1	2	yes	—
20	C₄×M₁₆	—	2	1	no	—
21	C₄×M₁₆	—	2	1	no	—
22	C₄×M₁₆	—	2	1	no	—
23	C₄×M₁₆	C₄	1	2	yes	—
24	C₄×M₁₆	C₂²	1	2	yes	—
25	C₂×#97	—	2	1	no	—
26	C₂×C₄×C₈	—	2	1	no	—
27	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆	—	2	2	no	—
31	C₂×M₁₆	—	2	8	no	—
32	C₂×M₁₆	—	2	2	no	—
33	C₂×M₁₆	—	2	2	no	—
34	C₂×M₁₆	—	2	2	no	—
35	C₂×M₁₆	—	4	4	no	—
36	C₂×M₁₆	C₂×C₄	1	4	yes	—
37	C₂²×C₈	—	2	2	no	—
38	C₂²×C₈	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	D₄	1	1	yes	characteristic
43	C₂×C₄²	D₄	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	—	2	2	no	—
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	—	2	2	no	—
57	C₂²⋊C₈	—	2	2	no	—
58	C₂²⋊C₈	C₂×C₄	1	4	yes	—
59	#97	—	2	2	no	—
60	#97	—	2	2	no	—
61	#97	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	2	2	no	—
65	C₂⁴	C₄²	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	—	4	2	no	—
70	C₂²×C₄	C₄²	1	1	yes	characteristic
71	C₂²×C₄	C₄²	1	1	yes	characteristic
72	C₂²×C₄	C₄²	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄	C₂²⋊C₄	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	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
88	M₁₆	—	2	8	no	—
89	M₁₆	—	2	4	no	—
90	M₁₆	—	4	4	no	—
91	M₁₆	—	4	4	no	—
92	M₁₆	—	4	4	no	—
93	M₁₆	M₁₆	1	8	yes	complemented
94	C₂×C₈	—	2	8	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	4	4	no	—
99	C₄²	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²	C₂²⋊C₄	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	C₄⋊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	C₂×D₄	1	1	yes	characteristic
112	C₄²	C₂×D₄	1	1	yes	characteristic
113	C₄²	C₂×D₄	1	1	yes	characteristic
114	C₄²	C₂×Q₈	1	1	yes	characteristic
115	C₂³	—	2	2	no	—
116	C₂³	—	2	2	no	—
117	C₂³	—	4	1	no	—
118	C₂³	#95	1	1	yes	characteristic
119	C₂³	#95	1	1	yes	characteristic
120	C₂³	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
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	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	#95	1	1	yes	characteristic
140	C₂×C₄	#95	1	1	yes	characteristic
141	C₂×C₄	#95	1	1	yes	characteristic
142	C₂×C₄	#95	1	1	yes	characteristic
143	C₂×C₄	#95	1	1	yes	characteristic
144	C₂×C₄	#95	1	1	yes	characteristic
145	C₂×C₄	#100	1	2	yes	—
146	C₂×C₄	#101	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄	C₂×M₁₆	1	4	yes	—
154	C₈	—	2	8	no	—
155	C₈	—	4	4	no	—
156	C₈	—	4	4	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	4	4	no	—
165	C₂²	#343	1	2	yes	—
166	C₂²	C₂×#95	1	1	yes	characteristic
167	C₂²	C₄×M₁₆	1	2	yes	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#343	1	2	yes	—
176	C₄	#344	1	2	yes	—
177	C₄	#407	1	2	yes	—
178	C₄	C₂×#100	1	1	yes	characteristic
179	C₄	C₂×#101	1	1	yes	characteristic
180	C₄	#423	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	—	2	2	no	—
183	C₂	—	4	1	no	—
184	C₂	#1728	1	1	yes	characteristic
185	C₂	C₂×#343	1	1	yes	characteristic
186	C₂	C₂×#344	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	C₂×#95
prime radical	C₂⁴	C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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. #1283	C₂	semidirect
93. M₁₆	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2920	2
C₂×#423	2
C₂×C₄×M₁₆	2
#1283	2