Subgroups — #23172

Order 256.

313 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3162	C₂	1	1	yes	maximal, characteristic
2	#2962	C₂	1	2	yes	maximal
3	#1986	C₂	1	2	yes	maximal
4	#1946	C₂	1	1	yes	maximal, characteristic
5	#1945	C₂	1	1	yes	maximal, characteristic
6	#1922	C₂	1	2	yes	maximal
7	#1922	C₂	1	2	yes	maximal
8	#1778	C₂	1	1	yes	maximal, characteristic
9	#1773	C₂	1	1	yes	maximal, characteristic
10	#1772	C₂	1	1	yes	maximal, characteristic
11	#1771	C₂	1	1	yes	maximal, characteristic
12	#533	C₂²	1	2	yes	—
13	#501	C₄	1	2	yes	complemented
14	#501	C₄	1	2	yes	complemented
15	#501	C₄	1	2	yes	complemented
16	#501	C₄	1	2	yes	complemented
17	#446	—	2	2	no	—
18	#446	—	2	2	no	—
19	#446	—	2	2	no	—
20	C₈×Q₈	—	2	2	no	—
21	C₂×#108	C₂²	1	1	yes	characteristic
22	C₂×#108	C₂²	1	1	yes	characteristic
23	#427	C₂²	1	2	yes	—
24	#427	C₂²	1	1	yes	characteristic
25	#427	C₂²	1	1	yes	characteristic
26	#424	C₂²	1	2	yes	—
27	#424	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	#405	C₂²	1	2	yes	—
30	#405	C₂²	1	2	yes	—
31	C₄×M₁₆	C₂²	1	2	yes	—
32	C₂×#97	C₂²	1	1	yes	characteristic
33	C₂×#97	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	#341	C₂²	1	2	yes	—
37	#339	C₂²	1	2	yes	—
38	#339	C₂²	1	2	yes	—
39	#339	C₂²	1	2	yes	—
40	#339	C₂²	1	2	yes	—
41	#336	C₂²	1	1	yes	characteristic
42	#336	C₂²	1	1	yes	characteristic
43	C₂×M₁₆	C₂³	1	2	yes	—
44	C₂×M₁₆	C₂³	1	2	yes	—
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈	C₂×C₄	1	2	yes	—
50	C₄⋊Q₈	C₂×C₄	1	2	yes	—
51	#125	C₂×C₄	1	2	yes	—
52	#125	C₂×C₄	1	2	yes	—
53	C₂²⋊Q₈	—	2	2	no	—
54	C₂²⋊Q₈	—	2	2	no	—
55	C₄×Q₈	—	2	2	no	—
56	C₄×Q₈	—	2	2	no	—
57	#117	C₂³	1	2	yes	—
58	#117	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄	—	2	1	no	—
60	C₂×C₄²	C₂³	1	1	yes	characteristic
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	C₂×C₄²	C₂³	1	1	yes	characteristic
63	#108	—	2	2	no	—
64	#108	Q₈	1	4	yes	complemented
65	#107	C₂×C₄	1	2	yes	—
66	#107	C₂×C₄	1	2	yes	—
67	#107	C₂×C₄	1	1	yes	characteristic
68	#107	C₂×C₄	1	1	yes	characteristic
69	#107	C₂×C₄	1	1	yes	characteristic
70	#107	C₂×C₄	1	1	yes	characteristic
71	#106	C₂×C₄	1	2	yes	—
72	#106	C₂×C₄	1	2	yes	—
73	#106	C₂×C₄	1	2	yes	—
74	#106	C₂×C₄	1	2	yes	—
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	—	2	2	no	—
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	—	2	2	no	—
79	C₄⋊C₈	D₄	1	2	yes	—
80	C₄⋊C₈	D₄	1	2	yes	—
81	#97	—	2	2	no	—
82	#97	—	2	2	no	—
83	#97	—	2	2	no	—
84	#97	—	2	2	no	—
85	#97	C₂×C₄	1	2	yes	—
86	#97	C₂×C₄	1	2	yes	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	#95	—	2	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	#41	1	1	yes	characteristic
97	C₂²×C₄	#41	1	1	yes	characteristic
98	C₂²×C₄	C₂⁴	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	2	2	no	—
101	M₁₆	—	2	2	no	—
102	M₁₆	C₂×Q₈	1	4	yes	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	C₂²×C₄	1	2	yes	—
112	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈	C₂×Q₈	1	2	yes	—
118	C₂×C₈	#41	1	2	yes	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄	C₂²×C₄	1	2	yes	—
128	C₄⋊C₄	C₂²×C₄	1	2	yes	—
129	C₄⋊C₄	C₂²×C₄	1	2	yes	—
130	C₂²⋊C₄	—	2	2	no	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	C₂²×C₄	1	2	yes	—
139	C₄²	C₂×D₄	1	2	yes	—
140	C₂³	C₂×#41	1	1	yes	characteristic
141	Q₈	—	4	4	no	—
142	Q₈	—	8	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	C₄×D₄	1	2	yes	—
156	C₂×C₄	C₄×D₄	1	2	yes	—
157	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄	C₂×#41	1	1	yes	characteristic
161	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
162	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	2	2	no	—
167	C₈	—	2	4	no	—
168	C₈	—	4	2	no	—
169	C₈	C₄×Q₈	1	2	yes	—
170	C₈	C₄×Q₈	1	2	yes	—
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂²	#556	1	1	yes	characteristic
175	C₂²	#557	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	1	no	—
184	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
185	C₄	#519	1	1	yes	characteristic
186	C₄	#547	1	1	yes	characteristic
187	C₄	D₄×Q₈	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	#2365	1	1	yes	characteristic
191	C₂	#2955	1	1	yes	characteristic
192	C₂	#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂³	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
13. #501	C₄	semidirect
14. #501	C₄	semidirect
15. #501	C₄	semidirect
16. #501	C₄	semidirect
64. #108	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3162	2
#2962	2
#1986	2
#1946	2
#1945	2
#1922	2
#1778	2
#1773	2
#1772	2
#1771	2