Subgroups — #20446

Order 256.

505 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	C₂²×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×#427	C₂	1	1	yes	maximal, characteristic
3	#2873	C₂	1	2	yes	maximal
4	C₂×#405	C₂	1	1	yes	maximal, characteristic
5	#1773	C₂	1	4	yes	maximal, complemented
6	#1772	C₂	1	4	yes	maximal, complemented
7	#1733	C₂	1	2	yes	maximal
8	C₂²×M₁₆	C₂²	1	2	yes	—
9	C₂²×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈	C₂²	1	1	yes	characteristic
11	C₂×#117	C₂²	1	2	yes	—
12	C₂×#117	C₂²	1	1	yes	characteristic
13	C₂×#108	C₄	1	8	yes	complemented
14	C₂×#108	C₂²	1	2	yes	—
15	C₂×#108	C₂²	1	2	yes	—
16	#427	C₄	1	2	yes	complemented
17	#427	C₄	1	2	yes	complemented
18	#427	C₂²	1	2	yes	—
19	#427	C₂²	1	2	yes	—
20	C₂×#107	C₄	1	2	yes	complemented
21	C₂×#106	C₄	1	2	yes	complemented
22	#405	—	2	4	no	—
23	#405	C₄	1	8	yes	complemented
24	#405	C₂²	1	8	yes	—
25	C₄×M₁₆	—	2	4	no	—
26	C₂×#97	C₂²	1	2	yes	—
27	C₂×C₄×C₈	C₂²	1	2	yes	—
28	#339	C₂²	1	8	yes	—
29	C₂³×C₄	C₂³	1	1	yes	characteristic
30	C₂×M₁₆	—	2	2	no	—
31	C₂×M₁₆	—	2	4	no	—
32	C₂×M₁₆	C₂×C₄	1	8	yes	—
33	C₂×M₁₆	C₂×C₄	1	4	yes	—
34	C₂×M₁₆	D₄	1	2	yes	—
35	C₂×M₁₆	D₄	1	2	yes	—
36	C₂×M₁₆	Q₈	1	4	yes	—
37	C₂×M₁₆	C₂³	1	4	yes	—
38	C₂²×C₈	—	2	2	no	—
39	C₂²×C₈	C₂×C₄	1	4	yes	—
40	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂×C₄	1	4	yes	—
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₈	C₂³	1	2	yes	—
48	#117	—	2	2	no	—
49	#117	—	2	2	no	—
50	#117	—	2	2	no	—
51	#117	C₂×C₄	1	4	yes	—
52	#117	C₂×C₄	1	4	yes	—
53	#117	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄	—	2	1	no	—
55	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
56	C₂×C₂²⋊C₄	—	2	1	no	—
57	C₂×C₂²⋊C₄	—	2	2	no	—
58	C₂×C₄²	—	2	2	no	—
59	C₂×C₄²	C₂³	1	2	yes	—
60	#108	C₂×C₄	1	8	yes	complemented
61	#108	C₂×C₄	1	8	yes	complemented
62	#107	C₂×C₄	1	4	yes	—
63	#107	C₂×C₄	1	4	yes	—
64	#106	C₂×C₄	1	4	yes	—
65	#106	C₂×C₄	1	4	yes	—
66	#97	—	2	8	no	—
67	#97	C₂×C₄	1	8	yes	complemented
68	C₄×C₈	—	2	8	no	—
69	C₄×C₈	C₂×C₄	1	8	yes	complemented
70	C₂⁴	#41	1	1	yes	characteristic
71	C₂²×C₄	—	2	2	no	—
72	C₂²×C₄	—	2	4	no	—
73	C₂²×C₄	C₂²×C₄	1	2	yes	—
74	C₂²×C₄	C₂²×C₄	1	2	yes	—
75	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄	C₂×Q₈	1	2	yes	—
80	C₂²×C₄	#41	1	1	yes	characteristic
81	C₂²×C₄	#41	1	2	yes	—
82	C₂²×C₄	C₂⁴	1	1	yes	characteristic
83	M₁₆	—	2	8	no	—
84	M₁₆	—	2	4	no	—
85	M₁₆	C₄⋊C₄	1	8	yes	—
86	M₁₆	C₄⋊C₄	1	8	yes	—
87	M₁₆	C₂²×C₄	1	8	yes	—
88	C₂×C₈	—	2	8	no	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	C₄²	1	4	yes	—
91	C₂×C₈	C₄²	1	4	yes	—
92	C₂×C₈	C₄²	1	4	yes	—
93	C₂×C₈	C₄²	1	4	yes	—
94	C₂×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈	C₂²×C₄	1	2	yes	—
101	C₂×C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₈	C₂²×C₄	1	4	yes	—
103	C₂×C₈	C₂×D₄	1	4	yes	—
104	C₂×C₈	C₂×Q₈	1	4	yes	—
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	C₂²×C₄	1	8	yes	—
108	C₂²⋊C₄	—	2	4	no	—
109	C₂²⋊C₄	—	2	4	no	—
110	C₂²⋊C₄	—	2	4	no	—
111	C₄²	—	2	4	no	—
112	C₄²	—	2	4	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²×C₄	1	4	yes	—
115	C₂³	—	4	2	no	—
116	C₂³	#117	1	2	yes	—
117	C₂³	C₄×D₄	1	1	yes	characteristic
118	C₂³	C₄×D₄	1	1	yes	characteristic
119	C₂³	C₄×Q₈	1	2	yes	—
120	C₂³	C₂×#41	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	8	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	—	8	4	no	—
125	C₂×C₄	C₂×C₄²	1	2	yes	—
126	C₂×C₄	C₂×C₄²	1	2	yes	—
127	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄	#117	1	2	yes	—
130	C₂×C₄	C₄×D₄	1	2	yes	—
131	C₂×C₄	C₄×D₄	1	2	yes	—
132	C₂×C₄	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄	C₄×Q₈	1	2	yes	—
135	C₂×C₄	C₄×Q₈	1	2	yes	—
136	C₂×C₄	C₄×Q₈	1	2	yes	—
137	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
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	C₂×C₄	C₂×#41	1	1	yes	characteristic
143	C₈	—	2	8	no	—
144	C₈	C₂×C₄²	1	2	yes	—
145	C₈	C₂×C₄²	1	2	yes	—
146	C₈	C₂×C₄²	1	2	yes	—
147	C₈	C₂×C₄²	1	2	yes	—
148	C₈	C₂×C₄⋊C₄	1	8	yes	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	4	2	no	—
154	C₂²	C₄×C₄⋊C₄	1	2	yes	—
155	C₂²	C₄×C₄⋊C₄	1	2	yes	—
156	C₂²	C₂×#117	1	1	yes	characteristic
157	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
159	C₄	—	4	4	no	—
160	C₄	—	4	8	no	—
161	C₄	C₄×C₄⋊C₄	1	2	yes	—
162	C₄	C₄×C₄⋊C₄	1	2	yes	—
163	C₄	C₂²×C₄²	1	1	yes	characteristic
164	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₄	C₂×C₄×D₄	1	1	yes	characteristic
166	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	2	2	no	—
170	C₂	—	2	2	no	—
171	C₂	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
172	C₂	#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂⁴	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #1773	C₂	semidirect
6. #1772	C₂	semidirect
13. C₂×#108	C₄	semidirect
16. #427	C₄	semidirect
17. #427	C₄	semidirect
20. C₂×#107	C₄	semidirect
21. C₂×#106	C₄	semidirect
23. #405	C₄	semidirect
60. #108	C₂×C₄	semidirect
61. #108	C₂×C₄	semidirect
67. #97	C₂×C₄	semidirect
69. C₄×C₈	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#108	2
C₂×#427	2
#2873	2
C₂×#405	2
#1773	2
#1772	2
#1733	2