Subgroups — #33422

Order 256.

300 conjugacy classes of subgroups.

The subgroup lattice has 169 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405	C₂	1	1	yes	maximal, characteristic
2	#2164	C₂	1	4	yes	maximal, complemented
3	#2163	C₂	1	2	yes	maximal, complemented
4	#2162	C₂	1	2	yes	maximal, complemented
5	#2151	C₂	1	2	yes	maximal, complemented
6	#2112	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	#2108	C₂	1	2	yes	maximal, complemented
9	C₂²×M₁₆	C₄	1	1	yes	characteristic
10	C₂³×C₈	C₂²	1	1	yes	characteristic
11	C₂×#117	C₄	1	1	yes	characteristic
12	C₂×M₃₂	C₂²	1	2	yes	—
13	C₂²×C₁₆	C₂²	1	2	yes	—
14	#405	C₄	1	1	yes	characteristic
15	#405	C₄	1	2	yes	—
16	#405	C₄	1	1	yes	characteristic
17	#405	C₂²	1	1	yes	characteristic
18	#405	C₂²	1	1	yes	characteristic
19	#405	C₂²	1	2	yes	—
20	C₂×#97	C₄	1	2	yes	—
21	C₂×C₄×C₈	C₂²	1	2	yes	—
22	C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₄⋊C₁₆	C₂²	1	4	yes	complemented
24	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₂²⋊C₁₆	C₂²	1	2	yes	—
29	#346	C₂²	1	4	yes	complemented
30	C₄×C₁₆	C₂²	1	4	yes	complemented
31	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆	—	2	4	no	—
33	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆	C₂×C₄	1	2	yes	—
35	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈	—	2	4	no	—
37	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈	C₂×C₄	1	2	yes	—
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	#117	C₈	1	1	yes	characteristic
45	#117	C₈	1	1	yes	characteristic
46	#117	C₈	1	2	yes	—
47	#117	C₂×C₄	1	2	yes	—
48	#117	C₂×C₄	1	1	yes	characteristic
49	#117	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
51	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
52	C₂×C₂²⋊C₄	C₈	1	2	yes	—
53	C₂×C₄²	C₂×C₄	1	2	yes	—
54	M₃₂	—	2	4	no	—
55	C₂×C₁₆	—	2	4	no	—
56	C₂×C₁₆	C₂³	1	2	yes	—
57	C₂×C₁₆	C₂³	1	4	yes	—
58	C₂×C₁₆	C₂³	1	2	yes	—
59	#97	C₂×C₄	1	2	yes	—
60	#97	C₂×C₄	1	2	yes	—
61	#97	C₂×C₄	1	2	yes	—
62	#97	C₂×C₄	1	2	yes	—
63	C₄×C₈	C₂×C₄	1	2	yes	—
64	C₄×C₈	C₂×C₄	1	2	yes	—
65	C₄×C₈	C₂³	1	2	yes	—
66	C₄×C₈	C₂³	1	2	yes	—
67	C₂⁴	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	2	2	no	—
70	C₂²×C₄	C₂×C₈	1	2	yes	—
71	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄	C₂×C₈	1	2	yes	—
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	M₁₆	—	2	2	no	—
79	M₁₆	—	2	4	no	—
80	M₁₆	M₁₆	1	4	yes	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₈	#41	1	4	yes	—
93	C₂×C₈	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄	C₂×C₈	1	2	yes	—
95	C₄⋊C₄	C₂×C₈	1	2	yes	—
96	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
97	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
98	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
99	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
100	C₂²⋊C₄	C₂×C₈	1	2	yes	—
101	C₂²⋊C₄	C₂×C₈	1	2	yes	—
102	C₂²⋊C₄	C₂×C₈	1	2	yes	—
103	C₂²⋊C₄	C₂×C₈	1	2	yes	—
104	C₄²	C₂×C₈	1	2	yes	—
105	C₄²	C₂×C₈	1	2	yes	—
106	C₄²	C₂²×C₄	1	2	yes	—
107	C₄²	C₂²×C₄	1	2	yes	—
108	C₁₆	—	2	2	no	—
109	C₁₆	—	2	2	no	—
110	C₁₆	—	2	4	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	—	4	1	no	—
114	C₂³	C₂²×C₈	1	1	yes	characteristic
115	C₂³	C₂²×C₈	1	1	yes	characteristic
116	C₂³	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	#117	1	2	yes	—
127	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄	C₂²×C₈	1	2	yes	—
129	C₂×C₄	C₂²×C₈	1	2	yes	—
130	C₂×C₄	C₂²×C₈	1	2	yes	—
131	C₂×C₄	C₂²×C₈	1	2	yes	—
132	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄	C₂×M₁₆	1	2	yes	—
135	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	2	no	—
139	C₈	#117	1	2	yes	—
140	C₈	C₂×M₁₆	1	4	yes	—
141	C₈	C₂×#41	1	2	yes	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	4	4	no	—
150	C₂²	#431	1	2	yes	—
151	C₂²	#504	1	2	yes	—
152	C₂²	C₂³×C₈	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	2	2	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#431	1	2	yes	—
160	C₄	C₂×#117	1	1	yes	characteristic
161	C₄	C₂²×M₁₆	1	1	yes	characteristic
162	C₂	—	2	2	no	—
163	C₂	—	2	2	no	—
164	C₂	—	4	1	no	—
165	C₂	C₂×#431	1	1	yes	characteristic
166	C₂	C₂×#504	1	1	yes	characteristic
167	C₂	#3407	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. #2164	C₂	semidirect
3. #2163	C₂	semidirect
4. #2162	C₂	semidirect
5. #2151	C₂	semidirect
8. #2108	C₂	semidirect
22. C₄⋊C₁₆	C₂²	semidirect
23. C₄⋊C₁₆	C₂²	semidirect
29. #346	C₂²	semidirect
30. C₄×C₁₆	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#405	2
#2164	2
#2163	2
#2162	2
#2151	2
#2112	2
C₂×C₂²⋊C₁₆	2
#2108	2