Subgroups — #33423

Order 256.

295 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#131	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	#2150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#2113	C₂	1	2	yes	maximal
8	#2108	C₂	1	2	yes	maximal, complemented
9	C₂×#131	C₄	1	1	yes	characteristic
10	C₂×#131	C₂²	1	1	yes	characteristic
11	C₄×#41	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	C₂²	1	1	yes	characteristic
17	#405	C₂²	1	2	yes	—
18	C₄×M₁₆	C₄	1	2	yes	—
19	C₄×M₁₆	C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂²	1	2	yes	—
22	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₄⋊C₁₆	C₂²	1	2	yes	—
25	C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₄⋊C₁₆	C₂²	1	2	yes	—
27	C₂²⋊C₁₆	C₂²	1	4	yes	—
28	C₂²⋊C₁₆	C₂²	1	4	yes	—
29	#346	C₂²	1	4	yes	—
30	C₄×C₁₆	C₂²	1	4	yes	—
31	C₂×#41	C₂×C₄	1	1	yes	characteristic
32	#131	—	2	4	no	—
33	#131	—	2	2	no	—
34	#131	—	2	2	no	—
35	C₂×M₁₆	C₂×C₄	1	2	yes	—
36	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×M₁₆	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	2	yes	—
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₄×Q₈	C₈	1	1	yes	characteristic
44	C₄×Q₈	C₈	1	1	yes	characteristic
45	C₄×D₄	C₈	1	2	yes	—
46	C₄×D₄	C₈	1	1	yes	characteristic
47	C₄×D₄	C₈	1	2	yes	—
48	C₄×D₄	C₈	1	1	yes	characteristic
49	#117	C₂×C₄	1	2	yes	—
50	#117	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²	C₂×C₄	1	2	yes	—
52	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
53	M₃₂	—	2	4	no	—
54	C₂×C₁₆	—	2	4	no	—
55	C₂×C₁₆	C₂³	1	2	yes	—
56	C₂×C₁₆	C₂³	1	4	yes	—
57	C₂×C₁₆	C₂³	1	2	yes	—
58	#97	C₂×C₄	1	2	yes	—
59	#97	C₂×C₄	1	2	yes	—
60	#97	C₂×C₄	1	2	yes	—
61	C₄×C₈	C₂×C₄	1	2	yes	—
62	C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈	C₂×C₄	1	2	yes	—
64	C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈	C₂³	1	2	yes	—
66	C₄×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₈	C₂³	1	1	yes	characteristic
68	#41	—	2	2	no	—
69	#41	—	2	2	no	—
70	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
71	C₂×D₄	C₂×C₈	1	2	yes	—
72	C₂×D₄	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄	C₂×C₈	1	2	yes	—
74	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄	C₂×C₈	1	2	yes	—
76	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄	C₂²×C₄	1	2	yes	—
78	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	4	no	—
81	M₁₆	—	2	4	no	—
82	M₁₆	M₁₆	1	4	yes	—
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	2	yes	—
93	C₂×C₈	#41	1	2	yes	—
94	C₂×C₈	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄	C₂×C₈	1	2	yes	—
96	C₄⋊C₄	C₂×C₈	1	2	yes	—
97	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
98	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
99	C₂²⋊C₄	C₂×C₈	1	2	yes	—
100	C₂²⋊C₄	C₂×C₈	1	2	yes	—
101	C₂²⋊C₄	C₂×C₈	1	2	yes	—
102	C₄²	C₂×C₈	1	2	yes	—
103	C₄²	C₂×C₈	1	1	yes	characteristic
104	C₄²	C₂×C₈	1	2	yes	—
105	C₄²	C₂×C₈	1	1	yes	characteristic
106	C₄²	C₂²×C₄	1	1	yes	characteristic
107	C₄²	C₂²×C₄	1	2	yes	—
108	C₄²	C₂²×C₄	1	1	yes	characteristic
109	C₁₆	—	2	2	no	—
110	C₁₆	—	2	2	no	—
111	C₁₆	—	2	4	no	—
112	C₂³	C₂²×C₈	1	2	yes	—
113	C₂³	C₂²×C₈	1	1	yes	characteristic
114	Q₈	—	2	2	no	—
115	D₄	—	2	4	no	—
116	D₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	#117	1	2	yes	—
124	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄	C₂²×C₈	1	2	yes	—
126	C₂×C₄	C₂²×C₈	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	1	yes	characteristic
130	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄	C₂×M₁₆	1	2	yes	—
133	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	#117	1	2	yes	—
138	C₈	C₂×M₁₆	1	4	yes	—
139	C₈	C₂×#41	1	1	yes	characteristic
140	C₈	C₂×#41	1	1	yes	characteristic
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	#431	1	2	yes	—
147	C₂²	C₂³×C₈	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	2	no	—
153	C₄	—	2	2	no	—
154	C₄	#431	1	2	yes	—
155	C₄	#504	1	2	yes	—
156	C₄	C₂×#117	1	1	yes	characteristic
157	C₄	C₂²×M₁₆	1	1	yes	characteristic
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	C₂×#431	1	1	yes	characteristic
161	C₂	C₂×#504	1	1	yes	characteristic
162	C₂	#3407	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	C₂×#41	C₂×C₄
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. #2150	C₂	semidirect
6. C₂×C₄⋊C₁₆	C₂	semidirect
8. #2108	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×#131	2
#2164	2
#2163	2
#2162	2
#2150	2
C₂×C₄⋊C₁₆	2
#2113	2
#2108	2