Subgroups — #33483

Order 256.

258 conjugacy classes of subgroups.

The subgroup lattice has 171 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2966	C₂	1	1	yes	maximal, characteristic
2	#2183	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#2169	C₂	1	1	yes	maximal, characteristic, complemented
5	#2169	C₂	1	2	yes	maximal, complemented
6	#2168	C₂	1	2	yes	maximal, complemented
7	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#2164	C₂	1	2	yes	maximal, complemented
9	#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	#2108	C₂	1	2	yes	maximal, complemented
11	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	#529	C₄	1	1	yes	characteristic
13	C₂×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×M₃₂	C₂²	1	2	yes	—
16	C₂²×C₁₆	C₂²	1	2	yes	—
17	#446	C₄	1	1	yes	characteristic
18	C₈×Q₈	C₂²	1	1	yes	characteristic
19	#436	C₄	1	1	yes	characteristic
20	#435	C₄	1	2	yes	—
21	C₈×D₄	C₂²	1	1	yes	characteristic
22	C₈×D₄	C₂²	1	2	yes	—
23	#433	C₄	1	2	yes	—
24	#432	C₄	1	1	yes	characteristic
25	#405	C₂²	1	2	yes	—
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₂²⋊C₁₆	C₂²	1	2	yes	—
34	#346	C₂²	1	1	yes	characteristic
35	#346	C₂²	1	1	yes	characteristic
36	#346	C₂²	1	2	yes	—
37	C₄×C₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₁₆	C₂²	1	1	yes	characteristic
39	C₄×C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆	C₂²	1	2	yes	—
41	C₂×M₁₆	C₂×C₄	1	2	yes	—
42	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	2	yes	—
45	#126	C₈	1	2	yes	—
46	#125	C₈	1	1	yes	characteristic
47	#124	C₈	1	1	yes	characteristic
48	#123	C₈	1	2	yes	—
49	C₂²⋊Q₈	C₈	1	1	yes	characteristic
50	#121	C₈	1	1	yes	characteristic
51	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
52	C₄×D₄	C₂×C₄	1	2	yes	—
53	C₄×D₄	C₂×C₄	1	1	yes	characteristic
54	#117	C₂×C₄	1	2	yes	—
55	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
56	M₃₂	—	2	2	no	—
57	M₃₂	—	2	2	no	—
58	M₃₂	—	2	4	no	—
59	C₂×C₁₆	—	2	4	no	—
60	C₂×C₁₆	C₂³	1	2	yes	—
61	C₂×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₁₆	C₂³	1	2	yes	—
64	C₂×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₁₆	C₂³	1	1	yes	characteristic
66	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₄⋊C₈	C₂×C₄	1	2	yes	—
69	C₄⋊C₈	C₂×C₄	1	2	yes	—
70	C₂²⋊C₈	C₂×C₄	1	2	yes	—
71	C₂²⋊C₈	C₂×C₄	1	2	yes	—
72	C₂²⋊C₈	C₂×C₄	1	2	yes	—
73	#97	C₂×C₄	1	2	yes	—
74	#97	C₂×C₄	1	2	yes	—
75	C₄×C₈	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₈	C₂³	1	2	yes	—
80	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
81	C₂×D₄	C₂×C₈	1	1	yes	characteristic
82	C₂×D₄	C₂×C₈	1	2	yes	—
83	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄	C₂²×C₄	1	2	yes	—
87	M₁₆	—	2	4	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄	C₂×C₈	1	2	yes	—
100	C₄⋊C₄	C₂×C₈	1	2	yes	—
101	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
102	C₄⋊C₄	C₂×C₈	1	2	yes	—
103	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
104	C₄⋊C₄	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄	C₂×C₈	1	2	yes	—
106	C₂²⋊C₄	C₂×C₈	1	2	yes	—
107	C₂²⋊C₄	C₂×C₈	1	2	yes	—
108	C₂²⋊C₄	C₂×C₈	1	2	yes	—
109	C₂²⋊C₄	C₂×C₈	1	2	yes	—
110	C₄²	C₂×C₈	1	1	yes	characteristic
111	C₄²	C₂×C₈	1	1	yes	characteristic
112	C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₄²	C₂²×C₄	1	1	yes	characteristic
114	C₄²	C₂²×C₄	1	2	yes	—
115	C₁₆	—	2	2	no	—
116	C₁₆	—	2	1	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	—	2	2	no	—
119	C₁₆	#41	1	2	yes	—
120	C₁₆	#41	1	2	yes	—
121	C₂³	C₂²×C₈	1	2	yes	—
122	C₂³	C₂²×C₈	1	1	yes	characteristic
123	Q₈	—	2	2	no	—
124	D₄	—	2	2	no	—
125	D₄	—	4	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄	C₂²×C₈	1	2	yes	—
137	C₂×C₄	C₂²×C₈	1	2	yes	—
138	C₂×C₄	C₂²×C₈	1	2	yes	—
139	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₈	#131	1	2	yes	—
146	C₈	C₂×#41	1	1	yes	characteristic
147	C₈	C₂×#41	1	1	yes	characteristic
148	C₂²	—	2	1	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	2	no	—
154	C₂²	C₂³×C₈	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	2	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	2	no	—
162	C₄	#504	1	2	yes	—
163	C₄	C₄×#41	1	1	yes	characteristic
164	C₄	C₂×#131	1	1	yes	characteristic
165	C₂	—	4	1	no	—
166	C₂	—	4	2	no	—
167	C₂	C₈×#41	1	1	yes	characteristic
168	C₂	C₂×#504	1	1	yes	characteristic
169	C₂	#3407	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	#121	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. #2183	C₂	semidirect
3. Q₈×C₁₆	C₂	semidirect
4. #2169	C₂	semidirect
5. #2169	C₂	semidirect
6. #2168	C₂	semidirect
7. D₄×C₁₆	C₂	semidirect
8. #2164	C₂	semidirect
9. #2162	C₂	semidirect
10. #2108	C₂	semidirect
11. C₄×M₃₂	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2966	2
#2183	2
Q₈×C₁₆	2
#2169	2
#2168	2
D₄×C₁₆	2
#2164	2
#2162	2
#2108	2
C₄×M₃₂	2