Subgroups — #17973

Order 256.

296 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3168	C₂	1	1	yes	maximal, characteristic
2	#2993	C₂	1	2	yes	maximal, complemented
3	#1592	C₂	1	2	yes	maximal, complemented
4	#1591	C₂	1	1	yes	maximal, characteristic, complemented
5	#1589	C₂	1	1	yes	maximal, characteristic, complemented
6	SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#1567	C₂	1	1	yes	maximal, characteristic, complemented
10	#1495	C₂	1	2	yes	maximal, complemented
11	#1450	C₂	1	1	yes	maximal, characteristic, complemented
12	#529	C₂²	1	2	yes	—
13	#499	C₄	1	1	yes	characteristic
14	#495	C₄	1	1	yes	characteristic
15	#488	C₄	1	2	yes	—
16	#467	C₄	1	1	yes	characteristic
17	#466	C₄	1	1	yes	characteristic
18	#465	C₄	1	2	yes	—
19	#446	C₂²	1	2	yes	—
20	C₈×Q₈	C₂²	1	2	yes	—
21	C₄×Q₁₆	C₂²	1	1	yes	characteristic
22	C₄×SD₁₆	C₂²	1	2	yes	—
23	C₄×D₈	C₂²	1	1	yes	characteristic
24	#436	—	2	2	no	—
25	#436	C₂²	1	2	yes	—
26	#435	—	2	2	no	—
27	C₈×D₄	C₂²	1	2	yes	—
28	#433	—	2	2	no	—
29	#431	C₂²	1	1	yes	characteristic
30	#431	C₂²	1	1	yes	characteristic
31	#424	—	2	2	no	—
32	#423	C₂²	1	2	yes	—
33	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
34	#335	C₂²	1	1	yes	characteristic
35	#335	C₂²	1	1	yes	characteristic
36	#335	C₂²	1	2	yes	—
37	Q₈⋊C₈	C₂²	1	2	yes	—
38	Q₈⋊C₈	C₂²	1	2	yes	—
39	D₄⋊C₈	C₂²	1	2	yes	—
40	D₄⋊C₈	C₂²	1	2	yes	—
41	#322	C₂²	1	1	yes	characteristic
42	#322	C₂²	1	1	yes	characteristic
43	C₈²	C₂²	1	1	yes	characteristic
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×SD₁₆	C₂×C₄	1	2	yes	—
47	C₂×D₈	C₂×C₄	1	1	yes	characteristic
48	C₂×M₁₆	—	2	2	no	—
49	C₂×M₁₆	—	2	2	no	—
50	C₂²×C₈	—	2	2	no	—
51	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
53	#126	—	2	2	no	—
54	#125	C₂×C₄	1	2	yes	—
55	#124	C₂×C₄	1	2	yes	—
56	#123	—	2	2	no	—
57	C₂²⋊Q₈	C₂×C₄	1	2	yes	—
58	#121	C₂×C₄	1	2	yes	—
59	C₄×Q₈	C₂³	1	2	yes	—
60	C₄×D₄	—	2	2	no	—
61	C₄×D₄	C₂³	1	2	yes	—
62	#117	—	2	2	no	—
63	C₂×C₄²	C₂³	1	1	yes	characteristic
64	#107	C₂×C₄	1	1	yes	characteristic
65	#107	C₂×C₄	1	1	yes	characteristic
66	#106	C₂×C₄	1	2	yes	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	—	2	2	no	—
69	C₄⋊C₈	—	2	2	no	—
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₄⋊C₈	D₄	1	1	yes	characteristic
73	C₄⋊C₈	C₂³	1	2	yes	—
74	C₄⋊C₈	C₂³	1	2	yes	—
75	#103	C₂×C₄	1	2	yes	—
76	#103	C₂×C₄	1	2	yes	—
77	#102	C₂×C₄	1	2	yes	—
78	#102	C₂×C₄	1	2	yes	—
79	C₂²⋊C₈	—	2	1	no	—
80	C₂²⋊C₈	—	2	2	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	C₂²⋊C₈	D₄	1	1	yes	characteristic
83	C₂²⋊C₈	D₄	1	1	yes	characteristic
84	#97	—	2	2	no	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	C₂×C₄	1	1	yes	characteristic
87	C₄×C₈	C₂×C₄	1	1	yes	characteristic
88	C₄×C₈	C₂³	1	1	yes	characteristic
89	C₄×C₈	C₂³	1	1	yes	characteristic
90	C₄×C₈	C₂³	1	1	yes	characteristic
91	C₄×C₈	C₂³	1	1	yes	characteristic
92	C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₈	C₂³	1	1	yes	characteristic
94	C₂×Q₈	C₂²×C₄	1	2	yes	—
95	C₂×D₄	—	2	2	no	—
96	C₂×D₄	C₂²×C₄	1	2	yes	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄	#41	1	1	yes	characteristic
101	Q₁₆	—	2	2	no	—
102	SD₁₆	—	2	4	no	—
103	D₈	—	2	2	no	—
104	M₁₆	—	8	2	no	—
105	M₁₆	—	8	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	8	2	no	—
113	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈	C₂×D₄	1	1	yes	characteristic
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	C₂²×C₄	1	2	yes	—
126	C₄⋊C₄	C₂²×C₄	1	2	yes	—
127	C₂²⋊C₄	—	2	2	no	—
128	C₂²⋊C₄	—	2	2	no	—
129	C₂²⋊C₄	—	2	2	no	—
130	C₂²⋊C₄	—	2	2	no	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	C₂²×C₄	1	1	yes	characteristic
134	C₄²	C₂×D₄	1	1	yes	characteristic
135	C₄²	#41	1	1	yes	characteristic
136	C₄²	C₂⁴	1	1	yes	characteristic
137	C₂³	—	2	2	no	—
138	C₂³	C₄×D₄	1	1	yes	characteristic
139	Q₈	—	2	2	no	—
140	Q₈	—	2	2	no	—
141	D₄	—	2	2	no	—
142	D₄	—	2	2	no	—
143	D₄	—	8	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄	C₂×#41	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	4	1	no	—
167	C₈	—	4	1	no	—
168	C₈	—	4	2	no	—
169	C₈	—	4	2	no	—
170	C₈	#131	1	2	yes	—
171	C₈	#131	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	4	2	no	—
177	C₂²	—	4	2	no	—
178	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	C₂×#131	1	1	yes	characteristic
187	C₄	C₂×#131	1	1	yes	characteristic
188	C₄	#576	1	1	yes	characteristic
189	C₄	#578	1	1	yes	characteristic
190	C₂	—	4	1	no	—
191	C₂	—	8	2	no	—
192	C₂	#2931	1	1	yes	characteristic
193	C₂	#2951	1	1	yes	characteristic
194	C₂	#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	#466	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2993	C₂	semidirect
3. #1592	C₂	semidirect
4. #1591	C₂	semidirect
5. #1589	C₂	semidirect
6. SD₁₆⋊C₈	C₂	semidirect
7. C₈×Q₁₆	C₂	semidirect
8. C₈×D₈	C₂	semidirect
9. #1567	C₂	semidirect
10. #1495	C₂	semidirect
11. #1450	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3168	2
#2993	2
#1592	2
#1591	2
#1589	2
SD₁₆⋊C₈	2
C₈×Q₁₆	2
C₈×D₈	2
#1567	2
#1495	2
#1450	2