Subgroups — #17982

Order 256.

303 conjugacy classes of subgroups.

The subgroup lattice has 185 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	#2966	C₂	1	2	yes	maximal, complemented
3	#1588	C₂	1	2	yes	maximal, complemented
4	#1587	C₂	1	2	yes	maximal, complemented
5	#1586	C₂	1	2	yes	maximal, complemented
6	#1585	C₂	1	2	yes	maximal, complemented
7	#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	#1494	C₂	1	2	yes	maximal, complemented
9	C₂×#322	C₂	1	1	yes	maximal, characteristic, complemented
10	#529	C₂²	1	2	yes	—
11	#499	C₄	1	1	yes	characteristic
12	#495	C₄	1	1	yes	characteristic
13	#488	C₄	1	2	yes	—
14	#467	C₄	1	1	yes	characteristic
15	#466	C₄	1	1	yes	characteristic
16	#465	C₄	1	2	yes	—
17	#446	C₂²	1	2	yes	—
18	C₈×Q₈	C₂²	1	2	yes	—
19	C₄×Q₁₆	C₂²	1	1	yes	characteristic
20	C₄×SD₁₆	C₂²	1	2	yes	—
21	C₄×D₈	C₂²	1	1	yes	characteristic
22	#436	C₂²	1	2	yes	—
23	#435	—	2	2	no	—
24	C₈×D₄	—	2	2	no	—
25	C₈×D₄	C₂²	1	2	yes	—
26	#433	—	2	2	no	—
27	#432	C₂²	1	2	yes	—
28	#405	—	2	2	no	—
29	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	#335	C₂²	1	1	yes	characteristic
33	#335	C₂²	1	1	yes	characteristic
34	#334	C₂²	1	2	yes	—
35	Q₈⋊C₈	C₂²	1	2	yes	—
36	Q₈⋊C₈	C₂²	1	2	yes	—
37	D₄⋊C₈	C₂²	1	2	yes	—
38	D₄⋊C₈	C₂²	1	2	yes	—
39	#322	C₂²	1	2	yes	—
40	#322	C₂²	1	2	yes	—
41	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂×SD₁₆	C₂×C₄	1	2	yes	—
43	C₂×D₈	C₂×C₄	1	1	yes	characteristic
44	C₂×M₁₆	—	2	2	no	—
45	C₂²×C₈	—	2	1	no	—
46	C₂²×C₈	—	2	2	no	—
47	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈	D₄	1	1	yes	characteristic
50	C₂²×C₈	D₄	1	1	yes	characteristic
51	#126	—	2	2	no	—
52	#125	C₂×C₄	1	2	yes	—
53	#124	C₂×C₄	1	2	yes	—
54	#123	—	2	2	no	—
55	C₂²⋊Q₈	C₂×C₄	1	2	yes	—
56	#121	C₂×C₄	1	2	yes	—
57	C₄×Q₈	C₂³	1	2	yes	—
58	C₄×D₄	—	2	2	no	—
59	C₄×D₄	C₂³	1	2	yes	—
60	#117	—	2	2	no	—
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	#107	C₂×C₄	1	1	yes	characteristic
63	#107	C₂×C₄	1	1	yes	characteristic
64	#106	C₂×C₄	1	2	yes	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	C₂³	1	2	yes	—
68	C₄⋊C₈	C₂³	1	2	yes	—
69	#103	C₂×C₄	1	2	yes	—
70	#103	C₂×C₄	1	2	yes	—
71	#102	C₂×C₄	1	2	yes	—
72	#102	C₂×C₄	1	2	yes	—
73	C₂²⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	—	2	2	no	—
75	C₂²⋊C₈	—	2	2	no	—
76	#97	—	2	2	no	—
77	#97	—	2	2	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	C₂×C₄	1	1	yes	characteristic
81	C₄×C₈	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈	D₄	1	2	yes	—
83	C₄×C₈	C₂³	1	2	yes	—
84	C₄×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₈	C₂³	1	1	yes	characteristic
86	C₄×C₈	C₂³	1	2	yes	—
87	C₂×Q₈	C₂²×C₄	1	2	yes	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	C₂²×C₄	1	2	yes	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄	#41	1	1	yes	characteristic
94	Q₁₆	—	4	1	no	—
95	SD₁₆	—	4	2	no	—
96	D₈	—	4	1	no	—
97	M₁₆	—	4	4	no	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	—	4	4	no	—
107	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈	C₂×D₄	1	2	yes	—
112	C₂×C₈	C₂×D₄	1	2	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	C₂²×C₄	1	2	yes	—
118	C₄⋊C₄	C₂²×C₄	1	2	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	C₂²×C₄	1	1	yes	characteristic
126	C₄²	C₂×D₄	1	1	yes	characteristic
127	C₄²	#41	1	1	yes	characteristic
128	C₄²	C₂⁴	1	1	yes	characteristic
129	C₂³	—	2	2	no	—
130	C₂³	C₄×D₄	1	1	yes	characteristic
131	Q₈	—	2	2	no	—
132	Q₈	—	2	2	no	—
133	D₄	—	2	2	no	—
134	D₄	—	2	2	no	—
135	D₄	—	4	4	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄	C₂×#41	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	2	no	—
158	C₈	—	2	2	no	—
159	C₈	—	4	2	no	—
160	C₈	—	4	2	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	2	no	—
165	C₂²	—	4	2	no	—
166	C₂²	#443	1	2	yes	—
167	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	#568	1	1	yes	characteristic
176	C₄	#568	1	1	yes	characteristic
177	C₄	#575	1	1	yes	characteristic
178	C₄	#575	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	—	8	2	no	—
181	C₂	#2932	1	1	yes	characteristic
182	C₂	#2943	1	1	yes	characteristic
183	C₂	C₂×#443	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 ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2966	C₂	semidirect
3. #1588	C₂	semidirect
4. #1587	C₂	semidirect
5. #1586	C₂	semidirect
6. #1585	C₂	semidirect
7. #1568	C₂	semidirect
8. #1494	C₂	semidirect
9. C₂×#322	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
#2966	2
#1588	2
#1587	2
#1586	2
#1585	2
#1568	2
#1494	2
C₂×#322	2