Subgroups — #17824

Order 256.

286 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2989	C₂	1	2	yes	maximal
2	#2922	C₂	1	1	yes	maximal, characteristic
3	#1574	C₂	1	2	yes	maximal, complemented
4	#1573	C₂	1	2	yes	maximal, complemented
5	#1572	C₂	1	2	yes	maximal, complemented
6	#1570	C₂	1	2	yes	maximal, complemented
7	#1569	C₂	1	2	yes	maximal, complemented
8	#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	#1566	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#41	C₂²	1	1	yes	characteristic
11	#446	—	2	2	no	—
12	#436	—	2	2	no	—
13	#435	—	2	2	no	—
14	#435	—	2	2	no	—
15	#433	C₄	1	4	yes	—
16	#432	C₂²	1	2	yes	—
17	#432	C₂²	1	2	yes	—
18	#431	C₂²	1	2	yes	—
19	#431	C₂²	1	2	yes	—
20	#431	C₂²	1	2	yes	—
21	#424	C₄	1	2	yes	—
22	#423	C₂²	1	2	yes	—
23	#423	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈	C₂²	1	2	yes	—
25	#408	C₄	1	2	yes	—
26	C₄×M₁₆	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
28	#335	C₂²	1	2	yes	—
29	#335	C₂²	1	2	yes	—
30	#335	C₂²	1	2	yes	—
31	#334	C₂²	1	2	yes	—
32	#334	C₂²	1	1	yes	characteristic
33	#334	C₂²	1	1	yes	characteristic
34	#334	C₂²	1	1	yes	characteristic
35	#334	C₂²	1	1	yes	characteristic
36	#334	C₂²	1	2	yes	—
37	#334	C₂²	1	2	yes	—
38	C₂×#41	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆	—	2	2	no	—
40	C₂×M₁₆	—	2	2	no	—
41	C₂×M₁₆	C₂×C₄	1	2	yes	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	C₂×C₄	1	2	yes	—
44	C₄×Q₈	D₄	1	1	yes	characteristic
45	C₄×Q₈	Q₈	1	1	yes	characteristic
46	C₄×D₄	D₄	1	1	yes	characteristic
47	C₄×D₄	D₄	1	2	yes	—
48	C₄×D₄	Q₈	1	1	yes	characteristic
49	C₄×D₄	Q₈	1	2	yes	—
50	#117	C₂×C₄	1	1	yes	characteristic
51	#117	C₂×C₄	1	2	yes	—
52	C₂×C₄²	C₂³	1	1	yes	characteristic
53	C₂×C₄²	C₂³	1	2	yes	—
54	C₄⋊C₈	—	2	2	no	—
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	—	2	2	no	—
57	C₄⋊C₈	C₂×C₄	1	4	yes	—
58	C₄⋊C₈	C₂×C₄	1	4	yes	—
59	C₄⋊C₈	C₂³	1	2	yes	—
60	C₄⋊C₈	C₂³	1	2	yes	—
61	C₄⋊C₈	C₂³	1	2	yes	—
62	C₄⋊C₈	C₂³	1	2	yes	—
63	C₂²⋊C₈	—	2	2	no	—
64	C₂²⋊C₈	—	2	2	no	—
65	C₂²⋊C₈	—	2	2	no	—
66	C₂²⋊C₈	C₂×C₄	1	4	yes	—
67	C₂²⋊C₈	C₂×C₄	1	4	yes	—
68	#97	—	2	2	no	—
69	#97	—	2	2	no	—
70	#97	C₂×C₄	1	2	yes	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	C₂×C₄	1	2	yes	—
73	C₄×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₈	C₂³	1	1	yes	characteristic
75	C₄×C₈	C₂³	1	2	yes	—
76	#41	—	4	2	no	—
77	C₂×Q₈	C₄⋊C₄	1	1	yes	characteristic
78	C₂×D₄	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₄	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄	C₂²×C₄	1	2	yes	—
81	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₄	1	2	yes	—
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄	C₂×Q₈	1	2	yes	—
86	M₁₆	—	4	2	no	—
87	M₁₆	—	8	2	no	—
88	M₁₆	—	8	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	—	8	2	no	—
95	C₂×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₈	C₂²×C₄	1	4	yes	—
98	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
99	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
100	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
102	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
103	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
104	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
105	C₄²	C₂²×C₄	1	1	yes	characteristic
106	C₄²	C₂²×C₄	1	2	yes	—
107	C₄²	C₂×D₄	1	2	yes	—
108	C₄²	C₂×D₄	1	1	yes	characteristic
109	C₄²	C₂×Q₈	1	1	yes	characteristic
110	C₄²	C₂×Q₈	1	2	yes	—
111	C₄²	C₂⁴	1	1	yes	characteristic
112	C₂³	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
114	Q₈	—	4	1	no	—
115	D₄	—	4	1	no	—
116	D₄	—	4	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	2	no	—
135	C₈	—	2	4	no	—
136	C₈	—	4	2	no	—
137	C₈	—	4	2	no	—
138	C₈	—	4	2	no	—
139	C₈	—	4	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	1	no	—
146	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	2	no	—
154	C₄	#568	1	2	yes	—
155	C₄	#577	1	2	yes	—
156	C₂	—	4	1	no	—
157	C₂	—	4	2	no	—
158	C₂	#2905	1	1	yes	characteristic
159	C₂	#2912	1	1	yes	characteristic
160	C₂	#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂²×C₄⋊C₄
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1574	C₂	semidirect
4. #1573	C₂	semidirect
5. #1572	C₂	semidirect
6. #1570	C₂	semidirect
7. #1569	C₂	semidirect
8. #1568	C₂	semidirect
9. #1566	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2989	2
#2922	2
#1574	2
#1573	2
#1572	2
#1570	2
#1569	2
#1568	2
#1566	2