Subgroups — #17834

Order 256.

308 conjugacy classes of subgroups.

The subgroup lattice has 167 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2990	C₂	1	2	yes	maximal
2	C₈×#41	C₂	1	1	yes	maximal, characteristic
3	#1574	C₂	1	2	yes	maximal, complemented
4	#1573	C₂	1	4	yes	maximal, complemented
5	#1568	C₂	1	2	yes	maximal, complemented
6	#1566	C₂	1	1	yes	maximal, characteristic, complemented
7	#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#334	C₂	1	2	yes	maximal, complemented
9	C₄×#41	C₂²	1	1	yes	characteristic
10	#446	—	2	2	no	—
11	C₈×Q₈	C₄	1	1	yes	characteristic
12	C₈×Q₈	C₄	1	1	yes	characteristic
13	#435	—	2	4	no	—
14	C₈×D₄	—	2	2	no	—
15	C₈×D₄	C₄	1	2	yes	—
16	C₈×D₄	C₄	1	1	yes	characteristic
17	C₈×D₄	C₄	1	2	yes	—
18	C₈×D₄	C₄	1	1	yes	characteristic
19	#432	C₂²	1	4	yes	—
20	#431	C₂²	1	2	yes	—
21	#431	C₂²	1	2	yes	—
22	#431	C₂²	1	1	yes	characteristic
23	#423	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈	C₂²	1	4	yes	—
25	C₂×C₄×C₈	C₂²	1	2	yes	—
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	#335	C₂²	1	2	yes	—
28	#335	C₂²	1	4	yes	—
29	#334	C₂²	1	1	yes	characteristic
30	#334	C₂²	1	2	yes	—
31	#334	C₂²	1	4	yes	—
32	#334	C₂²	1	1	yes	characteristic
33	#334	C₂²	1	2	yes	—
34	C₂×#41	Q₈	1	1	yes	characteristic
35	C₂×M₁₆	—	2	2	no	—
36	C₂²×C₈	—	2	4	no	—
37	C₂²×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈	C₂×C₄	1	2	yes	—
40	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
41	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
42	C₄×Q₈	D₄	1	1	yes	characteristic
43	C₄×D₄	C₂×C₄	1	2	yes	—
44	C₄×D₄	C₂×C₄	1	1	yes	characteristic
45	C₄×D₄	D₄	1	1	yes	characteristic
46	C₄×D₄	D₄	1	2	yes	—
47	#117	Q₈	1	1	yes	characteristic
48	#117	Q₈	1	2	yes	—
49	C₂×C₄²	C₂³	1	2	yes	—
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	C₄⋊C₈	—	2	4	no	—
52	C₄⋊C₈	—	2	2	no	—
53	C₄⋊C₈	C₂×C₄	1	2	yes	—
54	C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₈	C₂³	1	2	yes	—
58	C₄⋊C₈	C₂³	1	4	yes	—
59	C₄⋊C₈	C₂³	1	2	yes	—
60	C₂²⋊C₈	—	2	2	no	—
61	C₂²⋊C₈	—	2	4	no	—
62	C₂²⋊C₈	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈	C₂×C₄	1	2	yes	—
64	C₂²⋊C₈	C₂×C₄	1	2	yes	—
65	#97	—	2	4	no	—
66	C₄×C₈	—	2	2	no	—
67	C₄×C₈	C₂×C₄	1	2	yes	—
68	C₄×C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈	C₂×C₄	1	2	yes	—
70	C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₈	C₂³	1	2	yes	—
73	C₄×C₈	C₂³	1	1	yes	characteristic
74	#41	—	2	4	no	—
75	C₂×Q₈	C₄⋊C₄	1	1	yes	characteristic
76	C₂×D₄	C₄⋊C₄	1	2	yes	—
77	C₂×D₄	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄	C₂²×C₄	1	2	yes	—
80	C₂²×C₄	C₂×D₄	1	2	yes	—
81	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄	C₂×Q₈	1	2	yes	—
83	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
84	M₁₆	—	8	2	no	—
85	C₂×C₈	—	2	4	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	4	2	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	—	8	4	no	—
93	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈	C₂²×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	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
103	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
104	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
105	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
106	C₄²	C₂²×C₄	1	2	yes	—
107	C₄²	C₂²×C₄	1	1	yes	characteristic
108	C₄²	C₂×D₄	1	1	yes	characteristic
109	C₄²	C₂×D₄	1	2	yes	—
110	C₄²	C₂×Q₈	1	2	yes	—
111	C₄²	C₂×Q₈	1	1	yes	characteristic
112	C₄²	C₂⁴	1	1	yes	characteristic
113	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂³	C₂×C₄⋊C₄	1	2	yes	—
115	Q₈	—	2	2	no	—
116	D₄	—	2	4	no	—
117	D₄	#108	1	4	yes	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	2	no	—
139	C₈	—	2	1	no	—
140	C₈	—	4	2	no	—
141	C₈	—	4	2	no	—
142	C₈	—	4	4	no	—
143	C₈	#131	1	2	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	C₂×#108	1	4	yes	—
150	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	2	no	—
157	C₄	C₂×#108	1	2	yes	—
158	C₄	C₂×#131	1	1	yes	characteristic
159	C₄	#568	1	1	yes	characteristic
160	C₄	#577	1	2	yes	—
161	C₂	—	2	4	no	—
162	C₂	—	4	1	no	—
163	C₂	#2906	1	1	yes	characteristic
164	C₂	#2912	1	1	yes	characteristic
165	C₂	C₂²×#108	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	Q₈
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. #1568	C₂	semidirect
6. #1566	C₂	semidirect
7. #1564	C₂	semidirect
8. C₂×#334	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2990	2
C₈×#41	2
#1574	2
#1573	2
#1568	2
#1566	2
#1564	2
C₂×#334	2