Subgroups — #17811

Order 256.

301 conjugacy classes of subgroups.

The subgroup lattice has 181 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2989	C₂	1	1	yes	maximal, characteristic
2	#2988	C₂	1	1	yes	maximal, characteristic
3	#2922	C₂	1	1	yes	maximal, characteristic
4	#1574	C₂	1	2	yes	maximal, complemented
5	#1573	C₂	1	2	yes	maximal, complemented
6	#1572	C₂	1	2	yes	maximal, complemented
7	#1570	C₂	1	2	yes	maximal, complemented
8	#1565	C₂	1	1	yes	maximal, characteristic, complemented
9	#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	#1564	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#334	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#41	C₂²	1	1	yes	characteristic
13	#446	—	2	1	no	—
14	C₈×Q₈	—	2	1	no	—
15	#436	—	2	1	no	—
16	#435	—	2	2	no	—
17	C₈×D₄	—	2	2	no	—
18	C₈×D₄	—	2	1	no	—
19	#433	C₄	1	4	yes	—
20	#432	C₂²	1	2	yes	—
21	#431	C₂²	1	1	yes	characteristic
22	#431	C₂²	1	1	yes	characteristic
23	#431	C₂²	1	2	yes	—
24	#431	C₂²	1	2	yes	—
25	#431	C₂²	1	2	yes	—
26	#424	C₄	1	2	yes	—
27	#423	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈	C₂²	1	2	yes	—
31	#408	C₄	1	2	yes	—
32	C₄×M₁₆	C₂²	1	1	yes	characteristic
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	#335	C₂²	1	2	yes	—
38	#334	C₂²	1	2	yes	—
39	#334	C₂²	1	2	yes	—
40	#334	C₂²	1	1	yes	characteristic
41	#334	C₂²	1	1	yes	characteristic
42	#334	C₂²	1	2	yes	—
43	#334	C₂²	1	2	yes	—
44	C₂×#41	C₂×C₄	1	1	yes	characteristic
45	C₂×M₁₆	—	2	2	no	—
46	C₂×M₁₆	C₂×C₄	1	2	yes	—
47	C₂²×C₈	—	2	1	no	—
48	C₂²×C₈	—	2	2	no	—
49	C₂²×C₈	—	2	1	no	—
50	C₂²×C₈	C₂×C₄	1	2	yes	—
51	C₄×Q₈	D₄	1	1	yes	characteristic
52	C₄×Q₈	Q₈	1	1	yes	characteristic
53	C₄×D₄	D₄	1	2	yes	—
54	C₄×D₄	D₄	1	1	yes	characteristic
55	C₄×D₄	Q₈	1	2	yes	—
56	C₄×D₄	Q₈	1	1	yes	characteristic
57	#117	C₂×C₄	1	1	yes	characteristic
58	#117	C₂×C₄	1	2	yes	—
59	C₂×C₄²	C₂³	1	1	yes	characteristic
60	C₂×C₄²	C₂³	1	2	yes	—
61	C₄⋊C₈	—	2	1	no	—
62	C₄⋊C₈	—	2	2	no	—
63	C₄⋊C₈	—	2	1	no	—
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	C₂×C₄	1	4	yes	—
66	C₄⋊C₈	C₂×C₄	1	4	yes	—
67	C₄⋊C₈	C₂³	1	2	yes	—
68	C₄⋊C₈	C₂³	1	1	yes	characteristic
69	C₄⋊C₈	C₂³	1	2	yes	—
70	C₄⋊C₈	C₂³	1	1	yes	characteristic
71	C₄⋊C₈	C₂³	1	1	yes	characteristic
72	C₄⋊C₈	C₂³	1	1	yes	characteristic
73	C₂²⋊C₈	—	2	1	no	—
74	C₂²⋊C₈	—	2	2	no	—
75	C₂²⋊C₈	—	2	1	no	—
76	C₂²⋊C₈	—	2	2	no	—
77	C₂²⋊C₈	C₂×C₄	1	4	yes	—
78	C₂²⋊C₈	C₂×C₄	1	4	yes	—
79	#97	—	2	2	no	—
80	#97	C₂×C₄	1	2	yes	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	—	2	2	no	—
83	C₄×C₈	—	2	1	no	—
84	C₄×C₈	C₂×C₄	1	2	yes	—
85	C₄×C₈	C₂³	1	1	yes	characteristic
86	C₄×C₈	C₂³	1	1	yes	characteristic
87	C₄×C₈	C₂³	1	2	yes	—
88	#41	—	2	4	no	—
89	C₂×Q₈	C₄⋊C₄	1	1	yes	characteristic
90	C₂×D₄	C₄⋊C₄	1	2	yes	—
91	C₂×D₄	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄	C₂²×C₄	1	2	yes	—
93	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₄	1	2	yes	—
96	C₂²×C₄	C₂×Q₈	1	2	yes	—
97	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
98	M₁₆	—	2	4	no	—
99	M₁₆	—	8	2	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	8	1	no	—
108	C₂×C₈	—	8	2	no	—
109	C₂×C₈	—	8	1	no	—
110	C₂×C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₈	C₂²×C₄	1	2	yes	—
112	C₂×C₈	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
115	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
116	C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
117	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
118	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
119	C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
120	C₄²	C₂²×C₄	1	1	yes	characteristic
121	C₄²	C₂²×C₄	1	2	yes	—
122	C₄²	C₂×D₄	1	1	yes	characteristic
123	C₄²	C₂×D₄	1	2	yes	—
124	C₄²	C₂×Q₈	1	2	yes	—
125	C₄²	C₂×Q₈	1	1	yes	characteristic
126	C₄²	C₂⁴	1	1	yes	characteristic
127	C₂³	C₂×C₄⋊C₄	1	2	yes	—
128	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
129	Q₈	—	2	2	no	—
130	D₄	—	2	4	no	—
131	D₄	—	2	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄	#131	1	4	yes	—
146	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	2	2	no	—
151	C₈	—	2	4	no	—
152	C₈	—	4	1	no	—
153	C₈	—	4	1	no	—
154	C₈	—	4	1	no	—
155	C₈	—	4	2	no	—
156	C₈	—	4	2	no	—
157	C₈	—	4	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	#424	1	2	yes	—
164	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	2	no	—
171	C₄	#424	1	2	yes	—
172	C₄	C₂×#131	1	2	yes	—
173	C₄	#577	1	1	yes	characteristic
174	C₄	#577	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	4	2	no	—
177	C₂	C₂×#424	1	1	yes	characteristic
178	C₂	#2912	1	1	yes	characteristic
179	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
4. #1574	C₂	semidirect
5. #1573	C₂	semidirect
6. #1572	C₂	semidirect
7. #1570	C₂	semidirect
8. #1565	C₂	semidirect
9. #1565	C₂	semidirect
10. #1564	C₂	semidirect
11. 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
#2989	2
#2988	2
#2922	2
#1574	2
#1573	2
#1572	2
#1570	2
#1565	2
#1564	2
C₂×#334	2