Subgroups — #20521

Order 256.

433 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	#2295	C₂	1	1	yes	maximal, characteristic
4	#1771	C₂	1	2	yes	maximal
5	#1770	C₂	1	2	yes	maximal
6	#1770	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	#1768	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
10	#1726	C₂	1	2	yes	maximal
11	C₄×#97	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
13	#446	C₄	1	8	yes	complemented
14	C₈×Q₈	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×#97	C₂²	1	1	yes	characteristic
20	C₂×#97	C₂²	1	1	yes	characteristic
21	C₂×#97	C₂²	1	1	yes	characteristic
22	C₂×#97	C₂²	1	2	yes	—
23	C₂×#97	C₂²	1	2	yes	—
24	C₂×#97	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂²	1	2	yes	—
26	C₂×C₄×C₈	C₂²	1	2	yes	—
27	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂²	1	2	yes	—
29	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
30	#391	C₄	1	2	yes	—
31	#389	C₄	1	2	yes	—
32	#387	C₄	1	4	yes	—
33	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄	C₂²	1	2	yes	—
35	#376	C₂²	1	2	yes	—
36	C₄³	C₂²	1	1	yes	characteristic
37	#336	C₂²	1	2	yes	—
38	#336	C₂²	1	2	yes	—
39	#336	C₂²	1	2	yes	—
40	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	2	yes	—
42	C₂²×C₈	C₂³	1	2	yes	—
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	C₄×Q₈	C₂×C₄	1	4	yes	—
48	C₄×Q₈	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²	C₂×C₄	1	2	yes	—
54	C₂×C₄²	C₂×C₄	1	2	yes	—
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	C₂×C₄²	C₂³	1	2	yes	—
57	C₂×C₄²	C₂³	1	1	yes	characteristic
58	C₂×C₄²	C₂³	1	2	yes	—
59	C₂×C₄²	C₂³	1	1	yes	characteristic
60	C₄⋊C₈	C₂×C₄	1	4	yes	—
61	C₄⋊C₈	C₂×C₄	1	8	yes	—
62	C₄⋊C₈	C₂×C₄	1	4	yes	—
63	C₄⋊C₈	C₂×C₄	1	8	yes	—
64	#97	—	2	2	no	—
65	#97	—	2	2	no	—
66	#97	—	2	4	no	—
67	#97	—	2	2	no	—
68	#97	C₂×C₄	1	8	yes	—
69	#97	Q₈	1	4	yes	complemented
70	C₄×C₈	—	2	4	no	—
71	C₄×C₈	—	2	4	no	—
72	C₄×C₈	C₂×C₄	1	4	yes	—
73	C₄×C₈	C₂×C₄	1	8	yes	—
74	C₄×C₈	C₂×C₄	1	4	yes	—
75	#95	C₂×C₄	1	4	yes	—
76	#95	C₂×C₄	1	4	yes	—
77	#95	C₂×C₄	1	2	yes	—
78	#95	C₂×C₄	1	4	yes	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	C₄²	1	4	yes	—
81	C₂²×C₄	C₂²×C₄	1	4	yes	—
82	C₂²×C₄	C₂²×C₄	1	2	yes	—
83	C₂²×C₄	C₂²×C₄	1	2	yes	—
84	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄	C₂²×C₄	1	2	yes	—
86	C₂²×C₄	C₂²×C₄	1	2	yes	—
87	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	C₂²×C₄	1	4	yes	—
98	C₂×C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₈	C₂²×C₄	1	2	yes	—
103	C₂×C₈	C₂×Q₈	1	2	yes	—
104	C₂×C₈	C₂×Q₈	1	2	yes	—
105	C₂×C₈	#41	1	2	yes	—
106	C₂×C₈	#41	1	2	yes	—
107	C₄⋊C₄	—	2	4	no	—
108	C₄⋊C₄	—	4	4	no	—
109	C₄⋊C₄	C₄²	1	8	yes	—
110	C₄⋊C₄	C₄²	1	4	yes	—
111	C₄²	—	2	4	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	4	2	no	—
117	C₄²	C₂²×C₄	1	4	yes	—
118	C₄²	C₂²×C₄	1	2	yes	—
119	C₄²	C₂²×C₄	1	2	yes	—
120	C₄²	C₂²×C₄	1	4	yes	—
121	C₄²	C₂×Q₈	1	2	yes	—
122	C₄²	#41	1	2	yes	—
123	C₂³	C₂³×C₄	1	1	yes	characteristic
124	Q₈	—	2	8	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	C₂×C₄²	1	2	yes	—
137	C₂×C₄	C₂×C₄²	1	4	yes	—
138	C₂×C₄	C₂×C₄²	1	4	yes	—
139	C₂×C₄	C₂×C₄²	1	2	yes	—
140	C₂×C₄	C₄×Q₈	1	4	yes	—
141	C₂×C₄	#131	1	4	yes	—
142	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄	C₂×#41	1	1	yes	characteristic
146	C₂×C₄	C₂×#41	1	1	yes	characteristic
147	C₂×C₄	C₂×#41	1	1	yes	characteristic
148	C₈	—	2	4	no	—
149	C₈	—	2	4	no	—
150	C₈	—	2	2	no	—
151	C₈	—	2	2	no	—
152	C₈	C₄×Q₈	1	4	yes	—
153	C₈	C₄×Q₈	1	4	yes	—
154	C₂²	C₂²×C₄²	1	1	yes	characteristic
155	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₂²	C₄×#41	1	1	yes	characteristic
157	C₂²	C₂×#131	1	2	yes	—
158	C₂²	#568	1	2	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	2	2	no	—
161	C₄	—	2	4	no	—
162	C₄	—	2	4	no	—
163	C₄	—	4	2	no	—
164	C₄	—	4	4	no	—
165	C₄	#405	1	4	yes	—
166	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄	C₄×#41	1	1	yes	characteristic
169	C₄	C₄×#41	1	1	yes	characteristic
170	C₂	C₄²×Q₈	1	1	yes	characteristic
171	C₂	C₂×#405	1	1	yes	characteristic
172	C₂	#2875	1	1	yes	characteristic
173	C₂	#2962	1	2	yes	—
174	C₂	#2969	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
13. #446	C₄	semidirect
14. C₈×Q₈	C₄	semidirect
69. #97	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446	2
C₂×C₈×Q₈	2
#2295	2
#1771	2
#1770	2
C₈×C₄⋊C₄	2
#1768	2
C₄×C₄⋊C₈	2
#1726	2
C₄×#97	2