Finite Group Databasebeta

Subgroups — #28023

Order 256.

305 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2514	C₂	1	2	yes	maximal
2	#2074	C₂	1	2	yes	maximal
3	#2068	C₂	1	2	yes	maximal
4	#1971	C₂	1	1	yes	maximal, characteristic
5	#1971	C₂	1	1	yes	maximal, characteristic
6	#1969	C₂	1	1	yes	maximal, characteristic
7	#1944	C₂	1	1	yes	maximal, characteristic
8	#1928	C₂	1	2	yes	maximal
9	#1846	C₂	1	1	yes	maximal, characteristic
10	#1776	C₂	1	1	yes	maximal, characteristic
11	#1771	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×#107	C₂²	1	1	yes	characteristic
14	C₂×#107	C₂²	1	2	yes	—
15	C₂×#106	C₂²	1	1	yes	characteristic
16	C₂×#106	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×#103	C₂²	1	2	yes	—
19	C₂×#103	C₂²	1	2	yes	—
20	C₂×#103	C₂²	1	2	yes	—
21	C₂×#97	C₂²	1	1	yes	characteristic
22	C₂×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	#401	—	2	2	no	—
25	#400	—	2	2	no	—
26	#391	C₂²	1	2	yes	—
27	#391	C₂²	1	2	yes	—
28	#391	C₂²	1	2	yes	—
29	#389	C₂²	1	2	yes	—
30	#387	—	2	2	no	—
31	#383	—	2	2	no	—
32	#382	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	#340	C₂²	1	2	yes	—
36	#340	C₂²	1	2	yes	—
37	#336	C₂²	1	1	yes	characteristic
38	#336	C₂²	1	1	yes	characteristic
39	C₂²×Q₈	C₂³	1	2	yes	—
40	C₂×Q₁₆	—	2	2	no	—
41	C₂×Q₁₆	—	2	2	no	—
42	C₂×Q₁₆	—	2	2	no	—
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	C₂×C₄⋊C₄	—	2	2	no	—
48	C₂×C₄⋊C₄	D₄	1	2	yes	—
49	C₂×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₄²	—	2	2	no	—
53	C₂×C₄²	C₂³	1	1	yes	characteristic
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	#107	—	2	2	no	—
57	#107	—	4	2	no	—
58	#106	—	2	2	no	—
59	#106	—	2	2	no	—
60	C₄⋊C₈	—	2	2	no	—
61	#103	—	2	4	no	—
62	#103	—	4	2	no	—
63	#103	—	4	2	no	—
64	#97	—	2	2	no	—
65	#97	—	2	2	no	—
66	C₄×C₈	—	2	2	no	—
67	#95	—	2	2	no	—
68	#95	—	2	2	no	—
69	#95	—	2	2	no	—
70	#95	—	2	2	no	—
71	#95	—	2	2	no	—
72	#95	—	2	2	no	—
73	#95	—	2	2	no	—
74	#95	—	2	2	no	—
75	#95	—	2	2	no	—
76	#95	—	2	2	no	—
77	#95	D₄	1	2	yes	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	—	2	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₄	1	2	yes	—
90	C₂²×C₄	C₂×D₄	1	2	yes	—
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	C₂⁴	1	1	yes	characteristic
93	Q₁₆	—	4	4	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	#41	1	2	yes	—
103	C₂×C₈	#41	1	2	yes	—
104	C₂×C₈	#41	1	2	yes	—
105	C₂×C₈	#41	1	2	yes	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	4	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	4	4	no	—
121	C₂³	C₂²×D₄	1	1	yes	characteristic
122	Q₈	—	4	2	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	4	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	C₂×#41	1	1	yes	characteristic
149	C₂×C₄	C₂×#41	1	1	yes	characteristic
150	C₂×C₄	C₂×#41	1	1	yes	characteristic
151	C₂×C₄	C₂×#41	1	1	yes	characteristic
152	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	4	1	no	—
156	C₈	—	4	1	no	—
157	C₈	#126	1	4	yes	—
158	C₂²	#532	1	1	yes	characteristic
159	C₂²	#547	1	1	yes	characteristic
160	C₂²	#550	1	1	yes	characteristic
161	C₂²	#575	1	2	yes	—
162	C₂²	#578	1	2	yes	—
163	C₄	—	4	2	no	—
164	C₄	—	4	2	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	8	2	no	—
168	C₄	—	8	2	no	—
169	C₄	—	8	2	no	—
170	C₄	—	8	2	no	—
171	C₄	C₂×#126	1	1	yes	characteristic
172	C₄	#542	1	1	yes	characteristic
173	C₄	#553	1	1	yes	characteristic
174	C₄	#556	1	1	yes	characteristic
175	C₂	#2512	1	1	yes	characteristic
176	C₂	#3177	1	1	yes	characteristic
177	C₂	#3183	1	1	yes	characteristic
178	C₂	#3349	1	2	yes	—
179	C₂	#3402	1	2	yes	—

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

Maximal subgroups

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

subgroup	index
#2514	2
#2074	2
#2068	2
#1971	2
#1969	2
#1944	2
#1928	2
#1846	2
#1776	2
#1771	2