Finite Group Databasebeta

Subgroups — #12561

Order 256.

285 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 159 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	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×#427	C₂	1	1	yes	maximal, characteristic
4	#1382	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆	—	2	1	no	—
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂×#117	—	2	1	no	—
8	C₂×M₃₂	—	2	1	no	—
9	C₂×M₃₂	—	2	1	no	—
10	C₂×M₃₂	—	2	2	no	—
11	C₂×M₃₂	C₄	1	4	yes	complemented
12	C₂²×C₁₆	C₂²	1	2	yes	—
13	C₂×#108	—	2	1	no	—
14	C₂×#108	—	2	2	no	—
15	C₂×#108	C₄	1	4	yes	complemented
16	C₂×#108	C₂²	1	2	yes	—
17	#427	—	2	1	no	—
18	#427	—	2	1	no	—
19	#427	C₄	1	2	yes	—
20	#427	C₂²	1	2	yes	—
21	C₂×#107	C₄	1	2	yes	—
22	C₂×#106	—	2	1	no	—
23	C₂³×C₄	D₄	1	1	yes	characteristic
24	C₂×M₁₆	—	2	4	no	—
25	C₂×M₁₆	—	2	2	no	—
26	C₂×M₁₆	—	4	1	no	—
27	C₂×M₁₆	—	4	2	no	—
28	C₂²×C₈	—	4	1	no	—
29	C₂²×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₈	D₄	1	2	yes	—
37	C₂²×C₈	Q₈	1	2	yes	—
38	C₂²×C₈	C₂³	1	1	yes	characteristic
39	#117	—	2	2	no	—
40	#117	—	2	2	no	—
41	#117	—	4	1	no	—
42	#117	—	4	1	no	—
43	C₂×C₄⋊C₄	—	2	2	no	—
44	C₂×C₂²⋊C₄	—	4	1	no	—
45	C₂×C₄²	—	4	1	no	—
46	M₃₂	—	2	4	no	—
47	M₃₂	—	2	4	no	—
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	C₂×C₄	1	4	yes	complemented
51	#108	—	2	4	no	—
52	#108	—	2	2	no	—
53	#108	C₂×C₄	1	4	yes	complemented
54	#107	—	2	2	no	—
55	#107	C₂×C₄	1	4	yes	—
56	#106	—	2	2	no	—
57	#106	—	2	2	no	—
58	C₂⁴	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄	—	4	2	no	—
60	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄	D₈	1	1	yes	characteristic
65	C₂²×C₄	D₈	1	1	yes	characteristic
66	C₂²×C₄	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄	SD₁₆	1	2	yes	—
68	C₂²×C₄	Q₁₆	1	2	yes	—
69	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
70	M₁₆	—	2	4	no	—
71	M₁₆	—	4	4	no	—
72	M₁₆	—	4	2	no	—
73	C₂×C₈	—	4	4	no	—
74	C₂×C₈	—	4	2	no	—
75	C₂×C₈	C₄²	1	2	yes	—
76	C₂×C₈	C₄²	1	2	yes	—
77	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈	C₄⋊C₄	1	2	yes	—
83	C₂×C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₈	C₄⋊C₄	1	2	yes	—
85	C₂×C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₈	C₄⋊C₄	1	2	yes	—
87	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₂²⋊C₄	—	4	2	no	—
96	C₂²⋊C₄	—	4	2	no	—
97	C₄²	—	4	2	no	—
98	C₄²	—	4	2	no	—
99	C₁₆	—	2	4	no	—
100	C₂³	—	4	2	no	—
101	C₂³	#102	1	1	yes	characteristic
102	C₂³	#102	1	1	yes	characteristic
103	C₂³	#103	1	2	yes	—
104	C₂³	#106	1	2	yes	—
105	C₂³	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄	—	4	4	no	—
107	C₂×C₄	—	8	4	no	—
108	C₂×C₄	#95	1	2	yes	—
109	C₂×C₄	#95	1	2	yes	—
110	C₂×C₄	#102	1	1	yes	characteristic
111	C₂×C₄	#102	1	1	yes	characteristic
112	C₂×C₄	#102	1	2	yes	—
113	C₂×C₄	#102	1	2	yes	—
114	C₂×C₄	#103	1	2	yes	—
115	C₂×C₄	#103	1	2	yes	—
116	C₂×C₄	#103	1	2	yes	—
117	C₂×C₄	#106	1	2	yes	—
118	C₂×C₄	#107	1	2	yes	—
119	C₂×C₄	#107	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
126	C₈	—	4	4	no	—
127	C₈	#95	1	2	yes	—
128	C₈	#95	1	2	yes	—
129	C₈	C₂×C₄²	1	1	yes	characteristic
130	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂²	—	2	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	4	2	no	—
137	C₂²	—	4	2	no	—
138	C₂²	#340	1	2	yes	—
139	C₂²	#340	1	2	yes	—
140	C₂²	C₂×#102	1	1	yes	characteristic
141	C₂²	C₂×#103	1	1	yes	characteristic
142	C₂²	C₂×#106	1	1	yes	characteristic
143	C₄	—	8	4	no	—
144	C₄	#340	1	2	yes	—
145	C₄	#340	1	2	yes	—
146	C₄	C₂×#95	1	1	yes	characteristic
147	C₄	C₂×#102	1	1	yes	characteristic
148	C₄	C₂×#103	1	1	yes	characteristic
149	C₄	C₂×#107	1	1	yes	characteristic
150	C₂	—	2	2	no	—
151	C₂	—	2	2	no	—
152	C₂	—	2	1	no	—
153	C₂	—	2	1	no	—
154	C₂	C₂×#340	1	1	yes	characteristic
155	C₂	#2143	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×#102
prime radical	C₂⁴	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1382	C₂	semidirect
11. C₂×M₃₂	C₄	semidirect
15. C₂×#108	C₄	semidirect
50. C₂×C₁₆	C₂×C₄	semidirect
53. #108	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂	2
C₂²×#108	2
C₂×#427	2
#1382	2