Finite Group Databasebeta

Subgroups — #21697

Order 256.

619 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 619 conjugacy classes of subgroups — too many to draw. Only its 195 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	C₂×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×#442	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₈	C₂	1	2	yes	maximal, complemented
4	#2328	C₂	1	2	yes	maximal, complemented
5	#1972	C₂	1	2	yes	maximal, complemented
6	#1970	C₂	1	2	yes	maximal, complemented
7	#1961	C₂	1	2	yes	maximal, complemented
8	#1846	C₂	1	1	yes	maximal, characteristic, complemented
9	#1726	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈	C₂²	1	2	yes	—
11	C₂²×D₈	C₂²	1	2	yes	—
12	C₂×#127	C₂²	1	2	yes	—
13	C₂×C₄×D₄	C₂²	1	4	yes	—
14	#493	C₄	1	8	yes	complemented
15	#442	—	2	8	no	—
16	C₄×D₈	—	2	8	no	—
17	C₂×#107	C₂²	1	2	yes	—
18	C₂×#106	C₂²	1	2	yes	—
19	C₂×#102	C₂²	1	4	yes	—
20	C₂×#102	C₂²	1	4	yes	—
21	C₂×#97	C₂²	1	1	yes	characteristic
22	C₂×#97	C₂²	1	1	yes	characteristic
23	C₂×#97	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	#390	C₂²	1	4	yes	—
28	#386	—	2	8	no	—
29	#382	C₂²	1	2	yes	—
30	#376	C₂²	1	1	yes	characteristic
31	C₂²×D₄	—	2	4	no	—
32	C₂²×D₄	C₂³	1	4	yes	—
33	C₂³×C₄	—	2	4	no	—
34	C₂×D₈	—	2	4	no	—
35	C₂×D₈	—	2	4	no	—
36	C₂×D₈	—	2	4	no	—
37	C₂×D₈	—	2	4	no	—
38	C₂×D₈	C₂×C₄	1	8	yes	complemented
39	C₂×D₈	C₂×C₄	1	8	yes	complemented
40	C₂²×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	#127	—	2	4	no	—
45	#127	C₂×C₄	1	8	yes	complemented
46	C₄×D₄	—	2	8	no	—
47	C₄×D₄	—	2	8	no	—
48	C₂×C₄⋊C₄	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄	—	2	4	no	—
50	C₂×C₂²⋊C₄	—	2	8	no	—
51	C₂×C₄²	C₂³	1	1	yes	characteristic
52	C₂×C₄²	C₂³	1	1	yes	characteristic
53	C₂×C₄²	C₂³	1	1	yes	characteristic
54	#107	—	2	4	no	—
55	#106	—	2	4	no	—
56	#102	—	2	8	no	—
57	#102	—	2	8	no	—
58	#97	—	2	2	no	—
59	#97	—	2	2	no	—
60	#97	D₄	1	4	yes	complemented
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	C₂×C₄	1	4	yes	complemented
63	C₄×C₈	D₄	1	4	yes	complemented
64	#95	—	2	4	no	—
65	#95	D₄	1	4	yes	—
66	C₂⁴	—	2	4	no	—
67	C₂×D₄	—	2	16	no	—
68	C₂×D₄	—	2	4	no	—
69	C₂×D₄	—	2	4	no	—
70	C₂×D₄	—	2	4	no	—
71	C₂×D₄	—	2	4	no	—
72	C₂×D₄	—	2	4	no	—
73	C₂×D₄	—	8	8	no	—
74	C₂×D₄	C₂²×C₄	1	8	yes	—
75	C₂²×C₄	—	2	4	no	—
76	C₂²×C₄	—	4	8	no	—
77	C₂²×C₄	—	8	8	no	—
78	C₂²×C₄	C₂×D₄	1	2	yes	—
79	C₂²×C₄	C₂×D₄	1	2	yes	—
80	C₂²×C₄	C₂×D₄	1	2	yes	—
81	C₂²×C₄	C₂⁴	1	1	yes	characteristic
82	D₈	—	2	16	no	—
83	D₈	—	2	16	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₈	C₂²×C₄	1	2	yes	—
90	C₂×C₈	C₂×D₄	1	2	yes	—
91	C₂×C₈	C₂×D₄	1	2	yes	—
92	C₂×C₈	C₂×D₄	1	2	yes	—
93	C₂×C₈	C₂×D₄	1	2	yes	—
94	C₂×C₈	#41	1	2	yes	—
95	C₂×C₈	#41	1	2	yes	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	4	no	—
98	C₂²⋊C₄	—	4	8	no	—
99	C₂²⋊C₄	—	8	8	no	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₄²	C₂²×C₄	1	2	yes	—
103	C₄²	C₂×D₄	1	2	yes	—
104	C₄²	C₂×D₄	1	2	yes	—
105	C₄²	#41	1	2	yes	—
106	C₂³	—	2	8	no	—
107	C₂³	—	4	4	no	—
108	C₂³	—	4	4	no	—
109	C₂³	—	4	8	no	—
110	C₂³	—	4	8	no	—
111	C₂³	C₂²×D₄	1	1	yes	characteristic
112	D₄	—	2	8	no	—
113	D₄	—	2	8	no	—
114	D₄	—	4	4	no	—
115	D₄	—	4	4	no	—
116	D₄	—	8	16	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	4	4	no	—
122	C₂×C₄	—	4	8	no	—
123	C₂×C₄	—	8	16	no	—
124	C₂×C₄	C₄×D₄	1	4	yes	—
125	C₂×C₄	#127	1	4	yes	—
126	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄	C₂×#41	1	1	yes	characteristic
130	C₂×C₄	C₂×#41	1	1	yes	characteristic
131	C₂×C₄	C₂×#41	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	2	2	no	—
134	C₈	C₄×D₄	1	4	yes	—
135	C₈	C₄×D₄	1	4	yes	—
136	C₂²	—	4	8	no	—
137	C₂²	—	4	16	no	—
138	C₂²	—	8	4	no	—
139	C₂²	—	8	4	no	—
140	C₂²	—	8	8	no	—
141	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²	C₂×#127	1	1	yes	characteristic
143	C₂²	#532	1	1	yes	characteristic
144	C₂²	#576	1	4	yes	—
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	—	4	2	no	—
148	C₄	—	8	4	no	—
149	C₄	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄	#532	1	1	yes	characteristic
152	C₄	#532	1	1	yes	characteristic
153	C₂	—	8	8	no	—
154	C₂	C₄×#127	1	1	yes	characteristic
155	C₂	#2948	1	2	yes	—
156	C₂	#3155	1	2	yes	—
157	C₂	#3176	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂×#493	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#442	C₂	semidirect
3. C₂×C₄×D₈	C₂	semidirect
4. #2328	C₂	semidirect
5. #1972	C₂	semidirect
6. #1970	C₂	semidirect
7. #1961	C₂	semidirect
8. #1846	C₂	semidirect
9. #1726	C₂	semidirect
14. #493	C₄	semidirect
38. C₂×D₈	C₂×C₄	semidirect
39. C₂×D₈	C₂×C₄	semidirect
45. #127	C₂×C₄	semidirect
60. #97	D₄	semidirect
62. C₄×C₈	C₂×C₄	semidirect
63. C₄×C₈	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493	2
C₂×#442	2
C₂×C₄×D₈	2
#2328	2
#1972	2
#1970	2
#1961	2
#1846	2
#1726	2