Finite Group Databasebeta

Subgroups — #19267

Order 256.

435 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 127 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, complemented
2	#3038	C₂	1	2	yes	maximal, complemented
3	C₈×#41	C₂	1	1	yes	maximal, characteristic, complemented
4	#1668	C₂	1	4	yes	maximal, complemented
5	#1667	C₂	1	4	yes	maximal, complemented
6	#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	#1545	C₂	1	2	yes	maximal, complemented
8	C₂×AGL(1,ℤ/8)	—	2	2	no	—
9	C₂²×D₈	—	2	2	no	—
10	C₂×#127	C₂²	1	2	yes	—
11	C₄×#41	C₂²	1	1	yes	characteristic
12	#493	—	2	1	no	—
13	#493	—	2	1	no	—
14	#493	C₂²	1	4	yes	—
15	#460	—	2	2	no	—
16	#460	C₂²	1	4	yes	complemented
17	#459	—	2	2	no	—
18	#459	C₂²	1	4	yes	complemented
19	C₈×Q₈	C₂²	1	2	yes	complemented
20	C₈×D₄	—	2	2	no	—
21	C₈×D₄	C₂²	1	2	yes	complemented
22	#431	—	2	1	no	—
23	#431	C₂²	1	1	yes	characteristic
24	#423	C₂²	1	2	yes	—
25	#418	—	2	2	no	—
26	C₂×C₄×C₈	—	2	1	no	—
27	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
28	#335	C₂²	1	4	yes	complemented
29	#331	C₂²	1	8	yes	complemented
30	C₂×#41	D₄	1	1	yes	characteristic
31	C₂²×D₄	—	2	2	no	—
32	C₂²×D₄	—	2	2	no	—
33	C₂²×D₄	—	2	2	no	—
34	AGL(1,ℤ/8)	—	4	8	no	—
35	C₂×SD₁₆	—	2	4	no	—
36	C₂×D₈	—	2	4	no	—
37	C₂×D₈	—	2	4	no	—
38	C₂×D₈	—	2	4	no	—
39	C₂×D₈	—	4	8	no	—
40	C₂×M₁₆	—	2	2	no	—
41	C₂²×C₈	—	2	2	no	—
42	C₂²×C₈	D₄	1	2	yes	—
43	#127	—	2	2	no	—
44	#127	—	2	2	no	—
45	#127	—	2	2	no	—
46	#127	C₂³	1	4	yes	—
47	C₄×Q₈	D₄	1	1	yes	characteristic
48	C₄×Q₈	C₂³	1	1	yes	characteristic
49	C₄×D₄	—	2	1	no	—
50	C₄×D₄	—	2	1	no	—
51	C₄×D₄	D₄	1	1	yes	characteristic
52	C₄×D₄	C₂³	1	1	yes	characteristic
53	#117	—	2	1	no	—
54	#117	D₄	1	1	yes	characteristic
55	C₂×C₄²	—	2	1	no	—
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	2	2	no	—
59	C₄⋊C₈	C₂³	1	2	yes	—
60	C₄⋊C₈	C₂³	1	4	yes	—
61	#102	—	2	4	no	—
62	#102	—	2	4	no	—
63	C₂²⋊C₈	—	2	1	no	—
64	C₂²⋊C₈	—	2	2	no	—
65	C₄×C₈	—	2	2	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	D₄	1	1	yes	characteristic
68	C₄×C₈	D₄	1	1	yes	characteristic
69	C₄×C₈	C₂³	1	2	yes	—
70	C₂⁴	—	4	2	no	—
71	#41	—	2	2	no	—
72	#41	D₈	1	4	yes	complemented
73	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
74	C₂×D₄	—	2	1	no	—
75	C₂×D₄	—	2	4	no	—
76	C₂×D₄	—	2	4	no	—
77	C₂×D₄	—	2	4	no	—
78	C₂×D₄	—	4	8	no	—
79	C₂×D₄	—	4	2	no	—
80	C₂×D₄	—	4	2	no	—
81	C₂×D₄	—	4	2	no	—
82	C₂×D₄	—	4	4	no	—
83	C₂×D₄	—	4	4	no	—
84	C₂×D₄	—	8	4	no	—
85	C₂×D₄	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄	#41	1	1	yes	characteristic
92	SD₁₆	—	4	8	no	—
93	D₈	—	4	8	no	—
94	D₈	—	4	8	no	—
95	D₈	—	4	8	no	—
96	M₁₆	—	4	4	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	4	4	no	—
101	C₂×C₈	C₂×D₄	1	2	yes	—
102	C₂×C₈	C₂×D₄	1	2	yes	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
107	C₂²⋊C₄	—	2	1	no	—
108	C₂²⋊C₄	—	2	1	no	—
109	C₂²⋊C₄	—	2	1	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	C₂×D₄	1	1	yes	characteristic
114	C₄²	C₂×D₄	1	1	yes	characteristic
115	C₄²	#41	1	1	yes	characteristic
116	C₄²	C₂⁴	1	1	yes	characteristic
117	C₂³	—	2	1	no	—
118	C₂³	—	4	4	no	—
119	C₂³	—	8	4	no	—
120	C₂³	—	8	4	no	—
121	C₂³	—	8	4	no	—
122	C₂³	#121	1	1	yes	characteristic
123	Q₈	—	2	1	no	—
124	Q₈	C₂×D₈	1	2	yes	complemented
125	D₄	—	2	4	no	—
126	D₄	—	2	1	no	—
127	D₄	—	4	4	no	—
128	D₄	—	4	4	no	—
129	D₄	—	4	8	no	—
130	D₄	—	8	4	no	—
131	D₄	—	8	4	no	—
132	D₄	—	8	4	no	—
133	D₄	—	8	4	no	—
134	D₄	C₂×D₈	1	2	yes	complemented
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	#121	1	1	yes	characteristic
144	C₂×C₄	#121	1	1	yes	characteristic
145	C₂×C₄	#121	1	1	yes	characteristic
146	C₂×C₄	C₂×D₈	1	2	yes	—
147	C₂×C₄	#135	1	2	yes	—
148	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄	C₂×#41	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	4	4	no	—
153	C₈	—	4	2	no	—
154	C₈	AGL(1,ℤ/8)	1	4	yes	complemented
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	8	4	no	—
161	C₂²	—	8	4	no	—
162	C₂²	—	8	4	no	—
163	C₂²	—	16	8	no	—
164	C₂²	#466	1	2	yes	—
165	C₂²	C₂×#121	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	1	no	—
171	C₄	#466	1	2	yes	—
172	C₄	#472	1	2	yes	—
173	C₄	C₂²×D₈	1	1	yes	characteristic
174	C₄	C₂×#135	1	1	yes	characteristic
175	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
176	C₂	—	2	2	no	—
177	C₂	—	4	2	no	—
178	C₂	—	16	4	no	—
179	C₂	#3038	1	1	yes	characteristic
180	C₂	C₂×#466	1	1	yes	characteristic
181	C₂	C₂×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#493	C₂	semidirect
2. #3038	C₂	semidirect
3. C₈×#41	C₂	semidirect
4. #1668	C₂	semidirect
5. #1667	C₂	semidirect
6. #1567	C₂	semidirect
7. #1545	C₂	semidirect
16. #460	C₂²	semidirect
18. #459	C₂²	semidirect
19. C₈×Q₈	C₂²	semidirect
21. C₈×D₄	C₂²	semidirect
28. #335	C₂²	semidirect
29. #331	C₂²	semidirect
72. #41	D₈	semidirect
124. Q₈	C₂×D₈	semidirect
134. D₄	C₂×D₈	semidirect
154. C₈	AGL(1,ℤ/8)	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
#3038	2
C₈×#41	2
#1668	2
#1667	2
#1567	2
#1545	2