Finite Group Databasebeta

Subgroups — #19187

Order 256.

421 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 123 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	#3205	C₂	1	2	yes	maximal, complemented
2	C₂×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	#2931	C₂	1	1	yes	maximal, characteristic, complemented
4	#1687	C₂	1	4	yes	maximal, complemented
5	#1667	C₂	1	4	yes	maximal, complemented
6	#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	#1545	C₂	1	2	yes	maximal, complemented
8	C₂²×D₈	—	2	1	no	—
9	C₂²×D₈	—	2	1	no	—
10	C₂×#131	—	2	1	no	—
11	#532	C₂²	1	1	yes	characteristic
12	C₂×#127	C₂²	1	2	yes	—
13	#493	—	2	2	no	—
14	#493	—	2	1	no	—
15	#493	—	2	1	no	—
16	#493	C₂²	1	2	yes	—
17	#493	C₂²	1	2	yes	—
18	#486	—	2	2	no	—
19	#459	C₂²	1	8	yes	complemented
20	#447	—	2	4	no	—
21	#436	—	2	1	no	—
22	#436	C₂²	1	2	yes	complemented
23	C₈×D₄	—	2	1	no	—
24	C₈×D₄	C₂²	1	2	yes	complemented
25	#431	C₂²	1	2	yes	—
26	#423	C₂²	1	1	yes	characteristic
27	#408	—	2	1	no	—
28	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
29	#335	C₂²	1	2	yes	complemented
30	#334	C₂²	1	2	yes	complemented
31	#331	C₂²	1	8	yes	complemented
32	C₂×#41	—	2	1	no	—
33	C₂²×D₄	—	2	2	no	—
34	C₂²×D₄	—	2	2	no	—
35	C₂²×D₄	—	2	2	no	—
36	C₂×D₈	—	2	8	no	—
37	C₂×D₈	—	2	2	no	—
38	C₂×D₈	—	2	2	no	—
39	C₂×D₈	—	2	2	no	—
40	C₂×D₈	—	2	2	no	—
41	C₂×D₈	—	4	4	no	—
42	C₂×D₈	—	4	4	no	—
43	#131	—	4	4	no	—
44	C₂×M₁₆	—	2	1	no	—
45	C₂×M₁₆	—	2	2	no	—
46	C₂²×C₈	—	2	2	no	—
47	C₂²×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₈	D₄	1	1	yes	characteristic
49	C₄⋊Q₈	D₄	1	1	yes	characteristic
50	#127	—	2	2	no	—
51	#127	—	2	2	no	—
52	#127	—	2	2	no	—
53	#127	D₄	1	1	yes	characteristic
54	#127	C₂³	1	4	yes	—
55	#124	—	2	1	no	—
56	#121	—	2	1	no	—
57	#121	D₄	1	2	yes	—
58	C₄×D₄	—	2	1	no	—
59	C₄×D₄	C₂³	1	2	yes	—
60	C₂×C₄²	C₂³	1	1	yes	characteristic
61	C₄⋊C₈	—	2	1	no	—
62	C₄⋊C₈	C₂³	1	2	yes	—
63	C₄⋊C₈	C₂³	1	4	yes	—
64	#102	—	2	8	no	—
65	C₂²⋊C₈	—	2	2	no	—
66	C₂²⋊C₈	—	2	2	no	—
67	C₂²⋊C₈	—	2	2	no	—
68	C₄×C₈	—	2	2	no	—
69	C₄×C₈	D₄	1	1	yes	characteristic
70	C₄×C₈	D₄	1	1	yes	characteristic
71	C₄×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₈	C₂³	1	1	yes	characteristic
73	C₂⁴	—	4	2	no	—
74	#41	—	4	2	no	—
75	#41	—	4	2	no	—
76	C₂×Q₈	—	2	1	no	—
77	C₂×D₄	—	2	2	no	—
78	C₂×D₄	—	2	1	no	—
79	C₂×D₄	—	2	1	no	—
80	C₂×D₄	—	2	4	no	—
81	C₂×D₄	—	2	4	no	—
82	C₂×D₄	—	2	4	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂×D₄	—	4	2	no	—
85	C₂×D₄	—	4	2	no	—
86	C₂×D₄	—	4	4	no	—
87	C₂×D₄	—	4	4	no	—
88	C₂×D₄	—	8	4	no	—
89	C₂×D₄	—	8	4	no	—
90	C₂×D₄	C₂×D₄	1	2	yes	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄	#41	1	1	yes	characteristic
95	D₈	—	4	4	no	—
96	D₈	—	4	4	no	—
97	D₈	—	4	4	no	—
98	D₈	—	4	4	no	—
99	D₈	—	8	8	no	—
100	M₁₆	—	4	2	no	—
101	M₁₆	—	4	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	4	4	no	—
106	C₂×C₈	D₈	1	4	yes	complemented
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₄²	C₂×D₄	1	1	yes	characteristic
116	C₄²	C₂×D₄	1	1	yes	characteristic
117	C₄²	#41	1	1	yes	characteristic
118	C₄²	C₂⁴	1	1	yes	characteristic
119	C₂³	—	2	2	no	—
120	C₂³	—	4	4	no	—
121	C₂³	—	8	4	no	—
122	C₂³	—	8	4	no	—
123	C₂³	—	8	4	no	—
124	C₂³	#121	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	4	4	no	—
128	D₄	—	4	4	no	—
129	D₄	—	4	4	no	—
130	D₄	—	4	8	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	8	4	no	—
134	D₄	—	8	4	no	—
135	D₄	—	8	4	no	—
136	D₄	—	8	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	#121	1	1	yes	characteristic
143	C₂×C₄	#121	1	1	yes	characteristic
144	C₂×C₄	#121	1	1	yes	characteristic
145	C₂×C₄	C₂×D₈	1	2	yes	—
146	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄	C₂×#41	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	4	2	no	—
153	C₈	—	4	4	no	—
154	C₈	C₂×D₈	1	2	yes	—
155	C₈	C₂×D₈	1	2	yes	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	2	no	—
161	C₂²	—	8	4	no	—
162	C₂²	—	8	4	no	—
163	C₂²	—	8	4	no	—
164	C₂²	—	16	8	no	—
165	C₂²	#459	1	2	yes	—
166	C₂²	C₂×#121	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	2	no	—
170	C₄	#459	1	2	yes	—
171	C₄	#472	1	2	yes	—
172	C₄	C₂²×D₈	1	1	yes	characteristic
173	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₄	#576	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	8	2	no	—
177	C₂	—	16	4	no	—
178	C₂	C₂×#459	1	1	yes	characteristic
179	C₂	#3058	1	1	yes	characteristic
180	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. #3205	C₂	semidirect
2. C₂×#493	C₂	semidirect
3. #2931	C₂	semidirect
4. #1687	C₂	semidirect
5. #1667	C₂	semidirect
6. #1564	C₂	semidirect
7. #1545	C₂	semidirect
19. #459	C₂²	semidirect
22. #436	C₂²	semidirect
24. C₈×D₄	C₂²	semidirect
29. #335	C₂²	semidirect
30. #334	C₂²	semidirect
31. #331	C₂²	semidirect
106. C₂×C₈	D₈	semidirect

Maximal subgroups

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

subgroup	index
#3205	2
C₂×#493	2
#2931	2
#1687	2
#1667	2
#1564	2
#1545	2