Finite Group Databasebeta

Subgroups — #18066

Order 256.

361 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 163 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	#3169	C₂	1	1	yes	maximal, characteristic
2	#2935	C₂	1	2	yes	maximal, complemented
3	#1589	C₂	1	2	yes	maximal, complemented
4	#1581	C₂	1	4	yes	maximal, complemented
5	C₈×D₈	C₂	1	2	yes	maximal, complemented
6	C₂×#335	C₂	1	1	yes	maximal, characteristic, complemented
7	#1480	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131	—	2	2	no	—
10	#532	C₂²	1	2	yes	—
11	#500	C₄	1	1	yes	characteristic
12	#493	C₄	1	1	yes	characteristic
13	#486	C₄	1	2	yes	—
14	#466	C₄	1	2	yes	—
15	#466	C₄	1	2	yes	—
16	C₄×D₈	C₂²	1	2	yes	—
17	C₄×D₈	C₂²	1	2	yes	—
18	#436	C₂²	1	4	yes	—
19	#435	—	2	4	no	—
20	C₈×D₄	C₂²	1	4	yes	—
21	#431	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈	C₂²	1	2	yes	—
23	#408	—	2	2	no	—
24	C₄×M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	#335	C₂²	1	1	yes	characteristic
27	#335	C₂²	1	2	yes	—
28	#335	C₂²	1	1	yes	characteristic
29	D₄⋊C₈	C₂²	1	4	yes	—
30	D₄⋊C₈	C₂²	1	4	yes	—
31	#322	C₂²	1	2	yes	—
32	C₈²	C₂²	1	2	yes	—
33	C₂×#41	—	2	2	no	—
34	C₂×D₈	C₂×C₄	1	2	yes	—
35	C₂×D₈	C₂×C₄	1	2	yes	—
36	#131	—	4	8	no	—
37	C₂×M₁₆	—	2	4	no	—
38	C₂×M₁₆	D₄	1	1	yes	characteristic
39	C₂×M₁₆	D₄	1	1	yes	characteristic
40	C₂²×C₈	—	2	2	no	—
41	C₂²×C₈	—	2	4	no	—
42	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄⋊Q₈	C₂×C₄	1	2	yes	—
45	#127	C₂×C₄	1	2	yes	—
46	#124	—	2	2	no	—
47	#121	—	2	2	no	—
48	#121	C₂×C₄	1	4	yes	—
49	C₄×D₄	—	2	2	no	—
50	C₄×D₄	C₂³	1	4	yes	—
51	C₂×C₄²	C₂³	1	1	yes	characteristic
52	#107	C₂×C₄	1	2	yes	—
53	#107	C₂×C₄	1	2	yes	—
54	C₄⋊C₈	—	2	2	no	—
55	C₄⋊C₈	—	2	1	no	—
56	C₄⋊C₈	C₂³	1	2	yes	—
57	C₄⋊C₈	C₂³	1	2	yes	—
58	#102	C₂×C₄	1	4	yes	—
59	#102	C₂×C₄	1	4	yes	—
60	C₂²⋊C₈	—	2	1	no	—
61	C₂²⋊C₈	—	2	4	no	—
62	C₂²⋊C₈	—	2	4	no	—
63	#97	D₄	1	2	yes	—
64	C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈	C₂³	1	2	yes	—
67	C₄×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₈	C₂³	1	2	yes	—
70	#41	—	4	4	no	—
71	#41	—	4	4	no	—
72	C₂×Q₈	—	2	2	no	—
73	C₂×D₄	—	2	4	no	—
74	C₂×D₄	—	2	2	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	C₂²×C₄	1	4	yes	—
77	C₂²×C₄	—	2	4	no	—
78	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄	#41	1	1	yes	characteristic
81	D₈	—	2	4	no	—
82	D₈	—	4	2	no	—
83	M₁₆	—	2	2	no	—
84	M₁₆	—	4	8	no	—
85	M₁₆	D₈	1	4	yes	complemented
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	4	4	no	—
92	C₂×C₈	—	4	8	no	—
93	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₈	C₂×D₄	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	C₂²×C₄	1	4	yes	—
101	C₂²⋊C₄	—	2	4	no	—
102	C₂²⋊C₄	—	2	4	no	—
103	C₄²	C₂²×C₄	1	1	yes	characteristic
104	C₄²	C₂×D₄	1	1	yes	characteristic
105	C₄²	#41	1	1	yes	characteristic
106	C₄²	C₂⁴	1	1	yes	characteristic
107	C₂³	—	2	4	no	—
108	C₂³	C₄×D₄	1	1	yes	characteristic
109	Q₈	—	4	4	no	—
110	D₄	—	2	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	4	8	no	—
113	D₄	—	4	4	no	—
114	D₄	—	8	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	4	4	no	—
121	C₂×C₄	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄	C₂×D₈	1	2	yes	—
125	C₂×C₄	#135	1	2	yes	—
126	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄	C₂×#41	1	1	yes	characteristic
129	C₈	—	2	1	no	—
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	2	2	no	—
133	C₈	—	4	2	no	—
134	C₈	—	4	2	no	—
135	C₈	—	4	2	no	—
136	C₈	#131	1	2	yes	complemented
137	C₈	C₂×D₈	1	4	yes	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	4	no	—
143	C₂²	C₄×D₈	1	2	yes	—
144	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	4	no	—
149	C₄	C₄×D₈	1	2	yes	—
150	C₄	C₂×#131	1	1	yes	characteristic
151	C₄	#568	1	1	yes	characteristic
152	C₄	C₂²×D₈	1	1	yes	characteristic
153	C₄	C₂×#135	1	1	yes	characteristic
154	C₂	—	2	2	no	—
155	C₂	—	8	4	no	—
156	C₂	#2935	1	1	yes	characteristic
157	C₂	C₂×C₄×D₈	1	1	yes	characteristic
158	C₂	#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2935	C₂	semidirect
3. #1589	C₂	semidirect
4. #1581	C₂	semidirect
5. C₈×D₈	C₂	semidirect
6. C₂×#335	C₂	semidirect
7. #1480	C₂	semidirect
8. C₈×M₁₆	C₂	semidirect
85. M₁₆	D₈	semidirect
136. C₈	#131	semidirect

Maximal subgroups

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

subgroup	index
#3169	2
#2935	2
#1589	2
#1581	2
C₈×D₈	2
C₂×#335	2
#1480	2
C₈×M₁₆	2