Finite Group Databasebeta

Subgroups — #13026

Order 256.

243 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 69 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×#510	C₂	1	1	yes	maximal, characteristic
2	#2943	C₂	1	1	yes	maximal, characteristic
3	C₂×#360	C₂	1	1	yes	maximal, characteristic
4	#2141	C₂	1	1	yes	maximal, characteristic, complemented
5	#1971	C₂	1	1	yes	maximal, characteristic, complemented
6	#1386	C₂	1	1	yes	maximal, characteristic, complemented
7	#1377	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
10	C₄×#41	—	2	1	no	—
11	#510	—	2	2	no	—
12	#510	C₄	1	4	yes	complemented
13	C₂×Q₃₂	—	2	1	no	—
14	C₂×SD₃₂	—	2	1	no	—
15	C₂×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×M₃₂	C₂²	1	1	yes	characteristic
17	#442	—	2	1	no	—
18	#441	—	2	1	no	—
19	#440	—	2	2	no	—
20	#427	C₂²	1	1	yes	characteristic
21	C₂×#106	C₂²	1	1	yes	characteristic
22	#416	—	2	1	no	—
23	C₂×#103	—	2	1	no	—
24	C₂×#97	C₂²	1	1	yes	characteristic
25	#391	—	2	1	no	—
26	#360	—	2	2	no	—
27	#358	—	2	2	no	—
28	C₂×#41	—	2	1	no	—
29	C₂²×Q₈	—	2	1	no	—
30	#135	—	2	1	no	—
31	#135	—	2	1	no	—
32	#135	C₂×C₄	1	2	yes	—
33	#135	D₄	1	2	yes	—
34	C₂×Q₁₆	—	2	1	no	—
35	C₂×Q₁₆	—	2	1	no	—
36	C₂×Q₁₆	—	2	1	no	—
37	C₂×Q₁₆	—	2	1	no	—
38	C₂×Q₁₆	—	2	1	no	—
39	C₂×Q₁₆	C₂×C₄	1	2	yes	complemented
40	C₂×Q₁₆	D₄	1	1	yes	characteristic
41	C₂×SD₁₆	—	2	1	no	—
42	C₂×D₈	D₄	1	1	yes	characteristic
43	C₂²×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₄×Q₈	—	2	2	no	—
46	C₄×D₄	—	2	2	no	—
47	C₄×D₄	—	4	2	no	—
48	#117	—	2	1	no	—
49	#117	—	4	1	no	—
50	C₂×C₄⋊C₄	—	2	1	no	—
51	C₂×C₄²	—	4	1	no	—
52	C₂×C₄²	D₄	1	1	yes	characteristic
53	Q₃₂	—	2	4	no	—
54	SD₃₂	—	2	4	no	—
55	M₃₂	—	2	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	—	2	1	no	—
58	M₃₂	C₂×C₄	1	2	yes	complemented
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	2	1	no	—
61	#107	D₄	1	2	yes	—
62	#106	—	2	1	no	—
63	#106	—	2	1	no	—
64	#106	—	2	1	no	—
65	#103	—	2	2	no	—
66	#103	—	4	2	no	—
67	#102	—	2	2	no	—
68	#97	—	2	1	no	—
69	#97	—	2	1	no	—
70	#95	—	4	2	no	—
71	#41	—	2	2	no	—
72	#41	—	2	2	no	—
73	#41	—	4	1	no	—
74	#41	—	4	1	no	—
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	—	4	1	no	—
78	C₂×Q₈	—	4	1	no	—
79	C₂×Q₈	—	4	1	no	—
80	C₂×Q₈	—	4	1	no	—
81	C₂×Q₈	—	4	1	no	—
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	—	4	1	no	—
84	C₂²×C₄	—	4	2	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	—	4	1	no	—
87	C₂²×C₄	—	4	1	no	—
88	C₂²×C₄	D₈	1	2	yes	—
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	Q₁₆	—	2	2	no	—
91	Q₁₆	—	2	2	no	—
92	Q₁₆	—	2	1	no	—
93	Q₁₆	—	4	1	no	—
94	Q₁₆	—	4	1	no	—
95	Q₁₆	C₂²⋊C₄	1	2	yes	—
96	SD₁₆	—	2	2	no	—
97	SD₁₆	—	2	2	no	—
98	D₈	—	2	1	no	—
99	D₈	C₂²⋊C₄	1	2	yes	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	#41	1	1	yes	characteristic
108	C₂×C₈	#41	1	1	yes	characteristic
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₂²⋊C₄	—	4	2	no	—
114	C₂²⋊C₄	—	4	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	4	2	no	—
118	C₄²	—	4	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	4	1	no	—
121	C₂³	—	4	1	no	—
122	C₂³	C₂×D₈	1	1	yes	characteristic
123	Q₈	—	2	2	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	4	2	no	—
126	Q₈	—	4	2	no	—
127	Q₈	—	8	1	no	—
128	Q₈	—	8	1	no	—
129	D₄	—	2	2	no	—
130	D₄	—	4	2	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	8	4	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	8	1	no	—
145	C₂×C₄	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄	#120	1	1	yes	characteristic
147	C₂×C₄	#121	1	1	yes	characteristic
148	C₂×C₄	#135	1	1	yes	characteristic
149	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₈	—	4	1	no	—
152	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₈	C₄×D₄	1	1	yes	characteristic
154	C₈	#121	1	1	yes	characteristic
155	C₈	#123	1	1	yes	characteristic
156	C₂²	—	2	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	8	2	no	—
159	C₂²	C₄×D₈	1	1	yes	characteristic
160	C₂²	#447	1	1	yes	characteristic
161	C₂²	#459	1	1	yes	characteristic
162	C₄	—	4	2	no	—
163	C₄	—	4	2	no	—
164	C₄	—	8	2	no	—
165	C₄	—	8	2	no	—
166	C₄	—	16	1	no	—
167	C₄	#386	1	1	yes	characteristic
168	C₄	#442	1	1	yes	characteristic
169	C₄	#449	1	1	yes	characteristic
170	C₄	#463	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	8	2	no	—
174	C₂	#1879	1	1	yes	characteristic
175	C₂	#2191	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#447
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ 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. #2141	C₂	semidirect
5. #1971	C₂	semidirect
6. #1386	C₂	semidirect
7. #1377	C₂	semidirect
12. #510	C₄	semidirect
39. C₂×Q₁₆	C₂×C₄	semidirect
58. M₃₂	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#510	2
#2943	2
C₂×#360	2
#2141	2
#1971	2
#1386	2
#1377	2