Finite Group Databasebeta

Subgroups — #13040

Order 256.

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

normal subgroups only

This group has 254 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×#509	C₂	1	1	yes	maximal, characteristic
2	C₂×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#360	C₂	1	1	yes	maximal, characteristic, complemented
5	#1972	C₂	1	1	yes	maximal, characteristic, complemented
6	#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	C₂²	1	1	yes	characteristic
9	C₂²×D₈	C₂²	1	1	yes	characteristic
10	C₂×#131	—	2	1	no	—
11	#509	—	2	2	no	—
12	#509	—	2	2	no	—
13	C₂×SD₃₂	C₄	1	2	yes	complemented
14	C₂×D₁₆	C₄	1	2	yes	complemented
15	C₂×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×M₃₂	C₂²	1	1	yes	characteristic
17	#444	—	2	2	no	—
18	#444	—	2	2	no	—
19	C₂×#108	C₂²	1	1	yes	characteristic
20	C₂×#106	C₂²	1	1	yes	characteristic
21	C₂×#104	—	2	1	no	—
22	C₂×#102	—	2	1	no	—
23	C₂×#97	C₂²	1	1	yes	characteristic
24	#390	—	2	1	no	—
25	#361	—	2	2	no	—
26	#360	—	2	2	no	—
27	C₂×#41	—	2	1	no	—
28	C₂²×D₄	—	2	1	no	—
29	#135	—	2	1	no	—
30	#135	—	2	1	no	—
31	#135	D₄	1	2	yes	—
32	#135	D₄	1	2	yes	—
33	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×SD₁₆	—	2	1	no	—
35	C₂×D₈	—	2	1	no	—
36	C₂×D₈	—	2	1	no	—
37	C₂×D₈	—	2	1	no	—
38	C₂×D₈	—	2	1	no	—
39	C₂×D₈	—	2	1	no	—
40	C₂×D₈	C₂×C₄	1	2	yes	—
41	C₂×D₈	C₂×C₄	1	1	yes	characteristic
42	#131	—	2	4	no	—
43	#131	—	4	2	no	—
44	C₂×M₁₆	—	2	1	no	—
45	C₂×M₁₆	—	4	1	no	—
46	C₂²×C₈	—	4	1	no	—
47	C₂²×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄	—	2	1	no	—
50	C₂×C₂²⋊C₄	—	4	2	no	—
51	C₂×C₄²	D₄	1	1	yes	characteristic
52	SD₃₂	—	2	4	no	—
53	D₁₆	—	2	4	no	—
54	M₃₂	—	2	1	no	—
55	M₃₂	—	2	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	—	2	1	no	—
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	C₂×C₄	1	2	yes	—
60	#108	—	2	1	no	—
61	#108	D₄	1	2	yes	—
62	#106	—	2	1	no	—
63	#106	—	2	1	no	—
64	#104	—	2	4	no	—
65	#102	—	4	2	no	—
66	#97	—	2	1	no	—
67	#97	—	2	1	no	—
68	C₂⁴	—	4	1	no	—
69	#41	—	2	2	no	—
70	#41	—	2	2	no	—
71	#41	—	4	1	no	—
72	#41	—	4	1	no	—
73	C₂×Q₈	—	2	1	no	—
74	C₂×D₄	—	2	1	no	—
75	C₂×D₄	—	2	2	no	—
76	C₂×D₄	—	4	1	no	—
77	C₂×D₄	—	4	1	no	—
78	C₂×D₄	—	4	1	no	—
79	C₂×D₄	—	4	1	no	—
80	C₂×D₄	—	4	1	no	—
81	C₂×D₄	—	4	1	no	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	4	1	no	—
84	C₂²×C₄	—	4	1	no	—
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	Q₁₆	—	2	1	no	—
87	Q₁₆	C₂²⋊C₄	1	2	yes	complemented
88	SD₁₆	—	2	2	no	—
89	SD₁₆	—	2	2	no	—
90	D₈	—	2	2	no	—
91	D₈	—	2	1	no	—
92	D₈	—	2	2	no	—
93	D₈	—	4	1	no	—
94	D₈	—	4	1	no	—
95	D₈	C₂²⋊C₄	1	2	yes	complemented
96	M₁₆	—	2	2	no	—
97	M₁₆	—	4	4	no	—
98	M₁₆	—	4	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	4	4	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	D₈	1	2	yes	—
104	C₂×C₈	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	#41	1	1	yes	characteristic
109	C₂×C₈	#41	1	1	yes	characteristic
110	C₄⋊C₄	—	4	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₂²⋊C₄	—	8	4	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	1	no	—
115	C₁₆	—	2	2	no	—
116	C₁₆	—	4	1	no	—
117	C₂³	—	4	1	no	—
118	C₂³	—	4	2	no	—
119	C₂³	—	8	2	no	—
120	C₂³	—	8	2	no	—
121	C₂³	—	8	1	no	—
122	C₂³	—	8	1	no	—
123	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
124	Q₈	—	2	2	no	—
125	Q₈	—	4	1	no	—
126	D₄	—	2	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	1	no	—
131	D₄	—	4	2	no	—
132	D₄	—	8	1	no	—
133	D₄	—	8	1	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	1	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	8	1	no	—
141	C₂×C₄	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄	#120	1	1	yes	characteristic
143	C₂×C₄	#121	1	1	yes	characteristic
144	C₂×C₄	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄	#135	1	1	yes	characteristic
146	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	4	2	no	—
149	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₈	C₄×D₄	1	1	yes	characteristic
151	C₈	#121	1	1	yes	characteristic
152	C₈	#123	1	1	yes	characteristic
153	C₂²	—	2	2	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	—	8	4	no	—
158	C₂²	—	16	2	no	—
159	C₂²	—	16	1	no	—
160	C₂²	—	16	1	no	—
161	C₂²	#442	1	1	yes	characteristic
162	C₂²	#449	1	1	yes	characteristic
163	C₂²	#459	1	1	yes	characteristic
164	C₄	—	4	2	no	—
165	C₄	—	8	1	no	—
166	C₄	—	16	1	no	—
167	C₄	#386	1	1	yes	characteristic
168	C₄	C₄×D₈	1	1	yes	characteristic
169	C₄	#447	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	—	16	2	no	—
175	C₂	#1879	1	1	yes	characteristic
176	C₂	AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#442
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#509	C₂
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
2. C₂×#444	C₂	semidirect
3. C₂×#361	C₂	semidirect
4. C₂×#360	C₂	semidirect
5. #1972	C₂	semidirect
6. #1383	C₂	semidirect
7. #1375	C₂	semidirect
13. C₂×SD₃₂	C₄	semidirect
14. C₂×D₁₆	C₄	semidirect
87. Q₁₆	C₂²⋊C₄	semidirect
95. D₈	C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#509	2
C₂×#444	2
C₂×#361	2
C₂×#360	2
#1972	2
#1383	2
#1375	2