Finite Group Databasebeta

Subgroups — #13016

Order 256.

231 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 67 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×#508	C₂	1	1	yes	maximal, characteristic
2	#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	#2145	C₂	1	1	yes	maximal, characteristic, complemented
4	#2143	C₂	1	1	yes	maximal, characteristic, complemented
5	#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	C₂²	1	1	yes	characteristic
9	C₂×#135	C₂²	1	1	yes	characteristic
10	#568	—	2	1	no	—
11	#508	—	2	2	no	—
12	#508	—	2	2	no	—
13	C₂×Q₃₂	C₄	1	1	yes	characteristic, complemented
14	C₂×SD₃₂	C₄	1	1	yes	characteristic, complemented
15	C₂×SD₃₂	C₄	1	1	yes	characteristic, complemented
16	C₂×D₁₆	C₄	1	1	yes	characteristic, complemented
17	C₂×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆	C₂²	1	1	yes	characteristic
19	#444	—	2	2	no	—
20	#443	—	2	2	no	—
21	C₂×#108	C₂²	1	1	yes	characteristic
22	#427	C₂²	1	1	yes	characteristic
23	#421	—	2	1	no	—
24	#421	—	2	1	no	—
25	#410	—	2	1	no	—
26	#405	C₂²	1	1	yes	characteristic
27	#360	—	2	2	no	—
28	#359	—	2	2	no	—
29	C₂×#41	—	2	1	no	—
30	C₂×#41	—	2	1	no	—
31	#135	—	2	1	no	—
32	#135	—	2	1	no	—
33	#135	—	2	1	no	—
34	#135	—	2	1	no	—
35	#135	—	2	1	no	—
36	#135	—	2	1	no	—
37	#135	D₄	1	2	yes	—
38	#135	D₄	1	2	yes	—
39	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂×SD₁₆	—	2	1	no	—
42	C₂×SD₁₆	—	2	1	no	—
43	C₂×D₈	C₂×C₄	1	1	yes	characteristic
44	C₂×D₈	C₂×C₄	1	1	yes	characteristic
45	#131	—	2	4	no	—
46	#131	—	4	2	no	—
47	C₂×M₁₆	—	2	1	no	—
48	C₂×M₁₆	—	4	2	no	—
49	C₂×M₁₆	D₄	1	1	yes	characteristic
50	C₂²×C₈	C₂³	1	1	yes	characteristic
51	#117	—	2	1	no	—
52	#117	D₄	1	1	yes	characteristic
53	Q₃₂	—	2	2	no	—
54	SD₃₂	—	2	2	no	—
55	SD₃₂	—	2	2	no	—
56	D₁₆	—	2	2	no	—
57	M₃₂	—	2	1	no	—
58	M₃₂	—	2	1	no	—
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	2	1	no	—
61	C₂×C₁₆	—	2	1	no	—
62	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
63	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
64	#108	—	2	1	no	—
65	#108	D₄	1	2	yes	—
66	#107	—	2	1	no	—
67	#106	—	2	1	no	—
68	#104	—	2	4	no	—
69	#104	—	4	2	no	—
70	#99	—	4	2	no	—
71	#97	—	2	1	no	—
72	C₄×C₈	—	2	1	no	—
73	#41	—	2	2	no	—
74	#41	—	2	2	no	—
75	#41	—	4	1	no	—
76	#41	—	4	1	no	—
77	#41	—	4	1	no	—
78	#41	—	4	1	no	—
79	#41	—	4	1	no	—
80	#41	—	4	1	no	—
81	C₂×Q₈	—	2	1	no	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂×D₄	—	4	1	no	—
86	C₂×D₄	—	4	1	no	—
87	C₂²×C₄	—	4	1	no	—
88	C₂²×C₄	—	4	1	no	—
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	Q₁₆	—	2	1	no	—
91	Q₁₆	—	2	1	no	—
92	Q₁₆	—	2	1	no	—
93	Q₁₆	C₂²⋊C₄	1	2	yes	—
94	SD₁₆	—	2	2	no	—
95	SD₁₆	—	2	2	no	—
96	SD₁₆	—	4	1	no	—
97	SD₁₆	—	4	1	no	—
98	D₈	—	2	1	no	—
99	D₈	—	2	1	no	—
100	D₈	—	2	1	no	—
101	D₈	C₂²⋊C₄	1	2	yes	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	4	4	no	—
105	M₁₆	—	4	1	no	—
106	M₁₆	D₈	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	4	4	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈	#41	1	1	yes	characteristic
115	C₂×C₈	#41	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₂²⋊C₄	—	4	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	4	1	no	—
122	C₁₆	—	2	1	no	—
123	C₁₆	—	2	1	no	—
124	C₁₆	—	4	1	no	—
125	C₂³	—	4	1	no	—
126	C₂³	—	4	1	no	—
127	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	4	1	no	—
132	D₄	—	2	2	no	—
133	D₄	—	4	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	1	no	—
138	D₄	—	8	1	no	—
139	D₄	—	8	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	—	8	1	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄	#120	1	1	yes	characteristic
150	C₂×C₄	#121	1	1	yes	characteristic
151	C₂×C₄	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄	#135	1	1	yes	characteristic
153	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	4	2	no	—
156	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₈	C₄×D₄	1	1	yes	characteristic
158	C₈	#121	1	1	yes	characteristic
159	C₈	#123	1	1	yes	characteristic
160	C₂²	—	4	1	no	—
161	C₂²	—	4	2	no	—
162	C₂²	—	8	1	no	—
163	C₂²	—	16	1	no	—
164	C₂²	—	16	1	no	—
165	C₂²	#442	1	1	yes	characteristic
166	C₂²	#449	1	1	yes	characteristic
167	C₂²	#459	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	8	1	no	—
170	C₄	—	8	1	no	—
171	C₄	—	16	1	no	—
172	C₄	#386	1	1	yes	characteristic
173	C₄	C₄×D₈	1	1	yes	characteristic
174	C₄	#447	1	1	yes	characteristic
175	C₄	#463	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	—	8	2	no	—
180	C₂	—	16	1	no	—
181	C₂	#1879	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#447
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1879
prime radical	C₂×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ 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. #2957	C₂	semidirect
3. #2145	C₂	semidirect
4. #2143	C₂	semidirect
5. #1976	C₂	semidirect
6. #1382	C₂	semidirect
7. #1379	C₂	semidirect
13. C₂×Q₃₂	C₄	semidirect
14. C₂×SD₃₂	C₄	semidirect
15. C₂×SD₃₂	C₄	semidirect
16. C₂×D₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508	2
#2957	2
#2145	2
#2143	2
#1976	2
#1382	2
#1379	2