Finite Group Databasebeta

Subgroups — #33719

Order 256.

417 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 135 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	2	yes	maximal, complemented
2	C₂×#509	C₂	1	2	yes	maximal, complemented
3	#3147	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#495	C₂	1	1	yes	maximal, characteristic, complemented
5	#2251	C₂	1	2	yes	maximal, complemented
6	#2250	C₂	1	2	yes	maximal, complemented
7	#2249	C₂	1	4	yes	maximal, complemented
8	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135	—	2	1	no	—
10	C₂×#135	C₂²	1	2	yes	—
11	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂²×SD₁₆	—	2	1	no	—
13	C₂²×D₈	C₂²	1	1	yes	characteristic
14	#532	—	2	1	no	—
15	C₂×#124	—	2	1	no	—
16	#510	—	2	8	no	—
17	#509	—	2	8	no	—
18	C₂×Q₃₂	C₂²	1	4	yes	complemented
19	C₂×SD₃₂	C₂²	1	4	yes	—
20	C₂×SD₃₂	C₂²	1	4	yes	complemented
21	C₂×D₁₆	C₂²	1	4	yes	—
22	C₂×M₃₂	C₂²	1	2	yes	—
23	#495	—	2	1	no	—
24	#495	—	2	1	no	—
25	#495	—	2	1	no	—
26	#495	C₂²	1	2	yes	—
27	#495	C₂²	1	2	yes	—
28	#495	C₂²	1	2	yes	complemented
29	#494	C₂²	1	1	yes	characteristic, complemented
30	#493	C₂²	1	1	yes	characteristic, complemented
31	#492	—	2	1	no	—
32	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
33	#346	C₂²	1	2	yes	complemented
34	C₄×C₁₆	C₂²	1	2	yes	complemented
35	C₂×#41	—	2	2	no	—
36	C₂²×Q₈	—	2	1	no	—
37	C₂²×D₄	—	2	1	no	—
38	#135	—	2	4	no	—
39	#135	—	2	4	no	—
40	#135	—	4	4	no	—
41	C₂×Q₁₆	—	2	4	no	—
42	C₂×Q₁₆	—	2	1	no	—
43	C₂×Q₁₆	—	2	1	no	—
44	C₂×Q₁₆	C₂³	1	2	yes	—
45	C₂×Q₁₆	C₂³	1	2	yes	—
46	C₂×SD₁₆	—	2	2	no	—
47	C₂×SD₁₆	—	2	2	no	—
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×SD₁₆	—	2	2	no	—
50	C₂×SD₁₆	—	4	4	no	—
51	C₂×D₈	—	2	4	no	—
52	C₂×D₈	—	2	1	no	—
53	C₂×D₈	—	2	1	no	—
54	C₂×D₈	C₂³	1	2	yes	—
55	C₂×D₈	C₂³	1	2	yes	—
56	C₂²×C₈	D₄	1	1	yes	characteristic
57	C₂²×C₈	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈	—	2	1	no	—
59	#127	—	2	1	no	—
60	#124	—	2	2	no	—
61	#124	—	2	2	no	—
62	#124	—	2	2	no	—
63	#124	—	4	2	no	—
64	#121	—	4	2	no	—
65	C₄×D₄	—	4	2	no	—
66	C₂×C₂²⋊C₄	—	4	2	no	—
67	C₂×C₄²	D₄	1	1	yes	characteristic
68	Q₃₂	—	2	8	no	—
69	SD₃₂	—	2	8	no	—
70	SD₃₂	—	2	8	no	—
71	D₁₆	—	2	8	no	—
72	M₃₂	D₄	1	8	yes	complemented
73	C₂×C₁₆	C₂³	1	4	yes	—
74	C₄×C₈	D₄	1	1	yes	characteristic
75	C₄×C₈	D₄	1	1	yes	characteristic
76	C₄×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₈	C₂³	1	1	yes	characteristic
78	C₂⁴	—	4	1	no	—
79	#41	—	4	4	no	—
80	#41	—	4	4	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	4	1	no	—
84	C₂×Q₈	—	4	2	no	—
85	C₂×Q₈	—	4	2	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	4	2	no	—
89	C₂×D₄	—	4	2	no	—
90	C₂×D₄	—	4	2	no	—
91	C₂×D₄	—	4	2	no	—
92	C₂×D₄	—	4	1	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	D₈	1	2	yes	—
96	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
97	Q₁₆	—	2	2	no	—
98	Q₁₆	—	2	4	no	—
99	Q₁₆	—	2	2	no	—
100	Q₁₆	—	4	2	no	—
101	Q₁₆	—	4	2	no	—
102	SD₁₆	—	4	4	no	—
103	SD₁₆	—	4	4	no	—
104	SD₁₆	—	4	4	no	—
105	SD₁₆	—	4	4	no	—
106	D₈	—	2	2	no	—
107	D₈	—	2	4	no	—
108	D₈	—	2	2	no	—
109	D₈	—	4	2	no	—
110	D₈	—	4	2	no	—
111	C₂×C₈	D₈	1	4	yes	—
112	C₂×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈	C₂×D₄	1	4	yes	—
114	C₂×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄	—	4	2	no	—
118	C₂²⋊C₄	—	4	4	no	—
119	C₂²⋊C₄	—	4	4	no	—
120	C₂²⋊C₄	—	4	4	no	—
121	C₄²	D₈	1	2	yes	—
122	C₄²	C₂×D₄	1	1	yes	characteristic
123	C₄²	C₂×D₄	1	1	yes	characteristic
124	C₁₆	C₂×D₄	1	8	yes	—
125	C₂³	—	4	2	no	—
126	C₂³	—	4	2	no	—
127	C₂³	—	8	4	no	—
128	C₂³	—	8	2	no	—
129	C₂³	C₂×D₈	1	1	yes	characteristic
130	Q₈	—	4	2	no	—
131	Q₈	—	4	2	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	4	2	no	—
134	Q₈	—	8	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	2	no	—
137	D₄	—	4	2	no	—
138	D₄	—	4	2	no	—
139	D₄	—	8	4	no	—
140	D₄	—	8	4	no	—
141	D₄	—	8	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	8	4	no	—
147	C₂×C₄	#127	1	2	yes	—
148	C₂×C₄	C₂×D₈	1	2	yes	—
149	C₂×C₄	C₂×D₈	1	2	yes	—
150	C₂×C₄	C₂×D₈	1	2	yes	—
151	C₂×C₄	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
153	C₈	#127	1	2	yes	—
154	C₈	C₂×D₈	1	2	yes	—
155	C₈	C₂×D₈	1	2	yes	—
156	C₈	C₂²×D₄	1	2	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	—	8	2	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	8	4	no	—
161	C₂²	—	8	4	no	—
162	C₂²	—	16	4	no	—
163	C₂²	#493	1	2	yes	—
164	C₂²	C₂²×D₈	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	2	no	—
167	C₄	—	8	2	no	—
168	C₄	—	8	2	no	—
169	C₄	#493	1	2	yes	—
170	C₄	C₂×#127	1	1	yes	characteristic
171	C₄	C₂²×D₈	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	—	16	2	no	—
174	C₂	—	16	2	no	—
175	C₂	C₂×#493	1	1	yes	characteristic
176	C₂	#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#510	C₂	semidirect
2. C₂×#509	C₂	semidirect
3. #3147	C₂	semidirect
4. C₂×#495	C₂	semidirect
5. #2251	C₂	semidirect
6. #2250	C₂	semidirect
7. #2249	C₂	semidirect
8. C₄×M₃₂	C₂	semidirect
18. C₂×Q₃₂	C₂²	semidirect
20. C₂×SD₃₂	C₂²	semidirect
28. #495	C₂²	semidirect
29. #494	C₂²	semidirect
30. #493	C₂²	semidirect
33. #346	C₂²	semidirect
34. C₄×C₁₆	C₂²	semidirect
72. M₃₂	D₄	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
C₂×#509	2
#3147	2
C₂×#495	2
#2251	2
#2250	2
#2249	2
C₄×M₃₂	2