Finite Group Databasebeta

Subgroups — #33839

Order 256.

356 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 107 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	2	yes	maximal, complemented
2	#3280	C₂	1	2	yes	maximal, complemented
3	#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	#2220	C₂	1	2	yes	maximal, complemented
5	#2210	C₂	1	2	yes	maximal, complemented
6	#2189	C₂	1	2	yes	maximal, complemented
7	#2184	C₂	1	2	yes	maximal, complemented
8	C₄×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#2169	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135	C₂²	1	2	yes	—
11	C₂²×D₈	C₂²	1	2	yes	complemented
12	#546	—	2	2	no	—
13	#509	—	2	8	no	—
14	C₂×Q₃₂	C₂²	1	1	yes	characteristic, complemented
15	C₂×SD₃₂	C₂²	1	2	yes	complemented
16	C₂×SD₃₂	C₂²	1	2	yes	complemented
17	C₂×D₁₆	C₂²	1	2	yes	—
18	C₂×D₁₆	C₂²	1	1	yes	characteristic
19	C₂×D₁₆	C₂²	1	1	yes	characteristic
20	C₂×M₃₂	C₂²	1	2	yes	complemented
21	#495	C₂²	1	2	yes	complemented
22	#467	C₂²	1	2	yes	complemented
23	#466	C₂²	1	2	yes	—
24	#463	—	2	2	no	—
25	#450	—	2	2	no	—
26	#449	—	2	2	no	—
27	C₄×D₈	C₂²	1	2	yes	complemented
28	C₈×D₄	C₂²	1	1	yes	characteristic
29	#366	C₂²	1	1	yes	characteristic, complemented
30	C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
31	#358	C₂²	1	2	yes	complemented
32	#357	C₂²	1	2	yes	complemented
33	#357	C₂²	1	2	yes	complemented
34	C₂²⋊C₁₆	C₂²	1	2	yes	complemented
35	C₄×C₁₆	C₂²	1	1	yes	characteristic, complemented
36	C₂×#41	—	2	2	no	—
37	C₂²×D₄	—	2	2	no	—
38	#135	—	2	4	no	—
39	#135	—	4	2	no	—
40	C₂×Q₁₆	C₂³	1	2	yes	—
41	C₂×SD₁₆	—	2	2	no	—
42	C₂×SD₁₆	—	2	2	no	—
43	C₂×D₈	—	2	4	no	—
44	C₂×D₈	—	2	2	no	—
45	C₂×D₈	—	4	2	no	—
46	C₂×D₈	C₂³	1	2	yes	—
47	C₂×D₈	C₂³	1	2	yes	—
48	C₂²×C₈	C₂³	1	2	yes	—
49	#124	—	2	2	no	—
50	#123	—	4	2	no	—
51	C₂²⋊Q₈	—	2	2	no	—
52	#121	—	2	2	no	—
53	#121	—	4	2	no	—
54	#120	—	4	2	no	—
55	C₄×D₄	—	2	2	no	—
56	C₄×D₄	D₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄	—	4	2	no	—
58	Q₃₂	—	4	1	no	—
59	SD₃₂	—	2	4	no	—
60	SD₃₂	—	2	4	no	—
61	D₁₆	—	2	4	no	—
62	D₁₆	—	4	1	no	—
63	D₁₆	D₄	1	4	yes	complemented
64	M₃₂	—	2	4	no	—
65	C₂×C₁₆	C₂³	1	1	yes	characteristic
66	C₂×C₁₆	C₂³	1	2	yes	—
67	C₂×C₁₆	C₂³	1	1	yes	characteristic
68	#107	C₂³	1	2	yes	—
69	C₄⋊C₈	D₄	1	1	yes	characteristic
70	#103	—	2	2	no	—
71	#102	—	2	2	no	—
72	#102	—	2	2	no	—
73	C₂²⋊C₈	D₄	1	2	yes	—
74	C₄×C₈	C₂³	1	1	yes	characteristic
75	C₂⁴	—	4	2	no	—
76	#41	—	4	4	no	—
77	#41	—	8	2	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	—	4	4	no	—
82	C₂×D₄	—	4	2	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂×D₄	—	4	2	no	—
85	C₂×D₄	—	8	2	no	—
86	C₂×D₄	D₈	1	1	yes	characteristic
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	C₂×D₄	1	2	yes	—
90	Q₁₆	—	2	2	no	—
91	Q₁₆	—	2	2	no	—
92	SD₁₆	—	4	4	no	—
93	SD₁₆	—	8	2	no	—
94	D₈	—	2	2	no	—
95	D₈	—	2	2	no	—
96	D₈	—	2	2	no	—
97	D₈	—	4	4	no	—
98	D₈	C₂×D₄	1	4	yes	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈	C₂×D₄	1	2	yes	—
103	C₂×C₈	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	D₈	1	1	yes	characteristic
107	C₂²⋊C₄	—	4	2	no	—
108	C₂²⋊C₄	—	4	2	no	—
109	C₂²⋊C₄	—	4	2	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₂²⋊C₄	—	4	2	no	—
112	C₂²⋊C₄	D₈	1	2	yes	—
113	C₄²	C₂×D₄	1	1	yes	characteristic
114	C₁₆	—	2	2	no	—
115	C₁₆	—	2	1	no	—
116	C₁₆	C₂×D₄	1	2	yes	—
117	C₂³	—	4	2	no	—
118	C₂³	—	4	2	no	—
119	C₂³	—	8	4	no	—
120	C₂³	—	8	4	no	—
121	C₂³	—	16	2	no	—
122	C₂³	C₂×D₈	1	2	yes	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	D₄	—	2	4	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	8	4	no	—
131	D₄	—	8	4	no	—
132	D₄	—	16	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	8	4	no	—
138	C₂×C₄	—	16	2	no	—
139	C₂×C₄	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄	C₂×D₈	1	2	yes	—
142	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	C₂²×D₄	1	1	yes	characteristic
146	C₈	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	4	4	no	—
151	C₂²	—	8	2	no	—
152	C₂²	—	8	2	no	—
153	C₂²	—	8	2	no	—
154	C₂²	—	8	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	16	8	no	—
157	C₂²	C₂²×D₈	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	8	2	no	—
163	C₄	D₄²	1	1	yes	characteristic
164	C₄	#576	1	1	yes	characteristic
165	C₂	—	4	2	no	—
166	C₂	—	8	4	no	—
167	C₂	—	16	2	no	—
168	C₂	D₄×D₈	1	1	yes	characteristic
169	C₂	#3414	1	1	yes	characteristic
170	C₂	#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ 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×#509	C₂	semidirect
2. #3280	C₂	semidirect
3. #2249	C₂	semidirect
4. #2220	C₂	semidirect
5. #2210	C₂	semidirect
6. #2189	C₂	semidirect
7. #2184	C₂	semidirect
8. C₄×D₁₆	C₂	semidirect
9. #2169	C₂	semidirect
11. C₂²×D₈	C₂²	semidirect
14. C₂×Q₃₂	C₂²	semidirect
15. C₂×SD₃₂	C₂²	semidirect
16. C₂×SD₃₂	C₂²	semidirect
20. C₂×M₃₂	C₂²	semidirect
21. #495	C₂²	semidirect
22. #467	C₂²	semidirect
27. C₄×D₈	C₂²	semidirect
29. #366	C₂²	semidirect
30. C₄⋊C₁₆	C₂²	semidirect
31. #358	C₂²	semidirect
32. #357	C₂²	semidirect
33. #357	C₂²	semidirect
34. C₂²⋊C₁₆	C₂²	semidirect
35. C₄×C₁₆	C₂²	semidirect
63. D₁₆	D₄	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
#3280	2
#2249	2
#2220	2
#2210	2
#2189	2
#2184	2
C₄×D₁₆	2
#2169	2