Finite Group Databasebeta

Subgroups — #60626

Order 256.

1067 conjugacy classes of subgroups; only the 479 normal subgroups are shown.

normal subgroups only

This group has 1067 conjugacy classes of subgroups — too many to draw. Only its 479 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	#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	#3498	C₂	1	1	yes	maximal, characteristic
3	#3142	C₂	1	3	yes	maximal, complemented
4	#3138	C₂	1	3	yes	maximal, complemented
5	#3136	C₂	1	12	yes	maximal, complemented
6	#3135	C₂	1	3	yes	maximal, complemented
7	C₂×#490	C₂	1	3	yes	maximal, complemented
8	#2897	C₂	1	4	yes	maximal, complemented
9	#2875	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×H(5,𝔽₂)	C₂²	1	2	yes	—
11	C₂×#131	C₂²	1	2	yes	—
12	#557	—	2	2	no	—
13	#554	—	2	6	no	—
14	#550	—	2	2	no	—
15	D₄²	—	2	6	no	—
16	#544	C₂²	1	6	yes	complemented
17	#535	C₂²	1	6	yes	—
18	#533	C₂²	1	3	yes	complemented
19	#532	C₂²	1	3	yes	—
20	C₂×#127	C₂²	1	3	yes	—
21	C₂×#125	C₂²	1	3	yes	complemented
22	#520	C₂²	1	2	yes	—
23	C₄×#41	C₂²	1	1	yes	characteristic
24	#491	C₂²	1	1	yes	characteristic, complemented
25	#491	C₂²	1	3	yes	complemented
26	#490	C₂²	1	3	yes	complemented
27	#490	C₂²	1	12	yes	complemented
28	#490	C₂²	1	6	yes	complemented
29	#490	C₂²	1	3	yes	complemented
30	#489	C₂²	1	12	yes	complemented
31	#488	C₂²	1	12	yes	complemented
32	#486	C₂²	1	12	yes	complemented
33	#418	C₂²	1	12	yes	complemented
34	#417	C₂²	1	12	yes	complemented
35	#416	C₂²	1	12	yes	complemented
36	C₂×#102	C₂²	1	12	yes	complemented
37	#405	C₂²	1	6	yes	complemented
38	C₄×M₁₆	C₂²	1	3	yes	complemented
39	C₂×#97	C₂²	1	3	yes	complemented
40	H(5,𝔽₂)	—	2	16	no	—
41	C₂×#41	—	2	2	no	—
42	C₂×#41	C₂³	1	6	yes	—
43	C₂×#41	C₂³	1	1	yes	characteristic
44	C₂²×D₄	—	2	6	no	—
45	C₂²×D₄	—	2	6	no	—
46	C₂²×D₄	C₂³	1	6	yes	—
47	#131	—	2	8	no	—
48	C₂×M₁₆	C₂³	1	6	yes	—
49	C₂²×C₈	C₂³	1	6	yes	—
50	C₄⋊Q₈	—	2	6	no	—
51	C₄⋊Q₈	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈	C₂³	1	6	yes	—
53	#127	—	2	6	no	—
54	#127	C₂³	1	3	yes	—
55	#127	C₂³	1	6	yes	—
56	#125	—	2	6	no	—
57	#125	—	2	6	no	—
58	#125	C₂³	1	3	yes	—
59	#125	C₂³	1	6	yes	—
60	#125	C₂³	1	1	yes	characteristic
61	#124	C₂³	1	6	yes	—
62	C₂²⋊Q₈	—	2	6	no	—
63	C₂²⋊Q₈	—	2	12	no	—
64	#121	—	2	6	no	—
65	#121	—	2	12	no	—
66	#120	—	2	12	no	—
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	—	2	6	no	—
69	C₄×Q₈	D₄	1	2	yes	—
70	C₄×D₄	—	2	6	no	—
71	C₄×D₄	—	2	6	no	—
72	C₄×D₄	D₄	1	6	yes	—
73	#117	C₂³	1	6	yes	—
74	#117	C₂³	1	3	yes	—
75	C₂×C₄⋊C₄	—	2	6	no	—
76	C₂×C₄⋊C₄	C₂³	1	6	yes	—
77	C₂×C₄²	C₂³	1	3	yes	—
78	#103	C₂³	1	4	yes	—
79	#103	C₂³	1	12	yes	—
80	#102	C₂³	1	12	yes	—
81	#102	C₂³	1	24	yes	—
82	#102	C₂³	1	12	yes	—
83	#97	C₂³	1	1	yes	characteristic
84	#97	C₂³	1	3	yes	—
85	#97	C₂³	1	6	yes	—
86	C₄×C₈	C₂³	1	6	yes	—
87	C₂⁴	—	2	6	no	—
88	#41	—	2	24	no	—
89	#41	—	4	8	no	—
90	#41	#41	1	8	yes	—
91	C₂×Q₈	—	2	6	no	—
92	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
93	C₂×Q₈	C₂⁴	1	2	yes	—
94	C₂×D₄	—	2	6	no	—
95	C₂×D₄	—	2	6	no	—
96	C₂×D₄	—	2	6	no	—
97	C₂×D₄	—	2	12	no	—
98	C₂×D₄	—	2	24	no	—
99	C₂×D₄	—	4	24	no	—
100	C₂×D₄	—	8	12	no	—
101	C₂×D₄	C₂×D₄	1	3	yes	—
102	C₂×D₄	C₂⁴	1	6	yes	—
103	C₂²×C₄	—	2	6	no	—
104	C₂²×C₄	—	2	6	no	—
105	C₂²×C₄	C₂×D₄	1	6	yes	—
106	C₂²×C₄	C₂⁴	1	3	yes	—
107	M₁₆	—	2	12	no	—
108	C₂×C₈	—	2	12	no	—
109	C₂×C₈	C₂⁴	1	6	yes	—
110	C₂×C₈	C₂⁴	1	2	yes	—
111	C₄⋊C₄	—	2	6	no	—
112	C₄⋊C₄	—	2	12	no	—
113	C₄⋊C₄	—	2	6	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	6	no	—
117	C₄⋊C₄	C₂×D₄	1	6	yes	—
118	C₄⋊C₄	C₂⁴	1	6	yes	—
119	C₄⋊C₄	C₂⁴	1	2	yes	—
120	C₂²⋊C₄	—	2	6	no	—
121	C₂²⋊C₄	—	2	12	no	—
122	C₂²⋊C₄	C₂×D₄	1	6	yes	—
123	C₄²	—	2	6	no	—
124	C₄²	C₂×D₄	1	6	yes	—
125	C₄²	C₂⁴	1	3	yes	—
126	C₄²	C₂⁴	1	1	yes	characteristic
127	C₂³	—	2	6	no	—
128	C₂³	—	4	12	no	—
129	C₂³	—	8	12	no	—
130	C₂³	C₂²×D₄	1	3	yes	—
131	Q₈	—	2	4	no	—
132	Q₈	—	8	6	no	—
133	Q₈	#124	1	4	yes	—
134	D₄	—	2	12	no	—
135	D₄	—	4	12	no	—
136	D₄	—	4	12	no	—
137	D₄	—	4	12	no	—
138	D₄	—	8	12	no	—
139	D₄	#124	1	12	yes	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	6	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	6	no	—
144	C₂×C₄	—	4	12	no	—
145	C₂×C₄	—	8	6	no	—
146	C₂×C₄	C₂²×D₄	1	2	yes	—
147	C₂×C₄	C₂²×D₄	1	6	yes	—
148	C₂×C₄	C₂²×D₄	1	3	yes	—
149	C₂×C₄	C₂×#41	1	12	yes	—
150	C₂×C₄	C₂⁵	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	6	no	—
153	C₂²	—	2	6	no	—
154	C₂²	—	4	12	no	—
155	C₂²	—	4	6	no	—
156	C₂²	—	16	12	no	—
157	C₂²	C₂×#124	1	6	yes	—
158	C₂²	C₂³×D₄	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	6	no	—
161	C₄	—	4	6	no	—
162	C₄	—	4	2	no	—
163	C₄	—	4	2	no	—
164	C₄	C₂×#124	1	6	yes	—
165	C₄	C₂²×#41	1	2	yes	—
166	C₂	—	2	6	no	—
167	C₂	—	8	6	no	—
168	C₂	C₂²×#124	1	1	yes	characteristic
169	C₂	#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3504	C₂	semidirect
3. #3142	C₂	semidirect
4. #3138	C₂	semidirect
5. #3136	C₂	semidirect
6. #3135	C₂	semidirect
7. C₂×#490	C₂	semidirect
8. #2897	C₂	semidirect
9. #2875	C₂	semidirect
16. #544	C₂²	semidirect
18. #533	C₂²	semidirect
21. C₂×#125	C₂²	semidirect
24. #491	C₂²	semidirect
25. #491	C₂²	semidirect
26. #490	C₂²	semidirect
27. #490	C₂²	semidirect
28. #490	C₂²	semidirect
29. #490	C₂²	semidirect
30. #489	C₂²	semidirect
31. #488	C₂²	semidirect
32. #486	C₂²	semidirect
33. #418	C₂²	semidirect
34. #417	C₂²	semidirect
35. #416	C₂²	semidirect
36. C₂×#102	C₂²	semidirect
37. #405	C₂²	semidirect
38. C₄×M₁₆	C₂²	semidirect
39. C₂×#97	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3504	2
#3498	2
#3142	2
#3138	2
#3136	2
#3135	2
C₂×#490	2
#2897	2
#2875	2