Finite Group Databasebeta

Subgroups — #19749

Order 256.

411 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 119 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	#3185	C₂	1	1	yes	maximal, characteristic, complemented
2	#3182	C₂	1	2	yes	maximal, complemented
3	#3151	C₂	1	1	yes	maximal, characteristic
4	#3138	C₂	1	2	yes	maximal, complemented
5	#1722	C₂	1	1	yes	maximal, characteristic, complemented
6	#1719	C₂	1	2	yes	maximal, complemented
7	#1717	C₂	1	2	yes	maximal, complemented
8	#1715	C₂	1	1	yes	maximal, characteristic, complemented
9	#1709	C₂	1	1	yes	maximal, characteristic, complemented
10	#1708	C₂	1	1	yes	maximal, characteristic, complemented
11	#1455	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)	—	2	2	no	—
13	#533	C₂²	1	2	yes	—
14	#532	C₂²	1	1	yes	characteristic
15	C₂×#127	C₂²	1	1	yes	characteristic
16	#500	C₂²	1	2	yes	complemented
17	#498	C₂²	1	1	yes	characteristic, complemented
18	#496	—	2	2	no	—
19	#494	C₂²	1	1	yes	characteristic, complemented
20	#493	C₂²	1	1	yes	characteristic
21	#493	C₂²	1	1	yes	characteristic
22	#492	C₂²	1	1	yes	characteristic
23	#492	C₂²	1	1	yes	characteristic
24	#492	C₂²	1	2	yes	complemented
25	#490	—	2	4	no	—
26	#487	C₂²	1	2	yes	complemented
27	#487	C₂²	1	2	yes	complemented
28	#486	C₂²	1	2	yes	—
29	#486	C₂²	1	2	yes	—
30	#486	C₂²	1	2	yes	—
31	#486	C₂²	1	2	yes	—
32	#486	C₂²	1	2	yes	complemented
33	#470	—	2	2	no	—
34	#468	—	2	4	no	—
35	#441	—	2	2	no	—
36	#440	—	2	4	no	—
37	#418	—	2	4	no	—
38	C₄×M₁₆	C₂²	1	2	yes	—
39	C₄×M₁₆	C₂²	1	1	yes	characteristic
40	#322	C₂²	1	1	yes	characteristic, complemented
41	#322	C₂²	1	2	yes	complemented
42	C₈²	C₂²	1	1	yes	characteristic, complemented
43	C₂×#41	—	2	1	no	—
44	C₂²×D₄	—	2	1	no	—
45	C₂²×D₄	—	2	2	no	—
46	AGL(1,ℤ/8)	—	4	8	no	—
47	C₂×Q₁₆	—	2	2	no	—
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×SD₁₆	—	2	2	no	—
50	C₂×SD₁₆	—	2	4	no	—
51	C₂×D₈	—	2	2	no	—
52	C₂×D₈	—	2	2	no	—
53	C₂×M₁₆	D₄	1	2	yes	—
54	C₂×M₁₆	D₄	1	4	yes	—
55	C₄⋊Q₈	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈	C₂³	1	2	yes	—
57	C₄⋊Q₈	C₂³	1	2	yes	—
58	#127	—	2	2	no	—
59	#127	—	2	1	no	—
60	#127	C₂³	1	1	yes	characteristic
61	#127	C₂³	1	1	yes	characteristic
62	#127	C₂³	1	1	yes	characteristic
63	#125	—	2	2	no	—
64	#124	—	2	1	no	—
65	C₂²⋊Q₈	—	2	2	no	—
66	C₂²⋊Q₈	—	2	2	no	—
67	#121	—	2	2	no	—
68	C₄×Q₈	—	2	2	no	—
69	C₄×Q₈	—	2	2	no	—
70	C₄×D₄	—	2	2	no	—
71	#117	—	2	2	no	—
72	C₂×C₄²	C₂³	1	1	yes	characteristic
73	#107	—	2	4	no	—
74	#106	—	2	2	no	—
75	#103	—	2	4	no	—
76	#103	—	2	4	no	—
77	#102	—	2	4	no	—
78	#102	—	2	4	no	—
79	#102	—	2	4	no	—
80	#102	—	2	4	no	—
81	#102	—	2	4	no	—
82	#97	D₄	1	4	yes	—
83	#97	D₄	1	2	yes	—
84	C₄×C₈	C₂³	1	2	yes	—
85	C₄×C₈	C₂³	1	1	yes	characteristic
86	C₄×C₈	C₂³	1	2	yes	—
87	C₄×C₈	C₂³	1	1	yes	characteristic
88	C₂⁴	—	4	1	no	—
89	#41	—	4	4	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	—	2	2	no	—
96	C₂×D₄	—	2	1	no	—
97	C₂×D₄	—	2	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	4	4	no	—
100	C₂×D₄	—	4	8	no	—
101	C₂×D₄	—	4	1	no	—
102	C₂×D₄	—	4	2	no	—
103	C₂×D₄	—	4	1	no	—
104	C₂²×C₄	—	4	1	no	—
105	C₂²×C₄	C₂×D₄	1	2	yes	—
106	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
107	Q₁₆	—	4	4	no	—
108	SD₁₆	—	4	8	no	—
109	SD₁₆	—	4	4	no	—
110	SD₁₆	—	4	4	no	—
111	D₈	—	4	4	no	—
112	D₈	—	4	4	no	—
113	M₁₆	—	2	4	no	—
114	M₁₆	—	4	4	no	—
115	C₂×C₈	C₂×D₄	1	4	yes	—
116	C₂×C₈	C₂×D₄	1	4	yes	—
117	C₂×C₈	C₂×D₄	1	2	yes	—
118	C₂×C₈	C₂×D₄	1	2	yes	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₂²⋊C₄	—	4	2	no	—
127	C₂²⋊C₄	—	4	2	no	—
128	C₄²	—	4	2	no	—
129	C₄²	C₂×D₄	1	2	yes	—
130	C₄²	C₂×D₄	1	1	yes	characteristic
131	C₄²	C₂⁴	1	1	yes	characteristic
132	C₂³	—	4	1	no	—
133	C₂³	—	4	1	no	—
134	C₂³	—	4	1	no	—
135	C₂³	—	8	2	no	—
136	C₂³	—	8	4	no	—
137	C₂³	#127	1	1	yes	characteristic
138	Q₈	—	4	2	no	—
139	Q₈	—	4	4	no	—
140	Q₈	—	4	4	no	—
141	D₄	—	4	2	no	—
142	D₄	—	4	2	no	—
143	D₄	—	4	2	no	—
144	D₄	—	4	4	no	—
145	D₄	—	4	4	no	—
146	D₄	—	4	4	no	—
147	D₄	—	8	2	no	—
148	D₄	—	8	4	no	—
149	D₄	—	8	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	#127	1	2	yes	—
157	C₂×C₄	#127	1	1	yes	characteristic
158	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄	C₂²×D₄	1	2	yes	—
160	C₈	—	2	2	no	—
161	C₈	—	2	4	no	—
162	C₈	—	2	4	no	—
163	C₈	AGL(1,ℤ/8)	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	8	1	no	—
167	C₂²	—	8	1	no	—
168	C₂²	—	8	1	no	—
169	C₂²	—	8	2	no	—
170	C₂²	—	8	2	no	—
171	C₂²	—	8	2	no	—
172	C₂²	—	16	4	no	—
173	C₂²	C₂×#127	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	2	no	—
176	C₄	—	8	1	no	—
177	C₄	—	8	2	no	—
178	C₄	—	8	2	no	—
179	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
180	C₄	#575	1	4	yes	—
181	C₂	—	4	1	no	—
182	C₂	—	16	1	no	—
183	C₂	—	16	1	no	—
184	C₂	—	16	1	no	—
185	C₂	#3151	1	1	yes	characteristic
186	C₂	#3153	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3151	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3185	C₂	semidirect
2. #3182	C₂	semidirect
4. #3138	C₂	semidirect
5. #1722	C₂	semidirect
6. #1719	C₂	semidirect
7. #1717	C₂	semidirect
8. #1715	C₂	semidirect
9. #1709	C₂	semidirect
10. #1708	C₂	semidirect
11. #1455	C₂	semidirect
16. #500	C₂²	semidirect
17. #498	C₂²	semidirect
19. #494	C₂²	semidirect
24. #492	C₂²	semidirect
26. #487	C₂²	semidirect
27. #487	C₂²	semidirect
32. #486	C₂²	semidirect
40. #322	C₂²	semidirect
41. #322	C₂²	semidirect
42. C₈²	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3185	2
#3182	2
#3151	2
#3138	2
#1722	2
#1719	2
#1717	2
#1715	2
#1709	2
#1708	2
#1455	2