Finite Group Databasebeta

Subgroups — #33702

Order 256.

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

normal subgroups only

This group has 301 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	#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	#3155	C₂	1	1	yes	maximal, characteristic
3	#2244	C₂	1	2	yes	maximal, complemented
4	#2243	C₂	1	2	yes	maximal, complemented
5	#2242	C₂	1	2	yes	maximal, complemented
6	#2240	C₂	1	2	yes	maximal, complemented
7	#2140	C₂	1	1	yes	maximal, characteristic, complemented
8	#2139	C₂	1	1	yes	maximal, characteristic, complemented
9	#2138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#357	C₂	1	1	yes	maximal, characteristic, complemented
11	#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)	—	2	1	no	—
13	C₂×#135	C₂²	1	1	yes	characteristic
14	C₂²×D₈	C₂²	1	1	yes	characteristic
15	#544	—	2	1	no	—
16	#535	—	2	1	no	—
17	C₂×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆	C₂²	1	1	yes	characteristic
19	#501	—	2	2	no	—
20	#500	C₂²	1	2	yes	complemented
21	#499	C₂²	1	2	yes	complemented
22	#496	—	2	2	no	—
23	#495	C₂²	1	2	yes	—
24	#493	C₂²	1	2	yes	—
25	#428	—	2	1	no	—
26	#427	C₂²	1	1	yes	characteristic
27	C₂×#107	C₂²	1	1	yes	characteristic
28	#405	C₂²	1	1	yes	characteristic
29	#358	C₂²	1	1	yes	characteristic, complemented
30	#358	C₂²	1	1	yes	characteristic, complemented
31	#358	C₂²	1	2	yes	complemented
32	#357	C₂²	1	2	yes	—
33	#357	C₂²	1	2	yes	—
34	#357	C₂²	1	2	yes	—
35	#357	C₂²	1	2	yes	—
36	#357	C₂²	1	1	yes	characteristic, complemented
37	#357	C₂²	1	1	yes	characteristic, complemented
38	#357	C₂²	1	2	yes	complemented
39	#346	C₂²	1	2	yes	complemented
40	C₄×C₁₆	C₂²	1	2	yes	complemented
41	C₂×#41	—	2	1	no	—
42	C₂²×D₄	—	2	1	no	—
43	AGL(1,ℤ/8)	—	4	4	no	—
44	#135	—	2	2	no	—
45	#135	—	2	2	no	—
46	C₂×Q₁₆	C₂³	1	1	yes	characteristic
47	C₂×SD₁₆	—	2	1	no	—
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×D₈	—	2	1	no	—
50	C₂×D₈	—	2	2	no	—
51	C₂×D₈	—	2	2	no	—
52	C₂×D₈	—	2	2	no	—
53	C₂×D₈	C₂³	1	2	yes	—
54	C₂×D₈	C₂³	1	1	yes	characteristic
55	C₂×M₁₆	D₄	1	1	yes	characteristic
56	C₂²×C₈	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈	—	2	2	no	—
58	C₄⋊Q₈	—	4	1	no	—
59	#127	—	2	2	no	—
60	#125	—	2	2	no	—
61	#125	—	4	1	no	—
62	#124	—	2	2	no	—
63	C₂²⋊Q₈	—	4	2	no	—
64	#121	—	4	1	no	—
65	#121	—	4	1	no	—
66	#120	—	4	2	no	—
67	#117	—	2	1	no	—
68	#117	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄	—	2	1	no	—
70	M₃₂	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₁₆	C₂³	1	2	yes	—
74	C₂×C₁₆	C₂³	1	1	yes	characteristic
75	#107	—	2	1	no	—
76	#107	—	2	2	no	—
77	#107	C₂³	1	2	yes	—
78	#107	C₂³	1	1	yes	characteristic
79	#107	C₂³	1	1	yes	characteristic
80	#106	—	2	1	no	—
81	#106	—	2	2	no	—
82	#97	D₄	1	2	yes	—
83	C₄×C₈	C₂³	1	2	yes	—
84	C₂⁴	—	4	1	no	—
85	#41	—	4	2	no	—
86	#41	—	4	2	no	—
87	C₂×Q₈	—	2	1	no	—
88	C₂×Q₈	—	4	2	no	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	—	2	1	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	4	2	no	—
93	C₂×D₄	—	4	1	no	—
94	C₂×D₄	—	4	2	no	—
95	C₂×D₄	—	4	2	no	—
96	C₂×D₄	—	4	2	no	—
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	—	4	1	no	—
99	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
100	Q₁₆	—	2	1	no	—
101	Q₁₆	—	2	1	no	—
102	SD₁₆	—	4	2	no	—
103	SD₁₆	—	4	4	no	—
104	D₈	—	2	2	no	—
105	D₈	—	2	2	no	—
106	D₈	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	4	2	no	—
109	D₈	—	4	4	no	—
110	M₁₆	SD₁₆	1	4	yes	—
111	C₂×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈	C₂×D₄	1	2	yes	—
113	C₂×C₈	#41	1	2	yes	—
114	C₂×C₈	#41	1	2	yes	—
115	C₂×C₈	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	D₈	1	1	yes	characteristic
124	C₄⋊C₄	D₈	1	1	yes	characteristic
125	C₂²⋊C₄	—	4	1	no	—
126	C₂²⋊C₄	—	4	2	no	—
127	C₂²⋊C₄	—	4	2	no	—
128	C₂²⋊C₄	D₈	1	2	yes	—
129	C₄²	—	4	1	no	—
130	C₄²	C₂×D₄	1	2	yes	—
131	C₁₆	—	2	1	no	—
132	C₁₆	—	2	1	no	—
133	C₁₆	—	2	2	no	—
134	C₂³	—	4	2	no	—
135	C₂³	—	4	1	no	—
136	C₂³	—	8	2	no	—
137	C₂³	—	16	1	no	—
138	C₂³	—	16	1	no	—
139	C₂³	C₂×D₈	1	1	yes	characteristic
140	Q₈	—	4	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	16	2	no	—
143	D₄	—	4	2	no	—
144	D₄	—	4	1	no	—
145	D₄	—	4	1	no	—
146	D₄	—	4	2	no	—
147	D₄	—	8	2	no	—
148	D₄	—	8	2	no	—
149	D₄	—	16	2	no	—
150	D₄	—	16	2	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	16	1	no	—
157	C₂×C₄	#124	1	2	yes	—
158	C₂×C₄	C₂×D₈	1	2	yes	—
159	C₂×C₄	C₂×D₈	1	2	yes	—
160	C₂×C₄	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄	C₂×SD₁₆	1	2	yes	—
162	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
163	C₈	#124	1	2	yes	—
164	C₈	C₂×SD₁₆	1	4	yes	—
165	C₈	C₂×#41	1	1	yes	characteristic
166	C₈	C₂×#41	1	1	yes	characteristic
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	8	1	no	—
170	C₂²	—	8	2	no	—
171	C₂²	—	8	1	no	—
172	C₂²	—	8	2	no	—
173	C₂²	—	8	1	no	—
174	C₂²	—	8	2	no	—
175	C₂²	—	16	4	no	—
176	C₂²	#486	1	2	yes	—
177	C₂²	C₂²×D₈	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	2	2	no	—
180	C₄	—	8	1	no	—
181	C₄	—	8	2	no	—
182	C₄	—	8	1	no	—
183	C₄	—	8	1	no	—
184	C₄	#486	1	2	yes	—
185	C₄	C₂×#124	1	1	yes	characteristic
186	C₄	C₂²×SD₁₆	1	1	yes	characteristic
187	C₂	—	2	2	no	—
188	C₂	—	16	1	no	—
189	C₂	—	16	2	no	—
190	C₂	C₂×#486	1	1	yes	characteristic
191	C₂	#3415	1	1	yes	characteristic
192	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3155	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ 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. #3164	C₂	semidirect
3. #2244	C₂	semidirect
4. #2243	C₂	semidirect
5. #2242	C₂	semidirect
6. #2240	C₂	semidirect
7. #2140	C₂	semidirect
8. #2139	C₂	semidirect
9. #2138	C₂	semidirect
10. C₂×#357	C₂	semidirect
11. #2108	C₂	semidirect
20. #500	C₂²	semidirect
21. #499	C₂²	semidirect
29. #358	C₂²	semidirect
30. #358	C₂²	semidirect
31. #358	C₂²	semidirect
36. #357	C₂²	semidirect
37. #357	C₂²	semidirect
38. #357	C₂²	semidirect
39. #346	C₂²	semidirect
40. C₄×C₁₆	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3164	2
#3155	2
#2244	2
#2243	2
#2242	2
#2240	2
#2140	2
#2139	2
#2138	2
C₂×#357	2
#2108	2