Finite Group Databasebeta

Subgroups — #33788

Order 256.

297 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 99 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	#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	#3155	C₂	1	1	yes	maximal, characteristic
3	#2251	C₂	1	1	yes	maximal, characteristic, complemented
4	#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	#2244	C₂	1	1	yes	maximal, characteristic, complemented
6	#2242	C₂	1	1	yes	maximal, characteristic, complemented
7	#2211	C₂	1	2	yes	maximal, complemented
8	#2207	C₂	1	2	yes	maximal, complemented
9	#2189	C₂	1	2	yes	maximal, complemented
10	#2187	C₂	1	2	yes	maximal, complemented
11	#2164	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	#541	—	2	1	no	—
16	#535	—	2	1	no	—
17	C₂×Q₃₂	C₂²	1	2	yes	complemented
18	C₂×SD₃₂	C₂²	1	2	yes	—
19	C₂×SD₃₂	C₂²	1	2	yes	—
20	C₂×SD₃₂	C₂²	1	2	yes	complemented
21	#499	C₂²	1	1	yes	characteristic, complemented
22	#496	—	2	1	no	—
23	#496	—	2	1	no	—
24	#495	C₂²	1	1	yes	characteristic
25	#495	C₂²	1	1	yes	characteristic
26	#494	C₂²	1	1	yes	characteristic, complemented
27	#493	C₂²	1	1	yes	characteristic
28	#493	C₂²	1	1	yes	characteristic
29	#491	—	2	1	no	—
30	#470	—	2	1	no	—
31	#467	C₂²	1	2	yes	—
32	#441	—	2	1	no	—
33	C₄×Q₁₆	C₂²	1	2	yes	complemented
34	#405	C₂²	1	1	yes	characteristic
35	C₄⋊C₁₆	C₂²	1	2	yes	complemented
36	#358	C₂²	1	2	yes	complemented
37	#357	C₂²	1	2	yes	—
38	#357	C₂²	1	2	yes	—
39	#357	C₂²	1	2	yes	complemented
40	C₂²⋊C₁₆	C₂²	1	2	yes	—
41	#346	C₂²	1	1	yes	characteristic, complemented
42	C₄×C₁₆	C₂²	1	1	yes	characteristic, complemented
43	C₂×#41	—	2	1	no	—
44	C₂²×D₄	—	2	1	no	—
45	AGL(1,ℤ/8)	—	4	4	no	—
46	#135	—	4	2	no	—
47	C₂×Q₁₆	—	2	1	no	—
48	C₂×Q₁₆	C₂³	1	1	yes	characteristic
49	C₂×Q₁₆	C₂³	1	2	yes	—
50	C₂×SD₁₆	—	2	1	no	—
51	C₂×SD₁₆	—	2	1	no	—
52	C₂×SD₁₆	—	2	1	no	—
53	C₂×D₈	—	2	1	no	—
54	C₂×D₈	—	2	1	no	—
55	C₂×D₈	—	2	1	no	—
56	C₂×D₈	—	4	2	no	—
57	C₂×D₈	C₂³	1	1	yes	characteristic
58	C₂×D₈	C₂³	1	1	yes	characteristic
59	C₂×D₈	C₂³	1	1	yes	characteristic
60	C₂×M₁₆	D₄	1	1	yes	characteristic
61	C₂²×C₈	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈	—	2	1	no	—
63	#127	—	2	1	no	—
64	#127	—	2	1	no	—
65	#126	—	4	2	no	—
66	#125	—	2	1	no	—
67	#125	—	4	2	no	—
68	#124	—	2	1	no	—
69	#124	—	2	1	no	—
70	C₂²⋊Q₈	—	2	2	no	—
71	#121	—	4	2	no	—
72	#120	—	4	2	no	—
73	C₄×Q₈	—	2	2	no	—
74	#117	D₄	1	1	yes	characteristic
75	Q₃₂	—	4	2	no	—
76	SD₃₂	—	4	2	no	—
77	SD₃₂	—	4	2	no	—
78	SD₃₂	—	4	2	no	—
79	C₂×C₁₆	C₂³	1	2	yes	—
80	C₂×C₁₆	C₂³	1	2	yes	—
81	#107	C₂³	1	2	yes	—
82	#106	—	2	1	no	—
83	#103	—	2	2	no	—
84	#103	—	2	2	no	—
85	#97	D₄	1	1	yes	characteristic
86	#97	D₄	1	1	yes	characteristic
87	C₄×C₈	C₂³	1	1	yes	characteristic
88	C₄×C₈	C₂³	1	1	yes	characteristic
89	C₂⁴	—	4	1	no	—
90	#41	—	4	2	no	—
91	#41	—	4	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	C₂×D₄	—	4	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	—	4	1	no	—
100	C₂×D₄	—	4	1	no	—
101	C₂×D₄	—	4	2	no	—
102	C₂×D₄	—	4	2	no	—
103	C₂²×C₄	—	4	1	no	—
104	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
105	Q₁₆	—	2	2	no	—
106	Q₁₆	—	2	2	no	—
107	Q₁₆	—	2	2	no	—
108	Q₁₆	—	4	2	no	—
109	SD₁₆	—	4	2	no	—
110	SD₁₆	—	4	2	no	—
111	SD₁₆	—	4	2	no	—
112	D₈	—	2	2	no	—
113	D₈	—	2	2	no	—
114	D₈	—	2	2	no	—
115	D₈	—	4	2	no	—
116	D₈	—	4	2	no	—
117	D₈	—	4	2	no	—
118	M₁₆	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈	C₂⁴	1	1	yes	characteristic
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	D₈	1	2	yes	—
130	C₂²⋊C₄	—	4	2	no	—
131	C₂²⋊C₄	—	4	2	no	—
132	C₂²⋊C₄	—	4	2	no	—
133	C₂²⋊C₄	D₈	1	2	yes	—
134	C₄²	—	4	2	no	—
135	C₄²	C₂×D₄	1	1	yes	characteristic
136	C₄²	C₂×D₄	1	1	yes	characteristic
137	C₁₆	—	2	2	no	—
138	C₁₆	—	2	2	no	—
139	C₂³	—	4	1	no	—
140	C₂³	—	4	1	no	—
141	C₂³	—	4	1	no	—
142	C₂³	—	8	2	no	—
143	C₂³	—	16	2	no	—
144	C₂³	C₂×D₈	1	1	yes	characteristic
145	Q₈	—	4	1	no	—
146	Q₈	—	4	2	no	—
147	Q₈	—	4	1	no	—
148	Q₈	—	4	2	no	—
149	D₄	—	4	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	4	1	no	—
153	D₄	—	4	1	no	—
154	D₄	—	4	1	no	—
155	D₄	—	8	2	no	—
156	D₄	—	8	2	no	—
157	D₄	—	16	2	no	—
158	D₄	—	16	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	C₂×D₈	1	2	yes	—
166	C₂×C₄	C₂×D₈	1	2	yes	—
167	C₂×C₄	C₂×D₈	1	1	yes	characteristic
168	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	AGL(1,ℤ/8)	1	2	yes	—
172	C₈	H(5,𝔽₂)	1	2	yes	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	8	2	no	—
176	C₂²	—	8	2	no	—
177	C₂²	—	8	2	no	—
178	C₂²	—	8	1	no	—
179	C₂²	—	8	1	no	—
180	C₂²	—	8	1	no	—
181	C₂²	—	16	4	no	—
182	C₂²	C₂²×D₈	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	1	no	—
186	C₄	—	8	2	no	—
187	C₄	—	8	1	no	—
188	C₄	—	8	2	no	—
189	C₄	#535	1	1	yes	characteristic
190	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
191	C₂	—	4	1	no	—
192	C₂	—	16	1	no	—
193	C₂	—	16	1	no	—
194	C₂	—	16	1	no	—
195	C₂	#3205	1	1	yes	characteristic
196	C₂	#3416	1	2	yes	—

	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. #3177	C₂	semidirect
3. #2251	C₂	semidirect
4. #2248	C₂	semidirect
5. #2244	C₂	semidirect
6. #2242	C₂	semidirect
7. #2211	C₂	semidirect
8. #2207	C₂	semidirect
9. #2189	C₂	semidirect
10. #2187	C₂	semidirect
11. #2164	C₂	semidirect
17. C₂×Q₃₂	C₂²	semidirect
20. C₂×SD₃₂	C₂²	semidirect
21. #499	C₂²	semidirect
26. #494	C₂²	semidirect
33. C₄×Q₁₆	C₂²	semidirect
35. C₄⋊C₁₆	C₂²	semidirect
36. #358	C₂²	semidirect
39. #357	C₂²	semidirect
41. #346	C₂²	semidirect
42. 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
#3177	2
#3155	2
#2251	2
#2248	2
#2244	2
#2242	2
#2211	2
#2207	2
#2189	2
#2187	2
#2164	2