Finite Group Databasebeta

Subgroups — #20018

Order 256.

302 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 103 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	#3396	C₂	1	1	yes	maximal, characteristic, complemented
2	#3394	C₂	1	2	yes	maximal, complemented
3	#3342	C₂	1	1	yes	maximal, characteristic, complemented
4	#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	#1674	C₂	1	1	yes	maximal, characteristic, complemented
6	#1667	C₂	1	1	yes	maximal, characteristic
7	#1666	C₂	1	2	yes	maximal, complemented
8	#1640	C₂	1	2	yes	maximal, complemented
9	#1628	C₂	1	2	yes	maximal, complemented
10	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#559	C₂²	1	2	yes	—
13	#556	C₂²	1	2	yes	—
14	#500	C₂²	1	1	yes	characteristic
15	#495	—	2	1	no	—
16	#495	—	2	2	no	—
17	#493	C₂²	1	1	yes	characteristic
18	#492	C₂²	1	2	yes	—
19	#486	C₂²	1	2	yes	—
20	#485	—	2	2	no	—
21	#475	C₂²	1	2	yes	—
22	#467	—	2	1	no	—
23	#464	—	2	2	no	—
24	#463	—	2	2	no	—
25	#463	—	2	2	no	—
26	#462	C₂²	1	2	yes	—
27	#460	C₂²	1	2	yes	—
28	#459	C₂²	1	2	yes	—
29	C₈×Q₈	C₂²	1	1	yes	characteristic
30	C₈×Q₈	C₂²	1	2	yes	—
31	C₄×Q₁₆	C₂²	1	1	yes	characteristic
32	C₄×SD₁₆	—	2	2	no	—
33	C₄×SD₁₆	C₂²	1	1	yes	characteristic
34	C₄×D₈	—	2	1	no	—
35	C₈×D₄	C₂²	1	1	yes	characteristic
36	#427	—	2	1	no	—
37	#335	C₂²	1	1	yes	characteristic
38	#334	C₂²	1	1	yes	characteristic
39	#332	C₂²	1	2	yes	—
40	#331	C₂²	1	2	yes	—
41	Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈	C₂²	1	2	yes	—
43	D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×Q₁₆	D₄	1	1	yes	characteristic
47	C₂×SD₁₆	—	2	2	no	—
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×SD₁₆	—	2	2	no	—
50	C₂×SD₁₆	D₄	1	1	yes	characteristic
51	C₂×D₈	—	2	2	no	—
52	C₂×D₈	—	2	1	no	—
53	C₂×D₈	—	2	1	no	—
54	C₂²×C₈	—	2	1	no	—
55	C₄⋊Q₈	C₂³	1	2	yes	—
56	#127	C₂³	1	2	yes	—
57	#126	—	4	2	no	—
58	#124	—	2	2	no	—
59	#124	—	2	2	no	—
60	#124	—	2	2	no	—
61	#123	—	4	2	no	—
62	C₂²⋊Q₈	—	2	2	no	—
63	C₄×Q₈	—	2	2	no	—
64	C₄×Q₈	C₂³	1	2	yes	—
65	C₄×Q₈	C₂³	1	1	yes	characteristic
66	C₄×D₄	—	2	2	no	—
67	C₄×D₄	—	2	2	no	—
68	C₄×D₄	C₂³	1	1	yes	characteristic
69	#117	—	2	2	no	—
70	#107	—	2	1	no	—
71	#107	—	2	1	no	—
72	#107	D₄	1	1	yes	characteristic
73	#106	—	2	2	no	—
74	#106	D₄	1	1	yes	characteristic
75	C₄⋊C₈	—	2	1	no	—
76	C₄⋊C₈	—	2	2	no	—
77	C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊C₈	C₂³	1	2	yes	—
80	#103	—	2	2	no	—
81	#103	—	2	2	no	—
82	#103	D₄	1	1	yes	characteristic
83	#103	D₄	1	2	yes	—
84	#102	—	2	2	no	—
85	#102	—	2	2	no	—
86	#102	—	4	2	no	—
87	#102	D₄	1	1	yes	characteristic
88	C₂²⋊C₈	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	C₂³	1	1	yes	characteristic
92	C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₈	C₂³	1	1	yes	characteristic
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	C₂×D₄	1	2	yes	—
96	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
97	C₂×D₄	—	2	2	no	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂×D₄	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	4	2	no	—
103	Q₁₆	—	2	2	no	—
104	Q₁₆	—	8	2	no	—
105	SD₁₆	—	2	2	no	—
106	SD₁₆	—	4	4	no	—
107	SD₁₆	—	8	2	no	—
108	SD₁₆	—	8	2	no	—
109	D₈	—	4	2	no	—
110	D₈	—	8	1	no	—
111	D₈	—	8	2	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
127	C₄⋊C₄	C₂×D₄	1	2	yes	—
128	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
129	C₂²⋊C₄	—	2	1	no	—
130	C₂²⋊C₄	—	4	2	no	—
131	C₂²⋊C₄	—	4	2	no	—
132	C₂²⋊C₄	—	4	2	no	—
133	C₂²⋊C₄	—	4	2	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	2	no	—
138	C₄²	C₂⁴	1	1	yes	characteristic
139	C₂³	—	2	1	no	—
140	C₂³	—	4	2	no	—
141	Q₈	—	2	1	no	—
142	Q₈	—	2	2	no	—
143	Q₈	—	2	1	no	—
144	Q₈	—	2	2	no	—
145	Q₈	—	4	2	no	—
146	Q₈	—	4	2	no	—
147	D₄	—	2	1	no	—
148	D₄	—	2	1	no	—
149	D₄	—	4	2	no	—
150	D₄	—	4	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	2	no	—
153	D₄	—	8	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	16	2	no	—
163	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	4	1	no	—
169	C₈	—	4	1	no	—
170	C₈	—	4	2	no	—
171	C₈	—	4	1	no	—
172	C₈	#135	1	2	yes	—
173	C₈	#135	1	2	yes	—
174	C₂²	—	4	1	no	—
175	C₂²	—	4	1	no	—
176	C₂²	—	4	1	no	—
177	C₂²	—	8	2	no	—
178	C₂²	—	8	2	no	—
179	C₂²	—	8	2	no	—
180	C₂²	D₄²	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	4	1	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	1	no	—
187	C₄	—	8	2	no	—
188	C₄	C₂×#135	1	1	yes	characteristic
189	C₄	C₂×#135	1	1	yes	characteristic
190	C₄	#576	1	1	yes	characteristic
191	C₄	#577	1	1	yes	characteristic
192	C₂	—	8	1	no	—
193	C₂	—	16	2	no	—
194	C₂	#3284	1	1	yes	characteristic
195	C₂	#3287	1	1	yes	characteristic
196	C₂	#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#1667	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3396	C₂	semidirect
2. #3394	C₂	semidirect
3. #3342	C₂	semidirect
4. #1706	C₂	semidirect
5. #1674	C₂	semidirect
7. #1666	C₂	semidirect
8. #1640	C₂	semidirect
9. #1628	C₂	semidirect
10. C₈×Q₁₆	C₂	semidirect
11. C₈×SD₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3396	2
#3394	2
#3342	2
#1706	2
#1674	2
#1667	2
#1666	2
#1640	2
#1628	2
C₈×Q₁₆	2
C₈×SD₁₆	2