Finite Group Databasebeta

Subgroups — #19935

Order 256.

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

normal subgroups only

This group has 323 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	#3373	C₂	1	2	yes	maximal, complemented
3	D₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#1709	C₂	1	1	yes	maximal, characteristic, complemented
5	#1702	C₂	1	2	yes	maximal, complemented
6	#1670	C₂	1	1	yes	maximal, characteristic, complemented
7	#1667	C₂	1	1	yes	maximal, characteristic
8	#1640	C₂	1	2	yes	maximal, complemented
9	#1627	C₂	1	2	yes	maximal, complemented
10	#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆	—	2	1	no	—
13	#559	C₂²	1	2	yes	—
14	D₄×Q₈	C₂²	1	2	yes	—
15	#495	—	2	1	no	—
16	#494	C₂²	1	1	yes	characteristic
17	#493	C₂²	1	1	yes	characteristic
18	#489	—	2	2	no	—
19	#486	C₂²	1	2	yes	—
20	#486	C₂²	1	2	yes	—
21	#476	—	2	2	no	—
22	#475	C₂²	1	2	yes	—
23	#467	—	2	1	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	#451	—	2	2	no	—
30	#446	C₂²	1	2	yes	—
31	C₈×Q₈	C₂²	1	1	yes	characteristic
32	#442	—	2	2	no	—
33	C₄×Q₁₆	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆	C₂²	1	1	yes	characteristic
35	C₄×D₈	—	2	1	no	—
36	C₈×D₄	C₂²	1	1	yes	characteristic
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	C₂²×Q₈	—	2	2	no	—
46	C₂×Q₁₆	—	2	2	no	—
47	C₂×Q₁₆	—	2	1	no	—
48	C₂×Q₁₆	—	2	1	no	—
49	C₂×Q₁₆	—	4	4	no	—
50	C₂×Q₁₆	D₄	1	1	yes	characteristic
51	C₂×SD₁₆	—	2	2	no	—
52	C₂×SD₁₆	D₄	1	1	yes	characteristic
53	C₂×D₈	—	2	2	no	—
54	C₂×D₈	—	2	1	no	—
55	C₂×D₈	—	2	1	no	—
56	C₂²×C₈	—	2	1	no	—
57	C₄⋊Q₈	—	2	2	no	—
58	C₄⋊Q₈	C₂³	1	2	yes	—
59	#127	C₂³	1	2	yes	—
60	#124	—	2	2	no	—
61	#124	—	2	2	no	—
62	#123	—	4	2	no	—
63	C₂²⋊Q₈	—	2	2	no	—
64	C₂²⋊Q₈	—	4	2	no	—
65	C₄×Q₈	C₂³	1	2	yes	—
66	C₄×Q₈	C₂³	1	1	yes	characteristic
67	C₄×D₄	—	2	2	no	—
68	C₄×D₄	—	2	2	no	—
69	C₄×D₄	—	2	2	no	—
70	C₄×D₄	C₂³	1	1	yes	characteristic
71	#107	D₄	1	1	yes	characteristic
72	#106	—	2	2	no	—
73	#106	D₄	1	1	yes	characteristic
74	C₄⋊C₈	—	2	1	no	—
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊C₈	C₂³	1	2	yes	—
79	#103	—	2	2	no	—
80	#103	—	2	2	no	—
81	#103	D₄	1	1	yes	characteristic
82	#103	D₄	1	2	yes	—
83	#102	—	2	2	no	—
84	#102	—	2	2	no	—
85	#102	—	2	2	no	—
86	#102	—	2	2	no	—
87	#102	—	4	2	no	—
88	#102	D₄	1	1	yes	characteristic
89	C₂²⋊C₈	—	2	1	no	—
90	#97	—	2	2	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₈	C₂³	1	1	yes	characteristic
94	C₄×C₈	C₂³	1	1	yes	characteristic
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	4	2	no	—
97	C₂×Q₈	—	4	4	no	—
98	C₂×Q₈	—	8	2	no	—
99	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
100	C₂×Q₈	C₂×D₄	1	2	yes	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	4	2	no	—
107	C₂²×C₄	—	4	2	no	—
108	Q₁₆	—	2	2	no	—
109	Q₁₆	—	4	4	no	—
110	Q₁₆	—	4	2	no	—
111	Q₁₆	—	8	1	no	—
112	SD₁₆	—	2	2	no	—
113	SD₁₆	—	8	2	no	—
114	D₈	—	4	4	no	—
115	D₈	—	4	2	no	—
116	D₈	—	8	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄	C₂×D₄	1	2	yes	—
133	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
134	C₂²⋊C₄	—	2	1	no	—
135	C₂²⋊C₄	—	4	2	no	—
136	C₂²⋊C₄	—	4	2	no	—
137	C₂²⋊C₄	—	4	2	no	—
138	C₂²⋊C₄	—	4	2	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	2	no	—
141	C₄²	C₂⁴	1	1	yes	characteristic
142	C₂³	—	2	1	no	—
143	C₂³	—	4	2	no	—
144	Q₈	—	2	1	no	—
145	Q₈	—	2	2	no	—
146	Q₈	—	2	2	no	—
147	Q₈	—	2	1	no	—
148	Q₈	—	4	2	no	—
149	Q₈	—	4	2	no	—
150	Q₈	—	8	4	no	—
151	D₄	—	2	1	no	—
152	D₄	—	4	2	no	—
153	D₄	—	4	2	no	—
154	D₄	—	4	2	no	—
155	D₄	—	4	2	no	—
156	D₄	—	8	2	no	—
157	D₄	#135	1	2	yes	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	—	16	2	no	—
168	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	4	1	no	—
175	C₈	—	4	2	no	—
176	C₈	—	4	1	no	—
177	C₈	—	4	1	no	—
178	C₈	AGL(1,ℤ/8)	1	2	yes	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	1	no	—
181	C₂²	—	4	1	no	—
182	C₂²	—	8	2	no	—
183	C₂²	—	8	2	no	—
184	C₂²	—	8	2	no	—
185	C₂²	D₄²	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	1	no	—
190	C₄	—	4	2	no	—
191	C₄	—	4	1	no	—
192	C₄	—	8	2	no	—
193	C₄	C₂×#135	1	1	yes	characteristic
194	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₄	#576	1	1	yes	characteristic
196	C₄	#578	1	1	yes	characteristic
197	C₂	—	4	2	no	—
198	C₂	—	16	2	no	—
199	C₂	#3274	1	1	yes	characteristic
200	C₂	#3285	1	1	yes	characteristic
201	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. #3373	C₂	semidirect
3. D₄×Q₁₆	C₂	semidirect
4. #1709	C₂	semidirect
5. #1702	C₂	semidirect
6. #1670	C₂	semidirect
8. #1640	C₂	semidirect
9. #1627	C₂	semidirect
10. #1591	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
#3373	2
D₄×Q₁₆	2
#1709	2
#1702	2
#1670	2
#1667	2
#1640	2
#1627	2
#1591	2
C₈×SD₁₆	2