Finite Group Databasebeta

Subgroups — #28670

Order 256.

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

normal subgroups only

This group has 349 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	#3165	C₂	1	1	yes	maximal, characteristic, complemented
2	#3138	C₂	1	1	yes	maximal, characteristic
3	C₂×#486	C₂	1	1	yes	maximal, characteristic
4	#2960	C₂	1	1	yes	maximal, characteristic, complemented
5	#2063	C₂	1	4	yes	maximal, complemented
6	#1997	C₂	1	1	yes	maximal, characteristic, complemented
7	#1965	C₂	1	1	yes	maximal, characteristic
8	#1931	C₂	1	4	yes	maximal, complemented
9	C₄×#108	C₂	1	1	yes	maximal, characteristic, complemented
10	#533	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×#127	C₂²	1	1	yes	characteristic
13	#501	—	2	2	no	—
14	#500	—	2	1	no	—
15	#498	—	2	1	no	—
16	#490	—	2	2	no	—
17	#486	—	2	1	no	—
18	#486	—	2	1	no	—
19	#486	—	2	1	no	—
20	#486	—	2	1	no	—
21	#486	—	2	1	no	—
22	#486	—	2	1	no	—
23	#446	—	2	1	no	—
24	#446	—	2	1	no	—
25	C₈×Q₈	—	2	1	no	—
26	C₈×Q₈	—	2	1	no	—
27	C₂×#108	C₂²	1	2	yes	—
28	#428	C₂²	1	4	yes	—
29	#424	C₂²	1	2	yes	—
30	#423	C₂²	1	1	yes	characteristic
31	#418	C₂²	1	4	yes	—
32	C₂×#102	C₂²	1	2	yes	—
33	C₂×#102	C₂²	1	2	yes	—
34	C₂×#100	C₂²	1	4	yes	—
35	#405	C₂²	1	2	yes	—
36	C₄×M₁₆	C₂²	1	1	yes	characteristic
37	C₄×M₁₆	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
39	#343	C₂²	1	4	yes	—
40	#341	C₂²	1	2	yes	—
41	C₂²×Q₈	—	2	1	no	—
42	C₂²×D₄	—	2	1	no	—
43	C₂²×D₄	—	2	1	no	—
44	C₂²×D₄	C₂³	1	2	yes	—
45	C₂×M₁₆	C₂³	1	2	yes	—
46	C₂×M₁₆	C₂³	1	2	yes	—
47	C₂×M₁₆	C₂³	1	2	yes	—
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	C₄⋊Q₈	—	2	1	no	—
51	C₄⋊Q₈	—	2	1	no	—
52	C₄⋊Q₈	—	2	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	D₄	1	1	yes	characteristic
55	C₄⋊Q₈	D₄	1	1	yes	characteristic
56	#127	—	2	1	no	—
57	#127	—	2	1	no	—
58	#127	—	2	1	no	—
59	#127	—	2	1	no	—
60	#125	D₄	1	2	yes	—
61	C₂²⋊Q₈	—	2	1	no	—
62	C₂²⋊Q₈	—	2	1	no	—
63	C₄×Q₈	—	2	1	no	—
64	C₄×Q₈	—	2	1	no	—
65	#117	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄	—	2	1	no	—
67	C₂×C₄⋊C₄	C₂³	1	2	yes	—
68	C₂×C₄²	C₂³	1	1	yes	characteristic
69	#108	—	2	4	no	—
70	#107	—	2	4	no	—
71	#106	—	2	4	no	—
72	C₄⋊C₈	—	2	2	no	—
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	—	2	1	no	—
75	C₄⋊C₈	—	2	1	no	—
76	#102	—	2	2	no	—
77	#102	—	2	2	no	—
78	#102	—	2	2	no	—
79	#102	—	2	2	no	—
80	#102	—	2	4	no	—
81	#102	—	2	4	no	—
82	#100	—	2	8	no	—
83	#97	—	2	2	no	—
84	#97	—	2	1	no	—
85	#97	—	2	1	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	D₄	1	1	yes	characteristic
90	C₄×C₈	D₄	1	1	yes	characteristic
91	C₄×C₈	D₄	1	1	yes	characteristic
92	C₄×C₈	D₄	1	1	yes	characteristic
93	C₂⁴	—	2	2	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	4	1	no	—
97	C₂×Q₈	—	4	1	no	—
98	C₂×Q₈	—	8	2	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂×D₄	—	2	2	no	—
101	C₂×D₄	—	2	4	no	—
102	C₂×D₄	—	2	4	no	—
103	C₂×D₄	—	4	1	no	—
104	C₂×D₄	—	4	1	no	—
105	C₂×D₄	—	4	1	no	—
106	C₂×D₄	—	4	1	no	—
107	C₂×D₄	—	8	2	no	—
108	C₂×D₄	—	8	2	no	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	#41	1	1	yes	characteristic
111	C₂²×C₄	#41	1	1	yes	characteristic
112	C₂²×C₄	C₂⁴	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	4	no	—
115	M₁₆	—	2	2	no	—
116	M₁₆	—	2	2	no	—
117	M₁₆	#41	1	4	yes	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
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	C₂×C₈	C₂×D₄	1	1	yes	characteristic
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	4	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	C₂×D₄	1	2	yes	—
136	C₄⋊C₄	C₂×D₄	1	2	yes	—
137	C₂²⋊C₄	—	2	2	no	—
138	C₄²	—	2	2	no	—
139	C₄²	C₂×D₄	1	1	yes	characteristic
140	C₄²	C₂×D₄	1	1	yes	characteristic
141	C₄²	C₂×D₄	1	1	yes	characteristic
142	C₄²	C₂×D₄	1	1	yes	characteristic
143	C₂³	—	4	2	no	—
144	C₂³	—	4	4	no	—
145	C₂³	—	8	2	no	—
146	C₂³	—	8	2	no	—
147	C₂³	C₂×#41	1	1	yes	characteristic
148	Q₈	—	4	2	no	—
149	Q₈	—	4	2	no	—
150	Q₈	—	8	2	no	—
151	Q₈	—	8	2	no	—
152	D₄	—	4	2	no	—
153	D₄	—	4	2	no	—
154	D₄	—	4	4	no	—
155	D₄	—	8	2	no	—
156	D₄	—	8	2	no	—
157	D₄	—	8	2	no	—
158	D₄	—	8	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	#121	1	1	yes	characteristic
166	C₂×C₄	#121	1	1	yes	characteristic
167	C₂×C₄	#121	1	1	yes	characteristic
168	C₂×C₄	#121	1	1	yes	characteristic
169	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄	C₂×#41	1	2	yes	—
172	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	4	no	—
178	C₈	—	4	2	no	—
179	C₈	—	4	2	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	8	2	no	—
183	C₂²	—	8	2	no	—
184	C₂²	—	8	2	no	—
185	C₂²	—	16	4	no	—
186	C₂²	C₂×#121	1	1	yes	characteristic
187	C₂²	#556	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	4	2	no	—
191	C₄	—	4	2	no	—
192	C₄	—	8	2	no	—
193	C₄	#471	1	2	yes	—
194	C₄	#472	1	2	yes	—
195	C₄	#532	1	1	yes	characteristic
196	C₄	#536	1	1	yes	characteristic
197	C₄	#548	1	2	yes	—
198	C₂	—	2	2	no	—
199	C₂	—	16	2	no	—
200	C₂	#2545	1	1	yes	characteristic
201	C₂	C₂×#471	1	1	yes	characteristic
202	C₂	C₂×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3165	C₂	semidirect
4. #2960	C₂	semidirect
5. #2063	C₂	semidirect
6. #1997	C₂	semidirect
8. #1931	C₂	semidirect
9. C₄×#108	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3165	2
#3138	2
C₂×#486	2
#2960	2
#2063	2
#1997	2
#1965	2
#1931	2
C₄×#108	2