Finite Group Databasebeta

Subgroups — #28176

Order 256.

395 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 117 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	#3037	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#2931	C₂	1	1	yes	maximal, characteristic
4	#2078	C₂	1	2	yes	maximal, complemented
5	#2020	C₂	1	2	yes	maximal, complemented
6	#1908	C₂	1	4	yes	maximal, complemented
7	#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	#1801	C₂	1	2	yes	maximal, complemented
9	C₂×#137	C₂²	1	4	yes	—
10	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×#131	C₂²	1	2	yes	—
12	#532	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈	C₂²	1	2	yes	—
14	#464	—	2	4	no	—
15	#462	—	2	2	no	—
16	#460	—	2	2	no	—
17	C₄×Q₁₆	—	2	1	no	—
18	C₄×Q₁₆	—	2	1	no	—
19	C₄×Q₁₆	—	2	1	no	—
20	C₄×Q₁₆	—	2	1	no	—
21	#436	—	2	1	no	—
22	#436	—	2	1	no	—
23	C₈×D₄	—	2	1	no	—
24	C₈×D₄	—	2	1	no	—
25	C₂×#107	C₂²	1	1	yes	characteristic
26	#423	C₂²	1	1	yes	characteristic
27	#423	C₂²	1	2	yes	—
28	#417	C₂²	1	4	yes	—
29	C₂×#103	C₂²	1	2	yes	—
30	C₂×#101	C₂²	1	4	yes	—
31	#408	C₂²	1	2	yes	—
32	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
33	#343	C₂²	1	4	yes	—
34	#341	C₂²	1	2	yes	—
35	#341	C₂²	1	2	yes	—
36	C₂×#41	C₂³	1	2	yes	—
37	C₂²×Q₈	C₂³	1	2	yes	—
38	#137	—	2	8	no	—
39	#137	—	4	4	no	—
40	C₂×Q₁₆	—	2	4	no	—
41	C₂×Q₁₆	—	2	4	no	—
42	C₂×Q₁₆	—	2	1	no	—
43	C₂×Q₁₆	—	2	1	no	—
44	C₂×SD₁₆	—	2	4	no	—
45	#131	—	4	4	no	—
46	C₂×M₁₆	—	2	2	no	—
47	C₂×M₁₆	C₂³	1	2	yes	—
48	C₂×M₁₆	C₂³	1	4	yes	—
49	C₂²×C₈	—	2	2	no	—
50	C₂²×C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₈	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈	D₄	1	1	yes	characteristic
53	#127	D₄	1	1	yes	characteristic
54	#124	D₄	1	2	yes	—
55	#121	—	2	1	no	—
56	#121	—	2	1	no	—
57	C₄×Q₈	—	2	2	no	—
58	C₄×Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	C₄×Q₈	—	2	2	no	—
61	C₄×D₄	—	2	1	no	—
62	C₄×D₄	—	2	1	no	—
63	C₂×C₄⋊C₄	—	2	2	no	—
64	C₂×C₄⋊C₄	C₂³	1	2	yes	—
65	C₂×C₄²	—	2	2	no	—
66	C₂×C₄²	C₂³	1	1	yes	characteristic
67	#107	—	2	1	no	—
68	#107	—	2	1	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₄⋊C₈	—	2	2	no	—
73	#103	—	2	4	no	—
74	#103	—	2	2	no	—
75	#103	—	2	2	no	—
76	#102	—	2	4	no	—
77	#101	—	4	4	no	—
78	C₂²⋊C₈	—	2	2	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₄×C₈	D₄	1	1	yes	characteristic
81	C₄×C₈	D₄	1	1	yes	characteristic
82	C₄×C₈	D₄	1	1	yes	characteristic
83	C₄×C₈	D₄	1	1	yes	characteristic
84	#41	—	2	4	no	—
85	#41	—	2	4	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	C₂×D₄	1	2	yes	—
92	C₂×Q₈	#41	1	4	yes	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	C₂×D₄	1	2	yes	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	#41	1	1	yes	characteristic
101	C₂²×C₄	#41	1	1	yes	characteristic
102	C₂²×C₄	C₂⁴	1	1	yes	characteristic
103	Q₁₆	—	2	4	no	—
104	Q₁₆	—	2	4	no	—
105	Q₁₆	—	4	8	no	—
106	SD₁₆	—	4	8	no	—
107	M₁₆	—	2	4	no	—
108	M₁₆	—	2	4	no	—
109	M₁₆	—	2	4	no	—
110	M₁₆	—	4	4	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	4	4	no	—
116	C₂×C₈	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	4	2	no	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	2	2	no	—
128	C₄²	—	4	2	no	—
129	C₄²	—	4	2	no	—
130	C₄²	C₂×D₄	1	1	yes	characteristic
131	C₄²	C₂×D₄	1	1	yes	characteristic
132	C₄²	C₂×D₄	1	1	yes	characteristic
133	C₄²	C₂×D₄	1	1	yes	characteristic
134	C₂³	—	2	2	no	—
135	C₂³	C₂×#41	1	1	yes	characteristic
136	Q₈	—	2	4	no	—
137	Q₈	—	2	4	no	—
138	Q₈	—	2	4	no	—
139	Q₈	—	4	2	no	—
140	Q₈	—	4	2	no	—
141	D₄	—	2	4	no	—
142	D₄	—	4	2	no	—
143	D₄	—	4	4	no	—
144	D₄	—	4	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	#121	1	1	yes	characteristic
156	C₂×C₄	#121	1	1	yes	characteristic
157	C₂×C₄	#121	1	1	yes	characteristic
158	C₂×C₄	#121	1	1	yes	characteristic
159	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄	C₂×#41	1	2	yes	—
162	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	4	2	no	—
168	C₈	—	4	4	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	4	2	no	—
174	C₂²	C₂×#121	1	1	yes	characteristic
175	C₂²	#553	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	4	no	—
181	C₄	—	8	2	no	—
182	C₄	#473	1	2	yes	—
183	C₄	#525	1	1	yes	characteristic
184	C₄	#535	1	1	yes	characteristic
185	C₄	#546	1	2	yes	—
186	C₂	—	2	2	no	—
187	C₂	—	8	2	no	—
188	C₂	#2535	1	1	yes	characteristic
189	C₂	C₂×#473	1	1	yes	characteristic
190	C₂	#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	#532	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
2. C₂×C₄×Q₁₆	C₂	semidirect
4. #2078	C₂	semidirect
5. #2020	C₂	semidirect
6. #1908	C₂	semidirect
7. #1837	C₂	semidirect
8. #1801	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3037	2
C₂×C₄×Q₁₆	2
#2931	2
#2078	2
#2020	2
#1908	2
#1837	2
#1801	2