Finite Group Databasebeta

Subgroups — #28703

Order 256.

303 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 135 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	#3166	C₂	1	1	yes	maximal, characteristic
2	#3165	C₂	1	1	yes	maximal, characteristic
3	C₂×#501	C₂	1	1	yes	maximal, characteristic
4	#2961	C₂	1	1	yes	maximal, characteristic
5	#2090	C₂	1	2	yes	maximal
6	#2089	C₂	1	2	yes	maximal
7	#1998	C₂	1	1	yes	maximal, characteristic
8	#1997	C₂	1	1	yes	maximal, characteristic
9	#1952	C₂	1	2	yes	maximal
10	#1951	C₂	1	2	yes	maximal
11	#1778	C₂	1	1	yes	maximal, characteristic
12	#533	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×#125	C₂²	1	1	yes	characteristic
15	#501	—	2	1	no	—
16	#501	—	2	1	no	—
17	#501	—	2	2	no	—
18	#501	—	2	2	no	—
19	#501	—	2	2	no	—
20	#500	—	2	1	no	—
21	#499	—	2	2	no	—
22	#498	—	2	1	no	—
23	#446	—	2	2	no	—
24	#446	—	2	2	no	—
25	C₂×#108	C₂²	1	2	yes	—
26	#428	C₂²	1	2	yes	—
27	#428	C₂²	1	2	yes	—
28	#428	C₂²	1	2	yes	—
29	#428	C₂²	1	2	yes	—
30	C₂×#107	C₂²	1	1	yes	characteristic
31	C₂×#107	C₂²	1	1	yes	characteristic
32	C₂×#106	C₂²	1	1	yes	characteristic
33	C₂×#106	C₂²	1	1	yes	characteristic
34	#424	C₂²	1	2	yes	—
35	#423	C₂²	1	1	yes	characteristic
36	#405	C₂²	1	2	yes	—
37	C₄×M₁₆	C₂²	1	1	yes	characteristic
38	C₄×M₁₆	C₂²	1	1	yes	characteristic
39	C₂×#97	C₂²	1	1	yes	characteristic
40	#343	C₂²	1	2	yes	—
41	#343	C₂²	1	2	yes	—
42	#343	C₂²	1	2	yes	—
43	#343	C₂²	1	2	yes	—
44	#341	C₂²	1	2	yes	—
45	C₂²×Q₈	—	2	1	no	—
46	C₂×M₁₆	C₂³	1	2	yes	—
47	C₂×M₁₆	C₂³	1	2	yes	—
48	C₂×M₁₆	C₂³	1	2	yes	—
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₈	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈	—	2	1	no	—
52	C₄⋊Q₈	—	2	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	2	1	no	—
55	C₄⋊Q₈	D₄	1	1	yes	characteristic
56	C₄⋊Q₈	D₄	1	1	yes	characteristic
57	#125	—	2	1	no	—
58	#125	—	2	1	no	—
59	#125	—	2	1	no	—
60	#125	—	2	1	no	—
61	#125	D₄	1	2	yes	—
62	C₂²⋊Q₈	—	2	1	no	—
63	C₂²⋊Q₈	—	2	1	no	—
64	C₄×Q₈	—	2	1	no	—
65	C₄×Q₈	—	2	1	no	—
66	#117	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄	—	2	1	no	—
68	C₂×C₄⋊C₄	—	2	1	no	—
69	C₂×C₄⋊C₄	—	2	1	no	—
70	C₂×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂³	1	1	yes	characteristic
74	C₂×C₄²	C₂³	1	1	yes	characteristic
75	#108	Q₈	1	8	yes	complemented
76	#107	—	2	2	no	—
77	#107	—	2	2	no	—
78	#107	—	2	2	no	—
79	#107	—	2	2	no	—
80	#107	—	2	2	no	—
81	#107	—	2	2	no	—
82	#106	—	2	2	no	—
83	#106	—	2	2	no	—
84	#106	—	2	2	no	—
85	#106	—	2	2	no	—
86	#106	—	2	2	no	—
87	#106	—	2	2	no	—
88	C₄⋊C₈	—	2	4	no	—
89	C₄⋊C₈	—	2	1	no	—
90	C₄⋊C₈	—	2	1	no	—
91	#97	—	2	2	no	—
92	#97	—	2	1	no	—
93	#97	—	2	1	no	—
94	#97	D₄	1	2	yes	—
95	#97	D₄	1	2	yes	—
96	C₄×C₈	—	2	2	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	4	1	no	—
102	C₂×Q₈	—	4	1	no	—
103	C₂×Q₈	—	8	2	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	#41	1	1	yes	characteristic
109	C₂²×C₄	#41	1	1	yes	characteristic
110	C₂²×C₄	C₂⁴	1	1	yes	characteristic
111	M₁₆	—	2	2	no	—
112	M₁₆	—	2	2	no	—
113	M₁₆	—	2	4	no	—
114	M₁₆	C₂×Q₈	1	4	yes	—
115	M₁₆	C₂×Q₈	1	4	yes	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	C₂×D₄	1	2	yes	—
121	C₂×C₈	C₂×D₄	1	2	yes	—
122	C₂×C₈	C₂×Q₈	1	4	yes	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	4	1	no	—
139	C₄⋊C₄	—	4	1	no	—
140	C₄⋊C₄	—	4	1	no	—
141	C₄⋊C₄	C₂×D₄	1	2	yes	—
142	C₄⋊C₄	C₂×D₄	1	2	yes	—
143	C₂²⋊C₄	—	2	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	C₂×D₄	1	1	yes	characteristic
146	C₄²	C₂×D₄	1	1	yes	characteristic
147	C₄²	C₂×D₄	1	1	yes	characteristic
148	C₄²	C₂×D₄	1	1	yes	characteristic
149	C₂³	C₂×#41	1	1	yes	characteristic
150	Q₈	—	8	1	no	—
151	Q₈	—	8	1	no	—
152	Q₈	—	8	2	no	—
153	Q₈	—	8	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	8	1	no	—
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	#121	1	1	yes	characteristic
170	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄	C₂²×Q₈	1	2	yes	—
173	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	—	2	4	no	—
176	C₈	—	2	4	no	—
177	C₈	—	4	2	no	—
178	C₈	C₄⋊Q₈	1	4	yes	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	C₂×#121	1	1	yes	characteristic
182	C₂²	#557	1	2	yes	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	8	1	no	—
190	C₄	—	8	1	no	—
191	C₄	—	8	1	no	—
192	C₄	—	8	1	no	—
193	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
194	C₄	#536	1	1	yes	characteristic
195	C₄	D₄×Q₈	1	2	yes	—
196	C₂	—	2	2	no	—
197	C₂	#2546	1	1	yes	characteristic
198	C₂	#3068	1	1	yes	characteristic
199	C₂	#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂³	C₂×#41
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
75. #108	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3166	2
#3165	2
C₂×#501	2
#2961	2
#2090	2
#2089	2
#1998	2
#1997	2
#1952	2
#1951	2
#1778	2