Finite Group Databasebeta

Subgroups — #28669

Order 256.

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

normal subgroups only

This group has 323 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
2	#3139	C₂	1	1	yes	maximal, characteristic
3	C₂×#487	C₂	1	1	yes	maximal, characteristic
4	#2960	C₂	1	1	yes	maximal, characteristic
5	#2064	C₂	1	4	yes	maximal
6	#1997	C₂	1	1	yes	maximal, characteristic
7	#1966	C₂	1	1	yes	maximal, characteristic
8	#1930	C₂	1	4	yes	maximal
9	C₄×#108	C₂	1	1	yes	maximal, characteristic
10	#533	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	#501	—	2	2	no	—
14	#500	—	2	1	no	—
15	#498	—	2	1	no	—
16	#491	—	2	2	no	—
17	#487	—	2	1	no	—
18	#487	—	2	1	no	—
19	#487	—	2	1	no	—
20	#487	—	2	1	no	—
21	#487	—	2	1	no	—
22	#487	—	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	#419	C₂²	1	4	yes	—
32	C₂×#103	C₂²	1	2	yes	—
33	C₂×#103	C₂²	1	2	yes	—
34	C₂×#101	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×Q₈	C₂³	1	2	yes	—
43	C₂×M₁₆	C₂³	1	2	yes	—
44	C₂×M₁₆	C₂³	1	2	yes	—
45	C₂×M₁₆	C₂³	1	2	yes	—
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₈	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	2	1	no	—
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	—	2	1	no	—
55	C₄⋊Q₈	—	2	1	no	—
56	C₄⋊Q₈	D₄	1	1	yes	characteristic
57	C₄⋊Q₈	D₄	1	1	yes	characteristic
58	#125	D₄	1	2	yes	—
59	C₂²⋊Q₈	—	2	1	no	—
60	C₂²⋊Q₈	—	2	1	no	—
61	C₄×Q₈	—	2	1	no	—
62	C₄×Q₈	—	2	1	no	—
63	#117	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄⋊C₄	—	2	1	no	—
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	#103	—	2	2	no	—
77	#103	—	2	2	no	—
78	#103	—	2	2	no	—
79	#103	—	2	2	no	—
80	#103	—	2	4	no	—
81	#103	—	2	4	no	—
82	#101	—	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	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	4	1	no	—
100	C₂×Q₈	—	4	1	no	—
101	C₂×Q₈	—	8	2	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	#41	1	1	yes	characteristic
105	C₂²×C₄	#41	1	1	yes	characteristic
106	C₂²×C₄	C₂⁴	1	1	yes	characteristic
107	M₁₆	—	2	2	no	—
108	M₁₆	—	2	4	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	#41	1	4	yes	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	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	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	4	1	no	—
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	—	4	1	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄⋊C₄	C₂×D₄	1	2	yes	—
134	C₄⋊C₄	C₂×D₄	1	2	yes	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	C₂×D₄	1	1	yes	characteristic
138	C₄²	C₂×D₄	1	1	yes	characteristic
139	C₄²	C₂×D₄	1	1	yes	characteristic
140	C₄²	C₂×D₄	1	1	yes	characteristic
141	C₂³	C₂×#41	1	1	yes	characteristic
142	Q₈	—	4	2	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	4	2	no	—
145	Q₈	—	4	2	no	—
146	Q₈	—	4	4	no	—
147	Q₈	—	8	2	no	—
148	Q₈	—	8	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	#121	1	1	yes	characteristic
158	C₂×C₄	#121	1	1	yes	characteristic
159	C₂×C₄	#121	1	1	yes	characteristic
160	C₂×C₄	#121	1	1	yes	characteristic
161	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄	C₂×#41	1	2	yes	—
164	C₂×C₄	2₋¹⁺⁴	1	2	yes	—
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	4	no	—
170	C₈	—	4	2	no	—
171	C₈	—	4	2	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	C₂×#121	1	1	yes	characteristic
175	C₂²	#556	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	4	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	8	2	no	—
181	C₄	—	8	2	no	—
182	C₄	#471	1	2	yes	—
183	C₄	#473	1	2	yes	—
184	C₄	#532	1	1	yes	characteristic
185	C₄	#536	1	1	yes	characteristic
186	C₄	#548	1	2	yes	—
187	C₂	—	2	2	no	—
188	C₂	#2545	1	1	yes	characteristic
189	C₂	C₂×#471	1	1	yes	characteristic
190	C₂	C₂×#473	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

Maximal subgroups

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

subgroup	index
#3165	2
#3139	2
C₂×#487	2
#2960	2
#2064	2
#1997	2
#1966	2
#1930	2
C₄×#108	2