Finite Group Databasebeta

Subgroups — #9553

Order 256.

207 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 91 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	#1984	C₂	1	1	yes	maximal, characteristic
2	#1939	C₂	1	1	yes	maximal, characteristic
3	C₄×#103	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
6	#1279	C₂	1	1	yes	maximal, characteristic
7	#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈	—	2	1	no	—
10	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#103	C₄	1	1	yes	characteristic
13	C₂×#103	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	#340	C₄	1	1	yes	characteristic
19	#340	C₄	1	1	yes	characteristic
20	#336	—	2	1	no	—
21	#336	—	2	1	no	—
22	#336	—	2	1	no	—
23	Q₈⋊C₈	—	2	2	no	—
24	C₈²	—	2	2	no	—
25	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈	—	2	1	no	—
27	C₂²×C₈	—	2	1	no	—
28	C₂²×C₈	—	2	1	no	—
29	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₈	D₄	1	1	yes	characteristic
33	C₄×Q₈	—	2	1	no	—
34	C₄×Q₈	—	2	1	no	—
35	C₄×Q₈	—	2	1	no	—
36	C₄×Q₈	—	2	1	no	—
37	C₂×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²	—	2	1	no	—
42	C₂×C₄²	—	2	1	no	—
43	C₂×C₄²	C₂³	1	1	yes	characteristic
44	C₄⋊C₈	—	2	1	no	—
45	C₄⋊C₈	—	2	1	no	—
46	C₄⋊C₈	—	2	1	no	—
47	C₄⋊C₈	—	2	1	no	—
48	C₄⋊C₈	—	4	2	no	—
49	#103	—	2	2	no	—
50	#103	C₈	1	4	yes	complemented
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	—	2	1	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	—	2	1	no	—
56	C₄×C₈	D₄	1	2	yes	—
57	#95	—	2	1	no	—
58	C₂×Q₈	—	2	1	no	—
59	C₂×Q₈	—	2	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	2	1	no	—
63	C₂×Q₈	C₂×C₈	1	2	yes	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	2	1	no	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄	#41	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	1	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	C₂×C₈	1	2	yes	—
89	C₂×C₈	M₁₆	1	2	yes	—
90	C₂×C₈	SD₁₆	1	2	yes	—
91	C₂×C₈	Q₁₆	1	2	yes	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	4	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄⋊C₄	C₂×C₈	1	2	yes	—
100	C₄²	—	4	1	no	—
101	C₄²	—	4	1	no	—
102	C₄²	—	4	1	no	—
103	C₄²	—	4	1	no	—
104	C₄²	C₂×D₄	1	1	yes	characteristic
105	C₄²	#41	1	1	yes	characteristic
106	C₄²	#41	1	1	yes	characteristic
107	C₄²	#41	1	1	yes	characteristic
108	C₂³	C₄×D₄	1	1	yes	characteristic
109	Q₈	—	2	2	no	—
110	Q₈	—	2	2	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	#117	1	1	yes	characteristic
128	C₂×C₄	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄	#121	1	1	yes	characteristic
130	C₂×C₄	#123	1	1	yes	characteristic
131	C₂×C₄	#124	1	1	yes	characteristic
132	C₂×C₄	#126	1	1	yes	characteristic
133	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
135	C₂×C₄	#131	1	1	yes	characteristic
136	C₂×C₄	#131	1	1	yes	characteristic
137	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
139	C₂×C₄	#135	1	1	yes	characteristic
140	C₂×C₄	#135	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	8	1	no	—
147	C₈	—	8	1	no	—
148	C₂²	#388	1	1	yes	characteristic
149	C₂²	C₈×D₄	1	1	yes	characteristic
150	C₂²	#436	1	1	yes	characteristic
151	C₂²	C₄×SD₁₆	1	1	yes	characteristic
152	C₂²	C₄×Q₁₆	1	1	yes	characteristic
153	C₂²	#443	1	1	yes	characteristic
154	C₂²	#443	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	4	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	8	1	no	—
160	C₄	—	8	1	no	—
161	C₄	#431	1	1	yes	characteristic
162	C₄	#433	1	1	yes	characteristic
163	C₄	C₈×D₄	1	1	yes	characteristic
164	C₄	#435	1	1	yes	characteristic
165	C₄	#465	1	1	yes	characteristic
166	C₄	#467	1	1	yes	characteristic
167	C₄	#487	1	1	yes	characteristic
168	C₄	#488	1	1	yes	characteristic
169	C₂	C₈×SD₁₆	1	1	yes	characteristic
170	C₂	C₈×Q₁₆	1	1	yes	characteristic
171	C₂	#1590	1	1	yes	characteristic
172	C₂	#1591	1	1	yes	characteristic
173	C₂	#1923	1	1	yes	characteristic
174	C₂	#1933	1	1	yes	characteristic
175	C₂	#1937	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
50. #103	C₈	semidirect

Maximal subgroups

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

subgroup	index
#1984	2
#1939	2
C₄×#103	2
C₂×Q₈⋊C₈	2
C₂×C₈²	2
#1279	2
#1277	2