Finite Group Databasebeta

Subgroups — #10521

Order 256.

210 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 57 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	#3246	C₂	1	1	yes	maximal, characteristic
2	#1994	C₂	1	1	yes	maximal, characteristic, complemented
3	#1962	C₂	1	1	yes	maximal, characteristic
4	#1831	C₂	1	1	yes	maximal, characteristic, complemented
5	#1550	C₂	1	1	yes	maximal, characteristic, complemented
6	#1547	C₂	1	1	yes	maximal, characteristic
7	#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	#533	C₂²	1	1	yes	characteristic
9	#532	C₂²	1	1	yes	characteristic
10	#476	—	2	1	no	—
11	#475	—	2	1	no	—
12	#465	—	2	2	no	—
13	#461	—	2	1	no	—
14	#460	—	2	1	no	—
15	#423	C₂²	1	1	yes	characteristic
16	#423	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×#104	—	2	1	no	—
20	C₂×#104	—	2	1	no	—
21	C₄³	C₂²	1	1	yes	characteristic
22	#339	—	2	1	no	—
23	#339	—	2	1	no	—
24	#336	—	2	2	no	—
25	#333	—	2	1	no	—
26	#332	—	2	1	no	—
27	#332	—	2	1	no	—
28	#331	—	2	1	no	—
29	C₂×#41	—	2	1	no	—
30	C₂×SD₁₆	—	2	2	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂×M₁₆	—	2	1	no	—
33	C₂²×C₈	—	2	1	no	—
34	C₂²×C₈	D₄	1	2	yes	—
35	C₄⋊Q₈	D₄	1	1	yes	characteristic
36	C₄⋊Q₈	D₄	1	1	yes	characteristic
37	C₄⋊Q₈	D₄	1	1	yes	characteristic
38	#127	D₄	1	1	yes	characteristic
39	#125	—	2	1	no	—
40	#124	—	2	1	no	—
41	C₂²⋊Q₈	—	2	1	no	—
42	C₂²⋊Q₈	—	2	1	no	—
43	#121	—	2	1	no	—
44	#121	—	2	1	no	—
45	C₄×Q₈	—	2	1	no	—
46	C₄×Q₈	—	2	1	no	—
47	C₄×D₄	—	2	1	no	—
48	C₄×D₄	—	2	1	no	—
49	#117	—	2	1	no	—
50	C₂×C₄²	—	2	1	no	—
51	C₂×C₄²	—	2	1	no	—
52	C₂×C₄²	D₄	1	1	yes	characteristic
53	C₂×C₄²	D₄	1	1	yes	characteristic
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	#106	—	2	2	no	—
56	C₄⋊C₈	—	2	1	no	—
57	C₄⋊C₈	—	2	1	no	—
58	C₄⋊C₈	—	2	1	no	—
59	C₄⋊C₈	—	2	1	no	—
60	C₄⋊C₈	—	2	1	no	—
61	C₄⋊C₈	—	2	1	no	—
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	2	1	no	—
64	#104	—	4	2	no	—
65	#104	—	4	2	no	—
66	#103	—	2	2	no	—
67	#102	—	2	2	no	—
68	#41	—	4	1	no	—
69	#41	—	4	1	no	—
70	#41	—	4	1	no	—
71	#41	—	4	1	no	—
72	C₂×Q₈	—	2	1	no	—
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	2	1	no	—
75	C₂×D₄	—	2	1	no	—
76	C₂×D₄	—	2	1	no	—
77	C₂×D₄	—	2	1	no	—
78	C₂×D₄	—	4	1	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	4	1	no	—
82	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄	#41	1	1	yes	characteristic
85	SD₁₆	—	8	2	no	—
86	M₁₆	—	4	1	no	—
87	M₁₆	—	4	1	no	—
88	M₁₆	—	4	1	no	—
89	M₁₆	—	4	1	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	—	4	2	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	4	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	4	1	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₂²⋊C₄	—	4	1	no	—
105	C₂²⋊C₄	—	4	1	no	—
106	C₂²⋊C₄	—	4	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	2	1	no	—
110	C₄²	—	4	1	no	—
111	C₄²	—	4	1	no	—
112	C₄²	—	4	2	no	—
113	C₄²	—	4	1	no	—
114	C₄²	SD₁₆	1	2	yes	complemented
115	C₄²	C₂×D₄	1	1	yes	characteristic
116	C₄²	C₂×D₄	1	1	yes	characteristic
117	C₄²	#41	1	1	yes	characteristic
118	C₄²	#41	1	1	yes	characteristic
119	C₂³	—	4	1	no	—
120	C₂³	#121	1	1	yes	characteristic
121	Q₈	—	4	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	8	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	8	1	no	—
131	D₄	—	8	1	no	—
132	D₄	—	8	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	—	8	1	no	—
147	C₂×C₄	#120	1	1	yes	characteristic
148	C₂×C₄	#121	1	1	yes	characteristic
149	C₂×C₄	#121	1	1	yes	characteristic
150	C₂×C₄	#123	1	1	yes	characteristic
151	C₂×C₄	#123	1	1	yes	characteristic
152	C₂×C₄	#124	1	1	yes	characteristic
153	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
154	C₂×C₄	#135	1	1	yes	characteristic
155	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄	#137	1	1	yes	characteristic
157	C₈	—	4	2	no	—
158	C₈	—	8	1	no	—
159	C₈	—	8	1	no	—
160	C₈	—	8	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	8	1	no	—
164	C₂²	—	8	1	no	—
165	C₂²	—	8	1	no	—
166	C₂²	#394	1	1	yes	characteristic
167	C₂²	#465	1	1	yes	characteristic
168	C₂²	#468	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	2	no	—
174	C₄	—	8	1	no	—
175	C₄	—	8	1	no	—
176	C₄	—	8	1	no	—
177	C₄	#454	1	1	yes	characteristic
178	C₄	#455	1	1	yes	characteristic
179	C₄	#460	1	1	yes	characteristic
180	C₄	#461	1	1	yes	characteristic
181	C₄	#463	1	1	yes	characteristic
182	C₄	#464	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	16	1	no	—
186	C₂	#2040	1	1	yes	characteristic
187	C₂	#2050	1	1	yes	characteristic
188	C₂	#2055	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #1994	C₂	semidirect
4. #1831	C₂	semidirect
5. #1550	C₂	semidirect
7. #1292	C₂	semidirect
114. C₄²	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3246	2
#1994	2
#1962	2
#1831	2
#1550	2
#1547	2
#1292	2