Finite Group Databasebeta

Subgroups — #13380

Order 256.

249 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 65 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	C₂×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×#366	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#357	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#357	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#1808	C₂	1	1	yes	maximal, characteristic, complemented
7	#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈	C₂²	1	1	yes	characteristic
9	C₂³×C₈	C₂²	1	1	yes	characteristic
10	C₂×#121	—	2	1	no	—
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆	C₂²	1	1	yes	characteristic
13	#466	—	2	2	no	—
14	#466	—	2	2	no	—
15	C₂×#107	C₂²	1	1	yes	characteristic
16	C₂×#107	C₂²	1	1	yes	characteristic
17	C₂×#107	C₂²	1	1	yes	characteristic
18	C₂×#102	—	2	1	no	—
19	#380	—	2	1	no	—
20	#366	—	2	2	no	—
21	#357	—	2	2	no	—
22	#357	—	2	2	no	—
23	C₂²⋊C₁₆	—	2	2	no	—
24	#340	—	2	1	no	—
25	C₂²×D₄	—	2	1	no	—
26	C₂²×D₄	—	4	1	no	—
27	C₂³×C₄	D₄	1	1	yes	characteristic
28	C₂×D₈	—	2	1	no	—
29	C₂×D₈	—	2	1	no	—
30	C₂×D₈	—	2	1	no	—
31	C₂×D₈	—	2	1	no	—
32	C₂×D₈	—	2	1	no	—
33	C₂×D₈	D₄	1	2	yes	—
34	C₂²×C₈	—	2	1	no	—
35	C₂²×C₈	—	2	1	no	—
36	C₂²×C₈	—	2	1	no	—
37	C₂²×C₈	—	2	1	no	—
38	C₂²×C₈	—	2	1	no	—
39	C₂²×C₈	—	2	1	no	—
40	C₂²×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	#121	—	2	4	no	—
43	#121	—	4	2	no	—
44	C₂×C₄⋊C₄	—	2	1	no	—
45	C₂×C₄⋊C₄	—	2	1	no	—
46	C₂×C₄⋊C₄	—	2	1	no	—
47	C₂×C₂²⋊C₄	—	4	1	no	—
48	C₂×C₂²⋊C₄	—	4	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	D₄	1	2	yes	complemented
55	#107	—	2	1	no	—
56	#107	—	2	1	no	—
57	#107	—	2	1	no	—
58	#107	—	2	1	no	—
59	#107	—	2	1	no	—
60	#107	D₄	1	2	yes	—
61	#102	—	2	4	no	—
62	#95	—	4	1	no	—
63	C₂⁴	—	4	1	no	—
64	C₂⁴	D₈	1	1	yes	characteristic
65	C₂×D₄	—	2	2	no	—
66	C₂×D₄	—	4	4	no	—
67	C₂×D₄	—	4	1	no	—
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	4	1	no	—
70	C₂×D₄	—	4	1	no	—
71	C₂×D₄	—	4	1	no	—
72	C₂×D₄	—	8	2	no	—
73	C₂×D₄	—	8	2	no	—
74	C₂²×C₄	—	2	1	no	—
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	4	1	no	—
81	C₂²×C₄	—	4	1	no	—
82	C₂²×C₄	—	4	1	no	—
83	C₂²×C₄	D₈	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	D₈	—	2	2	no	—
86	D₈	—	2	2	no	—
87	D₈	—	4	1	no	—
88	D₈	—	4	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	4	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈	#41	1	1	yes	characteristic
100	C₂×C₈	#41	1	1	yes	characteristic
101	C₂×C₈	#41	1	1	yes	characteristic
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	4	1	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	1	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₂²⋊C₄	—	4	4	no	—
109	C₂²⋊C₄	—	8	2	no	—
110	C₁₆	—	2	2	no	—
111	C₁₆	—	4	1	no	—
112	C₂³	—	2	1	no	—
113	C₂³	—	2	1	no	—
114	C₂³	—	2	1	no	—
115	C₂³	—	2	1	no	—
116	C₂³	—	2	1	no	—
117	C₂³	—	2	1	no	—
118	C₂³	—	4	2	no	—
119	C₂³	—	8	1	no	—
120	C₂³	—	8	1	no	—
121	C₂³	—	8	1	no	—
122	C₂³	—	8	1	no	—
123	C₂³	—	8	1	no	—
124	C₂³	—	8	1	no	—
125	C₂³	D₁₆	1	2	yes	—
126	C₂³	C₂×D₈	1	1	yes	characteristic
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	8	8	no	—
130	D₄	—	8	1	no	—
131	D₄	—	8	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	8	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	8	1	no	—
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	—	8	1	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	#120	1	1	yes	characteristic
149	C₂×C₄	#121	1	1	yes	characteristic
150	C₂×C₄	#123	1	1	yes	characteristic
151	C₂×C₄	#135	1	1	yes	characteristic
152	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₄	#137	1	1	yes	characteristic
154	C₈	—	4	1	no	—
155	C₈	#121	1	1	yes	characteristic
156	C₈	#121	1	1	yes	characteristic
157	C₈	#123	1	1	yes	characteristic
158	C₈	#124	1	1	yes	characteristic
159	C₂²	—	2	2	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	4	1	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	—	8	2	no	—
167	C₂²	—	8	2	no	—
168	C₂²	—	8	2	no	—
169	C₂²	—	16	1	no	—
170	C₂²	—	16	1	no	—
171	C₂²	—	16	1	no	—
172	C₂²	—	16	1	no	—
173	C₂²	#447	1	1	yes	characteristic
174	C₂²	#466	1	1	yes	characteristic
175	C₂²	#480	1	1	yes	characteristic
176	C₂²	C₂×D₁₆	1	1	yes	characteristic
177	C₂²	#508	1	1	yes	characteristic
178	C₂²	#509	1	1	yes	characteristic
179	C₂²	#510	1	1	yes	characteristic
180	C₄	—	4	1	no	—
181	C₄	—	8	2	no	—
182	C₄	—	16	1	no	—
183	C₄	—	16	1	no	—
184	C₄	#394	1	1	yes	characteristic
185	C₄	#452	1	1	yes	characteristic
186	C₄	#468	1	1	yes	characteristic
187	C₄	#482	1	1	yes	characteristic
188	C₂	—	4	2	no	—
189	C₂	—	16	2	no	—
190	C₂	#2033	1	1	yes	characteristic
191	C₂	#2184	1	1	yes	characteristic
192	C₂	#2189	1	1	yes	characteristic
193	C₂	#2215	1	1	yes	characteristic
194	C₂	#2220	1	1	yes	characteristic
195	C₂	#2232	1	1	yes	characteristic
196	C₂	#2234	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#466	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#366	C₂	semidirect
3. C₂×#357	C₂	semidirect
4. C₂×#357	C₂	semidirect
5. C₂×C₂²⋊C₁₆	C₂	semidirect
6. #1808	C₂	semidirect
7. #1380	C₂	semidirect
54. C₂×C₁₆	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#466	2
C₂×#366	2
C₂×#357	2
C₂×C₂²⋊C₁₆	2
#1808	2
#1380	2