Finite Group Databasebeta

Subgroups — #12921

Order 256.

221 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 83 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×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×#367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#1808	C₂	1	1	yes	maximal, characteristic
6	#1380	C₂	1	1	yes	maximal, characteristic, complemented
7	#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₈	C₂²	1	1	yes	characteristic
9	C₂×#117	—	2	1	no	—
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	#427	—	2	2	no	—
13	#427	—	2	2	no	—
14	C₂×#107	C₂²	1	1	yes	characteristic
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×#106	—	2	1	no	—
19	#380	—	2	1	no	—
20	#367	—	2	2	no	—
21	#366	—	2	2	no	—
22	C₂²⋊C₁₆	C₄	1	4	yes	complemented
23	#340	—	2	1	no	—
24	C₂³×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₈	—	2	1	no	—
26	C₂²×C₈	—	2	1	no	—
27	C₂²×C₈	—	2	1	no	—
28	C₂²×C₈	—	2	1	no	—
29	C₂²×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₈	Q₈	1	2	yes	—
32	C₂²×C₈	C₂³	1	1	yes	characteristic
33	#117	—	2	4	no	—
34	#117	—	4	2	no	—
35	C₂×C₄⋊C₄	—	2	1	no	—
36	C₂×C₄⋊C₄	—	2	1	no	—
37	C₂×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄⋊C₄	—	2	1	no	—
39	C₂×C₂²⋊C₄	—	4	1	no	—
40	C₂×C₂²⋊C₄	—	4	1	no	—
41	C₂×C₄²	—	4	1	no	—
42	C₂×C₁₆	—	2	1	no	—
43	C₂×C₁₆	—	2	1	no	—
44	C₂×C₁₆	—	2	1	no	—
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	C₂×C₄	1	2	yes	—
47	C₂×C₁₆	C₂×C₄	1	2	yes	—
48	#107	—	2	1	no	—
49	#107	—	2	1	no	—
50	#107	—	2	1	no	—
51	#107	—	2	1	no	—
52	#107	—	2	1	no	—
53	#107	—	2	1	no	—
54	#107	D₄	1	2	yes	—
55	#107	D₄	1	2	yes	—
56	#106	—	2	4	no	—
57	#95	—	4	1	no	—
58	C₂⁴	D₈	1	1	yes	characteristic
59	C₂²×C₄	—	2	1	no	—
60	C₂²×C₄	—	2	1	no	—
61	C₂²×C₄	—	2	1	no	—
62	C₂²×C₄	—	2	1	no	—
63	C₂²×C₄	—	4	1	no	—
64	C₂²×C₄	—	4	1	no	—
65	C₂²×C₄	—	4	1	no	—
66	C₂²×C₄	—	4	1	no	—
67	C₂²×C₄	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄	D₈	1	1	yes	characteristic
69	C₂²×C₄	Q₁₆	1	2	yes	—
70	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	C₄⋊C₄	1	2	yes	—
78	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₈	#41	1	1	yes	characteristic
83	C₂×C₈	#41	1	1	yes	characteristic
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	4	1	no	—
87	C₄⋊C₄	—	4	1	no	—
88	C₄⋊C₄	—	4	1	no	—
89	C₄⋊C₄	—	4	1	no	—
90	C₄⋊C₄	—	4	1	no	—
91	C₄⋊C₄	—	4	1	no	—
92	C₂²⋊C₄	—	4	4	no	—
93	C₂²⋊C₄	—	8	2	no	—
94	C₄²	—	4	4	no	—
95	C₁₆	—	2	2	no	—
96	C₁₆	—	2	2	no	—
97	C₂³	—	2	1	no	—
98	C₂³	—	2	1	no	—
99	C₂³	—	2	1	no	—
100	C₂³	—	2	1	no	—
101	C₂³	—	2	1	no	—
102	C₂³	—	2	1	no	—
103	C₂³	#107	1	2	yes	—
104	C₂³	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	8	1	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	8	1	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	—	8	1	no	—
119	C₂×C₄	—	8	1	no	—
120	C₂×C₄	—	8	1	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	#107	1	2	yes	—
124	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄	#120	1	1	yes	characteristic
126	C₂×C₄	#123	1	1	yes	characteristic
127	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
128	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄	#137	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	C₄×D₄	1	1	yes	characteristic
132	C₈	C₄×D₄	1	1	yes	characteristic
133	C₈	C₂²⋊Q₈	1	1	yes	characteristic
134	C₈	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂²	—	2	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	C₂×#107	1	1	yes	characteristic
143	C₂²	#447	1	1	yes	characteristic
144	C₂²	#480	1	1	yes	characteristic
145	C₂²	#508	1	1	yes	characteristic
146	C₂²	#508	1	1	yes	characteristic
147	C₂²	#509	1	1	yes	characteristic
148	C₂²	#510	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	8	2	no	—
151	C₄	—	8	2	no	—
152	C₄	—	16	1	no	—
153	C₄	—	16	1	no	—
154	C₄	#385	1	1	yes	characteristic
155	C₄	#428	1	1	yes	characteristic
156	C₄	#451	1	1	yes	characteristic
157	C₄	#484	1	1	yes	characteristic
158	C₂	—	4	2	no	—
159	C₂	#1852	1	1	yes	characteristic
160	C₂	#2158	1	1	yes	characteristic
161	C₂	#2159	1	1	yes	characteristic
162	C₂	#2186	1	1	yes	characteristic
163	C₂	#2189	1	1	yes	characteristic
164	C₂	#2234	1	1	yes	characteristic
165	C₂	#2237	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂⁴	D₈
LCS	G ▷ C₂×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×#367	C₂	semidirect
3. C₂×#366	C₂	semidirect
6. #1380	C₂	semidirect
7. #1380	C₂	semidirect
22. C₂²⋊C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#427	2
C₂×#367	2
C₂×#366	2
C₂×C₂²⋊C₁₆	2
#1808	2
#1380	2