Finite Group Databasebeta

Subgroups — #19286

Order 256.

345 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 115 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	#3169	C₂	1	1	yes	maximal, characteristic
2	#3037	C₂	1	2	yes	maximal, complemented
3	C₂×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	#1702	C₂	1	4	yes	maximal, complemented
5	#1694	C₂	1	2	yes	maximal, complemented
6	#1688	C₂	1	2	yes	maximal, complemented
7	#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	#1547	C₂	1	2	yes	maximal, complemented
9	C₂×#137	—	2	2	no	—
10	#532	C₂²	1	2	yes	—
11	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
12	#500	C₂²	1	1	yes	characteristic
13	#493	C₂²	1	1	yes	characteristic
14	#486	C₂²	1	2	yes	—
15	#466	—	2	1	no	—
16	#466	—	2	1	no	—
17	#464	—	2	4	no	—
18	#462	C₂²	1	4	yes	—
19	#460	C₂²	1	4	yes	—
20	#446	—	2	1	no	—
21	#446	—	2	1	no	—
22	#446	C₂²	1	2	yes	—
23	#446	C₂²	1	2	yes	—
24	C₄×D₈	—	2	1	no	—
25	C₄×D₈	—	2	1	no	—
26	#423	C₂²	1	2	yes	—
27	C₂×C₄⋊C₈	—	2	1	no	—
28	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	#417	—	2	2	no	—
30	C₂×#97	—	2	1	no	—
31	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
32	#334	C₂²	1	2	yes	—
33	#334	C₂²	1	2	yes	—
34	#332	C₂²	1	4	yes	—
35	#331	C₂²	1	4	yes	—
36	C₂×#41	—	2	2	no	—
37	C₂²×Q₈	D₄	1	1	yes	characteristic
38	#137	—	4	8	no	—
39	C₂×Q₁₆	—	2	4	no	—
40	C₂×SD₁₆	—	2	4	no	—
41	C₂×D₈	—	2	1	no	—
42	C₂×D₈	—	2	1	no	—
43	C₂×M₁₆	—	2	2	no	—
44	C₂²×C₈	—	2	1	no	—
45	C₂²×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₈	D₄	1	1	yes	characteristic
47	C₄⋊Q₈	C₂³	1	2	yes	—
48	#127	C₂³	1	2	yes	—
49	#124	—	2	2	no	—
50	#121	—	2	2	no	—
51	#121	—	2	2	no	—
52	C₄×Q₈	—	2	1	no	—
53	C₄×Q₈	—	2	1	no	—
54	C₄×Q₈	D₄	1	2	yes	—
55	C₄×Q₈	C₂³	1	2	yes	—
56	C₄×D₄	—	2	2	no	—
57	C₄×D₄	—	2	2	no	—
58	C₂×C₄⋊C₄	—	2	1	no	—
59	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
60	C₂×C₄²	—	2	1	no	—
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	#107	—	2	1	no	—
63	#107	—	2	1	no	—
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	1	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	C₂³	1	2	yes	—
69	C₄⋊C₈	C₂³	1	4	yes	—
70	#103	—	2	4	no	—
71	#102	—	2	2	no	—
72	#102	—	2	2	no	—
73	#102	—	2	4	no	—
74	#97	—	2	2	no	—
75	#97	—	2	2	no	—
76	C₄×C₈	D₄	1	1	yes	characteristic
77	C₄×C₈	D₄	1	1	yes	characteristic
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₈	C₂³	1	1	yes	characteristic
80	#41	—	4	4	no	—
81	#41	—	4	4	no	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	2	4	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	C₂×D₄	1	2	yes	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	4	2	no	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	4	2	no	—
93	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄	#41	1	1	yes	characteristic
96	Q₁₆	—	4	8	no	—
97	SD₁₆	—	4	8	no	—
98	D₈	—	4	2	no	—
99	D₈	—	4	2	no	—
100	M₁₆	—	4	4	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈	C₂×D₄	1	1	yes	characteristic
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	C₂×D₄	1	2	yes	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₂²⋊C₄	—	4	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	2	2	no	—
120	C₄²	C₂×D₄	1	1	yes	characteristic
121	C₄²	C₂×D₄	1	1	yes	characteristic
122	C₄²	#41	1	1	yes	characteristic
123	C₄²	C₂⁴	1	1	yes	characteristic
124	C₂³	—	4	2	no	—
125	C₂³	#121	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	4	2	no	—
130	D₄	—	4	2	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	4	2	no	—
134	D₄	—	8	2	no	—
135	D₄	—	8	4	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	8	4	no	—
145	C₂×C₄	#121	1	1	yes	characteristic
146	C₂×C₄	#121	1	1	yes	characteristic
147	C₂×C₄	#121	1	1	yes	characteristic
148	C₂×C₄	AGL(1,ℤ/8)	1	4	yes	—
149	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄	C₂×#41	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	4	4	no	—
156	C₈	—	4	2	no	—
157	C₈	#137	1	2	yes	complemented
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	2	no	—
162	C₂²	—	8	2	no	—
163	C₂²	#469	1	2	yes	—
164	C₂²	C₂×#121	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	8	2	no	—
171	C₄	#469	1	2	yes	—
172	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
173	C₄	C₂×#137	1	1	yes	characteristic
174	C₄	#575	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	16	2	no	—
177	C₂	#3037	1	1	yes	characteristic
178	C₂	C₂×#469	1	1	yes	characteristic
179	C₂	#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #3037	C₂	semidirect
3. C₂×#446	C₂	semidirect
4. #1702	C₂	semidirect
5. #1694	C₂	semidirect
6. #1688	C₂	semidirect
7. #1566	C₂	semidirect
8. #1547	C₂	semidirect
157. C₈	#137	semidirect

Maximal subgroups

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

subgroup	index
#3169	2
#3037	2
C₂×#446	2
#1702	2
#1694	2
#1688	2
#1566	2
#1547	2