Finite Group Databasebeta

Subgroups — #33617

Order 256.

321 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 117 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×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	#3067	C₂	1	2	yes	maximal, complemented
4	#2223	C₂	1	4	yes	maximal, complemented
5	#2222	C₂	1	2	yes	maximal, complemented
6	#2221	C₂	1	2	yes	maximal, complemented
7	C₂×#368	C₂	1	1	yes	maximal, characteristic, complemented
8	#2148	C₂	1	2	yes	maximal, complemented
9	#575	—	2	2	no	—
10	C₂×#135	C₂²	1	2	yes	—
11	C₂×#131	C₂²	1	1	yes	characteristic
12	#508	—	2	1	no	—
13	#508	—	2	1	no	—
14	#508	—	2	1	no	—
15	#508	—	2	1	no	—
16	C₂×Q₃₂	C₂²	1	1	yes	characteristic, complemented
17	C₂×SD₃₂	C₂²	1	2	yes	complemented
18	C₂×D₁₆	C₂²	1	1	yes	characteristic, complemented
19	#504	—	2	1	no	—
20	#504	—	2	1	no	—
21	#504	C₂²	1	2	yes	—
22	#504	C₂²	1	2	yes	—
23	C₂×M₃₂	—	2	1	no	—
24	C₂×M₃₂	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆	—	2	1	no	—
26	C₂²×C₁₆	C₂²	1	1	yes	characteristic
27	#473	C₂²	1	4	yes	—
28	#472	C₂²	1	4	yes	—
29	#471	—	2	4	no	—
30	C₂×#108	C₂²	1	2	yes	—
31	#413	—	2	2	no	—
32	#368	C₂²	1	2	yes	complemented
33	#368	C₂²	1	2	yes	complemented
34	#362	C₂²	1	4	yes	complemented
35	#361	C₂²	1	4	yes	complemented
36	C₂×#41	—	2	2	no	—
37	C₂×#41	D₄	1	1	yes	characteristic
38	#137	—	2	4	no	—
39	#137	—	4	2	no	—
40	AGL(1,ℤ/8)	—	2	4	no	—
41	AGL(1,ℤ/8)	—	4	2	no	—
42	#135	—	2	2	no	—
43	#135	—	2	2	no	—
44	#135	—	2	2	no	—
45	#135	—	2	2	no	—
46	#135	—	4	4	no	—
47	C₂×Q₁₆	C₂³	1	2	yes	—
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×D₈	C₂³	1	2	yes	—
50	#131	—	2	1	no	—
51	#131	—	2	1	no	—
52	#131	D₄	1	2	yes	—
53	#131	C₂³	1	2	yes	—
54	C₂×M₁₆	—	2	1	no	—
55	C₂×M₁₆	—	2	2	no	—
56	C₂×M₁₆	D₄	1	1	yes	characteristic
57	C₂²×C₈	—	2	1	no	—
58	C₂²×C₈	C₂³	1	1	yes	characteristic
59	Q₃₂	—	2	1	no	—
60	Q₃₂	—	2	1	no	—
61	SD₃₂	—	2	2	no	—
62	SD₃₂	—	2	2	no	—
63	D₁₆	—	2	1	no	—
64	D₁₆	—	2	1	no	—
65	M₃₂	—	2	1	no	—
66	M₃₂	—	2	2	no	—
67	M₃₂	—	2	2	no	—
68	M₃₂	C₂³	1	2	yes	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	D₄	1	1	yes	characteristic
73	C₂×C₁₆	D₄	1	1	yes	characteristic
74	C₂×C₁₆	D₄	1	1	yes	characteristic
75	C₂×C₁₆	D₄	1	1	yes	characteristic
76	C₂×C₁₆	C₂³	1	1	yes	characteristic
77	C₂×C₁₆	C₂³	1	1	yes	characteristic
78	#108	—	2	2	no	—
79	#108	C₂³	1	4	yes	—
80	#101	—	2	4	no	—
81	#100	—	2	4	no	—
82	#41	—	2	1	no	—
83	#41	—	2	1	no	—
84	#41	—	4	2	no	—
85	#41	—	4	2	no	—
86	#41	—	4	2	no	—
87	#41	—	4	2	no	—
88	#41	D₈	1	2	yes	—
89	#41	C₂×D₄	1	2	yes	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	D₈	1	1	yes	characteristic
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	4	2	no	—
95	C₂×D₄	D₈	1	1	yes	characteristic
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
99	Q₁₆	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	—	4	4	no	—
102	SD₁₆	—	4	4	no	—
103	SD₁₆	—	4	4	no	—
104	SD₁₆	—	4	2	no	—
105	SD₁₆	—	4	2	no	—
106	D₈	—	2	2	no	—
107	D₈	—	2	2	no	—
108	D₈	—	4	4	no	—
109	M₁₆	—	2	1	no	—
110	M₁₆	—	2	2	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	2	4	no	—
113	M₁₆	—	4	2	no	—
114	M₁₆	C₂×D₄	1	2	yes	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈	#41	1	1	yes	characteristic
123	C₂×C₈	#41	1	1	yes	characteristic
124	C₂×C₈	C₂⁴	1	1	yes	characteristic
125	C₁₆	—	2	2	no	—
126	C₁₆	C₂×D₄	1	1	yes	characteristic
127	C₁₆	C₂×D₄	1	1	yes	characteristic
128	C₁₆	C₂×D₄	1	1	yes	characteristic
129	C₁₆	C₂×D₄	1	1	yes	characteristic
130	C₂³	—	2	1	no	—
131	C₂³	—	4	2	no	—
132	C₂³	#135	1	1	yes	characteristic
133	Q₈	—	2	1	no	—
134	Q₈	—	4	2	no	—
135	Q₈	—	4	2	no	—
136	Q₈	C₂×D₈	1	2	yes	—
137	D₄	—	2	2	no	—
138	D₄	—	2	2	no	—
139	D₄	—	2	1	no	—
140	D₄	—	4	2	no	—
141	D₄	—	4	2	no	—
142	D₄	—	8	2	no	—
143	D₄	—	8	2	no	—
144	D₄	C₂×D₈	1	2	yes	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	#121	1	1	yes	characteristic
152	C₂×C₄	#121	1	1	yes	characteristic
153	C₂×C₄	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄	#135	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	4	4	no	—
159	C₈	#121	1	1	yes	characteristic
160	C₈	#121	1	1	yes	characteristic
161	C₈	C₂²×D₄	1	1	yes	characteristic
162	C₈	C₂×#41	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	—	4	1	no	—
165	C₂²	—	4	2	no	—
166	C₂²	—	8	2	no	—
167	C₂²	—	16	2	no	—
168	C₂²	#466	1	1	yes	characteristic
169	C₂²	#466	1	1	yes	characteristic
170	C₂²	C₂×#135	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	8	2	no	—
173	C₄	#466	1	1	yes	characteristic
174	C₄	#466	1	1	yes	characteristic
175	C₄	C₂×#121	1	1	yes	characteristic
176	C₄	C₂²×D₈	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	4	2	no	—
181	C₂	—	16	2	no	—
182	C₂	C₂×#466	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂	C₂×#466
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#508	C₂	semidirect
2. C₂×#504	C₂	semidirect
3. #3067	C₂	semidirect
4. #2223	C₂	semidirect
5. #2222	C₂	semidirect
6. #2221	C₂	semidirect
7. C₂×#368	C₂	semidirect
8. #2148	C₂	semidirect
16. C₂×Q₃₂	C₂²	semidirect
17. C₂×SD₃₂	C₂²	semidirect
18. C₂×D₁₆	C₂²	semidirect
32. #368	C₂²	semidirect
33. #368	C₂²	semidirect
34. #362	C₂²	semidirect
35. #361	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508	2
C₂×#504	2
#3067	2
#2223	2
#2222	2
#2221	2
C₂×#368	2
#2148	2