Finite Group Databasebeta

Subgroups — #33586

Order 256.

358 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 119 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×#509	C₂	1	1	yes	maximal, characteristic
2	C₂²×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#3156	C₂	1	1	yes	maximal, characteristic, complemented
4	#2948	C₂	1	1	yes	maximal, characteristic, complemented
5	#2210	C₂	1	4	yes	maximal, complemented
6	#2208	C₂	1	4	yes	maximal, complemented
7	#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	#2138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137	—	2	1	no	—
11	C₂×#135	C₂²	1	1	yes	characteristic
12	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂²×D₈	C₂²	1	1	yes	characteristic
14	#536	—	2	1	no	—
15	#518	—	2	1	no	—
16	#509	—	2	4	no	—
17	C₂×SD₃₂	—	2	4	no	—
18	C₂×SD₃₂	C₂²	1	4	yes	—
19	C₂×SD₃₂	C₂²	1	2	yes	—
20	C₂×D₁₆	C₂²	1	2	yes	—
21	C₂×M₃₂	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆	C₂²	1	1	yes	characteristic
23	#497	—	2	2	no	—
24	#495	C₂²	1	2	yes	complemented
25	#494	C₂²	1	2	yes	complemented
26	#442	—	2	2	no	—
27	C₄×D₈	C₂²	1	4	yes	—
28	#427	C₂²	1	1	yes	characteristic
29	#418	—	2	1	no	—
30	#405	C₂²	1	1	yes	characteristic
31	C₄⋊C₁₆	C₂²	1	4	yes	complemented
32	#358	C₂²	1	2	yes	complemented
33	#358	C₂²	1	4	yes	complemented
34	#357	C₂²	1	2	yes	complemented
35	C₂×#41	—	2	1	no	—
36	C₂²×Q₈	—	2	1	no	—
37	C₂²×D₄	—	2	1	no	—
38	#137	—	4	4	no	—
39	#135	—	2	2	no	—
40	#135	—	2	2	no	—
41	C₂×Q₁₆	—	2	2	no	—
42	C₂×Q₁₆	—	2	1	no	—
43	C₂×Q₁₆	—	2	2	no	—
44	C₂×Q₁₆	—	2	2	no	—
45	C₂×Q₁₆	C₂³	1	2	yes	—
46	C₂×Q₁₆	C₂³	1	1	yes	characteristic
47	C₂×SD₁₆	—	2	1	no	—
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×D₈	—	2	1	no	—
50	C₂×D₈	—	2	2	no	—
51	C₂×D₈	D₄	1	4	yes	—
52	C₂×D₈	C₂³	1	1	yes	characteristic
53	C₂×D₈	C₂³	1	2	yes	—
54	C₂×M₁₆	D₄	1	1	yes	characteristic
55	C₂²×C₈	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈	—	2	2	no	—
57	#124	—	2	2	no	—
58	#123	—	4	2	no	—
59	C₂²⋊Q₈	—	4	2	no	—
60	C₄×D₄	—	2	4	no	—
61	C₄×D₄	—	4	2	no	—
62	#117	—	2	1	no	—
63	#117	D₄	1	1	yes	characteristic
64	C₂×C₂²⋊C₄	—	4	2	no	—
65	SD₃₂	—	2	8	no	—
66	SD₃₂	—	2	4	no	—
67	D₁₆	—	2	4	no	—
68	M₃₂	—	2	2	no	—
69	C₂×C₁₆	—	2	2	no	—
70	C₂×C₁₆	C₂³	1	2	yes	—
71	C₂×C₁₆	C₂³	1	2	yes	—
72	#107	C₂³	1	2	yes	—
73	#106	—	2	1	no	—
74	#102	—	2	4	no	—
75	#97	D₄	1	2	yes	—
76	C₄×C₈	C₂³	1	2	yes	—
77	C₂⁴	—	4	1	no	—
78	#41	—	4	2	no	—
79	#41	—	4	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	1	no	—
82	C₂×Q₈	—	4	1	no	—
83	C₂×Q₈	—	4	2	no	—
84	C₂×Q₈	—	4	2	no	—
85	C₂×D₄	—	2	4	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	1	no	—
88	C₂×D₄	—	4	1	no	—
89	C₂×D₄	—	4	1	no	—
90	C₂×D₄	—	4	2	no	—
91	C₂²×C₄	—	4	1	no	—
92	C₂²×C₄	—	4	1	no	—
93	C₂²×C₄	—	4	4	no	—
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	Q₁₆	—	2	2	no	—
96	Q₁₆	—	2	1	no	—
97	Q₁₆	—	2	2	no	—
98	Q₁₆	—	2	1	no	—
99	Q₁₆	—	4	4	no	—
100	Q₁₆	—	4	2	no	—
101	SD₁₆	—	4	4	no	—
102	SD₁₆	—	4	2	no	—
103	D₈	—	2	4	no	—
104	D₈	—	2	1	no	—
105	D₈	—	2	2	no	—
106	D₈	—	2	1	no	—
107	D₈	C₂×D₄	1	4	yes	—
108	M₁₆	—	2	2	no	—
109	C₂×C₈	C₂×D₄	1	2	yes	—
110	C₂×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈	C₂×D₄	1	2	yes	—
112	C₂×C₈	#41	1	2	yes	—
113	C₂×C₈	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	4	4	no	—
116	C₄⋊C₄	D₈	1	2	yes	—
117	C₂²⋊C₄	—	4	1	no	—
118	C₂²⋊C₄	—	4	4	no	—
119	C₂²⋊C₄	—	4	4	no	—
120	C₂²⋊C₄	D₈	1	2	yes	—
121	C₄²	—	4	1	no	—
122	C₄²	C₂×D₄	1	2	yes	—
123	C₁₆	—	2	2	no	—
124	C₁₆	—	2	2	no	—
125	C₂³	—	4	2	no	—
126	C₂³	—	4	4	no	—
127	C₂³	—	4	1	no	—
128	C₂³	—	16	2	no	—
129	C₂³	C₂×D₈	1	1	yes	characteristic
130	Q₈	—	4	1	no	—
131	Q₈	—	4	2	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	8	2	no	—
135	D₄	—	2	4	no	—
136	D₄	—	4	1	no	—
137	D₄	—	4	2	no	—
138	D₄	—	4	1	no	—
139	D₄	—	4	4	no	—
140	D₄	—	8	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	16	4	no	—
147	C₂×C₄	#121	1	2	yes	—
148	C₂×C₄	C₂×D₈	1	4	yes	—
149	C₂×C₄	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
151	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
152	C₈	—	2	2	no	—
153	C₈	#121	1	2	yes	—
154	C₈	C₂²×D₄	1	1	yes	characteristic
155	C₈	C₂×#41	1	1	yes	characteristic
156	C₂²	—	2	2	no	—
157	C₂²	—	4	4	no	—
158	C₂²	—	8	4	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	8	1	no	—
161	C₂²	—	16	4	no	—
162	C₂²	#459	1	2	yes	—
163	C₂²	C₂²×D₈	1	1	yes	characteristic
164	C₄	—	2	4	no	—
165	C₄	—	8	2	no	—
166	C₄	—	8	1	no	—
167	C₄	—	8	2	no	—
168	C₄	#459	1	2	yes	—
169	C₄	C₂×#121	1	1	yes	characteristic
170	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₂	—	2	2	no	—
172	C₂	—	8	4	no	—
173	C₂	—	16	1	no	—
174	C₂	C₂×#459	1	1	yes	characteristic
175	C₂	#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×SD₃₂	C₂	semidirect
3. #3156	C₂	semidirect
4. #2948	C₂	semidirect
5. #2210	C₂	semidirect
6. #2208	C₂	semidirect
7. #2151	C₂	semidirect
8. #2141	C₂	semidirect
9. #2138	C₂	semidirect
24. #495	C₂²	semidirect
25. #494	C₂²	semidirect
31. C₄⋊C₁₆	C₂²	semidirect
32. #358	C₂²	semidirect
33. #358	C₂²	semidirect
34. #357	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#509	2
C₂²×SD₃₂	2
#3156	2
#2948	2
#2210	2
#2208	2
#2151	2
#2141	2
#2138	2