Finite Group Databasebeta

Subgroups — #33609

Order 256.

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

normal subgroups only

This group has 341 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	C₂²×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	#3059	C₂	1	1	yes	maximal, characteristic, complemented
4	#3058	C₂	1	1	yes	maximal, characteristic
5	#2220	C₂	1	4	yes	maximal, complemented
6	#2217	C₂	1	4	yes	maximal, complemented
7	#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	#2140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂²×D₈	C₂²	1	1	yes	characteristic
12	C₂×#131	C₂²	1	1	yes	characteristic
13	#536	—	2	1	no	—
14	#535	—	2	1	no	—
15	C₂×SD₃₂	—	2	4	no	—
16	C₂×SD₃₂	C₂²	1	2	yes	—
17	C₂×SD₃₂	C₂²	1	2	yes	—
18	#504	—	2	4	no	—
19	C₂×M₃₂	C₂²	1	1	yes	characteristic
20	C₂×M₃₂	C₂²	1	2	yes	complemented
21	C₂²×C₁₆	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆	C₂²	1	2	yes	complemented
23	#470	—	2	2	no	—
24	#469	—	2	2	no	—
25	#467	C₂²	1	4	yes	complemented
26	#466	C₂²	1	4	yes	—
27	#427	C₂²	1	1	yes	characteristic
28	#427	C₂²	1	1	yes	characteristic
29	#419	—	2	1	no	—
30	#418	—	2	1	no	—
31	#367	C₂²	1	2	yes	complemented
32	#366	C₂²	1	2	yes	complemented
33	#358	C₂²	1	4	yes	complemented
34	#357	C₂²	1	4	yes	—
35	C₂×#41	D₄	1	1	yes	characteristic
36	C₂²×Q₈	—	2	1	no	—
37	C₂²×D₄	—	2	1	no	—
38	C₂×Q₁₆	—	2	2	no	—
39	C₂×Q₁₆	—	2	1	no	—
40	C₂×Q₁₆	—	2	2	no	—
41	C₂×Q₁₆	C₂³	1	2	yes	—
42	C₂×D₈	—	2	2	no	—
43	C₂×D₈	—	2	1	no	—
44	C₂×D₈	—	2	2	no	—
45	C₂×D₈	C₂³	1	2	yes	—
46	#131	—	2	2	no	—
47	#131	—	2	2	no	—
48	C₂×M₁₆	D₄	1	1	yes	characteristic
49	C₂×M₁₆	D₄	1	2	yes	—
50	C₂²×C₈	C₂³	1	2	yes	—
51	C₂²×C₈	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈	—	4	1	no	—
53	#127	—	4	1	no	—
54	#124	—	4	1	no	—
55	#124	—	4	1	no	—
56	#123	—	4	2	no	—
57	C₂²⋊Q₈	—	2	4	no	—
58	#121	—	2	4	no	—
59	#120	—	4	2	no	—
60	#117	—	2	1	no	—
61	#117	—	2	1	no	—
62	SD₃₂	—	2	4	no	—
63	SD₃₂	—	2	4	no	—
64	M₃₂	—	2	2	no	—
65	M₃₂	—	2	4	no	—
66	C₂×C₁₆	—	2	4	no	—
67	C₂×C₁₆	D₄	1	4	yes	complemented
68	C₂×C₁₆	C₂³	1	2	yes	—
69	C₂×C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₁₆	C₂³	1	1	yes	characteristic
71	#107	C₂³	1	2	yes	—
72	#107	C₂³	1	2	yes	—
73	#106	—	2	1	no	—
74	#106	—	2	1	no	—
75	#103	—	2	4	no	—
76	#102	—	2	4	no	—
77	C₂⁴	—	4	1	no	—
78	#41	—	2	2	no	—
79	#41	—	2	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	4	1	no	—
82	C₂×Q₈	—	4	2	no	—
83	C₂×Q₈	—	4	2	no	—
84	C₂×Q₈	D₈	1	1	yes	characteristic
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	4	4	no	—
87	C₂×D₄	—	4	2	no	—
88	C₂×D₄	—	4	1	no	—
89	C₂×D₄	—	4	2	no	—
90	C₂×D₄	D₈	1	2	yes	—
91	C₂×D₄	D₈	1	1	yes	characteristic
92	C₂²×C₄	—	4	1	no	—
93	C₂²×C₄	C₂×D₄	1	2	yes	—
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	Q₁₆	—	2	2	no	—
96	Q₁₆	—	2	2	no	—
97	Q₁₆	—	4	2	no	—
98	D₈	—	2	2	no	—
99	D₈	—	2	2	no	—
100	D₈	—	4	2	no	—
101	M₁₆	—	2	2	no	—
102	M₁₆	—	2	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	C₂×D₄	1	2	yes	—
106	C₂×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₈	#41	1	2	yes	—
109	C₂×C₈	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	4	no	—
113	C₂²⋊C₄	—	4	1	no	—
114	C₂²⋊C₄	—	4	1	no	—
115	C₂²⋊C₄	—	4	4	no	—
116	C₂²⋊C₄	—	4	4	no	—
117	C₄²	—	4	1	no	—
118	C₄²	—	4	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	C₂×D₄	1	2	yes	—
121	C₁₆	C₂×D₄	1	2	yes	—
122	C₂³	—	4	2	no	—
123	C₂³	—	8	2	no	—
124	C₂³	—	16	2	no	—
125	C₂³	C₂×D₈	1	2	yes	—
126	C₂³	C₂×D₈	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	4	2	no	—
130	Q₈	—	8	2	no	—
131	D₄	—	2	4	no	—
132	D₄	—	2	2	no	—
133	D₄	—	4	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	8	2	no	—
136	D₄	—	16	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	#121	1	2	yes	—
144	C₂×C₄	C₂×D₈	1	2	yes	—
145	C₂×C₄	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄	#135	1	2	yes	—
147	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	#121	1	2	yes	—
150	C₈	C₂²×D₄	1	1	yes	characteristic
151	C₈	C₂×#41	1	1	yes	characteristic
152	C₂²	—	2	2	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	8	4	no	—
156	C₂²	—	8	2	no	—
157	C₂²	—	16	4	no	—
158	C₂²	#466	1	2	yes	—
159	C₂²	C₂²×D₈	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	8	2	no	—
163	C₄	—	8	2	no	—
164	C₄	#466	1	2	yes	—
165	C₄	C₂×#121	1	1	yes	characteristic
166	C₄	C₂×#135	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	—	4	2	no	—
169	C₂	—	16	2	no	—
170	C₂	C₂×#466	1	1	yes	characteristic
171	C₂	#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ 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. C₂²×SD₃₂	C₂	semidirect
2. C₂×#504	C₂	semidirect
3. #3059	C₂	semidirect
5. #2220	C₂	semidirect
6. #2217	C₂	semidirect
7. #2158	C₂	semidirect
8. #2141	C₂	semidirect
9. #2140	C₂	semidirect
20. C₂×M₃₂	C₂²	semidirect
22. C₂²×C₁₆	C₂²	semidirect
25. #467	C₂²	semidirect
31. #367	C₂²	semidirect
32. #366	C₂²	semidirect
33. #358	C₂²	semidirect
67. 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
C₂²×SD₃₂	2
C₂×#504	2
#3059	2
#3058	2
#2220	2
#2217	2
#2158	2
#2141	2
#2140	2