Finite Group Databasebeta

Subgroups — #19646

Order 256.

366 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 123 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	#3210	C₂	1	2	yes	maximal, complemented
2	#3182	C₂	1	2	yes	maximal, complemented
3	#3169	C₂	1	1	yes	maximal, characteristic
4	#3160	C₂	1	1	yes	maximal, characteristic, complemented
5	#1720	C₂	1	2	yes	maximal, complemented
6	#1714	C₂	1	2	yes	maximal, complemented
7	#1713	C₂	1	1	yes	maximal, characteristic, complemented
8	#1710	C₂	1	1	yes	maximal, characteristic, complemented
9	#1706	C₂	1	1	yes	maximal, characteristic, complemented
10	#1706	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#533	C₂²	1	2	yes	—
13	#532	C₂²	1	2	yes	—
14	#500	C₂²	1	1	yes	characteristic
15	#500	C₂²	1	1	yes	characteristic
16	#500	C₂²	1	1	yes	characteristic
17	#500	C₂²	1	2	yes	—
18	#499	—	2	2	no	—
19	#498	C₂²	1	2	yes	—
20	#494	C₂²	1	2	yes	—
21	#493	C₂²	1	1	yes	characteristic
22	#492	C₂²	1	2	yes	—
23	#492	C₂²	1	2	yes	—
24	#487	C₂²	1	2	yes	—
25	#487	C₂²	1	2	yes	—
26	#486	C₂²	1	2	yes	—
27	#486	C₂²	1	2	yes	—
28	#486	C₂²	1	2	yes	—
29	#468	—	2	4	no	—
30	#466	—	2	2	no	—
31	#464	—	2	4	no	—
32	#452	—	2	4	no	—
33	#440	—	2	4	no	—
34	C₄×D₈	—	2	2	no	—
35	#431	C₂²	1	1	yes	characteristic
36	#427	—	2	2	no	—
37	C₄×M₁₆	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
39	#322	C₂²	1	1	yes	characteristic
40	#322	C₂²	1	1	yes	characteristic
41	C₈²	C₂²	1	1	yes	characteristic
42	C₈²	C₂²	1	1	yes	characteristic
43	C₂×#41	—	2	2	no	—
44	C₂×Q₁₆	—	2	4	no	—
45	C₂×SD₁₆	—	2	4	no	—
46	C₂×SD₁₆	—	2	4	no	—
47	C₂×D₈	—	2	2	no	—
48	C₂×M₁₆	D₄	1	2	yes	—
49	C₂²×C₈	D₄	1	2	yes	—
50	C₄⋊Q₈	C₂³	1	2	yes	—
51	C₄⋊Q₈	C₂³	1	2	yes	—
52	C₄⋊Q₈	C₂³	1	2	yes	—
53	#127	C₂³	1	2	yes	—
54	#125	—	2	2	no	—
55	#124	—	2	2	no	—
56	C₂²⋊Q₈	—	2	2	no	—
57	C₂²⋊Q₈	—	2	2	no	—
58	#121	—	2	2	no	—
59	#121	—	2	2	no	—
60	C₄×Q₈	—	2	2	no	—
61	C₄×Q₈	—	2	2	no	—
62	C₄×D₄	—	2	2	no	—
63	C₄×D₄	—	2	2	no	—
64	#117	—	2	2	no	—
65	C₂×C₄²	C₂³	1	1	yes	characteristic
66	#107	—	2	2	no	—
67	#107	—	2	2	no	—
68	#107	—	2	2	no	—
69	#107	—	2	4	no	—
70	#106	—	2	4	no	—
71	C₄⋊C₈	D₄	1	2	yes	—
72	#103	—	2	4	no	—
73	#103	—	2	4	no	—
74	#102	—	2	4	no	—
75	#102	—	2	4	no	—
76	#102	—	2	4	no	—
77	C₂²⋊C₈	D₄	1	2	yes	—
78	#97	D₄	1	2	yes	—
79	C₄×C₈	D₄	1	2	yes	—
80	C₄×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₈	C₂³	1	1	yes	characteristic
83	C₄×C₈	C₂³	1	1	yes	characteristic
84	C₄×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₈	C₂³	1	1	yes	characteristic
86	#41	—	8	4	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
98	Q₁₆	—	8	4	no	—
99	SD₁₆	—	4	8	no	—
100	SD₁₆	—	8	4	no	—
101	D₈	—	4	4	no	—
102	M₁₆	—	2	4	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	C₂×D₄	1	2	yes	—
105	C₂×C₈	C₂×D₄	1	2	yes	—
106	C₂×C₈	C₂×D₄	1	2	yes	—
107	C₂×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₈	C₂×D₄	1	2	yes	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	4	2	no	—
118	C₂²⋊C₄	—	4	2	no	—
119	C₂²⋊C₄	—	4	2	no	—
120	C₂²⋊C₄	—	4	2	no	—
121	C₄²	—	4	2	no	—
122	C₄²	C₂×D₄	1	1	yes	characteristic
123	C₄²	C₂×D₄	1	1	yes	characteristic
124	C₄²	C₂×D₄	1	1	yes	characteristic
125	C₄²	C₂⁴	1	1	yes	characteristic
126	C₂³	—	4	2	no	—
127	C₂³	#127	1	1	yes	characteristic
128	Q₈	—	4	4	no	—
129	Q₈	—	4	4	no	—
130	Q₈	—	4	4	no	—
131	D₄	—	4	4	no	—
132	D₄	—	4	4	no	—
133	D₄	—	4	4	no	—
134	D₄	—	8	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	#127	1	1	yes	characteristic
143	C₂×C₄	#127	1	1	yes	characteristic
144	C₂×C₄	#127	1	1	yes	characteristic
145	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	2	no	—
150	C₈	—	2	2	no	—
151	C₈	—	2	2	no	—
152	C₈	#135	1	4	yes	—
153	C₈	#137	1	4	yes	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	8	2	no	—
158	C₂²	—	8	2	no	—
159	C₂²	—	8	2	no	—
160	C₂²	C₂×#127	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	8	2	no	—
165	C₄	—	8	2	no	—
166	C₄	—	8	2	no	—
167	C₄	C₂×#135	1	2	yes	—
168	C₄	C₂×#137	1	2	yes	—
169	C₄	#576	1	2	yes	—
170	C₂	—	4	1	no	—
171	C₂	—	16	2	no	—
172	C₂	#3147	1	1	yes	characteristic
173	C₂	#3152	1	1	yes	characteristic
174	C₂	#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
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
1. #3210	C₂	semidirect
2. #3182	C₂	semidirect
4. #3160	C₂	semidirect
5. #1720	C₂	semidirect
6. #1714	C₂	semidirect
7. #1713	C₂	semidirect
8. #1710	C₂	semidirect
9. #1706	C₂	semidirect
10. #1706	C₂	semidirect
11. C₈×M₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3210	2
#3182	2
#3169	2
#3160	2
#1720	2
#1714	2
#1713	2
#1710	2
#1706	2
C₈×M₁₆	2