Finite Group Databasebeta

Subgroups — #21659

Order 256.

361 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 163 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	#3131	C₂	1	1	yes	maximal, characteristic
2	#2961	C₂	1	1	yes	maximal, characteristic, complemented
3	#2932	C₂	1	1	yes	maximal, characteristic
4	#1995	C₂	1	1	yes	maximal, characteristic, complemented
5	#1963	C₂	1	1	yes	maximal, characteristic
6	#1938	C₂	1	2	yes	maximal, complemented
7	#1937	C₂	1	2	yes	maximal, complemented
8	#1837	C₂	1	1	yes	maximal, characteristic, complemented
9	#1765	C₂	1	2	yes	maximal, complemented
10	#1764	C₂	1	2	yes	maximal, complemented
11	#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131	C₂²	1	2	yes	—
13	#533	C₂²	1	1	yes	characteristic
14	#532	C₂²	1	1	yes	characteristic
15	#488	C₄	1	4	yes	complemented
16	#487	C₄	1	2	yes	complemented
17	#486	C₄	1	2	yes	complemented
18	#446	—	2	4	no	—
19	#435	—	2	4	no	—
20	#424	C₂²	1	2	yes	—
21	#423	C₂²	1	1	yes	characteristic
22	#423	C₂²	1	1	yes	characteristic
23	C₂×#104	C₂²	1	2	yes	—
24	C₂×#104	C₂²	1	2	yes	—
25	#416	C₂²	1	2	yes	—
26	#416	C₂²	1	2	yes	—
27	#408	C₂²	1	2	yes	—
28	#405	C₂²	1	2	yes	—
29	C₂×#97	C₂²	1	1	yes	characteristic
30	C₂×#97	C₂²	1	1	yes	characteristic
31	C₂×#97	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
35	#376	C₂²	1	1	yes	characteristic
36	#341	C₂²	1	2	yes	—
37	#341	C₂²	1	2	yes	—
38	#339	C₂²	1	2	yes	—
39	#339	C₂²	1	2	yes	—
40	C₂×#41	C₂³	1	2	yes	—
41	#131	—	2	4	no	—
42	#131	—	2	4	no	—
43	C₂×M₁₆	—	2	2	no	—
44	C₂×M₁₆	C₂³	1	2	yes	—
45	C₂×M₁₆	C₂³	1	2	yes	—
46	C₂²×C₈	—	2	2	no	—
47	C₂²×C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₈	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈	C₂×C₄	1	2	yes	—
53	#127	C₂×C₄	1	1	yes	characteristic
54	#125	C₂×C₄	1	2	yes	—
55	#124	C₂×C₄	1	2	yes	—
56	C₂²⋊Q₈	—	2	2	no	—
57	#121	—	2	2	no	—
58	C₄×Q₈	—	2	2	no	—
59	C₄×D₄	—	2	2	no	—
60	#117	C₂³	1	2	yes	—
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	C₂×C₄²	C₂³	1	1	yes	characteristic
63	C₂×C₄²	C₂³	1	1	yes	characteristic
64	C₄⋊C₈	—	2	4	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	#104	—	2	4	no	—
68	#104	—	2	4	no	—
69	#103	C₂×C₄	1	4	yes	—
70	#103	C₂×C₄	1	4	yes	—
71	#102	C₂×C₄	1	4	yes	—
72	#102	C₂×C₄	1	4	yes	—
73	C₂²⋊C₈	—	2	4	no	—
74	#97	—	2	2	no	—
75	#97	—	2	2	no	—
76	#97	—	2	2	no	—
77	#97	D₄	1	4	yes	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	C₂×C₄	1	2	yes	—
82	C₄×C₈	C₂×C₄	1	2	yes	—
83	#95	—	2	2	no	—
84	#41	—	2	2	no	—
85	#41	—	2	2	no	—
86	#41	—	2	2	no	—
87	#41	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	C₂²×C₄	1	2	yes	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	C₂²×C₄	1	2	yes	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#41	1	2	yes	—
97	C₂²×C₄	C₂⁴	1	1	yes	characteristic
98	M₁₆	—	2	2	no	—
99	M₁₆	—	2	2	no	—
100	M₁₆	—	2	2	no	—
101	M₁₆	—	2	2	no	—
102	M₁₆	—	4	4	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	4	no	—
112	C₂×C₈	C₂²×C₄	1	2	yes	—
113	C₂×C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₈	C₂×D₄	1	2	yes	—
115	C₂×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₈	#41	1	2	yes	—
117	C₂×C₈	#41	1	2	yes	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	C₂²×C₄	1	4	yes	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₂²⋊C₄	—	2	4	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	C₂²×C₄	1	1	yes	characteristic
128	C₄²	C₂²×C₄	1	1	yes	characteristic
129	C₄²	C₂×D₄	1	2	yes	—
130	C₄²	#41	1	2	yes	—
131	C₄²	#41	1	2	yes	—
132	C₂³	—	2	2	no	—
133	C₂³	C₂×#41	1	1	yes	characteristic
134	Q₈	—	2	2	no	—
135	Q₈	—	2	2	no	—
136	Q₈	—	8	2	no	—
137	D₄	—	2	2	no	—
138	D₄	—	2	2	no	—
139	D₄	—	4	2	no	—
140	D₄	—	4	2	no	—
141	D₄	—	8	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	C₄×D₄	1	2	yes	—
152	C₂×C₄	C₄×D₄	1	2	yes	—
153	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄	C₂×#41	1	1	yes	characteristic
156	C₂×C₄	C₂×#41	1	1	yes	characteristic
157	C₂×C₄	C₂×#41	1	1	yes	characteristic
158	C₂×C₄	C₂×#41	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	4	2	no	—
163	C₈	—	4	2	no	—
164	C₈	#124	1	4	yes	—
165	C₂²	—	2	2	no	—
166	C₂²	—	4	4	no	—
167	C₂²	—	4	2	no	—
168	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²	#529	1	1	yes	characteristic
170	C₂²	#529	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	—	4	4	no	—
176	C₄	—	4	2	no	—
177	C₄	C₄×#41	1	1	yes	characteristic
178	C₄	C₄×#41	1	1	yes	characteristic
179	C₄	C₂×#124	1	1	yes	characteristic
180	C₄	#532	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	—	8	2	no	—
184	C₂	C₄×#124	1	1	yes	characteristic
185	C₂	#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2961	C₂	semidirect
4. #1995	C₂	semidirect
6. #1938	C₂	semidirect
7. #1937	C₂	semidirect
8. #1837	C₂	semidirect
9. #1765	C₂	semidirect
10. #1764	C₂	semidirect
11. #1726	C₂	semidirect
15. #488	C₄	semidirect
16. #487	C₄	semidirect
17. #486	C₄	semidirect

Maximal subgroups

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

subgroup	index
#3131	2
#2961	2
#2932	2
#1995	2
#1963	2
#1938	2
#1937	2
#1837	2
#1765	2
#1764	2
#1726	2