Finite Group Databasebeta

Subgroups — #8255

Order 256.

235 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 97 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	#2931	C₂	1	1	yes	maximal, characteristic
2	#1770	C₂	1	1	yes	maximal, characteristic, complemented
3	#1540	C₂	1	1	yes	maximal, characteristic
4	#1502	C₂	1	1	yes	maximal, characteristic
5	#1450	C₂	1	1	yes	maximal, characteristic, complemented
6	#1303	C₂	1	1	yes	maximal, characteristic, complemented
7	#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131	—	2	1	no	—
9	#532	C₂²	1	1	yes	characteristic
10	#436	—	2	1	no	—
11	#436	C₄	1	2	yes	complemented
12	C₈×D₄	—	2	1	no	—
13	C₈×D₄	C₄	1	2	yes	complemented
14	#431	C₂²	1	1	yes	characteristic
15	#431	C₂²	1	1	yes	characteristic
16	#423	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	#408	—	2	1	no	—
19	C₂×#97	—	2	1	no	—
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	#336	—	2	1	no	—
23	#332	C₄	1	2	yes	complemented
24	#331	C₄	1	2	yes	complemented
25	#327	C₄	1	4	yes	—
26	#322	—	2	1	no	—
27	C₈²	—	2	1	no	—
28	C₂×#41	—	2	1	no	—
29	#131	—	4	2	no	—
30	#131	—	4	2	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂×M₁₆	—	2	2	no	—
33	C₂²×C₈	—	2	1	no	—
34	C₂²×C₈	—	2	2	no	—
35	C₂²×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₈	D₄	1	1	yes	characteristic
37	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	#127	C₂×C₄	1	1	yes	characteristic
39	#124	—	2	1	no	—
40	#121	—	2	1	no	—
41	#121	C₂×C₄	1	2	yes	—
42	C₄×D₄	—	2	1	no	—
43	C₄×D₄	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄	—	2	1	no	—
45	C₂×C₄²	—	2	1	no	—
46	C₂×C₄²	C₂³	1	1	yes	characteristic
47	C₄⋊C₈	—	2	1	no	—
48	C₄⋊C₈	—	2	1	no	—
49	C₄⋊C₈	—	2	1	no	—
50	C₄⋊C₈	C₂×C₄	1	2	yes	—
51	C₄⋊C₈	C₂×C₄	1	2	yes	—
52	C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₂²⋊C₈	—	2	1	no	—
54	C₂²⋊C₈	—	2	2	no	—
55	C₂²⋊C₈	—	2	2	no	—
56	C₂²⋊C₈	C₂×C₄	1	2	yes	—
57	#97	—	4	2	no	—
58	C₄×C₈	—	2	1	no	—
59	C₄×C₈	—	2	1	no	—
60	C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈	D₄	1	1	yes	characteristic
63	C₄×C₈	D₄	1	1	yes	characteristic
64	#95	C₂×C₄	1	2	yes	—
65	#41	—	4	1	no	—
66	#41	—	4	1	no	—
67	#41	—	4	1	no	—
68	#41	—	4	1	no	—
69	C₂×Q₈	—	2	1	no	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	—	2	1	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	C₄²	1	2	yes	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄	#41	1	1	yes	characteristic
79	M₁₆	—	4	1	no	—
80	M₁₆	—	4	1	no	—
81	M₁₆	—	4	4	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	4	1	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	—	4	4	no	—
91	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈	D₈	1	2	yes	—
96	C₂×C₈	SD₁₆	1	2	yes	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	C₄²	1	2	yes	—
100	C₂²⋊C₄	—	2	2	no	—
101	C₂²⋊C₄	—	2	2	no	—
102	C₄²	—	4	1	no	—
103	C₄²	—	4	1	no	—
104	C₄²	C₂²×C₄	1	1	yes	characteristic
105	C₄²	C₂²×C₄	1	1	yes	characteristic
106	C₄²	C₂×D₄	1	1	yes	characteristic
107	C₄²	#41	1	1	yes	characteristic
108	C₂³	—	2	2	no	—
109	C₂³	C₄×D₄	1	1	yes	characteristic
110	Q₈	—	4	1	no	—
111	Q₈	—	4	1	no	—
112	D₄	—	4	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	1	no	—
116	D₄	—	4	1	no	—
117	D₄	—	4	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄	#117	1	1	yes	characteristic
129	C₂×C₄	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄	#135	1	1	yes	characteristic
135	C₂×C₄	#135	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	4	2	no	—
139	C₈	—	4	2	no	—
140	C₈	—	4	2	no	—
141	C₈	—	4	2	no	—
142	C₈	#102	1	2	yes	—
143	C₈	#102	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	2	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²	C₄×D₈	1	1	yes	characteristic
151	C₂²	C₄×SD₁₆	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	2	no	—
156	C₄	—	8	2	no	—
157	C₄	C₂×#102	1	1	yes	characteristic
158	C₄	#416	1	1	yes	characteristic
159	C₄	C₄×D₈	1	1	yes	characteristic
160	C₄	C₄×SD₁₆	1	1	yes	characteristic
161	C₄	#444	1	2	yes	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	—	8	2	no	—
165	C₂	#1757	1	1	yes	characteristic
166	C₂	C₄×#102	1	1	yes	characteristic
167	C₂	#1765	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	C₄×C₂²⋊C₄
prime radical	#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #1770	C₂	semidirect
5. #1450	C₂	semidirect
6. #1303	C₂	semidirect
7. #1301	C₂	semidirect
11. #436	C₄	semidirect
13. C₈×D₄	C₄	semidirect
23. #332	C₄	semidirect
24. #331	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2931	2
#1770	2
#1540	2
#1502	2
#1450	2
#1303	2
#1301	2