Finite Group Databasebeta

Subgroups — #8250

Order 256.

200 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 81 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	#2966	C₂	1	1	yes	maximal, characteristic
2	#1746	C₂	1	1	yes	maximal, characteristic, complemented
3	#1524	C₂	1	1	yes	maximal, characteristic
4	#1500	C₂	1	1	yes	maximal, characteristic
5	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#1302	C₂	1	1	yes	maximal, characteristic, complemented
7	#1300	C₂	1	1	yes	maximal, characteristic, complemented
8	#529	C₂²	1	1	yes	characteristic
9	#446	C₄	1	1	yes	characteristic, complemented
10	C₈×Q₈	C₄	1	1	yes	characteristic, complemented
11	#436	C₄	1	1	yes	characteristic, complemented
12	#435	—	2	1	no	—
13	C₈×D₄	—	2	1	no	—
14	C₈×D₄	C₄	1	1	yes	characteristic, complemented
15	#433	—	2	1	no	—
16	#432	C₂²	1	1	yes	characteristic
17	#431	C₂²	1	1	yes	characteristic
18	#405	—	2	1	no	—
19	C₄×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×#97	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	#376	C₂²	1	1	yes	characteristic
23	#336	—	2	1	no	—
24	#336	—	2	1	no	—
25	#330	C₄	1	2	yes	complemented
26	#329	C₄	1	2	yes	complemented
27	#328	C₄	1	2	yes	—
28	#327	C₄	1	2	yes	—
29	#322	—	2	1	no	—
30	C₈²	—	2	1	no	—
31	C₂×M₁₆	—	2	1	no	—
32	C₂×M₁₆	—	2	1	no	—
33	C₂²×C₈	—	2	1	no	—
34	C₂²×C₈	—	2	1	no	—
35	C₂²×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₈	D₄	1	1	yes	characteristic
37	#126	—	2	1	no	—
38	#125	C₂×C₄	1	1	yes	characteristic
39	#124	C₂×C₄	1	1	yes	characteristic
40	#123	—	2	1	no	—
41	C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	#121	C₂×C₄	1	1	yes	characteristic
43	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
44	C₄×D₄	—	2	1	no	—
45	C₄×D₄	C₂×C₄	1	1	yes	characteristic
46	#117	—	2	1	no	—
47	C₂×C₄²	—	2	1	no	—
48	C₂×C₄²	C₂³	1	1	yes	characteristic
49	C₄⋊C₈	—	2	1	no	—
50	C₄⋊C₈	—	2	1	no	—
51	C₄⋊C₈	—	2	1	no	—
52	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂²⋊C₈	—	2	1	no	—
55	C₂²⋊C₈	—	2	1	no	—
56	C₂²⋊C₈	—	2	1	no	—
57	C₂²⋊C₈	C₂×C₄	1	2	yes	—
58	#97	—	2	1	no	—
59	#97	—	2	1	no	—
60	#97	—	2	1	no	—
61	#97	C₂×C₄	1	2	yes	—
62	#97	C₂×C₄	1	2	yes	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈	D₄	1	1	yes	characteristic
69	C₄×C₈	D₄	1	1	yes	characteristic
70	#95	—	2	1	no	—
71	#95	C₂×C₄	1	2	yes	—
72	C₂×Q₈	C₄²	1	1	yes	characteristic
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	C₄²	1	1	yes	characteristic
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	—	2	2	no	—
77	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄	#41	1	1	yes	characteristic
80	M₁₆	—	2	4	no	—
81	M₁₆	—	4	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	4	1	no	—
89	C₂×C₈	—	4	2	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	4	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	C₄²	1	1	yes	characteristic
101	C₄⋊C₄	C₄²	1	1	yes	characteristic
102	C₂²⋊C₄	—	2	1	no	—
103	C₂²⋊C₄	—	2	1	no	—
104	C₂²⋊C₄	—	2	1	no	—
105	C₂²⋊C₄	—	2	1	no	—
106	C₂²⋊C₄	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	4	1	no	—
109	C₄²	—	4	1	no	—
110	C₄²	C₂²×C₄	1	1	yes	characteristic
111	C₄²	C₂²×C₄	1	1	yes	characteristic
112	C₄²	C₂×D₄	1	1	yes	characteristic
113	C₄²	#41	1	1	yes	characteristic
114	C₂³	—	2	1	no	—
115	C₂³	C₄×D₄	1	1	yes	characteristic
116	Q₈	—	4	1	no	—
117	D₄	—	4	1	no	—
118	D₄	—	4	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄	#117	1	1	yes	characteristic
134	C₂×C₄	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄	C₄×D₄	1	1	yes	characteristic
137	C₈	—	2	4	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	4	2	no	—
143	C₈	—	4	2	no	—
144	C₈	—	4	1	no	—
145	C₈	—	4	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²	#443	1	2	yes	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	4	1	no	—
159	C₄	—	4	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	8	2	no	—
162	C₄	#421	1	1	yes	characteristic
163	C₄	#421	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	—	8	1	no	—
166	C₂	#1757	1	1	yes	characteristic
167	C₂	#1759	1	1	yes	characteristic
168	C₂	#1764	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	#121	C₂×C₄
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. #1746	C₂	semidirect
5. C₈×M₁₆	C₂	semidirect
6. #1302	C₂	semidirect
7. #1300	C₂	semidirect
9. #446	C₄	semidirect
10. C₈×Q₈	C₄	semidirect
11. #436	C₄	semidirect
14. C₈×D₄	C₄	semidirect
25. #330	C₄	semidirect
26. #329	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2966	2
#1746	2
#1524	2
#1500	2
C₈×M₁₆	2
#1302	2
#1300	2