Finite Group Databasebeta

Subgroups — #22672

Order 256.

525 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 207 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	C₂×#442	C₂	1	2	yes	maximal
2	C₂×C₄×D₈	C₂	1	1	yes	maximal, characteristic
3	#2349	C₂	1	2	yes	maximal, complemented
4	#1943	C₂	1	2	yes	maximal, complemented
5	#1862	C₂	1	4	yes	maximal, complemented
6	#1846	C₂	1	1	yes	maximal, characteristic, complemented
7	#1804	C₂	1	2	yes	maximal, complemented
8	#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄	C₂²	1	4	yes	—
11	C₂×C₄×D₄	C₂²	1	2	yes	—
12	#442	—	2	8	no	—
13	C₄×D₈	C₄	1	8	yes	complemented
14	C₂×#107	C₂²	1	2	yes	—
15	C₂×#107	C₂²	1	1	yes	characteristic
16	C₂×#106	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×#102	C₂²	1	4	yes	—
19	C₂×#102	C₂²	1	2	yes	—
20	C₂×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	#389	C₂²	1	4	yes	—
23	#385	—	2	4	no	—
24	#382	C₂²	1	2	yes	—
25	#380	—	2	2	no	—
26	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×#95	—	2	2	no	—
28	#340	C₂²	1	4	yes	—
29	#336	C₂²	1	2	yes	—
30	C₂²×D₄	C₂³	1	2	yes	—
31	C₂³×C₄	—	2	2	no	—
32	C₂³×C₄	—	2	4	no	—
33	C₂×D₈	C₂×C₄	1	4	yes	—
34	C₂×D₈	D₄	1	4	yes	—
35	C₂×D₈	Q₈	1	4	yes	—
36	C₂²×C₈	C₂³	1	2	yes	—
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₈	C₂³	1	1	yes	characteristic
39	C₄×D₄	—	2	8	no	—
40	C₄×D₄	—	2	4	no	—
41	C₄×D₄	—	4	4	no	—
42	C₄×D₄	C₂×C₄	1	8	yes	complemented
43	C₂×C₄⋊C₄	—	2	4	no	—
44	C₂×C₄⋊C₄	D₄	1	2	yes	—
45	C₂×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₂²⋊C₄	—	2	2	no	—
48	C₂×C₂²⋊C₄	—	2	4	no	—
49	C₂×C₄²	C₂³	1	2	yes	—
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	#107	—	4	2	no	—
52	#107	C₂×C₄	1	4	yes	complemented
53	#106	—	2	4	no	—
54	C₄⋊C₈	—	2	2	no	—
55	#102	—	2	8	no	—
56	#102	C₂×C₄	1	8	yes	complemented
57	#97	—	2	4	no	—
58	C₄×C₈	C₂×C₄	1	4	yes	complemented
59	#95	—	2	2	no	—
60	#95	—	2	4	no	—
61	#95	—	2	2	no	—
62	#95	D₄	1	2	yes	—
63	C₂⁴	—	2	2	no	—
64	C₂×D₄	—	2	2	no	—
65	C₂×D₄	—	2	2	no	—
66	C₂×D₄	—	2	2	no	—
67	C₂×D₄	C₂²×C₄	1	4	yes	—
68	C₂×D₄	C₂×D₄	1	4	yes	—
69	C₂×D₄	C₂×Q₈	1	4	yes	—
70	C₂²×C₄	—	2	2	no	—
71	C₂²×C₄	—	2	8	no	—
72	C₂²×C₄	—	2	4	no	—
73	C₂²×C₄	—	4	8	no	—
74	C₂²×C₄	—	4	8	no	—
75	C₂²×C₄	—	8	4	no	—
76	C₂²×C₄	—	8	4	no	—
77	C₂²×C₄	C₂×D₄	1	4	yes	—
78	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄	C₂⁴	1	1	yes	characteristic
81	D₈	C₄⋊C₄	1	16	yes	complemented
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	C₂²×C₄	1	2	yes	—
88	C₂×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₈	C₂×Q₈	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	8	4	no	—
96	C₄⋊C₄	C₂²×C₄	1	4	yes	—
97	C₂²⋊C₄	—	2	8	no	—
98	C₂²⋊C₄	—	4	8	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	C₂²×C₄	1	2	yes	—
102	C₄²	#41	1	2	yes	—
103	C₂³	—	2	4	no	—
104	C₂³	—	2	4	no	—
105	C₂³	—	2	4	no	—
106	C₂³	—	4	4	no	—
107	C₂³	—	4	4	no	—
108	C₂³	C₂²×D₄	1	1	yes	characteristic
109	D₄	—	2	4	no	—
110	D₄	—	2	4	no	—
111	D₄	—	2	4	no	—
112	D₄	C₂×C₄⋊C₄	1	8	yes	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	4	4	no	—
121	C₂×C₄	—	4	4	no	—
122	C₂×C₄	—	4	4	no	—
123	C₂×C₄	—	8	8	no	—
124	C₂×C₄	—	8	16	no	—
125	C₂×C₄	C₄×D₄	1	2	yes	—
126	C₂×C₄	C₄×D₄	1	2	yes	—
127	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄	C₂×#41	1	1	yes	characteristic
131	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	2	no	—
135	C₈	C₂×C₄⋊C₄	1	4	yes	—
136	C₂²	—	2	8	no	—
137	C₂²	—	4	8	no	—
138	C₂²	—	4	4	no	—
139	C₂²	—	4	4	no	—
140	C₂²	—	4	8	no	—
141	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²	D₄²	1	1	yes	characteristic
143	C₂²	D₄×Q₈	1	1	yes	characteristic
144	C₂²	#575	1	2	yes	—
145	C₂²	#576	1	2	yes	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	4	4	no	—
149	C₄	—	4	4	no	—
150	C₄	—	8	4	no	—
151	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄	#520	1	1	yes	characteristic
153	C₄	#547	1	1	yes	characteristic
154	C₄	#554	1	1	yes	characteristic
155	C₂	—	4	8	no	—
156	C₂	D₄×C₄⋊C₄	1	1	yes	characteristic
157	C₂	#2943	1	1	yes	characteristic
158	C₂	#2948	1	1	yes	characteristic
159	C₂	#3273	1	2	yes	—
160	C₂	#3389	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂²×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #2349	C₂	semidirect
4. #1943	C₂	semidirect
5. #1862	C₂	semidirect
6. #1846	C₂	semidirect
7. #1804	C₂	semidirect
8. #1770	C₂	semidirect
13. C₄×D₈	C₄	semidirect
42. C₄×D₄	C₂×C₄	semidirect
52. #107	C₂×C₄	semidirect
56. #102	C₂×C₄	semidirect
58. C₄×C₈	C₂×C₄	semidirect
81. D₈	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#442	2
C₂×C₄×D₈	2
#2349	2
#1943	2
#1862	2
#1846	2
#1804	2
#1770	2