Finite Group Databasebeta

Subgroups — #22692

Order 256.

431 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 199 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×#443	C₂	1	1	yes	maximal, characteristic
2	#2951	C₂	1	2	yes	maximal
3	#2913	C₂	1	2	yes	maximal, complemented
4	#1952	C₂	1	2	yes	maximal, complemented
5	#1859	C₂	1	4	yes	maximal, complemented
6	#1845	C₂	1	1	yes	maximal, characteristic, complemented
7	#1816	C₂	1	2	yes	maximal, complemented
8	#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135	C₂²	1	1	yes	characteristic
10	C₂×#131	C₂²	1	2	yes	—
11	#520	C₂²	1	4	yes	—
12	#443	C₄	1	8	yes	complemented
13	#440	—	2	4	no	—
14	C₄×Q₁₆	—	2	2	no	—
15	C₄×D₈	—	2	2	no	—
16	C₂×#108	C₂²	1	1	yes	characteristic
17	C₂×#108	C₂²	1	1	yes	characteristic
18	#428	—	2	4	no	—
19	#428	C₂²	1	4	yes	—
20	#427	—	2	2	no	—
21	#427	C₂²	1	1	yes	characteristic
22	C₂×#107	—	2	2	no	—
23	C₂×#107	C₂²	1	2	yes	—
24	C₂×#107	C₂²	1	2	yes	—
25	C₂×#104	C₂²	1	2	yes	—
26	#417	C₂²	1	4	yes	—
27	#405	C₂²	1	2	yes	—
28	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
29	#384	C₂²	1	2	yes	—
30	#343	C₂²	1	4	yes	—
31	#339	C₂²	1	2	yes	—
32	C₂×#41	C₂³	1	2	yes	—
33	#135	C₂×C₄	1	4	yes	—
34	#135	Q₈	1	4	yes	—
35	C₂×Q₁₆	D₄	1	1	yes	characteristic
36	C₂×SD₁₆	D₄	1	2	yes	—
37	C₂×D₈	D₄	1	1	yes	characteristic
38	#131	—	2	4	no	—
39	#131	C₂×C₄	1	8	yes	complemented
40	C₂×M₁₆	—	2	2	no	—
41	C₂×M₁₆	C₂³	1	2	yes	—
42	C₂×M₁₆	C₂³	1	2	yes	—
43	C₂²×C₈	—	2	2	no	—
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₄×Q₈	—	2	4	no	—
47	C₄×D₄	—	2	4	no	—
48	C₄×D₄	—	4	4	no	—
49	#117	—	2	4	no	—
50	#117	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄	—	2	4	no	—
52	C₂×C₄⋊C₄	—	4	2	no	—
53	C₂×C₄⋊C₄	C₂³	1	4	yes	—
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	#108	—	2	2	no	—
56	#108	C₂×C₄	1	4	yes	complemented
57	#107	—	2	2	no	—
58	#107	—	2	2	no	—
59	#107	—	2	2	no	—
60	#107	—	2	2	no	—
61	#107	—	2	4	no	—
62	#107	D₄	1	1	yes	characteristic
63	#107	D₄	1	1	yes	characteristic
64	#106	—	2	4	no	—
65	#106	D₄	1	2	yes	—
66	#104	C₂×C₄	1	8	yes	complemented
67	#103	—	2	4	no	—
68	#102	—	2	4	no	—
69	#97	—	2	2	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	C₂×C₄	1	4	yes	complemented
72	#41	—	2	2	no	—
73	#41	—	2	2	no	—
74	#41	C₂²×C₄	1	4	yes	—
75	#41	C₂×Q₈	1	4	yes	—
76	C₂×Q₈	C₂×D₄	1	2	yes	—
77	C₂×D₄	—	2	2	no	—
78	C₂×D₄	C₂×D₄	1	2	yes	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	2	4	no	—
82	C₂²×C₄	—	4	4	no	—
83	C₂²×C₄	C₂⁴	1	1	yes	characteristic
84	Q₁₆	C₄⋊C₄	1	4	yes	complemented
85	SD₁₆	C₄⋊C₄	1	8	yes	complemented
86	D₈	C₄⋊C₄	1	4	yes	complemented
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	8	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	2	2	no	—
91	M₁₆	C₂²×C₄	1	4	yes	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	8	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈	C₂×Q₈	1	2	yes	—
102	C₂×C₈	#41	1	2	yes	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	4	4	no	—
106	C₄⋊C₄	—	8	4	no	—
107	C₄⋊C₄	C₂×D₄	1	2	yes	—
108	C₄⋊C₄	C₂×D₄	1	2	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	4	4	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	4	4	no	—
114	C₄²	C₂²×C₄	1	2	yes	—
115	C₂³	—	2	2	no	—
116	C₂³	H(5,𝔽₂)	1	1	yes	characteristic
117	Q₈	—	2	2	no	—
118	Q₈	C₂×C₄⋊C₄	1	4	yes	—
119	D₄	—	2	4	no	—
120	D₄	—	2	4	no	—
121	D₄	—	2	2	no	—
122	D₄	C₂×C₄⋊C₄	1	4	yes	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	8	4	no	—
133	C₂×C₄	—	16	4	no	—
134	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄	C₂×#41	1	1	yes	characteristic
139	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
140	C₈	—	2	4	no	—
141	C₈	—	4	2	no	—
142	C₈	C₂×C₄⋊C₄	1	2	yes	—
143	C₈	C₂×C₄⋊C₄	1	2	yes	—
144	C₈	C₄×D₄	1	2	yes	—
145	C₈	C₄×D₄	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	4	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	#520	1	1	yes	characteristic
152	C₂²	D₄²	1	1	yes	characteristic
153	C₂²	#554	1	1	yes	characteristic
154	C₄	—	2	4	no	—
155	C₄	—	2	4	no	—
156	C₄	—	4	2	no	—
157	C₄	—	4	2	no	—
158	C₄	—	8	4	no	—
159	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₄	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄	#547	1	1	yes	characteristic
162	C₄	D₄×Q₈	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	4	4	no	—
166	C₂	D₄×C₄⋊C₄	1	1	yes	characteristic
167	C₂	#3292	1	1	yes	characteristic
168	C₂	#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×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. #2913	C₂	semidirect
4. #1952	C₂	semidirect
5. #1859	C₂	semidirect
6. #1845	C₂	semidirect
7. #1816	C₂	semidirect
8. #1772	C₂	semidirect
12. #443	C₄	semidirect
39. #131	C₂×C₄	semidirect
56. #108	C₂×C₄	semidirect
66. #104	C₂×C₄	semidirect
71. C₄×C₈	C₂×C₄	semidirect
84. Q₁₆	C₄⋊C₄	semidirect
85. SD₁₆	C₄⋊C₄	semidirect
86. 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×#443	2
#2951	2
#2913	2
#1952	2
#1859	2
#1845	2
#1816	2
#1772	2