Finite Group Databasebeta

Subgroups — #19965

Order 256.

295 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 101 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	#3400	C₂	1	2	yes	maximal, complemented
2	#3365	C₂	1	2	yes	maximal, complemented
3	#1706	C₂	1	1	yes	maximal, characteristic
4	#1686	C₂	1	2	yes	maximal, complemented
5	#1682	C₂	1	2	yes	maximal, complemented
6	#1648	C₂	1	2	yes	maximal, complemented
7	#1620	C₂	1	2	yes	maximal
8	#1591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#559	C₂²	1	2	yes	—
11	#557	C₂²	1	2	yes	—
12	#500	C₂²	1	1	yes	characteristic
13	#497	—	2	2	no	—
14	#493	C₂²	1	1	yes	characteristic
15	#492	C₂²	1	2	yes	—
16	#488	—	2	2	no	—
17	#486	C₂²	1	2	yes	—
18	#478	—	2	2	no	—
19	#475	C₂²	1	2	yes	—
20	#464	—	2	2	no	—
21	#464	—	2	2	no	—
22	#462	C₂²	1	2	yes	—
23	#461	—	2	2	no	—
24	#460	C₂²	1	2	yes	—
25	#460	C₂²	1	2	yes	—
26	#446	C₂²	1	2	yes	—
27	C₈×Q₈	C₂²	1	2	yes	—
28	#440	—	2	2	no	—
29	C₄×Q₁₆	C₂²	1	1	yes	characteristic
30	C₄×Q₁₆	C₂²	1	1	yes	characteristic
31	C₄×SD₁₆	—	2	2	no	—
32	#335	C₂²	1	1	yes	characteristic
33	#335	C₂²	1	1	yes	characteristic
34	#332	C₂²	1	2	yes	—
35	#331	C₂²	1	2	yes	—
36	Q₈⋊C₈	C₂²	1	2	yes	—
37	Q₈⋊C₈	C₂²	1	2	yes	—
38	C₈²	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆	—	2	2	no	—
40	C₂×Q₁₆	D₄	1	1	yes	characteristic
41	C₂×Q₁₆	D₄	1	1	yes	characteristic
42	C₂×SD₁₆	—	2	2	no	—
43	C₂×SD₁₆	—	2	2	no	—
44	C₂×SD₁₆	—	2	2	no	—
45	C₂×SD₁₆	—	2	2	no	—
46	C₂×D₈	—	4	1	no	—
47	C₄⋊Q₈	—	2	2	no	—
48	C₄⋊Q₈	C₂³	1	2	yes	—
49	#127	C₂³	1	2	yes	—
50	#125	—	2	2	no	—
51	#125	—	4	2	no	—
52	#124	—	2	2	no	—
53	#124	—	2	2	no	—
54	#123	—	4	2	no	—
55	C₄×Q₈	—	2	2	no	—
56	C₄×Q₈	—	2	2	no	—
57	C₄×Q₈	C₂³	1	2	yes	—
58	C₄×Q₈	C₂³	1	2	yes	—
59	C₄×D₄	—	2	2	no	—
60	C₄×D₄	—	2	2	no	—
61	#107	—	4	1	no	—
62	#107	D₄	1	1	yes	characteristic
63	#107	D₄	1	1	yes	characteristic
64	#106	—	2	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	C₂³	1	2	yes	—
68	C₄⋊C₈	C₂³	1	2	yes	—
69	#103	—	2	2	no	—
70	#103	—	2	2	no	—
71	#103	D₄	1	2	yes	—
72	#103	D₄	1	2	yes	—
73	#102	—	2	2	no	—
74	#102	—	2	2	no	—
75	#102	—	2	2	no	—
76	#102	—	2	2	no	—
77	#97	—	2	2	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	C₂×D₄	1	2	yes	—
84	C₂×Q₈	C₂×D₄	1	2	yes	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂²×C₄	—	4	2	no	—
89	Q₁₆	—	2	2	no	—
90	Q₁₆	—	4	1	no	—
91	Q₁₆	—	8	2	no	—
92	SD₁₆	—	4	4	no	—
93	SD₁₆	—	4	4	no	—
94	SD₁₆	—	8	2	no	—
95	SD₁₆	—	8	2	no	—
96	D₈	—	8	2	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	C₂×D₄	1	2	yes	—
112	C₄⋊C₄	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄	—	4	2	no	—
114	C₂²⋊C₄	—	4	2	no	—
115	C₂²⋊C₄	—	4	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	4	2	no	—
119	C₄²	C₂⁴	1	1	yes	characteristic
120	C₂³	—	4	2	no	—
121	Q₈	—	2	2	no	—
122	Q₈	—	2	2	no	—
123	Q₈	—	2	2	no	—
124	Q₈	—	2	2	no	—
125	Q₈	—	4	2	no	—
126	Q₈	—	4	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	2	no	—
131	D₄	—	8	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	16	2	no	—
138	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	4	2	no	—
146	C₈	—	4	2	no	—
147	C₈	#135	1	2	yes	—
148	C₂²	—	8	2	no	—
149	C₂²	—	8	2	no	—
150	C₂²	—	8	2	no	—
151	C₂²	D₄²	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	4	2	no	—
154	C₄	—	4	2	no	—
155	C₄	—	4	2	no	—
156	C₄	—	4	2	no	—
157	C₄	—	8	2	no	—
158	C₄	C₂×#135	1	1	yes	characteristic
159	C₄	#575	1	1	yes	characteristic
160	C₄	#576	1	1	yes	characteristic
161	C₄	#578	1	1	yes	characteristic
162	C₂	—	16	2	no	—
163	C₂	#3277	1	1	yes	characteristic
164	C₂	#3287	1	1	yes	characteristic
165	C₂	#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#493	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3400	C₂	semidirect
2. #3365	C₂	semidirect
4. #1686	C₂	semidirect
5. #1682	C₂	semidirect
6. #1648	C₂	semidirect
8. #1591	C₂	semidirect
9. C₈×Q₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3400	2
#3365	2
#1706	2
#1686	2
#1682	2
#1648	2
#1620	2
#1591	2
C₈×Q₁₆	2