Finite Group Databasebeta

Subgroups — #20008

Order 256.

277 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 111 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	#3383	C₂	1	1	yes	maximal, characteristic
2	Q₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	#3363	C₂	1	2	yes	maximal
4	#1710	C₂	1	1	yes	maximal, characteristic
5	#1684	C₂	1	2	yes	maximal
6	#1676	C₂	1	1	yes	maximal, characteristic
7	#1674	C₂	1	1	yes	maximal, characteristic
8	#1645	C₂	1	2	yes	maximal
9	#1636	C₂	1	2	yes	maximal
10	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
12	Q₈²	C₂²	1	2	yes	—
13	#557	C₂²	1	2	yes	—
14	#500	—	2	1	no	—
15	#500	C₂²	1	1	yes	characteristic
16	#500	C₂²	1	1	yes	characteristic
17	#499	—	2	1	no	—
18	#487	—	2	2	no	—
19	#487	C₂²	1	2	yes	—
20	#487	C₂²	1	2	yes	—
21	#479	—	2	2	no	—
22	#477	—	2	2	no	—
23	#477	—	2	2	no	—
24	#477	C₂²	1	2	yes	—
25	#477	C₂²	1	2	yes	—
26	#475	C₂²	1	2	yes	—
27	#462	—	2	2	no	—
28	#462	C₂²	1	2	yes	—
29	C₈×Q₈	C₂²	1	2	yes	—
30	C₈×Q₈	C₂²	1	1	yes	characteristic
31	C₈×Q₈	C₂²	1	1	yes	characteristic
32	C₄×Q₁₆	—	2	1	no	—
33	C₄×Q₁₆	—	2	1	no	—
34	C₄×Q₁₆	—	2	2	no	—
35	C₄×Q₁₆	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆	C₂²	1	1	yes	characteristic
37	#335	C₂²	1	1	yes	characteristic
38	#335	C₂²	1	1	yes	characteristic
39	#333	C₂²	1	2	yes	—
40	#332	C₂²	1	2	yes	—
41	Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈	C₂²	1	2	yes	—
43	Q₈⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×Q₁₆	D₄	1	1	yes	characteristic
47	C₂×Q₁₆	D₄	1	1	yes	characteristic
48	C₄⋊Q₈	—	2	2	no	—
49	C₄⋊Q₈	—	2	2	no	—
50	C₄⋊Q₈	—	2	2	no	—
51	C₄⋊Q₈	—	4	2	no	—
52	C₄⋊Q₈	C₂³	1	2	yes	—
53	C₄⋊Q₈	C₂³	1	2	yes	—
54	#125	—	2	2	no	—
55	#125	—	4	2	no	—
56	C₄×Q₈	—	2	2	no	—
57	C₄×Q₈	—	2	2	no	—
58	C₄×Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	C₄×Q₈	C₂³	1	1	yes	characteristic
61	C₄×Q₈	C₂³	1	2	yes	—
62	C₄×Q₈	C₂³	1	1	yes	characteristic
63	#107	—	2	1	no	—
64	#107	—	2	1	no	—
65	#107	—	2	1	no	—
66	#107	—	2	1	no	—
67	#107	—	2	2	no	—
68	#107	—	2	2	no	—
69	#107	D₄	1	1	yes	characteristic
70	#107	D₄	1	1	yes	characteristic
71	#106	—	2	2	no	—
72	C₄⋊C₈	—	2	2	no	—
73	C₄⋊C₈	—	2	1	no	—
74	C₄⋊C₈	—	2	1	no	—
75	C₄⋊C₈	C₂³	1	1	yes	characteristic
76	C₄⋊C₈	C₂³	1	2	yes	—
77	C₄⋊C₈	C₂³	1	1	yes	characteristic
78	#103	—	2	2	no	—
79	#103	—	2	2	no	—
80	#103	—	2	2	no	—
81	#103	—	2	2	no	—
82	#103	—	2	2	no	—
83	#103	—	2	2	no	—
84	#103	—	4	2	no	—
85	#103	D₄	1	1	yes	characteristic
86	#103	D₄	1	2	yes	—
87	#103	D₄	1	1	yes	characteristic
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₂³	1	1	yes	characteristic
92	C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₈	C₂³	1	1	yes	characteristic
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	4	2	no	—
97	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
98	C₂×Q₈	C₂×D₄	1	2	yes	—
99	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
100	Q₁₆	—	2	2	no	—
101	Q₁₆	—	4	4	no	—
102	Q₁₆	Q₁₆	1	4	yes	complemented
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	4	2	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄	C₂×D₄	1	2	yes	—
125	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
126	C₄²	—	2	1	no	—
127	C₄²	—	2	2	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	4	2	no	—
130	C₄²	—	4	2	no	—
131	C₄²	C₂⁴	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	2	1	no	—
135	Q₈	—	2	1	no	—
136	Q₈	—	4	2	no	—
137	Q₈	—	4	2	no	—
138	Q₈	—	8	2	no	—
139	Q₈	—	16	2	no	—
140	Q₈	C₂×Q₁₆	1	4	yes	—
141	Q₈	#135	1	2	yes	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	4	1	no	—
157	C₈	—	4	2	no	—
158	C₈	—	4	1	no	—
159	C₈	—	4	1	no	—
160	C₈	C₂×Q₁₆	1	2	yes	—
161	C₂²	D₄²	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	4	no	—
164	C₄	—	2	1	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	4	2	no	—
168	C₄	—	4	1	no	—
169	C₄	—	8	2	no	—
170	C₄	—	8	2	no	—
171	C₄	C₂²×Q₁₆	1	1	yes	characteristic
172	C₄	C₂×#135	1	1	yes	characteristic
173	C₄	#578	1	1	yes	characteristic
174	C₄	#578	1	1	yes	characteristic
175	C₂	#3285	1	1	yes	characteristic
176	C₂	D₄×Q₁₆	1	1	yes	characteristic
177	C₂	#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
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
102. Q₁₆	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#3383	2
Q₈×Q₁₆	2
#3363	2
#1710	2
#1684	2
#1676	2
#1674	2
#1645	2
#1636	2
C₈×Q₁₆	2