Finite Group Databasebeta

Subgroups — #19887

Order 256.

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

normal subgroups only

This group has 345 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	Q₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#3334	C₂	1	1	yes	maximal, characteristic
3	#3276	C₂	1	2	yes	maximal, complemented
4	#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	#1689	C₂	1	2	yes	maximal, complemented
6	#1674	C₂	1	1	yes	maximal, characteristic, complemented
7	#1673	C₂	1	1	yes	maximal, characteristic, complemented
8	#1649	C₂	1	2	yes	maximal, complemented
9	#1647	C₂	1	2	yes	maximal, complemented
10	#1589	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137	—	2	2	no	—
13	D₄×Q₈	C₂²	1	2	yes	—
14	#547	C₂²	1	2	yes	—
15	#500	—	2	1	no	—
16	#500	C₂²	1	1	yes	characteristic
17	#500	C₂²	1	1	yes	characteristic
18	#500	C₂²	1	2	yes	—
19	#494	C₂²	1	2	yes	—
20	#480	—	2	2	no	—
21	#474	—	2	2	no	—
22	#474	C₂²	1	2	yes	—
23	#474	C₂²	1	2	yes	—
24	#470	—	2	2	no	—
25	#466	—	2	1	no	—
26	#462	C₂²	1	2	yes	—
27	#461	C₂²	1	2	yes	—
28	#452	—	2	2	no	—
29	#451	—	2	2	no	—
30	C₈×Q₈	C₂²	1	1	yes	characteristic
31	C₄×SD₁₆	C₂²	1	1	yes	characteristic
32	C₄×D₈	—	2	1	no	—
33	C₄×D₈	C₂²	1	1	yes	characteristic
34	#436	C₂²	1	2	yes	—
35	C₈×D₄	C₂²	1	1	yes	characteristic
36	C₂×#107	—	2	1	no	—
37	#335	C₂²	1	1	yes	characteristic
38	#334	C₂²	1	1	yes	characteristic
39	#332	C₂²	1	2	yes	—
40	#332	C₂²	1	2	yes	—
41	Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	D₄⋊C₈	C₂²	1	1	yes	characteristic
43	D₄⋊C₈	C₂²	1	2	yes	—
44	C₈²	C₂²	1	1	yes	characteristic
45	C₂×#41	—	2	2	no	—
46	C₂²×Q₈	—	2	2	no	—
47	#137	—	4	8	no	—
48	C₂×Q₁₆	—	2	2	no	—
49	C₂×Q₁₆	—	2	2	no	—
50	C₂×Q₁₆	—	2	2	no	—
51	C₂×SD₁₆	—	2	2	no	—
52	C₂×SD₁₆	D₄	1	1	yes	characteristic
53	C₂×D₈	D₄	1	1	yes	characteristic
54	C₂×M₁₆	—	2	2	no	—
55	C₂²×C₈	—	2	1	no	—
56	C₄⋊Q₈	—	2	2	no	—
57	C₄⋊Q₈	C₂³	1	2	yes	—
58	C₄⋊Q₈	C₂³	1	2	yes	—
59	#123	—	4	2	no	—
60	C₂²⋊Q₈	—	2	2	no	—
61	C₂²⋊Q₈	—	2	2	no	—
62	C₂²⋊Q₈	—	4	2	no	—
63	#121	—	2	2	no	—
64	C₄×Q₈	C₂³	1	1	yes	characteristic
65	C₄×D₄	—	2	2	no	—
66	C₄×D₄	C₂³	1	1	yes	characteristic
67	C₄×D₄	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄	—	2	2	no	—
69	#107	—	2	1	no	—
70	#107	—	2	1	no	—
71	#107	—	2	1	no	—
72	#107	—	2	1	no	—
73	#107	—	2	2	no	—
74	#107	—	2	2	no	—
75	#107	—	4	2	no	—
76	#107	D₄	1	1	yes	characteristic
77	#106	D₄	1	1	yes	characteristic
78	C₄⋊C₈	—	2	1	no	—
79	C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₄⋊C₈	C₂³	1	2	yes	—
81	C₄⋊C₈	C₂³	1	1	yes	characteristic
82	#103	—	2	2	no	—
83	#103	—	2	2	no	—
84	#103	D₄	1	1	yes	characteristic
85	#102	—	2	2	no	—
86	#102	—	2	2	no	—
87	#102	D₄	1	2	yes	—
88	#102	D₄	1	1	yes	characteristic
89	C₂²⋊C₈	—	2	1	no	—
90	C₂²⋊C₈	—	2	2	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₈	C₂³	1	1	yes	characteristic
94	C₄×C₈	C₂³	1	1	yes	characteristic
95	#41	—	4	4	no	—
96	#41	—	8	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	—	8	2	no	—
102	C₂×Q₈	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄	—	4	2	no	—
104	C₂×D₄	C₂×D₄	1	2	yes	—
105	C₂×D₄	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	—	4	2	no	—
110	Q₁₆	—	4	4	no	—
111	Q₁₆	—	4	4	no	—
112	Q₁₆	—	8	2	no	—
113	SD₁₆	—	4	4	no	—
114	SD₁₆	D₈	1	4	yes	complemented
115	D₈	—	2	2	no	—
116	M₁₆	—	4	4	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄	C₂×D₄	1	2	yes	—
135	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
136	C₂²⋊C₄	—	2	2	no	—
137	C₂²⋊C₄	—	2	1	no	—
138	C₂²⋊C₄	—	4	2	no	—
139	C₂²⋊C₄	—	4	2	no	—
140	C₄²	—	2	1	no	—
141	C₄²	C₂⁴	1	1	yes	characteristic
142	C₂³	—	2	1	no	—
143	C₂³	—	2	2	no	—
144	Q₈	—	2	1	no	—
145	Q₈	—	4	2	no	—
146	Q₈	—	4	2	no	—
147	Q₈	—	4	2	no	—
148	Q₈	—	4	2	no	—
149	Q₈	—	8	4	no	—
150	Q₈	C₂×D₈	1	2	yes	—
151	D₄	—	2	2	no	—
152	D₄	—	2	1	no	—
153	D₄	—	2	2	no	—
154	D₄	—	8	4	no	—
155	D₄	C₂×D₈	1	2	yes	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	—	8	4	no	—
168	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
171	C₈	—	2	2	no	—
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	4	2	no	—
175	C₈	—	4	1	no	—
176	C₈	—	4	1	no	—
177	C₈	C₂×D₈	1	2	yes	—
178	C₈	#137	1	2	yes	complemented
179	C₂²	—	2	2	no	—
180	C₂²	—	4	1	no	—
181	C₂²	—	4	1	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	D₄²	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	1	no	—
191	C₄	—	8	2	no	—
192	C₄	—	8	2	no	—
193	C₄	C₂²×D₈	1	1	yes	characteristic
194	C₄	C₂×#137	1	1	yes	characteristic
195	C₄	#576	1	1	yes	characteristic
196	C₄	#578	1	1	yes	characteristic
197	C₂	—	4	2	no	—
198	C₂	—	8	2	no	—
199	C₂	#3276	1	1	yes	characteristic
200	C₂	D₄×D₈	1	1	yes	characteristic
201	C₂	#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3334	C₂
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. Q₈×D₈	C₂	semidirect
3. #3276	C₂	semidirect
4. #1713	C₂	semidirect
5. #1689	C₂	semidirect
6. #1674	C₂	semidirect
7. #1673	C₂	semidirect
8. #1649	C₂	semidirect
9. #1647	C₂	semidirect
10. #1589	C₂	semidirect
11. C₈×SD₁₆	C₂	semidirect
114. SD₁₆	D₈	semidirect
178. C₈	#137	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₈	2
#3334	2
#3276	2
#1713	2
#1689	2
#1674	2
#1673	2
#1649	2
#1647	2
#1589	2
C₈×SD₁₆	2