Finite Group Databasebeta

Subgroups — #19955

Order 256.

318 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 103 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	#3393	C₂	1	1	yes	maximal, characteristic, complemented
2	#3353	C₂	1	2	yes	maximal, complemented
3	#3283	C₂	1	1	yes	maximal, characteristic
4	#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	#1693	C₂	1	2	yes	maximal, complemented
6	#1670	C₂	1	1	yes	maximal, characteristic, complemented
7	#1669	C₂	1	1	yes	maximal, characteristic, complemented
8	#1650	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×#135	—	2	1	no	—
13	#556	C₂²	1	2	yes	—
14	#547	C₂²	1	2	yes	—
15	#500	C₂²	1	2	yes	—
16	#500	C₂²	1	2	yes	—
17	#495	—	2	1	no	—
18	#494	C₂²	1	1	yes	characteristic
19	#494	C₂²	1	1	yes	characteristic
20	#484	—	2	2	no	—
21	#482	—	2	2	no	—
22	#475	C₂²	1	2	yes	—
23	#474	C₂²	1	2	yes	—
24	#468	—	2	2	no	—
25	#467	—	2	1	no	—
26	#464	—	2	2	no	—
27	#461	C₂²	1	2	yes	—
28	#461	C₂²	1	2	yes	—
29	#452	—	2	2	no	—
30	C₈×Q₈	C₂²	1	1	yes	characteristic
31	C₄×Q₁₆	—	2	1	no	—
32	C₄×SD₁₆	C₂²	1	1	yes	characteristic
33	C₄×D₈	C₂²	1	1	yes	characteristic
34	#436	C₂²	1	2	yes	—
35	C₈×D₄	C₂²	1	1	yes	characteristic
36	#428	—	2	2	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	#135	—	4	4	no	—
47	C₂×Q₁₆	—	2	1	no	—
48	C₂×Q₁₆	—	2	1	no	—
49	C₂×Q₁₆	—	2	1	no	—
50	C₂×Q₁₆	—	2	1	no	—
51	C₂×SD₁₆	—	2	2	no	—
52	C₂×SD₁₆	—	2	2	no	—
53	C₂×SD₁₆	D₄	1	1	yes	characteristic
54	C₂×D₈	D₄	1	1	yes	characteristic
55	C₂×M₁₆	—	2	2	no	—
56	C₂²×C₈	—	2	1	no	—
57	C₄⋊Q₈	C₂³	1	2	yes	—
58	C₄⋊Q₈	C₂³	1	2	yes	—
59	#126	—	4	2	no	—
60	#124	—	2	2	no	—
61	#123	—	4	2	no	—
62	C₂²⋊Q₈	—	2	2	no	—
63	C₂²⋊Q₈	—	2	2	no	—
64	#121	—	2	2	no	—
65	C₄×Q₈	—	2	2	no	—
66	C₄×Q₈	C₂³	1	1	yes	characteristic
67	C₄×D₄	C₂³	1	1	yes	characteristic
68	C₄×D₄	C₂³	1	2	yes	—
69	#117	—	2	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	#107	—	2	2	no	—
72	#107	—	2	2	no	—
73	#107	—	2	2	no	—
74	#107	—	4	2	no	—
75	#107	D₄	1	1	yes	characteristic
76	#106	—	2	2	no	—
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	—	2	2	no	—
85	#103	D₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
100	C₂×D₄	—	4	2	no	—
101	C₂×D₄	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄	C₂×D₄	1	2	yes	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	4	2	no	—
106	Q₁₆	—	4	2	no	—
107	Q₁₆	—	4	2	no	—
108	Q₁₆	—	8	1	no	—
109	Q₁₆	—	8	1	no	—
110	SD₁₆	—	2	2	no	—
111	SD₁₆	—	4	4	no	—
112	SD₁₆	—	8	2	no	—
113	D₈	—	2	2	no	—
114	M₁₆	—	4	4	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄	—	2	1	no	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	C₂×D₄	1	2	yes	—
132	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
134	C₂²⋊C₄	—	2	1	no	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₂²⋊C₄	—	4	2	no	—
137	C₂²⋊C₄	—	4	2	no	—
138	C₂²⋊C₄	—	4	2	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	4	2	no	—
141	C₄²	C₂⁴	1	1	yes	characteristic
142	C₂³	—	2	2	no	—
143	C₂³	—	2	1	no	—
144	Q₈	—	2	1	no	—
145	Q₈	—	2	1	no	—
146	Q₈	—	4	2	no	—
147	Q₈	—	4	2	no	—
148	Q₈	—	4	2	no	—
149	Q₈	—	4	2	no	—
150	D₄	—	2	1	no	—
151	D₄	—	2	2	no	—
152	D₄	—	2	2	no	—
153	D₄	—	2	1	no	—
154	D₄	—	8	4	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	8	4	no	—
166	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	1	no	—
171	C₈	—	2	1	no	—
172	C₈	—	4	2	no	—
173	C₈	—	4	1	no	—
174	C₈	—	4	1	no	—
175	C₈	#135	1	2	yes	complemented
176	C₈	AGL(1,ℤ/8)	1	2	yes	—
177	C₂²	—	4	1	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	1	no	—
181	C₂²	—	4	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	D₄²	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	2	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	8	2	no	—
190	C₄	—	8	2	no	—
191	C₄	C₂×#135	1	1	yes	characteristic
192	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
193	C₄	#576	1	1	yes	characteristic
194	C₄	#578	1	1	yes	characteristic
195	C₂	—	8	1	no	—
196	C₂	—	8	2	no	—
197	C₂	#3274	1	1	yes	characteristic
198	C₂	#3283	1	1	yes	characteristic
199	C₂	#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3283	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. #3393	C₂	semidirect
2. #3353	C₂	semidirect
4. #1713	C₂	semidirect
5. #1693	C₂	semidirect
6. #1670	C₂	semidirect
7. #1669	C₂	semidirect
8. #1650	C₂	semidirect
9. #1647	C₂	semidirect
10. #1589	C₂	semidirect
11. C₈×SD₁₆	C₂	semidirect
175. C₈	#135	semidirect

Maximal subgroups

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

subgroup	index
#3393	2
#3353	2
#3283	2
#1713	2
#1693	2
#1670	2
#1669	2
#1650	2
#1647	2
#1589	2
C₈×SD₁₆	2