Finite Group Databasebeta

Subgroups — #19645

Order 256.

386 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 123 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	#3228	C₂	1	2	yes	maximal, complemented
2	#3182	C₂	1	2	yes	maximal, complemented
3	#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	#3147	C₂	1	1	yes	maximal, characteristic
5	#1719	C₂	1	2	yes	maximal, complemented
6	#1714	C₂	1	2	yes	maximal, complemented
7	#1713	C₂	1	1	yes	maximal, characteristic, complemented
8	#1709	C₂	1	1	yes	maximal, characteristic, complemented
9	#1707	C₂	1	1	yes	maximal, characteristic, complemented
10	#1706	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	—	2	2	no	—
13	#533	C₂²	1	2	yes	—
14	#532	C₂²	1	2	yes	—
15	#500	C₂²	1	1	yes	characteristic
16	#500	C₂²	1	2	yes	—
17	#500	C₂²	1	2	yes	—
18	#498	C₂²	1	2	yes	—
19	#495	—	2	2	no	—
20	#494	C₂²	1	1	yes	characteristic
21	#494	C₂²	1	1	yes	characteristic
22	#493	C₂²	1	1	yes	characteristic
23	#492	C₂²	1	2	yes	—
24	#492	C₂²	1	2	yes	—
25	#487	C₂²	1	2	yes	—
26	#487	C₂²	1	2	yes	—
27	#486	C₂²	1	2	yes	—
28	#486	C₂²	1	2	yes	—
29	#486	C₂²	1	2	yes	—
30	#482	—	2	4	no	—
31	#478	—	2	4	no	—
32	#468	—	2	4	no	—
33	#467	—	2	2	no	—
34	#440	—	2	4	no	—
35	C₄×Q₁₆	—	2	2	no	—
36	#431	C₂²	1	1	yes	characteristic
37	C₄×M₁₆	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
39	#322	C₂²	1	1	yes	characteristic
40	#322	C₂²	1	1	yes	characteristic
41	C₈²	C₂²	1	1	yes	characteristic
42	C₈²	C₂²	1	1	yes	characteristic
43	C₂×#41	—	2	2	no	—
44	#135	—	4	8	no	—
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×Q₁₆	—	2	2	no	—
47	C₂×SD₁₆	—	2	4	no	—
48	C₂×SD₁₆	—	2	4	no	—
49	C₂×D₈	—	2	2	no	—
50	C₂×M₁₆	D₄	1	2	yes	—
51	C₂²×C₈	D₄	1	2	yes	—
52	C₄⋊Q₈	C₂³	1	2	yes	—
53	C₄⋊Q₈	C₂³	1	2	yes	—
54	C₄⋊Q₈	C₂³	1	2	yes	—
55	#127	C₂³	1	2	yes	—
56	#125	—	2	2	no	—
57	#124	—	2	2	no	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₂²⋊Q₈	—	2	2	no	—
60	#121	—	2	2	no	—
61	#121	—	2	2	no	—
62	C₄×Q₈	—	2	2	no	—
63	C₄×Q₈	—	2	2	no	—
64	C₄×D₄	—	2	2	no	—
65	C₄×D₄	—	2	2	no	—
66	#117	—	2	2	no	—
67	C₂×C₄²	C₂³	1	1	yes	characteristic
68	#107	—	2	2	no	—
69	#107	—	2	4	no	—
70	#107	—	2	4	no	—
71	#106	—	2	4	no	—
72	C₄⋊C₈	D₄	1	2	yes	—
73	#103	—	2	4	no	—
74	#103	—	2	4	no	—
75	#102	—	2	4	no	—
76	#102	—	2	4	no	—
77	#102	—	2	4	no	—
78	C₂²⋊C₈	D₄	1	2	yes	—
79	#97	D₄	1	2	yes	—
80	C₄×C₈	D₄	1	2	yes	—
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₈	C₂³	1	1	yes	characteristic
83	C₄×C₈	C₂³	1	1	yes	characteristic
84	C₄×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₈	C₂³	1	1	yes	characteristic
86	C₄×C₈	C₂³	1	1	yes	characteristic
87	#41	—	4	8	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	4	2	no	—
95	C₂²×C₄	—	4	2	no	—
96	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
99	Q₁₆	—	4	4	no	—
100	Q₁₆	—	4	4	no	—
101	SD₁₆	—	4	8	no	—
102	SD₁₆	—	4	8	no	—
103	D₈	—	4	4	no	—
104	M₁₆	—	2	4	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	C₂×D₄	1	2	yes	—
107	C₂×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₈	C₂×D₄	1	2	yes	—
110	C₂×C₈	C₂×D₄	1	2	yes	—
111	C₂×C₈	C₂×D₄	1	2	yes	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	2	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	C₂²⋊C₄	—	4	2	no	—
121	C₂²⋊C₄	—	4	2	no	—
122	C₂²⋊C₄	—	4	2	no	—
123	C₄²	—	4	2	no	—
124	C₄²	C₂×D₄	1	1	yes	characteristic
125	C₄²	C₂×D₄	1	1	yes	characteristic
126	C₄²	C₂×D₄	1	1	yes	characteristic
127	C₄²	C₂⁴	1	1	yes	characteristic
128	C₂³	—	4	2	no	—
129	C₂³	#127	1	1	yes	characteristic
130	Q₈	—	4	4	no	—
131	Q₈	—	4	4	no	—
132	Q₈	—	4	4	no	—
133	D₄	—	4	4	no	—
134	D₄	—	4	4	no	—
135	D₄	—	4	4	no	—
136	D₄	—	8	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	#127	1	1	yes	characteristic
145	C₂×C₄	#127	1	1	yes	characteristic
146	C₂×C₄	#127	1	1	yes	characteristic
147	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	2	2	no	—
154	C₈	#135	1	4	yes	complemented
155	C₈	AGL(1,ℤ/8)	1	4	yes	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	8	2	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	2	no	—
162	C₂²	C₂×#127	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	8	2	no	—
167	C₄	—	8	2	no	—
168	C₄	—	8	2	no	—
169	C₄	C₂×#135	1	2	yes	—
170	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
171	C₄	#578	1	2	yes	—
172	C₂	—	4	1	no	—
173	C₂	—	16	2	no	—
174	C₂	#3147	1	1	yes	characteristic
175	C₂	#3151	1	1	yes	characteristic
176	C₂	#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3228	C₂	semidirect
2. #3182	C₂	semidirect
3. #3170	C₂	semidirect
5. #1719	C₂	semidirect
6. #1714	C₂	semidirect
7. #1713	C₂	semidirect
8. #1709	C₂	semidirect
9. #1707	C₂	semidirect
10. #1706	C₂	semidirect
11. C₈×M₁₆	C₂	semidirect
154. C₈	#135	semidirect

Maximal subgroups

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

subgroup	index
#3228	2
#3182	2
#3170	2
#3147	2
#1719	2
#1714	2
#1713	2
#1709	2
#1707	2
#1706	2
C₈×M₁₆	2