Finite Group Databasebeta

Subgroups — #13314

Order 256.

268 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 59 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	C₂²×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#473	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#472	C₂	1	1	yes	maximal, characteristic
4	C₂×#362	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#361	C₂	1	1	yes	maximal, characteristic, complemented
6	#2113	C₂	1	1	yes	maximal, characteristic, complemented
7	#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#137	—	2	1	no	—
9	C₂×AGL(1,ℤ/8)	—	2	1	no	—
10	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂²×D₈	C₂²	1	1	yes	characteristic
12	C₂×#131	C₂²	1	1	yes	characteristic
13	C₂×SD₃₂	—	2	2	no	—
14	C₂×SD₃₂	—	2	2	no	—
15	C₂×SD₃₂	—	2	2	no	—
16	C₂×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆	C₂²	1	1	yes	characteristic
18	#473	—	2	2	no	—
19	#473	—	2	2	no	—
20	#472	—	2	2	no	—
21	#472	—	2	2	no	—
22	C₂×#108	C₂²	1	1	yes	characteristic
23	C₂×#108	C₂²	1	1	yes	characteristic
24	C₂×#101	—	2	1	no	—
25	C₂×#100	—	2	1	no	—
26	#362	—	2	2	no	—
27	#361	—	2	2	no	—
28	C₂²⋊C₁₆	—	2	2	no	—
29	C₂×#41	D₄	1	1	yes	characteristic
30	C₂²×Q₈	—	2	1	no	—
31	C₂²×D₄	—	2	1	no	—
32	#137	—	2	4	no	—
33	#137	—	4	2	no	—
34	AGL(1,ℤ/8)	—	2	4	no	—
35	AGL(1,ℤ/8)	—	4	2	no	—
36	C₂×Q₁₆	—	2	1	no	—
37	C₂×Q₁₆	—	2	1	no	—
38	C₂×Q₁₆	—	2	1	no	—
39	C₂×Q₁₆	—	2	1	no	—
40	C₂×Q₁₆	—	2	1	no	—
41	C₂×Q₁₆	—	4	1	no	—
42	C₂×Q₁₆	D₄	1	2	yes	—
43	C₂×SD₁₆	—	4	1	no	—
44	C₂×SD₁₆	—	4	1	no	—
45	C₂×D₈	—	2	1	no	—
46	C₂×D₈	—	2	1	no	—
47	C₂×D₈	—	2	1	no	—
48	C₂×D₈	—	2	1	no	—
49	C₂×D₈	—	2	1	no	—
50	C₂×D₈	—	4	1	no	—
51	C₂×D₈	D₄	1	2	yes	—
52	#131	—	2	1	no	—
53	#131	—	2	1	no	—
54	#131	—	2	1	no	—
55	#131	—	2	1	no	—
56	C₂×M₁₆	—	2	1	no	—
57	C₂×M₁₆	—	2	1	no	—
58	C₂×M₁₆	—	2	1	no	—
59	C₂×M₁₆	D₄	1	1	yes	characteristic
60	C₂²×C₈	—	2	1	no	—
61	C₂²×C₈	C₂³	1	1	yes	characteristic
62	SD₃₂	—	2	8	no	—
63	M₃₂	—	2	1	no	—
64	M₃₂	—	2	1	no	—
65	C₂×C₁₆	—	2	1	no	—
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	—	2	1	no	—
68	C₂×C₁₆	D₄	1	2	yes	—
69	#108	—	2	1	no	—
70	#108	—	2	1	no	—
71	#108	D₄	1	2	yes	—
72	#108	D₄	1	2	yes	—
73	#101	—	2	4	no	—
74	#100	—	2	4	no	—
75	C₂⁴	—	4	1	no	—
76	#41	—	2	1	no	—
77	#41	—	2	1	no	—
78	#41	—	2	1	no	—
79	#41	—	2	1	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	4	1	no	—
82	C₂×Q₈	—	4	1	no	—
83	C₂×Q₈	—	4	1	no	—
84	C₂×Q₈	—	4	1	no	—
85	C₂×Q₈	—	4	1	no	—
86	C₂×Q₈	D₈	1	1	yes	characteristic
87	C₂×D₄	—	2	1	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	4	1	no	—
90	C₂×D₄	—	4	1	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	4	1	no	—
93	C₂×D₄	—	4	1	no	—
94	C₂×D₄	D₈	1	1	yes	characteristic
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	4	1	no	—
97	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
98	Q₁₆	—	2	2	no	—
99	Q₁₆	—	2	2	no	—
100	Q₁₆	—	4	4	no	—
101	Q₁₆	—	4	1	no	—
102	Q₁₆	—	4	1	no	—
103	SD₁₆	—	4	4	no	—
104	SD₁₆	—	4	4	no	—
105	D₈	—	2	2	no	—
106	D₈	—	2	2	no	—
107	D₈	—	4	4	no	—
108	D₈	—	4	1	no	—
109	D₈	—	4	1	no	—
110	M₁₆	—	2	1	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	2	1	no	—
113	M₁₆	—	2	2	no	—
114	M₁₆	—	4	1	no	—
115	M₁₆	—	4	1	no	—
116	M₁₆	—	4	1	no	—
117	M₁₆	—	4	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	4	1	no	—
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	C₂×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈	#41	1	1	yes	characteristic
129	C₁₆	—	2	2	no	—
130	C₁₆	—	4	1	no	—
131	C₂³	—	2	1	no	—
132	C₂³	—	4	2	no	—
133	C₂³	—	8	1	no	—
134	C₂³	—	8	1	no	—
135	C₂³	—	16	2	no	—
136	C₂³	#135	1	1	yes	characteristic
137	Q₈	—	2	1	no	—
138	Q₈	—	2	1	no	—
139	Q₈	—	4	2	no	—
140	Q₈	—	4	2	no	—
141	Q₈	—	8	1	no	—
142	Q₈	—	8	1	no	—
143	D₄	—	2	1	no	—
144	D₄	—	2	1	no	—
145	D₄	—	4	1	no	—
146	D₄	—	4	1	no	—
147	D₄	—	4	2	no	—
148	D₄	—	4	2	no	—
149	D₄	—	8	1	no	—
150	D₄	—	8	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	8	1	no	—
156	C₂×C₄	—	8	1	no	—
157	C₂×C₄	#120	1	1	yes	characteristic
158	C₂×C₄	#120	1	1	yes	characteristic
159	C₂×C₄	#121	1	1	yes	characteristic
160	C₂×C₄	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₈	—	4	1	no	—
164	C₈	—	4	2	no	—
165	C₈	—	4	2	no	—
166	C₈	#120	1	1	yes	characteristic
167	C₈	#121	1	1	yes	characteristic
168	C₈	#121	1	1	yes	characteristic
169	C₈	#127	1	1	yes	characteristic
170	C₂²	—	2	2	no	—
171	C₂²	—	4	2	no	—
172	C₂²	—	4	1	no	—
173	C₂²	—	8	2	no	—
174	C₂²	—	16	2	no	—
175	C₂²	—	16	2	no	—
176	C₂²	—	16	2	no	—
177	C₂²	#449	1	1	yes	characteristic
178	C₂²	#449	1	1	yes	characteristic
179	C₂²	#466	1	1	yes	characteristic
180	C₄	—	4	1	no	—
181	C₄	—	8	2	no	—
182	C₄	#392	1	1	yes	characteristic
183	C₄	#447	1	1	yes	characteristic
184	C₄	#447	1	1	yes	characteristic
185	C₄	#469	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	—	8	1	no	—
189	C₂	—	16	2	no	—
190	C₂	#1999	1	1	yes	characteristic
191	C₂	#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#472	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×SD₃₂	C₂	semidirect
2. C₂×#473	C₂	semidirect
4. C₂×#362	C₂	semidirect
5. C₂×#361	C₂	semidirect
6. #2113	C₂	semidirect
7. #1381	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SD₃₂	2
C₂×#473	2
C₂×#472	2
C₂×#362	2
C₂×#361	2
#2113	2
#1381	2