Finite Group Databasebeta

Subgroups — #13028

Order 256.

238 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂×#510	C₂	1	1	yes	maximal, characteristic
2	C₂×#443	C₂	1	1	yes	maximal, characteristic
3	C₂×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	#2139	C₂	1	1	yes	maximal, characteristic
5	#1969	C₂	1	1	yes	maximal, characteristic, complemented
6	#1387	C₂	1	1	yes	maximal, characteristic, complemented
7	#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×#131	—	2	1	no	—
11	#510	—	2	2	no	—
12	#510	C₄	1	4	yes	complemented
13	C₂×Q₃₂	—	2	1	no	—
14	C₂×SD₃₂	—	2	1	no	—
15	C₂×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×M₃₂	C₂²	1	1	yes	characteristic
17	#443	—	2	2	no	—
18	#443	—	2	2	no	—
19	C₂×#108	C₂²	1	1	yes	characteristic
20	C₂×#107	C₂²	1	1	yes	characteristic
21	C₂×#104	—	2	1	no	—
22	C₂×#103	—	2	1	no	—
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	#391	—	2	1	no	—
25	#362	—	2	2	no	—
26	#358	—	2	1	no	—
27	#357	—	2	1	no	—
28	C₂×#41	—	2	1	no	—
29	C₂²×Q₈	—	2	1	no	—
30	#135	—	2	1	no	—
31	#135	—	2	1	no	—
32	#135	C₂×C₄	1	2	yes	—
33	#135	D₄	1	2	yes	—
34	C₂×Q₁₆	—	2	1	no	—
35	C₂×Q₁₆	—	2	1	no	—
36	C₂×Q₁₆	—	2	1	no	—
37	C₂×Q₁₆	—	2	1	no	—
38	C₂×Q₁₆	—	2	1	no	—
39	C₂×Q₁₆	C₂×C₄	1	2	yes	—
40	C₂×Q₁₆	D₄	1	1	yes	characteristic
41	C₂×SD₁₆	—	2	1	no	—
42	C₂×D₈	D₄	1	1	yes	characteristic
43	#131	—	2	4	no	—
44	#131	—	4	2	no	—
45	C₂×M₁₆	—	2	1	no	—
46	C₂×M₁₆	—	4	1	no	—
47	C₂²×C₈	—	4	1	no	—
48	C₂²×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄	—	2	1	no	—
51	C₂×C₄²	D₄	1	1	yes	characteristic
52	Q₃₂	—	2	4	no	—
53	SD₃₂	—	2	4	no	—
54	M₃₂	—	2	1	no	—
55	M₃₂	—	2	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	C₂×C₄	1	2	yes	—
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	—	2	1	no	—
60	#108	—	2	1	no	—
61	#108	D₄	1	2	yes	—
62	#107	—	2	1	no	—
63	#107	—	2	1	no	—
64	#104	—	2	4	no	—
65	#103	—	4	1	no	—
66	#103	—	4	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	#95	—	4	2	no	—
70	#41	—	2	2	no	—
71	#41	—	2	2	no	—
72	#41	—	4	1	no	—
73	#41	—	4	1	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	1	no	—
76	C₂×Q₈	—	4	1	no	—
77	C₂×Q₈	—	4	1	no	—
78	C₂×Q₈	—	4	1	no	—
79	C₂×Q₈	—	4	1	no	—
80	C₂×Q₈	—	4	1	no	—
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	—	4	1	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	4	1	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	—	4	1	no	—
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	Q₁₆	—	2	2	no	—
89	Q₁₆	—	2	2	no	—
90	Q₁₆	—	2	1	no	—
91	Q₁₆	—	4	1	no	—
92	Q₁₆	—	4	1	no	—
93	Q₁₆	C₂²⋊C₄	1	2	yes	—
94	SD₁₆	—	2	2	no	—
95	SD₁₆	—	2	2	no	—
96	D₈	—	2	1	no	—
97	D₈	C₂²⋊C₄	1	2	yes	—
98	M₁₆	—	2	2	no	—
99	M₁₆	—	4	4	no	—
100	M₁₆	—	4	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	4	4	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	D₈	1	1	yes	characteristic
106	C₂×C₈	D₈	1	1	yes	characteristic
107	C₂×C₈	C₂²×C₄	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	C₂×C₈	#41	1	1	yes	characteristic
112	C₂×C₈	#41	1	1	yes	characteristic
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₁₆	—	2	2	no	—
118	C₁₆	—	4	1	no	—
119	C₂³	—	4	1	no	—
120	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
121	Q₈	—	2	2	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	8	1	no	—
126	Q₈	—	8	1	no	—
127	D₄	—	2	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	1	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄	#120	1	1	yes	characteristic
144	C₂×C₄	#121	1	1	yes	characteristic
145	C₂×C₄	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄	#135	1	1	yes	characteristic
147	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	4	2	no	—
151	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₈	C₄×D₄	1	1	yes	characteristic
153	C₈	#121	1	1	yes	characteristic
154	C₈	#123	1	1	yes	characteristic
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	8	1	no	—
159	C₂²	—	8	1	no	—
160	C₂²	#442	1	1	yes	characteristic
161	C₂²	#449	1	1	yes	characteristic
162	C₂²	#459	1	1	yes	characteristic
163	C₄	—	4	2	no	—
164	C₄	—	4	2	no	—
165	C₄	—	8	2	no	—
166	C₄	—	16	1	no	—
167	C₄	#386	1	1	yes	characteristic
168	C₄	C₄×D₈	1	1	yes	characteristic
169	C₄	#447	1	1	yes	characteristic
170	C₄	#463	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	8	2	no	—
174	C₂	#1879	1	1	yes	characteristic
175	C₂	#2212	1	1	yes	characteristic
176	C₂	#2214	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#459
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂×#362	C₂	semidirect
5. #1969	C₂	semidirect
6. #1387	C₂	semidirect
7. #1376	C₂	semidirect
12. #510	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#510	2
C₂×#443	2
C₂×#362	2
#2139	2
#1969	2
#1387	2
#1376	2