Finite Group Databasebeta

Subgroups — #33469

Order 256.

323 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 149 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	#3414	C₂	1	1	yes	maximal, characteristic
2	#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	#2952	C₂	1	1	yes	maximal, characteristic, complemented
4	AGL(1,ℤ/16)	C₂	1	4	yes	maximal, complemented
5	#2180	C₂	1	4	yes	maximal, complemented
6	C₂×#365	C₂	1	1	yes	maximal, characteristic, complemented
7	#2148	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#360	C₂	1	1	yes	maximal, characteristic, complemented
9	#2110	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135	C₂²	1	1	yes	characteristic
11	C₂×#135	C₂²	1	1	yes	characteristic
12	#568	—	2	1	no	—
13	#520	—	2	1	no	—
14	#510	C₄	1	2	yes	complemented
15	#510	C₂²	1	2	yes	—
16	#509	C₄	1	2	yes	complemented
17	#509	C₂²	1	2	yes	—
18	#508	C₄	1	4	yes	complemented
19	C₂×M₃₂	C₂²	1	1	yes	characteristic
20	C₂×M₃₂	C₂²	1	1	yes	characteristic
21	#444	C₂²	1	4	yes	—
22	#443	—	2	2	no	—
23	#442	C₂²	1	2	yes	complemented
24	#441	C₂²	1	2	yes	complemented
25	C₄×SD₁₆	—	2	2	no	—
26	C₂×#108	C₂²	1	1	yes	characteristic
27	C₂×#106	C₂²	1	1	yes	characteristic
28	#421	—	2	1	no	—
29	#417	—	2	1	no	—
30	#405	C₂²	1	1	yes	characteristic
31	#365	C₂²	1	4	yes	complemented
32	#362	C₂²	1	2	yes	complemented
33	#361	C₂²	1	2	yes	complemented
34	#360	C₂²	1	4	yes	—
35	#347	C₂²	1	4	yes	complemented
36	C₂×#41	—	2	1	no	—
37	C₂×#41	—	2	1	no	—
38	#135	—	2	1	no	—
39	#135	—	2	1	no	—
40	#135	—	2	1	no	—
41	#135	—	2	1	no	—
42	#135	C₂×C₄	1	2	yes	—
43	#135	C₂×C₄	1	2	yes	—
44	#135	C₂×C₄	1	2	yes	complemented
45	#135	C₂³	1	2	yes	—
46	C₂×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×Q₁₆	C₂³	1	1	yes	characteristic
48	C₂×SD₁₆	—	2	1	no	—
49	C₂×SD₁₆	—	2	1	no	—
50	C₂×D₈	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂×D₈	C₂³	1	1	yes	characteristic
52	#131	—	2	4	no	—
53	#131	—	4	2	no	—
54	C₂×M₁₆	—	2	1	no	—
55	C₂×M₁₆	—	4	2	no	—
56	C₂×M₁₆	D₄	1	1	yes	characteristic
57	C₂²×C₈	C₂³	1	1	yes	characteristic
58	C₄×Q₈	—	2	2	no	—
59	C₄×D₄	—	2	2	no	—
60	C₄×D₄	—	4	2	no	—
61	#117	—	4	1	no	—
62	#117	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄	—	2	1	no	—
64	C₂×C₄⋊C₄	—	4	1	no	—
65	Q₃₂	C₂×C₄	1	4	yes	complemented
66	SD₃₂	C₂×C₄	1	4	yes	complemented
67	SD₃₂	C₂×C₄	1	4	yes	complemented
68	D₁₆	C₂×C₄	1	4	yes	complemented
69	M₃₂	—	2	1	no	—
70	M₃₂	C₂×C₄	1	2	yes	complemented
71	M₃₂	C₂³	1	2	yes	—
72	M₃₂	C₂³	1	2	yes	—
73	C₂×C₁₆	—	2	1	no	—
74	C₂×C₁₆	C₂×C₄	1	2	yes	complemented
75	#108	—	2	1	no	—
76	#108	C₂³	1	2	yes	—
77	#106	—	2	1	no	—
78	#106	C₂³	1	2	yes	—
79	#104	—	2	4	no	—
80	#103	—	2	2	no	—
81	#102	—	2	2	no	—
82	#97	C₂³	1	2	yes	—
83	C₄×C₈	D₄	1	2	yes	—
84	#41	—	2	2	no	—
85	#41	—	2	2	no	—
86	#41	—	2	2	no	—
87	#41	—	2	2	no	—
88	#41	—	4	1	no	—
89	#41	—	4	1	no	—
90	#41	—	4	1	no	—
91	#41	—	4	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	4	1	no	—
97	C₂×D₄	—	4	1	no	—
98	C₂²×C₄	—	4	2	no	—
99	C₂²×C₄	—	4	1	no	—
100	C₂²×C₄	—	4	1	no	—
101	C₂²×C₄	—	4	1	no	—
102	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
103	Q₁₆	—	2	1	no	—
104	Q₁₆	—	2	1	no	—
105	Q₁₆	C₂²×C₄	1	2	yes	—
106	Q₁₆	C₂²×C₄	1	2	yes	—
107	SD₁₆	—	2	2	no	—
108	SD₁₆	—	2	2	no	—
109	SD₁₆	—	2	2	no	—
110	SD₁₆	—	2	2	no	—
111	D₈	—	2	1	no	—
112	D₈	—	2	1	no	—
113	D₈	C₂²×C₄	1	2	yes	—
114	D₈	C₂²×C₄	1	2	yes	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	2	2	no	—
118	M₁₆	—	4	4	no	—
119	M₁₆	—	4	1	no	—
120	C₂×C₈	—	4	4	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈	C₂×D₄	1	2	yes	—
125	C₂×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈	#41	1	1	yes	characteristic
127	C₂×C₈	#41	1	1	yes	characteristic
128	C₂×C₈	C₂⁴	1	1	yes	characteristic
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	1	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₄²	—	4	2	no	—
136	C₄²	C₂×D₄	1	2	yes	—
137	C₁₆	—	2	2	no	—
138	C₁₆	C₂²×C₄	1	4	yes	—
139	C₂³	—	4	1	no	—
140	C₂³	—	4	1	no	—
141	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
142	Q₈	—	2	2	no	—
143	Q₈	—	2	2	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	4	1	no	—
146	D₄	—	2	2	no	—
147	D₄	—	2	2	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	2	no	—
150	D₄	—	4	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	1	no	—
153	D₄	—	4	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	16	2	no	—
163	C₂×C₄	—	16	1	no	—
164	C₂×C₄	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	4	2	no	—
172	C₈	C₄×D₄	1	1	yes	characteristic
173	C₈	C₄×D₄	1	1	yes	characteristic
174	C₈	C₂³×C₄	1	1	yes	characteristic
175	C₈	C₂×#41	1	1	yes	characteristic
176	C₂²	—	4	1	no	—
177	C₂²	—	4	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	8	1	no	—
180	C₂²	—	8	1	no	—
181	C₂²	—	16	1	no	—
182	C₂²	#442	1	1	yes	characteristic
183	C₂²	#442	1	1	yes	characteristic
184	C₂²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₄	—	4	2	no	—
186	C₄	—	4	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	8	2	no	—
189	C₄	#442	1	1	yes	characteristic
190	C₄	#442	1	1	yes	characteristic
191	C₄	C₂×C₄×D₄	1	1	yes	characteristic
192	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	—	8	2	no	—
197	C₂	—	8	2	no	—
198	C₂	C₂×#442	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×#442
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3414	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2957	C₂	semidirect
3. #2952	C₂	semidirect
4. AGL(1,ℤ/16)	C₂	semidirect
5. #2180	C₂	semidirect
6. C₂×#365	C₂	semidirect
7. #2148	C₂	semidirect
8. C₂×#360	C₂	semidirect
9. #2110	C₂	semidirect
14. #510	C₄	semidirect
16. #509	C₄	semidirect
18. #508	C₄	semidirect
23. #442	C₂²	semidirect
24. #441	C₂²	semidirect
31. #365	C₂²	semidirect
32. #362	C₂²	semidirect
33. #361	C₂²	semidirect
35. #347	C₂²	semidirect
44. #135	C₂×C₄	semidirect
46. C₂×Q₁₆	C₂×C₄	semidirect
50. C₂×D₈	C₂×C₄	semidirect
65. Q₃₂	C₂×C₄	semidirect
66. SD₃₂	C₂×C₄	semidirect
67. SD₃₂	C₂×C₄	semidirect
68. D₁₆	C₂×C₄	semidirect
70. M₃₂	C₂×C₄	semidirect
74. C₂×C₁₆	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
#3414	2
#2957	2
#2952	2
AGL(1,ℤ/16)	2
#2180	2
C₂×#365	2
#2148	2
C₂×#360	2
#2110	2