Finite Group Databasebeta

Subgroups — #33612

Order 256.

347 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 119 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×#509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	#3061	C₂	1	1	yes	maximal, characteristic, complemented
4	#3058	C₂	1	1	yes	maximal, characteristic, complemented
5	#2220	C₂	1	2	yes	maximal, complemented
6	#2218	C₂	1	2	yes	maximal, complemented
7	#2217	C₂	1	2	yes	maximal, complemented
8	#2215	C₂	1	2	yes	maximal, complemented
9	#2159	C₂	1	1	yes	maximal, characteristic, complemented
10	#2138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#357	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	C₂²	1	1	yes	characteristic
13	C₂²×D₈	C₂²	1	1	yes	characteristic
14	C₂×#131	C₂²	1	1	yes	characteristic
15	#537	—	2	1	no	—
16	#535	—	2	1	no	—
17	#509	—	2	4	no	—
18	C₂×SD₃₂	C₂²	1	2	yes	—
19	C₂×D₁₆	C₂²	1	2	yes	—
20	#504	—	2	4	no	—
21	C₂×M₃₂	C₂²	1	1	yes	characteristic
22	C₂×M₃₂	C₂²	1	2	yes	complemented
23	C₂²×C₁₆	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆	C₂²	1	2	yes	complemented
25	#469	—	2	2	no	—
26	#468	—	2	2	no	—
27	#467	C₂²	1	2	yes	complemented
28	#466	C₂²	1	2	yes	—
29	#466	C₂²	1	2	yes	—
30	#466	C₂²	1	2	yes	complemented
31	#427	C₂²	1	1	yes	characteristic
32	C₂×#107	C₂²	1	1	yes	characteristic
33	#418	—	2	1	no	—
34	#417	—	2	1	no	—
35	#367	C₂²	1	2	yes	complemented
36	#366	C₂²	1	2	yes	complemented
37	#358	C₂²	1	1	yes	characteristic, complemented
38	#358	C₂²	1	1	yes	characteristic, complemented
39	#357	C₂²	1	2	yes	—
40	#357	C₂²	1	2	yes	—
41	#357	C₂²	1	1	yes	characteristic, complemented
42	#357	C₂²	1	1	yes	characteristic, complemented
43	C₂×#41	—	2	1	no	—
44	C₂×#41	D₄	1	1	yes	characteristic
45	C₂²×D₄	—	2	1	no	—
46	#135	—	2	2	no	—
47	#135	—	2	2	no	—
48	C₂×Q₁₆	C₂³	1	1	yes	characteristic
49	C₂×SD₁₆	—	2	1	no	—
50	C₂×D₈	—	2	2	no	—
51	C₂×D₈	—	2	1	no	—
52	C₂×D₈	—	2	2	no	—
53	C₂×D₈	C₂³	1	2	yes	—
54	C₂×D₈	C₂³	1	1	yes	characteristic
55	#131	—	2	2	no	—
56	#131	—	2	2	no	—
57	C₂×M₁₆	D₄	1	1	yes	characteristic
58	C₂×M₁₆	D₄	1	2	yes	—
59	C₂²×C₈	C₂³	1	2	yes	—
60	C₂²×C₈	C₂³	1	1	yes	characteristic
61	#127	—	4	1	no	—
62	#124	—	4	1	no	—
63	C₂²⋊Q₈	—	2	2	no	—
64	C₂²⋊Q₈	—	4	1	no	—
65	#121	—	2	2	no	—
66	#121	—	2	2	no	—
67	#121	—	2	2	no	—
68	#121	—	4	1	no	—
69	#121	—	4	2	no	—
70	#120	—	4	2	no	—
71	#117	—	2	1	no	—
72	C₂×C₄⋊C₄	—	2	1	no	—
73	SD₃₂	—	2	4	no	—
74	D₁₆	—	2	4	no	—
75	M₃₂	—	2	4	no	—
76	M₃₂	D₄	1	4	yes	complemented
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	2	4	no	—
79	C₂×C₁₆	C₂³	1	1	yes	characteristic
80	C₂×C₁₆	C₂³	1	1	yes	characteristic
81	C₂×C₁₆	C₂³	1	2	yes	—
82	#107	—	2	1	no	—
83	#107	C₂³	1	2	yes	—
84	#107	C₂³	1	1	yes	characteristic
85	#107	C₂³	1	1	yes	characteristic
86	#106	—	2	1	no	—
87	#103	—	2	2	no	—
88	#102	—	2	2	no	—
89	#102	—	2	2	no	—
90	#102	—	2	2	no	—
91	C₂⁴	—	4	1	no	—
92	#41	—	2	2	no	—
93	#41	—	2	2	no	—
94	#41	—	4	2	no	—
95	#41	—	4	2	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	D₈	1	1	yes	characteristic
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	4	1	no	—
101	C₂×D₄	—	4	2	no	—
102	C₂×D₄	—	4	2	no	—
103	C₂×D₄	—	4	2	no	—
104	C₂×D₄	—	4	2	no	—
105	C₂×D₄	—	4	1	no	—
106	C₂×D₄	—	4	2	no	—
107	C₂×D₄	D₈	1	2	yes	—
108	C₂×D₄	D₈	1	1	yes	characteristic
109	C₂²×C₄	—	4	1	no	—
110	C₂²×C₄	—	4	1	no	—
111	C₂²×C₄	C₂×D₄	1	2	yes	—
112	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
113	Q₁₆	—	2	1	no	—
114	Q₁₆	—	2	1	no	—
115	SD₁₆	—	4	2	no	—
116	D₈	—	2	1	no	—
117	D₈	—	2	2	no	—
118	D₈	—	2	2	no	—
119	D₈	—	2	1	no	—
120	D₈	—	4	2	no	—
121	M₁₆	—	2	2	no	—
122	M₁₆	—	2	4	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	C₂×D₄	1	2	yes	—
126	C₂×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈	C₂×D₄	1	2	yes	—
128	C₂×C₈	#41	1	2	yes	—
129	C₂×C₈	C₂⁴	1	1	yes	characteristic
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	4	1	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₂²⋊C₄	—	4	1	no	—
136	C₂²⋊C₄	—	4	2	no	—
137	C₂²⋊C₄	—	4	2	no	—
138	C₂²⋊C₄	—	4	2	no	—
139	C₂²⋊C₄	—	4	2	no	—
140	C₄²	—	4	1	no	—
141	C₁₆	—	2	1	no	—
142	C₁₆	—	2	1	no	—
143	C₁₆	C₂×D₄	1	4	yes	—
144	C₂³	—	4	1	no	—
145	C₂³	—	4	2	no	—
146	C₂³	—	8	2	no	—
147	C₂³	—	16	1	no	—
148	C₂³	—	16	1	no	—
149	C₂³	C₂×D₈	1	2	yes	—
150	C₂³	C₂×D₈	1	1	yes	characteristic
151	Q₈	—	2	2	no	—
152	Q₈	—	4	1	no	—
153	Q₈	—	4	1	no	—
154	D₄	—	2	4	no	—
155	D₄	—	2	2	no	—
156	D₄	—	4	2	no	—
157	D₄	—	4	1	no	—
158	D₄	—	4	2	no	—
159	D₄	—	4	1	no	—
160	D₄	—	8	2	no	—
161	D₄	—	8	2	no	—
162	D₄	—	16	2	no	—
163	D₄	—	16	2	no	—
164	D₄	—	16	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	16	1	no	—
173	C₂×C₄	#121	1	2	yes	—
174	C₂×C₄	C₂×D₈	1	2	yes	—
175	C₂×C₄	C₂×D₈	1	1	yes	characteristic
176	C₂×C₄	#135	1	2	yes	—
177	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	#121	1	2	yes	—
180	C₈	C₂²×D₄	1	1	yes	characteristic
181	C₈	C₂×#41	1	1	yes	characteristic
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	8	1	no	—
188	C₂²	—	8	2	no	—
189	C₂²	—	8	2	no	—
190	C₂²	—	8	1	no	—
191	C₂²	—	8	1	no	—
192	C₂²	—	8	2	no	—
193	C₂²	—	16	4	no	—
194	C₂²	#466	1	2	yes	—
195	C₂²	C₂²×D₈	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	8	2	no	—
198	C₄	—	8	1	no	—
199	C₄	—	8	1	no	—
200	C₄	—	8	1	no	—
201	C₄	#466	1	2	yes	—
202	C₄	C₂×#121	1	1	yes	characteristic
203	C₄	C₂×#135	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	—	4	2	no	—
206	C₂	—	16	1	no	—
207	C₂	—	16	2	no	—
208	C₂	C₂×#466	1	1	yes	characteristic
209	C₂	#3415	1	1	yes	characteristic
210	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#509	C₂	semidirect
2. C₂×#504	C₂	semidirect
3. #3061	C₂	semidirect
4. #3058	C₂	semidirect
5. #2220	C₂	semidirect
6. #2218	C₂	semidirect
7. #2217	C₂	semidirect
8. #2215	C₂	semidirect
9. #2159	C₂	semidirect
10. #2138	C₂	semidirect
11. C₂×#357	C₂	semidirect
22. C₂×M₃₂	C₂²	semidirect
24. C₂²×C₁₆	C₂²	semidirect
27. #467	C₂²	semidirect
30. #466	C₂²	semidirect
35. #367	C₂²	semidirect
36. #366	C₂²	semidirect
37. #358	C₂²	semidirect
38. #358	C₂²	semidirect
41. #357	C₂²	semidirect
42. #357	C₂²	semidirect
76. M₃₂	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#509	2
C₂×#504	2
#3061	2
#3058	2
#2220	2
#2218	2
#2217	2
#2215	2
#2159	2
#2138	2
C₂×#357	2