Finite Group Databasebeta

Subgroups — #33840

Order 256.

374 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 109 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	2	yes	maximal, complemented
2	#3287	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#2246	C₂	1	1	yes	maximal, characteristic, complemented
5	#2218	C₂	1	2	yes	maximal, complemented
6	#2207	C₂	1	1	yes	maximal, characteristic, complemented
7	#2206	C₂	1	1	yes	maximal, characteristic, complemented
8	#2186	C₂	1	2	yes	maximal, complemented
9	#2184	C₂	1	2	yes	maximal, complemented
10	C₄×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
11	#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135	C₂²	1	2	yes	complemented
13	C₂²×D₈	C₂²	1	2	yes	—
14	#550	—	2	1	no	—
15	D₄²	—	2	1	no	—
16	#509	—	2	8	no	—
17	C₂×SD₃₂	C₂²	1	1	yes	characteristic
18	C₂×SD₃₂	C₂²	1	2	yes	complemented
19	C₂×D₁₆	C₂²	1	2	yes	—
20	C₂×D₁₆	C₂²	1	2	yes	complemented
21	C₂×D₁₆	C₂²	1	2	yes	complemented
22	C₂×M₃₂	C₂²	1	2	yes	complemented
23	#493	C₂²	1	1	yes	characteristic
24	#493	C₂²	1	1	yes	characteristic, complemented
25	#466	C₂²	1	2	yes	—
26	#466	C₂²	1	2	yes	complemented
27	#460	—	2	1	no	—
28	#459	—	2	1	no	—
29	#449	—	2	2	no	—
30	#447	—	2	2	no	—
31	C₄×Q₁₆	C₂²	1	1	yes	characteristic, complemented
32	C₄×D₈	C₂²	1	1	yes	characteristic
33	C₈×D₄	C₂²	1	1	yes	characteristic
34	#367	C₂²	1	1	yes	characteristic, complemented
35	C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
36	#358	C₂²	1	1	yes	characteristic, complemented
37	#357	C₂²	1	1	yes	characteristic
38	#357	C₂²	1	2	yes	complemented
39	#357	C₂²	1	2	yes	complemented
40	C₂²⋊C₁₆	C₂²	1	2	yes	complemented
41	C₄×C₁₆	C₂²	1	1	yes	characteristic, complemented
42	C₂×#41	—	2	2	no	—
43	C₂²×D₄	—	2	2	no	—
44	C₂²×D₄	—	4	1	no	—
45	#135	—	2	4	no	—
46	#135	—	4	2	no	—
47	C₂×Q₁₆	C₂³	1	1	yes	characteristic
48	C₂×SD₁₆	—	2	2	no	—
49	C₂×D₈	—	2	4	no	—
50	C₂×D₈	—	2	2	no	—
51	C₂×D₈	—	2	1	no	—
52	C₂×D₈	—	2	1	no	—
53	C₂×D₈	—	4	2	no	—
54	C₂×D₈	C₂³	1	2	yes	—
55	C₂×D₈	C₂³	1	2	yes	—
56	C₂×D₈	C₂³	1	1	yes	characteristic
57	C₂²×C₈	C₂³	1	2	yes	—
58	#127	—	2	1	no	—
59	#127	—	2	1	no	—
60	#127	—	4	1	no	—
61	#121	—	2	2	no	—
62	#121	—	2	2	no	—
63	#121	—	4	1	no	—
64	#121	—	4	2	no	—
65	#120	—	4	2	no	—
66	C₄×Q₈	—	2	1	no	—
67	C₄×D₄	—	2	1	no	—
68	C₄×D₄	—	4	1	no	—
69	C₄×D₄	D₄	1	1	yes	characteristic
70	SD₃₂	—	2	4	no	—
71	SD₃₂	D₄	1	4	yes	complemented
72	D₁₆	—	2	4	no	—
73	D₁₆	—	2	4	no	—
74	D₁₆	—	4	2	no	—
75	M₃₂	—	2	4	no	—
76	C₂×C₁₆	C₂³	1	1	yes	characteristic
77	C₂×C₁₆	C₂³	1	2	yes	—
78	C₂×C₁₆	C₂³	1	1	yes	characteristic
79	#107	C₂³	1	1	yes	characteristic
80	#107	C₂³	1	1	yes	characteristic
81	C₄⋊C₈	D₄	1	1	yes	characteristic
82	#103	—	2	1	no	—
83	#102	—	2	1	no	—
84	#102	—	2	2	no	—
85	#102	—	2	2	no	—
86	C₂²⋊C₈	D₄	1	2	yes	—
87	C₄×C₈	C₂³	1	1	yes	characteristic
88	C₂⁴	—	4	2	no	—
89	#41	—	4	4	no	—
90	#41	—	8	2	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	—	4	4	no	—
96	C₂×D₄	—	4	2	no	—
97	C₂×D₄	—	4	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	—	4	1	no	—
100	C₂×D₄	—	4	2	no	—
101	C₂×D₄	—	4	1	no	—
102	C₂×D₄	—	4	1	no	—
103	C₂×D₄	—	4	1	no	—
104	C₂×D₄	—	8	2	no	—
105	C₂×D₄	—	8	4	no	—
106	C₂×D₄	D₈	1	1	yes	characteristic
107	C₂²×C₄	—	4	1	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₂×D₄	1	2	yes	—
110	Q₁₆	—	2	1	no	—
111	Q₁₆	C₂×D₄	1	2	yes	complemented
112	SD₁₆	—	4	4	no	—
113	D₈	—	2	2	no	—
114	D₈	—	2	2	no	—
115	D₈	—	2	1	no	—
116	D₈	—	2	2	no	—
117	D₈	—	2	2	no	—
118	D₈	—	4	4	no	—
119	D₈	—	8	1	no	—
120	D₈	—	8	1	no	—
121	D₈	C₂×D₄	1	2	yes	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈	C₂×D₄	1	2	yes	—
126	C₂×C₈	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄	—	2	1	no	—
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	D₈	1	1	yes	characteristic
131	C₂²⋊C₄	—	4	1	no	—
132	C₂²⋊C₄	—	4	2	no	—
133	C₂²⋊C₄	—	4	2	no	—
134	C₂²⋊C₄	D₈	1	2	yes	—
135	C₄²	—	4	1	no	—
136	C₄²	C₂×D₄	1	1	yes	characteristic
137	C₁₆	—	2	2	no	—
138	C₁₆	—	2	1	no	—
139	C₁₆	C₂×D₄	1	2	yes	complemented
140	C₂³	—	4	1	no	—
141	C₂³	—	4	2	no	—
142	C₂³	—	4	2	no	—
143	C₂³	—	8	4	no	—
144	C₂³	—	8	4	no	—
145	C₂³	—	16	2	no	—
146	C₂³	C₂×D₈	1	2	yes	—
147	Q₈	—	2	2	no	—
148	Q₈	—	4	1	no	—
149	D₄	—	2	2	no	—
150	D₄	—	2	2	no	—
151	D₄	—	4	2	no	—
152	D₄	—	4	2	no	—
153	D₄	—	4	1	no	—
154	D₄	—	4	2	no	—
155	D₄	—	4	2	no	—
156	D₄	—	8	4	no	—
157	D₄	—	8	4	no	—
158	D₄	—	8	4	no	—
159	D₄	—	8	2	no	—
160	D₄	—	8	2	no	—
161	D₄	—	16	1	no	—
162	D₄	—	16	1	no	—
163	D₄	—	16	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄	C₂×D₈	1	2	yes	—
173	C₂×C₄	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	1	no	—
177	C₈	C₂²×D₄	1	1	yes	characteristic
178	C₈	H(5,𝔽₂)	1	1	yes	characteristic
179	C₂²	—	2	2	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	8	1	no	—
184	C₂²	—	8	1	no	—
185	C₂²	—	8	2	no	—
186	C₂²	—	8	2	no	—
187	C₂²	—	8	2	no	—
188	C₂²	—	8	2	no	—
189	C₂²	—	8	2	no	—
190	C₂²	—	8	2	no	—
191	C₂²	—	16	8	no	—
192	C₂²	C₂²×D₈	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	8	1	no	—
197	C₄	—	8	1	no	—
198	C₄	#509	1	2	yes	complemented
199	C₄	D₄²	1	1	yes	characteristic
200	C₄	#576	1	1	yes	characteristic
201	C₂	—	4	2	no	—
202	C₂	—	8	2	no	—
203	C₂	—	16	2	no	—
204	C₂	—	16	2	no	—
205	C₂	D₄×D₈	1	1	yes	characteristic
206	C₂	C₂×#509	1	1	yes	characteristic
207	C₂	#3415	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 ◁ C₄×D₄ ◁ C₈×D₄ ◁ 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. #3287	C₂	semidirect
3. D₄×D₈	C₂	semidirect
4. #2246	C₂	semidirect
5. #2218	C₂	semidirect
6. #2207	C₂	semidirect
7. #2206	C₂	semidirect
8. #2186	C₂	semidirect
9. #2184	C₂	semidirect
10. C₄×SD₃₂	C₂	semidirect
11. #2169	C₂	semidirect
12. C₂×#135	C₂²	semidirect
18. C₂×SD₃₂	C₂²	semidirect
20. C₂×D₁₆	C₂²	semidirect
21. C₂×D₁₆	C₂²	semidirect
22. C₂×M₃₂	C₂²	semidirect
24. #493	C₂²	semidirect
26. #466	C₂²	semidirect
31. C₄×Q₁₆	C₂²	semidirect
34. #367	C₂²	semidirect
35. C₄⋊C₁₆	C₂²	semidirect
36. #358	C₂²	semidirect
38. #357	C₂²	semidirect
39. #357	C₂²	semidirect
40. C₂²⋊C₁₆	C₂²	semidirect
41. C₄×C₁₆	C₂²	semidirect
71. SD₃₂	D₄	semidirect
111. Q₁₆	C₂×D₄	semidirect
139. C₁₆	C₂×D₄	semidirect
198. C₄	#509	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
#3287	2
D₄×D₈	2
#2246	2
#2218	2
#2207	2
#2206	2
#2186	2
#2184	2
C₄×SD₃₂	2
#2169	2