Finite Group Databasebeta

Subgroups — #33862

Order 256.

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

normal subgroups only

This group has 311 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	#3338	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₈	C₂	1	1	yes	maximal, characteristic
3	#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	#2233	C₂	1	2	yes	maximal, complemented
5	#2226	C₂	1	1	yes	maximal, characteristic, complemented
6	#2217	C₂	1	2	yes	maximal, complemented
7	#2207	C₂	1	1	yes	maximal, characteristic, complemented
8	#2187	C₂	1	2	yes	maximal, complemented
9	C₄×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#357	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈	C₂²	1	2	yes	—
13	#554	—	2	1	no	—
14	D₄²	—	2	1	no	—
15	C₂×SD₃₂	C₂²	1	1	yes	characteristic
16	C₂×SD₃₂	C₂²	1	2	yes	complemented
17	C₂²×C₁₆	C₂²	1	2	yes	complemented
18	#500	C₂²	1	1	yes	characteristic, complemented
19	#493	C₂²	1	1	yes	characteristic
20	#484	—	2	2	no	—
21	#477	—	2	1	no	—
22	#467	C₂²	1	2	yes	complemented
23	#466	C₂²	1	2	yes	—
24	#459	—	2	1	no	—
25	#447	—	2	2	no	—
26	C₄×Q₁₆	C₂²	1	1	yes	characteristic, complemented
27	C₄×D₈	C₂²	1	1	yes	characteristic
28	C₈×D₄	C₂²	1	1	yes	characteristic
29	C₂×#107	C₂²	1	2	yes	complemented
30	#367	C₂²	1	1	yes	characteristic, complemented
31	#367	C₂²	1	2	yes	complemented
32	C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
33	#358	C₂²	1	1	yes	characteristic, complemented
34	#357	C₂²	1	1	yes	characteristic
35	#357	C₂²	1	2	yes	—
36	#357	C₂²	1	2	yes	complemented
37	#357	C₂²	1	2	yes	complemented
38	#357	C₂²	1	2	yes	complemented
39	C₂²⋊C₁₆	C₂²	1	2	yes	complemented
40	C₄×C₁₆	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₄	—	2	2	no	—
42	C₂²×D₄	—	4	1	no	—
43	C₂×Q₁₆	C₂³	1	1	yes	characteristic
44	C₂×D₈	—	2	4	no	—
45	C₂×D₈	—	2	2	no	—
46	C₂×D₈	—	2	1	no	—
47	C₂×D₈	—	4	2	no	—
48	C₂×D₈	C₂³	1	2	yes	—
49	C₂×D₈	C₂³	1	1	yes	characteristic
50	C₂²×C₈	C₂³	1	2	yes	—
51	C₄⋊Q₈	—	2	1	no	—
52	#127	—	2	1	no	—
53	#125	—	4	1	no	—
54	C₂²⋊Q₈	—	2	2	no	—
55	C₂²⋊Q₈	—	4	2	no	—
56	#121	—	2	2	no	—
57	#121	—	4	1	no	—
58	#120	—	4	2	no	—
59	C₄×Q₈	—	2	1	no	—
60	C₄×D₄	—	2	1	no	—
61	C₄×D₄	—	4	1	no	—
62	C₄×D₄	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄	—	2	2	no	—
64	SD₃₂	—	2	2	no	—
65	SD₃₂	—	4	2	no	—
66	C₂×C₁₆	—	4	2	no	—
67	C₂×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₆	C₂³	1	2	yes	—
70	#107	—	2	2	no	—
71	#107	—	2	1	no	—
72	#107	C₂³	1	1	yes	characteristic
73	#107	C₂³	1	2	yes	—
74	#107	C₂³	1	1	yes	characteristic
75	C₄⋊C₈	D₄	1	1	yes	characteristic
76	#103	—	2	1	no	—
77	#103	—	2	2	no	—
78	#102	—	2	1	no	—
79	#102	—	2	2	no	—
80	C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	C₂⁴	—	4	2	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	4	1	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	4	4	no	—
88	C₂×D₄	—	4	2	no	—
89	C₂×D₄	—	4	2	no	—
90	C₂×D₄	—	4	1	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	8	2	no	—
93	C₂×D₄	—	8	4	no	—
94	C₂×D₄	D₈	1	1	yes	characteristic
95	C₂²×C₄	—	4	2	no	—
96	C₂²×C₄	—	4	1	no	—
97	C₂²×C₄	C₂×D₄	1	2	yes	—
98	Q₁₆	—	2	1	no	—
99	Q₁₆	—	2	1	no	—
100	D₈	—	2	2	no	—
101	D₈	—	2	1	no	—
102	D₈	—	2	2	no	—
103	D₈	—	4	4	no	—
104	D₈	—	8	1	no	—
105	D₈	#41	1	2	yes	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈	C₂×D₄	1	2	yes	—
110	C₂×C₈	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	4	2	no	—
118	C₄⋊C₄	D₈	1	1	yes	characteristic
119	C₂²⋊C₄	—	4	2	no	—
120	C₂²⋊C₄	—	4	1	no	—
121	C₂²⋊C₄	—	4	2	no	—
122	C₂²⋊C₄	D₈	1	2	yes	—
123	C₄²	—	4	1	no	—
124	C₄²	C₂×D₄	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	2	1	no	—
127	C₁₆	—	2	2	no	—
128	C₂³	—	4	1	no	—
129	C₂³	—	4	2	no	—
130	C₂³	—	8	4	no	—
131	C₂³	—	8	4	no	—
132	C₂³	—	16	2	no	—
133	C₂³	C₂×D₈	1	2	yes	—
134	Q₈	—	4	1	no	—
135	Q₈	—	4	1	no	—
136	Q₈	—	16	1	no	—
137	D₄	—	2	2	no	—
138	D₄	—	4	2	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	2	no	—
141	D₄	—	8	4	no	—
142	D₄	—	8	4	no	—
143	D₄	—	8	2	no	—
144	D₄	—	16	1	no	—
145	D₄	SD₃₂	1	4	yes	complemented
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	16	2	no	—
154	C₂×C₄	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄	C₂×D₈	1	2	yes	—
156	C₂×C₄	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
158	C₈	—	2	2	no	—
159	C₈	—	2	1	no	—
160	C₈	C₂×#41	1	1	yes	characteristic
161	C₈	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	4	2	no	—
165	C₂²	—	8	1	no	—
166	C₂²	—	8	1	no	—
167	C₂²	—	8	2	no	—
168	C₂²	—	8	2	no	—
169	C₂²	—	8	2	no	—
170	C₂²	—	16	8	no	—
171	C₂²	C₂×SD₃₂	1	4	yes	—
172	C₂²	C₂²×D₈	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	2	no	—
175	C₄	—	8	1	no	—
176	C₄	—	8	1	no	—
177	C₄	—	8	2	no	—
178	C₄	—	8	1	no	—
179	C₄	C₂×SD₃₂	1	2	yes	—
180	C₄	#546	1	1	yes	characteristic
181	C₄	#577	1	1	yes	characteristic
182	C₂	—	2	4	no	—
183	C₂	—	8	2	no	—
184	C₂	—	16	2	no	—
185	C₂	#3294	1	1	yes	characteristic
186	C₂	C₂²×SD₃₂	1	1	yes	characteristic
187	C₂	#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	D₄×D₈	C₂
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. #3338	C₂	semidirect
3. #2240	C₂	semidirect
4. #2233	C₂	semidirect
5. #2226	C₂	semidirect
6. #2217	C₂	semidirect
7. #2207	C₂	semidirect
8. #2187	C₂	semidirect
9. C₄×SD₃₂	C₂	semidirect
10. D₄×C₁₆	C₂	semidirect
11. C₂×#357	C₂	semidirect
16. C₂×SD₃₂	C₂²	semidirect
17. C₂²×C₁₆	C₂²	semidirect
18. #500	C₂²	semidirect
22. #467	C₂²	semidirect
26. C₄×Q₁₆	C₂²	semidirect
29. C₂×#107	C₂²	semidirect
30. #367	C₂²	semidirect
31. #367	C₂²	semidirect
32. C₄⋊C₁₆	C₂²	semidirect
33. #358	C₂²	semidirect
36. #357	C₂²	semidirect
37. #357	C₂²	semidirect
38. #357	C₂²	semidirect
39. C₂²⋊C₁₆	C₂²	semidirect
40. C₄×C₁₆	C₂²	semidirect
145. D₄	SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
#3338	2
D₄×D₈	2
#2240	2
#2233	2
#2226	2
#2217	2
#2207	2
#2187	2
C₄×SD₃₂	2
D₄×C₁₆	2
C₂×#357	2