Finite Group Databasebeta

Subgroups — #33799

Order 256.

233 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 101 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	#3168	C₂	1	1	yes	maximal, characteristic
2	#3160	C₂	1	1	yes	maximal, characteristic, complemented
3	#2253	C₂	1	2	yes	maximal, complemented
4	#2242	C₂	1	2	yes	maximal, complemented
5	#2237	C₂	1	2	yes	maximal, complemented
6	#2234	C₂	1	2	yes	maximal
7	#2227	C₂	1	1	yes	maximal, characteristic, complemented
8	#2226	C₂	1	1	yes	maximal, characteristic, complemented
9	#2225	C₂	1	1	yes	maximal, characteristic, complemented
10	#2224	C₂	1	1	yes	maximal, characteristic, complemented
11	#2162	C₂	1	1	yes	maximal, characteristic, complemented
12	#533	—	2	1	no	—
13	#529	—	2	1	no	—
14	#500	C₂²	1	1	yes	characteristic
15	#500	C₂²	1	1	yes	characteristic
16	#499	—	2	1	no	—
17	#499	C₂²	1	2	yes	—
18	#499	C₂²	1	1	yes	characteristic
19	#498	—	2	1	no	—
20	#495	C₂²	1	1	yes	characteristic
21	#488	—	2	1	no	—
22	#467	C₂²	1	1	yes	characteristic
23	#466	C₂²	1	1	yes	characteristic
24	#465	—	2	1	no	—
25	C₄×Q₁₆	C₂²	1	1	yes	characteristic
26	C₄×SD₁₆	—	2	1	no	—
27	C₄×D₈	C₂²	1	1	yes	characteristic
28	#427	—	2	1	no	—
29	#427	C₂²	1	2	yes	—
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	#367	C₂²	1	2	yes	—
32	#367	C₂²	1	2	yes	—
33	#366	C₂²	1	2	yes	—
34	#366	C₂²	1	2	yes	—
35	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
37	#358	C₂²	1	2	yes	—
38	#358	C₂²	1	2	yes	—
39	#357	C₂²	1	2	yes	—
40	#357	C₂²	1	2	yes	—
41	C₂²⋊C₁₆	C₂²	1	2	yes	—
42	C₄×C₁₆	C₂²	1	2	yes	—
43	C₂×Q₁₆	C₂³	1	1	yes	characteristic
44	C₂×SD₁₆	—	2	1	no	—
45	C₂×D₈	C₂³	1	1	yes	characteristic
46	C₂²×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₈	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	2	1	no	—
50	#126	—	4	1	no	—
51	#125	—	2	1	no	—
52	#125	—	2	2	no	—
53	#124	—	2	1	no	—
54	#123	—	4	1	no	—
55	C₂²⋊Q₈	—	2	1	no	—
56	C₂²⋊Q₈	—	4	1	no	—
57	C₂²⋊Q₈	—	4	1	no	—
58	#121	—	2	1	no	—
59	C₄×Q₈	—	2	1	no	—
60	C₄×Q₈	—	4	1	no	—
61	C₄×Q₈	—	4	1	no	—
62	C₄×D₄	—	2	1	no	—
63	C₄×D₄	—	4	1	no	—
64	#117	—	2	2	no	—
65	#117	—	4	1	no	—
66	C₂×C₄²	D₄	1	1	yes	characteristic
67	C₂×C₁₆	C₂³	1	2	yes	—
68	C₂×C₁₆	C₂³	1	2	yes	—
69	#107	—	2	1	no	—
70	#107	—	2	1	no	—
71	#107	C₂³	1	2	yes	—
72	#107	C₂³	1	2	yes	—
73	#107	C₂³	1	1	yes	characteristic
74	#107	C₂³	1	1	yes	characteristic
75	#106	—	2	2	no	—
76	#106	—	2	2	no	—
77	#106	—	2	1	no	—
78	#103	—	2	1	no	—
79	#103	—	2	1	no	—
80	#102	—	2	1	no	—
81	#102	—	2	1	no	—
82	C₄×C₈	D₄	1	1	yes	characteristic
83	C₄×C₈	D₄	1	1	yes	characteristic
84	C₄×C₈	C₂³	1	1	yes	characteristic
85	C₄×C₈	C₂³	1	1	yes	characteristic
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	4	1	no	—
88	C₂×Q₈	—	4	1	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂×D₄	—	4	1	no	—
91	C₂²×C₄	—	4	1	no	—
92	C₂²×C₄	D₈	1	1	yes	characteristic
93	C₂²×C₄	D₈	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	2	1	no	—
97	SD₁₆	—	4	2	no	—
98	D₈	—	2	1	no	—
99	D₈	—	2	1	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	4	1	no	—
119	C₂²⋊C₄	—	4	2	no	—
120	C₂²⋊C₄	—	4	1	no	—
121	C₂²⋊C₄	—	4	1	no	—
122	C₂²⋊C₄	—	4	1	no	—
123	C₂²⋊C₄	—	4	1	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₄²	—	4	2	no	—
126	C₄²	—	4	1	no	—
127	C₄²	D₈	1	1	yes	characteristic
128	C₄²	D₈	1	1	yes	characteristic
129	C₄²	C₂×D₄	1	1	yes	characteristic
130	C₄²	C₂×D₄	1	1	yes	characteristic
131	C₁₆	—	2	2	no	—
132	C₁₆	—	2	2	no	—
133	C₂³	—	4	1	no	—
134	C₂³	C₂×D₈	1	1	yes	characteristic
135	Q₈	—	4	1	no	—
136	Q₈	—	4	1	no	—
137	Q₈	—	8	2	no	—
138	Q₈	—	8	2	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	8	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	#137	1	2	yes	—
161	C₈	2₋¹⁺⁴	1	1	yes	characteristic
162	C₈	2₋¹⁺⁴	1	1	yes	characteristic
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	8	1	no	—
167	C₂²	—	8	1	no	—
168	C₂²	—	8	1	no	—
169	C₂²	C₂²×D₈	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	8	1	no	—
175	C₄	—	8	2	no	—
176	C₄	—	8	1	no	—
177	C₄	—	8	2	no	—
178	C₄	—	8	1	no	—
179	C₄	#508	1	2	yes	—
180	C₄	#536	1	1	yes	characteristic
181	C₄	C₂×#137	1	1	yes	characteristic
182	C₂	—	4	1	no	—
183	C₂	—	16	1	no	—
184	C₂	#3226	1	1	yes	characteristic
185	C₂	C₂×#508	1	1	yes	characteristic
186	C₂	#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ 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. #3160	C₂	semidirect
3. #2253	C₂	semidirect
4. #2242	C₂	semidirect
5. #2237	C₂	semidirect
7. #2227	C₂	semidirect
8. #2226	C₂	semidirect
9. #2225	C₂	semidirect
10. #2224	C₂	semidirect
11. #2162	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3168	2
#3160	2
#2253	2
#2242	2
#2237	2
#2234	2
#2227	2
#2226	2
#2225	2
#2224	2
#2162	2