Finite Group Databasebeta

Subgroups — #33746

Order 256.

278 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 105 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	#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	#3168	C₂	1	1	yes	maximal, characteristic, complemented
3	#2255	C₂	1	1	yes	maximal, characteristic, complemented
4	#2250	C₂	1	1	yes	maximal, characteristic, complemented
5	#2244	C₂	1	1	yes	maximal, characteristic, complemented
6	#2243	C₂	1	1	yes	maximal, characteristic, complemented
7	#2217	C₂	1	2	yes	maximal, complemented
8	#2215	C₂	1	2	yes	maximal, complemented
9	#2177	C₂	1	2	yes	maximal, complemented
10	#2175	C₂	1	2	yes	maximal, complemented
11	C₂×#346	C₂	1	1	yes	maximal, characteristic, complemented
12	#532	—	2	1	no	—
13	#529	—	2	1	no	—
14	C₂×SD₃₂	C₂²	1	2	yes	—
15	C₂×D₁₆	C₂²	1	2	yes	—
16	C₂²×C₁₆	C₂²	1	2	yes	—
17	#500	C₂²	1	1	yes	characteristic
18	#499	C₂²	1	1	yes	characteristic
19	#495	C₂²	1	1	yes	characteristic
20	#493	C₂²	1	1	yes	characteristic
21	#488	—	2	1	no	—
22	#486	—	2	1	no	—
23	#467	C₂²	1	1	yes	characteristic
24	#466	—	2	1	no	—
25	#466	C₂²	1	1	yes	characteristic
26	#466	C₂²	1	2	yes	—
27	#465	—	2	1	no	—
28	C₄×Q₁₆	C₂²	1	1	yes	characteristic
29	C₄×SD₁₆	—	2	1	no	—
30	C₄×D₈	—	2	1	no	—
31	C₄×D₈	C₂²	1	1	yes	characteristic
32	C₄×D₈	C₂²	1	2	yes	—
33	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
34	#367	C₂²	1	2	yes	—
35	#366	C₂²	1	2	yes	—
36	#358	C₂²	1	2	yes	—
37	#357	C₂²	1	2	yes	—
38	#357	C₂²	1	2	yes	—
39	#357	C₂²	1	2	yes	—
40	#346	C₂²	1	2	yes	—
41	#346	C₂²	1	1	yes	characteristic
42	#346	C₂²	1	1	yes	characteristic
43	C₂×#41	—	4	1	no	—
44	C₂×Q₁₆	C₂³	1	1	yes	characteristic
45	C₂×SD₁₆	—	2	1	no	—
46	C₂×D₈	—	2	1	no	—
47	C₂×D₈	C₂³	1	2	yes	—
48	C₂×D₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈	D₄	1	1	yes	characteristic
50	C₂²×C₈	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈	—	2	1	no	—
52	#127	—	2	1	no	—
53	#126	—	4	1	no	—
54	#125	—	2	1	no	—
55	#124	—	2	1	no	—
56	#124	—	4	1	no	—
57	#123	—	4	1	no	—
58	C₂²⋊Q₈	—	2	1	no	—
59	#121	—	2	1	no	—
60	#121	—	2	2	no	—
61	#121	—	4	1	no	—
62	C₄×Q₈	—	2	1	no	—
63	C₄×D₄	—	2	1	no	—
64	C₄×D₄	—	2	2	no	—
65	C₄×D₄	—	4	1	no	—
66	C₄×D₄	—	4	1	no	—
67	#117	—	4	1	no	—
68	C₂×C₄²	D₄	1	1	yes	characteristic
69	SD₃₂	—	2	4	no	—
70	D₁₆	—	2	4	no	—
71	C₂×C₁₆	—	2	4	no	—
72	C₂×C₁₆	C₂³	1	2	yes	—
73	C₂×C₁₆	C₂³	1	2	yes	—
74	#107	—	2	1	no	—
75	#107	C₂³	1	1	yes	characteristic
76	#107	C₂³	1	1	yes	characteristic
77	#107	C₂³	1	2	yes	—
78	#106	—	2	1	no	—
79	#103	—	2	1	no	—
80	#103	—	2	1	no	—
81	#102	—	2	2	no	—
82	#102	—	2	1	no	—
83	#102	—	2	2	no	—
84	#102	—	2	1	no	—
85	C₄×C₈	D₄	1	1	yes	characteristic
86	C₄×C₈	D₄	1	1	yes	characteristic
87	C₄×C₈	C₂³	1	1	yes	characteristic
88	C₄×C₈	C₂³	1	1	yes	characteristic
89	#41	—	8	4	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	4	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	4	1	no	—
95	C₂×D₄	—	4	2	no	—
96	C₂×D₄	—	4	1	no	—
97	C₂×D₄	—	4	1	no	—
98	C₂²×C₄	—	4	1	no	—
99	C₂²×C₄	—	4	2	no	—
100	C₂²×C₄	D₈	1	1	yes	characteristic
101	C₂²×C₄	D₈	1	1	yes	characteristic
102	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
103	Q₁₆	—	2	2	no	—
104	SD₁₆	—	4	2	no	—
105	D₈	—	2	2	no	—
106	D₈	—	2	2	no	—
107	D₈	—	2	2	no	—
108	D₈	—	4	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	4	1	no	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₂²⋊C₄	—	4	1	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₂²⋊C₄	—	4	1	no	—
126	C₂²⋊C₄	—	4	2	no	—
127	C₂²⋊C₄	—	4	1	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₂²⋊C₄	—	4	1	no	—
130	C₄²	—	4	1	no	—
131	C₄²	D₈	1	1	yes	characteristic
132	C₄²	D₈	1	1	yes	characteristic
133	C₄²	C₂×D₄	1	1	yes	characteristic
134	C₄²	C₂×D₄	1	1	yes	characteristic
135	C₁₆	—	2	2	no	—
136	C₁₆	#41	1	4	yes	complemented
137	C₂³	—	4	1	no	—
138	C₂³	—	4	2	no	—
139	C₂³	C₂×D₈	1	1	yes	characteristic
140	Q₈	—	4	1	no	—
141	Q₈	—	4	1	no	—
142	Q₈	—	8	2	no	—
143	D₄	—	4	1	no	—
144	D₄	—	4	2	no	—
145	D₄	—	4	2	no	—
146	D₄	—	4	1	no	—
147	D₄	—	8	2	no	—
148	D₄	—	8	2	no	—
149	D₄	—	8	2	no	—
150	D₄	—	8	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	#135	1	2	yes	—
170	C₈	C₂×#41	1	2	yes	—
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	8	2	no	—
174	C₂²	—	8	1	no	—
175	C₂²	—	8	1	no	—
176	C₂²	—	8	2	no	—
177	C₂²	—	8	2	no	—
178	C₂²	—	8	1	no	—
179	C₂²	#509	1	2	yes	—
180	C₂²	C₂²×D₈	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	8	2	no	—
187	C₄	—	8	1	no	—
188	C₄	—	8	1	no	—
189	C₄	—	8	1	no	—
190	C₄	#532	1	1	yes	characteristic
191	C₄	C₂×#135	1	1	yes	characteristic
192	C₂	—	2	2	no	—
193	C₂	—	16	2	no	—
194	C₂	—	16	1	no	—
195	C₂	#3169	1	1	yes	characteristic
196	C₂	C₂×#509	1	1	yes	characteristic
197	C₂	#3414	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×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
1. #3169	C₂	semidirect
2. #3168	C₂	semidirect
3. #2255	C₂	semidirect
4. #2250	C₂	semidirect
5. #2244	C₂	semidirect
6. #2243	C₂	semidirect
7. #2217	C₂	semidirect
8. #2215	C₂	semidirect
9. #2177	C₂	semidirect
10. #2175	C₂	semidirect
11. C₂×#346	C₂	semidirect
136. C₁₆	#41	semidirect

Maximal subgroups

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

subgroup	index
#3169	2
#3168	2
#2255	2
#2250	2
#2244	2
#2243	2
#2217	2
#2215	2
#2177	2
#2175	2
C₂×#346	2