Finite Group Databasebeta

Subgroups — #33650

Order 256.

294 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 111 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	#3058	C₂	1	1	yes	maximal, characteristic
2	#2913	C₂	1	1	yes	maximal, characteristic, complemented
3	#2234	C₂	1	2	yes	maximal, complemented
4	#2234	C₂	1	2	yes	maximal, complemented
5	#2233	C₂	1	2	yes	maximal, complemented
6	#2232	C₂	1	2	yes	maximal, complemented
7	#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	#2140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#357	C₂	1	1	yes	maximal, characteristic, complemented
11	#2113	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈	C₂²	1	1	yes	characteristic
13	C₂×#131	C₂²	1	1	yes	characteristic
14	#535	—	2	1	no	—
15	#520	—	2	1	no	—
16	C₂×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆	C₂²	1	1	yes	characteristic
18	#469	—	2	2	no	—
19	#466	C₂²	1	2	yes	—
20	#466	C₂²	1	2	yes	—
21	#428	—	2	2	no	—
22	#428	—	2	1	no	—
23	#427	C₂²	1	1	yes	characteristic
24	#427	C₂²	1	1	yes	characteristic
25	#427	C₂²	1	2	yes	complemented
26	C₂×#107	C₂²	1	1	yes	characteristic
27	C₂×#107	C₂²	1	2	yes	complemented
28	#418	—	2	1	no	—
29	#367	C₂²	1	1	yes	characteristic, complemented
30	#367	C₂²	1	1	yes	characteristic, complemented
31	#367	C₂²	1	2	yes	complemented
32	#366	C₂²	1	1	yes	characteristic, complemented
33	#366	C₂²	1	1	yes	characteristic, complemented
34	#366	C₂²	1	2	yes	complemented
35	#357	C₂²	1	2	yes	—
36	#357	C₂²	1	2	yes	—
37	#357	C₂²	1	2	yes	—
38	#357	C₂²	1	2	yes	—
39	C₂²⋊C₁₆	C₂²	1	2	yes	complemented
40	C₂²⋊C₁₆	C₂²	1	2	yes	complemented
41	C₂×#41	D₄	1	1	yes	characteristic
42	C₂²×D₄	—	2	1	no	—
43	C₂×D₈	—	2	1	no	—
44	C₂×D₈	—	2	2	no	—
45	C₂×D₈	—	2	2	no	—
46	C₂×D₈	C₂³	1	2	yes	—
47	#131	—	2	2	no	—
48	#131	—	2	2	no	—
49	C₂×M₁₆	D₄	1	1	yes	characteristic
50	C₂×M₁₆	D₄	1	2	yes	—
51	C₂²×C₈	C₂³	1	2	yes	—
52	C₂²×C₈	C₂³	1	1	yes	characteristic
53	#127	—	4	1	no	—
54	#124	—	4	1	no	—
55	#121	—	2	2	no	—
56	#121	—	2	2	no	—
57	#120	—	4	2	no	—
58	C₄×Q₈	—	4	1	no	—
59	C₄×D₄	—	4	2	no	—
60	C₄×D₄	—	4	1	no	—
61	#117	—	2	1	no	—
62	#117	—	2	2	no	—
63	#117	—	2	1	no	—
64	C₂×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄⋊C₄	—	2	2	no	—
66	M₃₂	—	2	2	no	—
67	C₂×C₁₆	—	2	2	no	—
68	C₂×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₆	C₂³	1	2	yes	—
70	C₂×C₁₆	C₂³	1	1	yes	characteristic
71	#107	—	2	1	no	—
72	#107	—	2	2	no	—
73	#107	C₂³	1	2	yes	—
74	#107	C₂³	1	1	yes	characteristic
75	#107	C₂³	1	1	yes	characteristic
76	#107	C₂³	1	1	yes	characteristic
77	#107	C₂³	1	1	yes	characteristic
78	#106	—	2	1	no	—
79	#106	—	2	1	no	—
80	#106	—	2	2	no	—
81	#102	—	2	2	no	—
82	#102	—	2	2	no	—
83	C₂⁴	—	4	1	no	—
84	#41	—	2	2	no	—
85	#41	—	2	2	no	—
86	C₂×Q₈	D₈	1	1	yes	characteristic
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	4	2	no	—
89	C₂×D₄	—	4	2	no	—
90	C₂×D₄	—	4	2	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	4	2	no	—
93	C₂×D₄	D₈	1	2	yes	—
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	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
99	D₈	—	2	2	no	—
100	D₈	—	2	2	no	—
101	D₈	—	4	2	no	—
102	M₁₆	—	2	2	no	—
103	M₁₆	—	2	4	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	C₂×D₄	1	2	yes	—
107	C₂×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈	#41	1	2	yes	—
109	C₂×C₈	#41	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	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₂²⋊C₄	—	4	2	no	—
119	C₂²⋊C₄	—	4	1	no	—
120	C₂²⋊C₄	—	4	1	no	—
121	C₂²⋊C₄	—	4	2	no	—
122	C₂²⋊C₄	—	4	2	no	—
123	C₄²	—	4	2	no	—
124	C₄²	—	4	1	no	—
125	C₄²	—	4	1	no	—
126	C₁₆	—	2	1	no	—
127	C₁₆	—	2	1	no	—
128	C₁₆	—	2	2	no	—
129	C₂³	—	4	2	no	—
130	C₂³	—	8	2	no	—
131	C₂³	—	16	1	no	—
132	C₂³	—	16	1	no	—
133	C₂³	C₂×D₈	1	2	yes	—
134	C₂³	C₂×D₈	1	1	yes	characteristic
135	Q₈	—	2	2	no	—
136	D₄	—	2	4	no	—
137	D₄	—	2	2	no	—
138	D₄	—	4	2	no	—
139	D₄	—	4	2	no	—
140	D₄	—	8	2	no	—
141	D₄	—	16	2	no	—
142	D₄	—	16	2	no	—
143	C₂×C₄	—	2	2	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	1	no	—
150	C₂×C₄	—	16	1	no	—
151	C₂×C₄	—	16	2	no	—
152	C₂×C₄	#123	1	2	yes	—
153	C₂×C₄	C₂×D₈	1	2	yes	—
154	C₂×C₄	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄	#137	1	2	yes	—
156	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	#123	1	2	yes	—
159	C₈	C₂×#41	1	1	yes	characteristic
160	C₈	C₂×#41	1	1	yes	characteristic
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	8	2	no	—
167	C₂²	—	8	2	no	—
168	C₂²	—	8	2	no	—
169	C₂²	—	16	4	no	—
170	C₂²	#480	1	2	yes	—
171	C₂²	C₂²×D₈	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	8	2	no	—
174	C₄	—	8	1	no	—
175	C₄	—	8	1	no	—
176	C₄	—	8	1	no	—
177	C₄	—	8	1	no	—
178	C₄	#480	1	2	yes	—
179	C₄	C₂×#123	1	1	yes	characteristic
180	C₄	C₂×#137	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	—	4	2	no	—
183	C₂	—	16	2	no	—
184	C₂	C₂×#480	1	1	yes	characteristic
185	C₂	#3415	1	1	yes	characteristic
186	C₂	#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	#3058	C₂
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
2. #2913	C₂	semidirect
3. #2234	C₂	semidirect
4. #2234	C₂	semidirect
5. #2233	C₂	semidirect
6. #2232	C₂	semidirect
7. #2159	C₂	semidirect
8. #2158	C₂	semidirect
9. #2140	C₂	semidirect
10. C₂×#357	C₂	semidirect
11. #2113	C₂	semidirect
25. #427	C₂²	semidirect
27. C₂×#107	C₂²	semidirect
29. #367	C₂²	semidirect
30. #367	C₂²	semidirect
31. #367	C₂²	semidirect
32. #366	C₂²	semidirect
33. #366	C₂²	semidirect
34. #366	C₂²	semidirect
39. C₂²⋊C₁₆	C₂²	semidirect
40. C₂²⋊C₁₆	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3058	2
#2913	2
#2234	2
#2233	2
#2232	2
#2159	2
#2158	2
#2140	2
C₂×#357	2
#2113	2